a
a/
.I
a\'
'1
-,.1
{
fL
L
L
L
L;
L
fL
L
L
L
L
tfL
L
L
L
L
L
L
L
L
L
L
u
COURSE NIO. : AMLLL0
ENGilNEERilNG MEGHAffiIG$
By
tr- c."
eEr 4,E-ga
tj" i;iE''
"
''::\ '' . -.' -
',, 6X
P' C"D U t/riR
for finite bodiesoisize much greater
MECHANICS
'
..
'
wlid
Classical 1t{cchanicsis not valid for point rnasses (particles) but
shall cover in this course
than the atomic size moving with speed less than 1/10th of the speed of light' We
2. Axiorns and force systerns - 3' Rigid body dvnamics
i- fio"*"tics of i point and a rigid body
work aird stability
4- Equilibrium 5. tagrange's equations 6. Principle of virtual
1. KINEMATICS'
frame' A' rcfcretce fi'o'me
Kinematics is the study of geometry of mo0ion with respect to a reference
rlith--Euclidean geometry
them
between
is a set of locations in three dimensions having invariant distances
disLance betrveen any
the
which
in
being valid. Reference frames can be attached to any igid bodg (a body
axes ,' yt z can be imbedded in a
tso rrraterial poinis remains the sanre fcr all tinie). ,Rectangular Carcesian
be the unit vectors in the directions
given frame .F rvith its origin and orientation being arbitrary- Let !,j,k
to frame F is defined b}'
nith
respect
oy(r)i+ o,(t)k
:,;:r- The derivarive of a vector A(t) :
Ar
l"!r)L+
= dl.,ldtii-;-,y lizcd =
The rules of derivatives of vectors are the
i"i*
dci+
(I'l)
a'k'
same as those for qcalars excePt
that the order of the cr
F
terrrr should be Preserved'
L1 ICTNEMATICS OT'A POINT
point at lirnes t
],er P and' P' (F.ig.1.1) be the locations of frame r. occupied by a moving
are defined by
f'
frame
to
and t * Ar. The ietocity qr and acceleration q5..rvith respect
gr = jil. *-
=Ji:.
*
=
er
,
4r
-
4r= 4r '
--
-
-BRr
$flir:.ie
Frw
of the a-\es x' y' i
gr and qp depend on F but are independen! of the choice of origin olr the orientation
for simplicicl'' yields
embedded in the frame F. lniegrating (1.2) rv-r.t. t a1d dropping ( )tr
'[tf'
r(t):r(0)
s(t):s(o) + !o' {t')dt,
For r.niforrn accelerrrtiott e= 9t,eq(1-3). + u(t) = g(0) + srt,
+ Jrv1)dt'
(1"3)
dt) = f(0) +g(O)t + est?lz' (1'4)
path coordinates'
8
\Are derive expressions for u,g in Carrrsiau, cylindrical polar and
kt%rg
:o
1.1-1 Cartesian soordinates
"))1--:
Then-eft
Let a(t),y(t),2(t) (fig-1.2) be the Cartesian coordiuatesof the moriug point at tirne t'
F;s
Ii-. - ; --:
+ ,Ur:ii+vi+rk, +
r(t)=r(t)i+v(l)i+z(tl\,
t!"=i,
ar=i=i1r,
oc=i, lz=i,
*3{
s(t)=;i+ii+
or=y='.ity, ,:=i=t:.,.
(tls1
(1'6)
gstt)
. - L ,rg,r4*
the
llV f zor
movingpoiniaitimet.Lete..(t),g1(t),e,(t)betheunitvectors/ffi;N'*-.,^
1.1.2 Cylindrical polar coordinates
r,et r(r),{(r),2(r) (Fig.l.3) be the cyliudrical polar coordinates of
g
i
(z) */. ffi'o-.!
il".*"#;;;;;;;ti;i;:.:'#Xtiar
;t9$t F-+-../\.,
t'
,$j$*:1:g:19
,.r', \
l ''
(d) and axiar
i,Yr
triad rvith
positionThese
form
a
righi-handed
c.rordinate lines at this
e,(t) =
$tr = (-sindi+ cosdrd =
In.tlresequel, (') = ( )1r
is
%(t) =
4e+,
ir-nfilied. The position
g(t) = k'
fig. t.3
c*lr = g' (1'7)
g,
vector r is differentiated rv.r.t. F to obtaiu a using (l'7):
t, = i = igo * r9r + i9r,* 29.,
ecl,, =
g = ci-Y
b"Sg-, +
(
(-aosCi- sindj-)i =
aip *'t 6)ef+Zezt
1
I
,L
-6%,
(r.8)
+
=)
-sin{!*tosfi,
+
ffiq,,F1-
:, :
cosdi+sinCi
'z
t1.e)
t
1
!
i
l
:i
_;: _- l:
;t
a.--
:r
= t6..':,':,r,: r,., . .o: = i,
@d,
ar=2,
aadi ,
,
-2i6+t6,
;. i
and a, * i-, oo i*' nosince g., 94 arcnot constanc tectors in-i;rime +
Note that
"* = 1tr1"'6)
1-1-3 Path cooidinates
Let s(r) (Fig.l.a) be the coordinate along the trajectory of the moving point at
time t- A right-handed triad of rinit veciors g, gn, g! at the location s is defined
s=
dr
de. &r
Ar
Jf45=*
;-':>;Sg,
a:
Eo
yn
as:
= frri=
o :+ ,*';:0"
'T
I
t
,
'*r."+
I
QrX9.=9C,
g is the fangent l)ectortothe curve in the direction of increasing s since in the limit as As
of the chord Ar becomes tlre tangent and the magnitude of-@!" becon:es 1' Notc tlrat
-e,)
(r.1i )
95=Qxq'
= w=;%'
tq.Xel=9.3r
gCXq=9.'
;; = Jito a,'
ftr,
(1110),.,,-
ud
-'
(1.12)
*
0, the directiorr
dfr
d*re..
.=
'=
ueclor oi t'he cun't'Tlrerefore the unit vecto.r e.n.in thJ-iirectio,r of der/Jis defined as the princip al normal
q.
and-dirccled ioirards cen!rc of
[here being infinite set of unit vecrors normal to-the curvg ai P. en t
l/p, :urd
ccrtalarc C (f+ = g, #g^Lof the curvl. The mignitude of dglds is delined as the c-uarctare
the radius of crtta:,nnr- of the curve at P. The cross-product oll and lc" ftq* the first ts'o
p
iscalled
equations of (l-f2) yields
(
e5
ic is rrorrnal to both q and 9,"- The pliure of g, ard g, at P is called the
(1'12):
the plane through P which is closesi to the trajector-r'. \lte obtairr g, a using
is called thb.6itornral
osctlatitg plo*e. ltis
da
'- dt-
aectora-s
dzd.s
,-u=ie':i.a
-__!_<
-.
r_.
I*
fU
rU
frJ
U
u
U
IJ
dsdt
da ds
o=i=ia+6|]fi,
+
.+a- k
(1.1.1)
'a4
(l.lri
4=3St+fre'r
=>
Ua=u5=$,
ilr:i,
= 3..
=u: izl1r'
ac = 0.
directed torvards the centre of curvature. If the trajec0ory is gisen paranretrically as !=
d2r ,dr rz dr d2r
*,
dr d.r dr
ds
and (1.13) yields
For
01
77=77t7;)
- dt ds'
:p :t#*P*rr*r
or aT.'
dild, - i+ r'ii
(l-17) +
-. |
dg f
dx
= f'
,d9^, rdt
i : l;!l =- l;fl,
,
-
4z4d* = f"!,.
.rp.=
,
q = cos 1i! f
sin
s,.7 (dslds) I Ve,ft')=
For a circrlar trojcctorX of radius
fi
i.tci
tJrcn
-EE'
=t#*fitrrffr
(I
,
.l7i
and rvith
(
- (-
dSld"
sin
lii {
cos
*trt*
(-qi!{i+ ce/i}t*) / l;,g
$l = *t-rtn
i{.,
Oand
(Fig.1.6) i:-j;::
r
-
g =-sq
.
. i:l
i
rp!+ cos r;!)
*}:...
-ve sign to be used for point A
'-'r
fi'! 0;
&e--:ie r
-n
"/\
'G9
.
p=R, et=?' 9.=-9-:
"=E' :=0,..=* X=0, i:'i=0.s-
i
l. ts)
riyld!lds,
re'=t
si#iaity,
{t).
E +:Stzltl? 111uft
{'j,
nilrere *r,e sign to be used for point P rvhere
(
r= e. r,rerobtain pusing(l-17):
l(drldt) x (d?ildx1ll= lI"l.
l&,1&l- (1+ I''l't',
aplcte lrajeclory(Fig-1.5), r=ei *f(a)j,
taag =
1.13)
r-S
.a
* -9n = _nire,+
Itd%.
2
:L i.e
.9t e
p--t
L
,;.
t,'.':./
({
I
t
.*.j\-
fr:-
_ ..--
Define"rclotiacaclocitygeelp-andor.iu:acxlerutioitgg,.elrlfpointP.relativetopoint/
tfrtr
=
- irrlr
wzrlF = irelr; 6(re
-: :.:a. ;r, : i.
-d
t',ttr)
s.PlF-4.^lF'
aptlF = ip1r-- fr(urlr - 3e1r)lr=
;;"
t2ele'
pararrcters
of independeat
sysrem are defined as rhe nunrber
ot a rigid bodlr are
of
.o.*
of
freedom
orrreedom
a"g,*
-
,'"
(.".:f"l:f:::iT:'"',X$iJ::r#;.'J;;;;"-.
t'hree rotational)'
"t*
"_*;
,-
t."
and
1t1.." translational
1.1.4 Index-Notation
. ^- -r^-^ro rlrp rnn.
.Indevelopmen[oftheoryit,iscont'enierittouseintlexnotoliorr,i.e..todenotetlrecoordinatGZ,V':.tlre
drra?'ag (ai); 9t'91'9s'' (g); resPectit'ely-.
ttre unit t-"ttoo f'!,L by e,'a2'23 ('i);
aslayto.-and
components
indoi such as j i' a t'erm of '
following
the
"tgeu'"it
use
we
For brevity
"o-*ot;ori-'lono"or;on " ""p""tud
anexpressionimpIiestlratthesumof,t,.*;;;;';tobetakeuwitlr1.takingvalues1,2,3.Forexample
:+ 6; * dircr * dizcz * daca = a;
6.; * d;c; = a;
a2' b3+ dsrcl + dszcl'!"!!*ca = tl3 '
O:'*
=
0:'*-d1r
:
or''
d13ca
dP,-h
*
"' :
i-e-, 0r *.'drrcr
{riinde''c:Di,+d''rc&
,i:::'
The repeated i.dex
The
is
ind& ii". fr".
*igr**y
argebraic
repraced by other.co*r-enient
irdcr and.-ci* be
values 1'2'3 to obtain
rli-rrrutiiau
oo'u.'"*Jta,t'"oi;td
i-
:-q
:
=
ecu4lo"*
1n=e
q
:--
.5{'
" *::jilJ*.:.": :: ; :H:;r:jl:-T
":fflx"
q1t(t)' .@.\' &p.,
t
-
Le
is governed b1''
of rn relative to F at time
The rate of ctrange of orientatioa
-r
,\t:},{ I::
c191(t)*c:u,(t)+caq(r){a}
_ ,.\ 0rsr(r)*0"9(t)+0sss(t).
r - ,r\ r r-- /rl_r-fi-a-(rl= esls(r)
e.,-(r) = Clgr(ar-rc3=?\&/TLrEr-r\-,
salr(r)
=
oler(l)*c2sr(r)*a3s"(r),
erlr(r.) =
orthogonal uuit vectors'
independent since g are nrutually
not
aIe
oi,0f
-compo[ents
nine
The
(6)
'ci
cs=oSinilarly
:+ 2o1=$' i'e'' c1 =$'
er(r)-gr(t)=1. + 2grir(4'e1t-t):o,'
:+ or*6r =0'
'tJi"ty
q1(t).g{t)=0, + 'ertr(r)'g(t}+e1(t)-Q1r(t}=0'
a3
'
(c)
-.b3' = -cr'
1-- Thu-"
a-b-c-a<and tr-'2+J+cros-product
bL : -a?-
i.e-.
-02=0'
c2=
The sinrilarrelationshaveSeenobtainedbycyciicchanges (a) witht'e r'trs'expressed as a
ioatpu"a"ot' ll',e iewtite
only three compoDents o3, 63, c13te
using (b),(c):
x gt = I X 9t"
9r;r atgz'-ct9g = [619r * crgr *4?9s]
=
g?lF
=--439r
fogr ='cr9.1'
-
*
bs93
6391
=
:=
[0a9r
[63gi
*
crgu
*
t
a3gj] x 92 ='t'' x g''
x 93 = gi x k'
<
+ crgs + "'!b]
(r-'20)
Ltr
i-e
r's'r-
=g*x€i
g x (l'20):
far q
u by
bv forming
formi'g q'
'o for
factot g- we soh'e
The underlined ternrs are added to ger a common
(grx 9i)'= (ei'et)gl-Gi1er.)e= $'t-tt=2s' e, xitr:g;x
=uiei=a,rsr +u"s2+."3qj =-?ei.s =9.er + %-e2+" * =_::.:;r,"
since
Bence
Z
of franre rn relative to frame
is defined as the angula'r velocit'y 9L4p
!
e
.a- i
'
(121;'
<
1-
rr
:'
F'
i1- and i1r
f,etAr(tlando;(t)bethecomPonentsofvec[orA(t)retativeto!'andg:ta;(t)q'
=
4(t) = Ar(t)& + AI.Q)D_+ Ae(t)&-= or(tlsr +iz(r)er-+as(t)h:i:::'*
4^=4q="""'=@ig.ir
'
:
4r={[-=.o.rr--'{iE'
:
'
1.2.1 Relatiou Betweeo
I
,
!
i
q
d
-
-r
L
I
t _.---
-9
=-,
rt
-:..-t__*-;j,:
: .i
i,
4r=G-q.)ii.::"*cier:":_-.fi]?,;;,,,6.
r-€'r
*
-
hin;
Io particular, if
L-.
A.'ts
A'
!:+e-x
vector in m' then
"tinffi
":;:;
4*=Q^tand (1'22)
br--s.x
,.1
(1.22)
:
yieldg
A-
1t:]
r'-*'Consideraniiherodhogonalri8ht-handedunittriad{'
of ql' qmsrqef arel'e'
the definition
we ptove ihe co*srslcncy of
-
6r"d-;
we obtain
,rr- U=i"g (1'20) and (123)'
.g4d.l=f,tu-d=*
r x (e x el)l =
I 4lr : 5bl
.- :4x
|tt*'*1"- te
-
,4.' -
-
,
::x"ffi ':i:l:1::::":':ffi
Lra
le directiorrs of
9i in m.
It
clrcrge ot
is a mea-.ute of t.he rale.of
;::j,i}:.'iit!::::lJ*;:::;'i:ilJ;*
::::lIflll5f}fi ##*;.1*fffi
-t--
I
L:
|;#r*;l;H|;:j:.:j:;#.'ff-'l,1ll,*T.il,HrffH;,;;;Gsr+is+sv-rh€ierore;::,
\='il?*ee1r) x L
x i!=
3P +{llsp
4= 5{3T
x
L-.
=
=
a1, =
-
L-
We
r. respectivel,
t.-
u
\\te take ,n"
&l?'
x
f''*'l.'ni"i}'o i"i-"u
,r*"
(1.?4)
(I
jzs)
?
F
-f:E< v tLt
and
to P and A be fu:red to nr' Thus
&0) = &i(4+!p.{(r)'
telocitY relaticn:
F and use (l '22) tr) ;-ield
derivative of eq(a) rv'r't' frame
+; x lr'rl'
+
= 8rt" + Lpslr =jl,.,rt Er^1(t-264)
!e1r = 9e1r *t'r x I're * 9P1''''
to f ield acceleration relarion:
frame F and use (l'22)
t'
w'r'
eq(l'26a)
of
derivative
\te take the tirnb
t-...
tj
lJ
t&r,.
lrel,
-rtr
u
tj
L
1-.
erlr:9e1r +elrf I
t*t*ex
d
Writing
!p,,= {L,
eqs(126} can be--written
.e.p
g1i
f i' =
ipelr + ft(arr*)rr
as
V- ietgxtSflrt.
i
yu71;
.grlF-l 9 a;r +sgxtf +
IJ
n
&:
,
-f;*"t + l*gtr-trn +sr xsPlml'
x
= g.rlF 4gx rPa *gl [i,p,tllplm * Qr1'a'
gzlr = 4r1r *t't" x Lp,t{t'r x (grx LP;,l+?4x
l*.
L
L.
dt"lt =ezlr *{ii1z +:*lL
* *12 *&arg"t" *d gP|- & erlr and 4'fl13-2 ltela.tions betweea
w't't' franres f' and m by I
denote the position vectors
L_
f.--
=
*l'anriag: ritt,
L*
u
I:
*1r
4= Gf"*ql1: 4)+*lI
li.*g{qr x A= Az*g&lt x.'9lsrr
.
=9s1:*9lz1,
_--r_+
x
-,-.--< *
6\r.
erir+
*.iqr
---:(eblz)1r' Gil , = 6,,
(es1r)1s
(1.260)
(r.27ol
(1.276)
9Q'*(se'xff)+&g!ix1e1"+sef"
.J
,.
\,
I
-i
:'t..
. \..a:r
_
-
1.3 RIGID BODY KINTIMATICS
_
Eg-r.ro
.
FormaterialpointsPand.l{ofrigidbodym(Fig.f.l0),g,,l,a=Oeel--gandeqp(1-26),(l-27)vi:td
(r28a)
3 plr= 3 afp +co x lPa +S9X(Slx len)
! rfr'- ! e1.+ g') X lpR ,
epir = e71lr +a x .AP * s-x (sx AE)'
9-plr = 9,t1n * gx AP.
.i
1.4 KINEMATICS RELATTVE TO TRANSLATING FRAME jr
Let frame m translating w-r.t,. frame F be named as ?. Frame f is siid Lo tzvr.slatc
rv.r-t. P i[displacements of all its poiuts A,B,C,-.. are the same in f' (Fig-f .11)- Eence
F 9s-
B(1-2s6)
ll.
-
:lt
)Fi?
I
L&a
lsnlt
= ilat1r =
L-BIF
:+ &sr b a constant vector in f-
-rr1r:&
Hence the displacements, velocities, accelerations of all
of
T
are constant in .F. Therefore
gr=
(9..
x
g'lr -
ioints of T rv.r.t. F are same and all the line elernents
0 and eqs(1.21) aud (1-22) yield
{p)/,2=Q,
:+
gL=0,,', ()tr=()tr.
Using (129), the kinematic retatioas (L-26) [cr'a rnoviag point
!,te1r
=
u-aJp
* u.4a,
P and a point
gplF = g.a1p.* g,pp.
.d trted to
(t-2e)
f
becorne
(1.30)
1.5 ANGULAR VELOCITY OF m F cqr =Q.
A body 2 is said to have plate rnotion relative to f if all its n'raierial points have plane trajectories in
parallel planes. Choose g attached to rn in a direction norrnal to planes of trajectorl', then qj moves parallel
to itself and Slr = 0. A rigid body rotating about a 6xed a:iis and a rigid cylinder rolling rvithout slip dos'n
an inclined plane have plane motion relative to ground- '
The motion of a rigid nut on a fured screrv is not a plane motion. An a-tis q attached to the nut in the
direction of the screrv a-tis does not chaage its orientation wJ.t. ground F, i.e., islr = 0For such cases, we choose & : e. (Fig-1.12). Then et,g, have no cornponent along $" Let d(t): be.the
angle betrveen e1 iuid Er at time t:
pr,
e?(t) - - sin d(t)E, + .*O1r1qr,
ft(t) = 83,
*1r =Q!r1r=(-sin0E1+cos0E2jd=0e,
9:1r=(-cos0E, -sindQ)0- -r9.,,
eq(121) + sr=(e.- xett')/2: [9r xirlr*e:xiztr*ss x4'F1/2=[e1 x iez*92*(-der)+esxq]/2,
er(t) =
=)
cos
d(t) E, + sin d(t)
- - 0€t__ol-
.=0- a
e
t 1-31)
tr equals the rate of rotation of a (any) line g, (f fu) fixed in m relative to a (any) line E1 (a &) fuced in
f. In this special case t.l is the rate of change of a f.ritc iaglc. For general spatial motion of a rigid body t:
of a fi.nite angle. F;nite onqutcr dGplaternestt rt'.o. fr Pra"rirg t<<g:{.!*&
does not equa! r
cnl directr'trr, is not a tectov c3 thc! do sg't <.ae
-g(r<.o<dilg *ro g..^ra\\oft1rca L"o qf olAiH,rn one
o 4Aili.'^ \s nob <aqrutolri-e .
on
trg'l'ta
5r
STANTANEOUS AXIS OF ROTATION OF A RIGID BODY
A rigid body is said to have an r-nslcalcteous cris o! rctatiott at time t if all iis points (its material
pgints or points on its three-dimeusional rigid extension) on such,an a:iis have zero velocity at that in<tant.
.exists
if
It does rol always crisf. For given g-. of iG point A and u at t, the instantaneous a-ris of rotation
lve can find a point B subh thai
9a =9e *u)xAB =Q,'i.e., if utxAB = -9^, i-e.,if u1 =0org^ICf,sincetrx.4B isnogllaltoq
i.e., only if q. .!l = 0. Hence insiantaneous a-xis of rotation Liists onlyl/ u,t .r^, = 0.
1
.JJ
J
J
JJ
J,J
JJ
'j
:J
,.J
.J
j
-
,-I
-I
-.-I
":I
:iJ
1i
,l
,:.
the
instaniai-&gs.€ras- of
!&"tioo
ahvays exists for plane mocion
'but'ii'genei.l
:f " l*d..b:d1
silce ue
f
IA
t
(Pig-f -13):
!^: k +o, x IA= kellJAlsing0og = ld(I6)g
. \
for. plane
er'#9-
The velocity of auy point ,4 can be expresbed in tcrnb of its radial distancc f.A from
ls.,{l
l-gr
f
_ls.al =lgql =ulIB
IC I=I
Fi3't'r3
(1.32)
The magpit,ude of uo o[ a point A equals the product of lol and its radial distance Jrl from the-instaetaag'ousa-xig,of rotation through the instantaneous cent,re f, and i-s directed uornral to /A in lhe sense correspondingto r.r about f. For a rigid body in plane nrotion, if the directions of velocities of its trvo points /. I haviug
'
tle same plane of motion are knorrn at a giveu instant, then the iusLantaneous centre .[ can be located in
that plane at the intersec[ion of the nornrals to velocity veciors at ,{ and B (Fig.l.l3)- The instantaneous
centre of a rigid body in plane translation is at infinity. The instantaneous centres ft of bodies A are shorvn
in Fi9.1.14.
,;\t,
-.1
\s
ug
.\
t
o
vn
f,1
f,
Fg-t.t+
\,
>.d!
1.? ROLLTNG IvITHOUT SLIP
Let the material points P1 and ?2 of bodies t.id* rt.*a,t,tS),Ue i1
The kinematic contact conditioii^iie defined as follorvs
conraci ar time ,..
I
l- stip at, t ir ci:i'" f .si,tr;,
IJ
L-
s"jt."l t
inpty that slji]- =
3- in'rpending slip at I if sp:,iii
iil]
in contact,
L
L
L
L
r
l-
does uot hold at !f, for the corresponding points
i
L
L
L
L.
L
L
L.
L.
f..
= uili; at t,!,bu[ this equaliiy
r-e- !pr1g (tr*) rb"fil*;l€(t.')
no St"1'
I
I
}.:
etoi
Pins:{ and B move in a 6xed elliptical
tt-tt:'t
rvhrich translates
as in a straight slot
:
lvith
Exarnple 1.1.
and
pin" o.r-t. (I) ground re*rence
acceleration ay=_rozr6cosotasshg*rnr.il.Ji.r".Thepositio"ril"r""tr*rdresrotatt=0are-6and
tt
or
alcebraiion
a.d
(a) velociry
rate of change
" ".rrtrlr
from each other and't'he
zero, respectirelv. Finiihe
rshich- the ceirtres of pins ."0""*
at
from the
,"te
reactions
u.ar,
tui
bodv' (c) the
(2) the translating
trl.r.*"*,ng
(1;
.r-..t.
;;;;;y
of this rerarive ,0""o oiooararion
on the pin.,{ ornras.s..'.t
i11;"::tl'!-;:'*:t'""X,:::1*"':1lt{fff
::'"'i'":
smoorhsrors
of ihe
to the right of the centre
Q
--
{ri
\=__ ulyy'
L
-1.
t/g
((\)
Solution
jr#"
r--t-
!
€ig . tl
D'of the slot
z1 be the locatior r of rlre Point
?
"1
[[encc:'
Let (c,V) be the locatiou'of'ttre
u1(0) = 0. ar = -u)-:rcrGosrd'
*e'
e1(0)
lta*e
*'e
=
D'
poi*t
for
anrd
;;;i*a-u.rl)'
ul --
it
The coordinates
Jt
d' =
=ur(0) + l" "'(')
r,g
:r1= c1(0) *
-r^r*o siurat'
-t
.:
G)
l.i
cent-re
Nl
1t
ordl
!
=uocosut'
ar'
(t)
:
slot:
of the ellipse and tlre straight
are relatcd by the equiations
L_q
7+ F=
1,
y=
(" - cr)tau0
for z'Y' Relations bctrteen
Equations (2) arePotved
i'i'
Relat'ious'betsreu
i'i
(2)
= DE taln0l
t
eqs(2):
areobtaiued b''- dilferentiatiug
v=
?*H:u'
Equatitxrs (3) ate soh'ed for
i,g
IAE
(i - i1)lau
(3)
0'
eqs(3):
are obtait rred bY diltereutiating
1
(4)
I
+
Equations(a)aresolr,edro,;:'ri.FortJregivendataiattlrepositi:n"',,..o..]..
a;t
cm;' :+ ""cosurl = 0'6' + sirl =t0,'l'
l,r, = ?o cosurf, = 0;6ro = 6
:
Eq.tr}+..i1:-80(10x0.8)=-6{0cnr/s.}1=-(80)l(roxos)=-38e0G,cm/s2.
v = ('- 6)/5'
400'
Eqs-(2)
:l
,t
become
zo
t44s
into trre lsr equation yierds 1Br3 Substit,ting y from the 2nd equati,oo
of .A and B, respectively'
,iu
0.2269- These a."o
.ooo-"."
('?)
+4!2=
*
32
=
0 rvith roots 10'85
I
a'd
"*oo.dinates
Y-(10'85-6)/5=8'400'
+
> i =-53g.d cm/s. ! = L74-2cm!s,
640h/5'
:
0'
i
G+
-8'4v/100
*
eqs(3)- + 10-85t/400
i=(i+38400)r/5'
(539.43+10.8$)/400+(1?4.2?*8.4f)/100=0.
eqs(a)+
cnr/s?'
7= -38338 crn/s?; ! = 10?'99
+
t0?'99! cm/ir'
a1 = ii*li! = -3833Si +
cm/s'
1?4'2j
'
'+
g^ = t! rj = -539r{i*
(u)
For
P.oitt
A:
9c(2)
E
u
L
u
u
IJ
$qaily, Fy,.t*tii:*=o-zzo{canob.!\ yelocrtv yd acceleration'1d,3;.' ,",',,..
' ls =-onz-f i- i-6'tj cui/s, s; = -32654i+ 10125i cm/sz'
.
'
: 201'1 cm/s, 3r.= fiacrxs*A = 10?39coeec60o = l.2tL7 <arrls2'
tt':::--= 11691 cm/s?'
cwcl = l0l25cosecOb"
sB =
em/s. 3s
ia co:ecu:
=gB
= -3-Mcosec60" : -4'2 cmls.
,;
LAt? = 3'{ 9= 124J9cmfs2'
lelr =ir €:201-1ecm/s ,
gtlr =3n9= 11691 gcm/s?lalr = sp e = -4-29cmls,
;1 = yac&c i =
u
u,
u
L
u
L
39 = ggccec0
<'
+-.
174-2covec60o
flence
(b)Thepositionvector-of r{relatinetoIissgwiththeseparatio65='48=(y:-ys)coaec8-
i = (rir-tia)cosecg
r -- r= 201.1-(-{2) = 205'3 cm/s'
t:, --ii^lr
i= - (!a-!3)cosec0
= 124'7-11691
==
-1156? cnr/s3'
as (Jkt*a
eguirt s
\
w.r.i- ground reference are egual
-- ; c ..:n!lrtinr frar
The ra.tes of separation rv-r.t. translating frame f and
z-atis'
rvith
''
(") ftt the tangent io ihe eltipse at (2,t, ye) make-augle e
,.;'pao+y?/t00 =
L.
I :+
2e1480+zwl
fiaa.=0.
li
: (-)i''
+ { = avfl 7 -'{:f!"
Fig'El-lc' The cquation of nrosion of pin
The FBD (free body diagrarn) of the pin is shorvn in
3 is
031(-383-3Si+
I!: aal1 + Nr(-"c 30. i+sir 30'i) + IE(cos 7!-!e lasiu ?zrli) - 0'0rsj = 0'1089'
j : 0'55r * 03516N: =
+ i: -0.8660rVr +0-3o?4rY2: -3-8338,
1'0?99i)
direction of reactr2u nrz
N- The negative sigu ia If2 inrplies that the actual
o{:r
is opposite to that shou'n in Fig-El'1c'
screrv of pitch I cm rvith
e*"orple 1.2 A rod $,ith a threaded hub rotates on'a righfhand
the time { is in seconds. one side of rhe rod has' a telescopic
o - o-uz rad (Fig;El.2), rvhere
crrr/s3' s'irh its tip A at rest at
arm mounted on it which has an outr*ard acceleration of 0-4t
side of the roc] has a riglrt-ltand thread
a d-rstance of 20 cm frorn thc a-xis at t = 0. The o0fter
o!rto the.thread aL Lltc rate of r = 0'2t rad/s
o pitclr of 0.5 cru- A disc rotat€s relalive
t\ llu a
n-ith
the
o[
rotation
of
fronr the axis
and a point B on its a-xi! a0 | = o has a distarrce of 30 cnr
F'1"it'z
\9e
I = l0 s'' -a<
at
I
and
points'4
rod- Find the velocity aud acceletal-ion of
point
A:
cau be easilv co'rputed'
co*Pon€nts since r,i' i,i,6';'i
;" ;;
;;*;t;;
+ llr :
3.?65
1q,
ir1
- -rso4
"rtina.i"a
i=o-4tcm/s?, =) i(t) = i(0)+ !o'o*dt=o'zt|cm/s,
d=0.{r2rad,
+
+. i=0-8lra'd/s'
at'=20+0-2r3l3 cm'
r({) = r(0) +
lo'olzf
+ o=0'8rad/s3'
(2r), the rod adences arially by one pitch (2 cm)- lleuce
i=16.l2r -f7tr cnr, :+ '' i -- ]fu cn'tls' :+ ! = $l* cmls?'
.1
l.aft fi ;
air=t0sr'"=.i0'i0.2(ig)3/3=86.6?crn, i=0-2(10)3=20cm/s' i=0'4(10) =4cmls2'
For)o-ne rotatiron
+
- :+
'
*:t6'i"d/",,ja : OS rad/s?, i = tlr= 2.$,tG cnr/s. i = 0-8/r = 0.2546 crn1s?'
i
'')'-': --; I+t'+- il:is,+rCe++ i:g=20q.*G93.aeo+2.546so cm/s,
A=
'
O.t(rgt
,..:*.-;9t:,=-,ffr':aO]rl*f1z;O+r;)e++ieo =-55{3q'+389'3ea+0'2546scm/s2
.=.;{.='E.}o.4qrjeHp:}
.='
-.i..rp 1o.s€iii**Fg
:
where g., Cr, i ."!9 -,T-,:-!,9.1y" at pornt 4 in Fig'El2'
rr
a{ I
f,,
.'
ya cosec0'
i'. il,t :
gis theunit vector alongdB' rvith sr =
Rdafiretoframe?., the pcitionrectorof r{ is crtg' where
L
L
u
ru
u
ru
u
rU
L
u
u
U
U
u
Li
-'.-lr)
:;-:
u
aL
-
(0.5'cm)' Eeace for a rdta0ion
For one rotation 1irf1"1*a'O-r*, point B advznccs re.dially by one pitctr
by
given
A0 of the disc, the incredse Ar ia the'radial coordinate of I is
0's/t0/2r cm.
+- i -- o'si[z* =
-
0'5'.i2r-
+ i = 0'5dul2r cmfs2'
=
"*1"'
u=02rrad/s, + 6=L2rad/s?,r(t)="(0)+ I"'*n-30+ lr'o''Ly'or=30*o.a2ilt"ftcm.
* aLt- 10 s : r = 30.80 cmr o, : 2 tad/s, i = O-5(21/2n = 0.1592 cm/s, f = 0' (O'2)l2r =O'01592 cm/s3
A,r
lflre
rralues of
i.d,i.!
for poinc B are the
".*" ".
for point''t- Hence
!.s = igoa .Oe++ ;9* = 0-f5929 +246'4ea+ 2-5469. cnr,/s'
+(2;O+ "O)+ * !e, - -19?1 1*27'194+ 0'25a69, cm/s?'
ae = (i"d2)q'
,:.,-i"i; ffi:1:::l;::r;:
iu a
II:ll
8,,'r-"\-l
i
and
vettical plane' A pin is
Iy=r.ar'
"
consrrained io follorv the path:,r:(o) ='0-2+,0icosd rvith r€sPect I
\44'
ro the plate. Firrd the velocity atrd accelera[iou of point D and l, *ffi;l
the centre P of tlre piu relative to the,ground'at'the'instant 5t1en'n w--:T'r
loning parabolic paths.(Fig.El-3)
rv
en.
e
: l't
:
=-.i::.
:l' : 1,1 11 :'^
1,'.::', :T::. ::,:'; Js'
pression is 2 cm and the speed of -.{ is 0.2 nr/s rvhich is decreasingga\
rr
i,':t'-"*
rt'^
'f
Li
-}
-^.;-^f
attIretateo[10nr/s?.Firrdtherrrilrinrunrsti{Irlessoftlrespring<
; ;;;;i; *" ", m*, o-r kg does follo*' the sive' pat'n zt tlis ,/
'tt?are stnooth. 'tt?the curved sut)port of the piu arestnooth.
The slot.and thecurvedsut)portof
insranf,. Theslot.and
,:1T,.^----
JJ
dfr,,J 4, r-l
J
f\ , r-Z
| 'Z-.+W
'{ {.%_ff' :
:
.f
Lr
//i
2o
-d
\u.5"
rtrr+
t"t=f Fi3.€l-5
Fi3.
in nriud the type of data gir.en. rve use path coordinates ro find the acceleration of "1
of tlre plate' Since lrr
and cytindrical components to find tlre acceleration of P s'-r-t- trartslating franre ?
say i. jlv.r-t- ground is the sunr of these accelerations, u'e need to express each of rhem in a common bas'rs.
m/3, .i = -10 m/s?, .g"= L2t2, + g! = 2'42, :+ y'' = 2'4 m-l '
=
y{=tamc-1.2, y''=2-4tr-I. P=(l +itzftzlly*l=l'588m' o:50-lgo'
:+-arr=0.5rn;
(l)
ga = sg = O2er nr/s,
se = Se + (33/p)e. - -r0e. + 032'5t9g." m/s2.
(2)
e, - cosa!*sinai- 0-M02i+0-76S2j, 9. = -sinoi* cosc'j = -0'7682i+0'6402j:
o-. - -6.il2li-7.666j m/s?
Substituting (2) in (1) yields u.e = 0.1280i+ 0-1536j m/s'
0.2
t
satne :+ llo = !!e, ,., = o
^'
Forcomputing9.p17..6=0,ii'=i=3.rad1s'6=ij=2rad|s2,i=0.J=0and
For all poiuts of [ranslatiug lranr1 u a.ud g are
-J
,..:J
Solution Bearing
6
fi,J
1-l
m. :+ i= -0-lsind0 rn/s, .+ f :
r=0.2866m, i=-0-l5nr/s, f =-0-879'tm/s2'
+ atd=30o,
!
llplr = i9.- + r/'d +:g, = -0.15e.- + 0'&59Ego.trr/s,
gptr= (i- rC')9" +(?r:f +"d)%*?9" =-3.4599.. -0.32689o m/s3,
=+
rvhere g. = cosdl*siuqij :0-866i+0.5j,
ea - -sinpi+ cosoj = -0-5!+0'866j.
J
.J
JJ
Jil
Di
<J
Ir
ii-t-J
d
r=0.2*0.1cos0
Er
.tgt
(4)
(5)
"-J
J
f,r
v,j
.J
:J
Substituring (5) in (3) and (4) yietds ltptr= -0-5598it0-6696j nr/s. aptr -2.832i-2'013j mls?'
Hence w-r.t. ground,!.p = ge * !?lr = -0.4318i+ 0.S232j mf s,9.p = ge * g.e;r = -9'253i- 9'679j m/s3'
Tlre FBD of the pin is shown iu Fig.El.3, where F1 , .lV1 , N2 are spring force. normal reaction froin the slot
and normal reaciion fron'r the curved support,. For the given conditiou A'2 =0. Equationof motion of pin:
F
- ntap, :+ -Fre.aNr9+
q
-O.1si=0-l(-9.253i-9.6?9i)
(6)
;
_d
J
':
-!i
J
.J
J
be ottaincd
'^-:--' "q-'=
-'-;'(u)'
' - :oa llr can
irom L iegngonen3s
ilr
,1er,
r'l:
' t_.,
_ o.i1ssr11O5) :o-r(_9.253x0 s66)+(-9,?9xg.5[,
can be obtained
,r,,.u*t"*.,"o I
Exanrpre
1.4
by
of t1e springis given
&
=
F1l6
directly by fcrrnt
*, . & =9-z9ls
(a)
Apoinr-",T-1.:1:::Y:1+;,:Xf*::i,-#;:
It is A*.'ii"i
o ez
u
]- -,,
c6
x:
fZJl.
':ff-e"
Ir
Gtg-Er-.+
r = i' =(-Rsinder + Eccd
"qe)d'
* $=-al{R2 +"}'/"
+"'l=7iii;+ct}r/2',
6,'
(E?+c"}r/?
1'={gl=
.
- Itsindee)d?'
a"o'de'+.':)i;i!;:/c:sesr
*
(-Esincqr
ir
tr= g: +
'
qqs(r)_(3)
+
:=11}|?J,ff5J
(b) We express 1in
*
tetms
r=
of e'"
e':
R. t - o'
"::{:i;;_r,r;,
r = Rcosdel
*
eqs(a)-(6)
(1)
(2)
(3)
+ c?),
_
+ (_Rc6de, nsin431u:11.4i
Rsin 4gz+ c|ps = R9'-
* cae-'
: = cQ'
"r'
"f.'
,'^=
'.=
^t,=
+ c2)Lt2'
+-";.t't''
:^:"''o'
+ Roea+i cde''
*
ie'
+
-R62
r6)e'
=
(z;d
+
+
ot
;: ;j(;7.u,f';l'*r;."!"i'n'
-= ;- i"*
:
=O:g4&;p'02=:11'+l/:'
- -'
+
Esin
;:ffif J#;;;;;;"rr""qtexpres'onsdirectrv
i, o, = -;;""u e=' R.cs 6s, +
soturion
..
+t')t/''
=+ g= (Ecd+ce')u/(82
;7
(4)
[]
al
ii
1
!
+ ce',lol(R" +
"')t/t'
e= -ll.o2l(R2+ "')lg. - lft+
.
+
+ Rcosegr +c93)dE/ds
(-Rsir{g'
drldsq
o,
>
(c) Given d$ld.s
1, + d$lds --.l!ot * "t)'/''
le,l = (E' a "'1't21a61<lsl =
(7)
g=(-Esindg.r*Ecos@g+ce.)/(R;*t'f'.;-
D ac;o,-'siroq,)
- rlsin og.\l(R'+'=)'
i)'t' - (-Rcosee'
I?sia{q'
Xiii''itt'U+
(-.R'cos{e1
<Iqlds=
p=(R?+'"\/n'
rlp=ldgldsl=R/('R?+c2) +
'
g,= P(tgrlds\= 1-cosdgr -sinS92) = -9"
+tt)] 9"'
g - -oq +{u'lp)e^='aga+[r'?t?7141
u = ua'
.
'*n';,: :1,iil',X$":1tr|,:l lJtrl?;'
one roratiou
berween tr," t"ng"ni ono
ocis: p =2c.tr.Tlre angle
cosa
=q'
9d
-
Rl(Rz+
,.
t')'/''
1=
1
(8)
(9)
pitch p or trre
the axis is carre-d' trre
angle o:
l:.:1.'0"'
i *'oer) is called the helix
-.inrJ,
=
tauo = cl R= pl2rB-
Helix is
pri'cipal radius of currtaturea.d
augre
rre*x
pitcrr,
of
constant varues
The given circurar herix has
animportantcurvewhiclrisusedirrdrills.scrervdrives'scrervfeeds.etc.i
ru
,l
-.-
u
U.
4o
i
:a
i
l
I
:,
REMARK
egit€ oft€u in connected systemilbodies in 3D motion, the information is provided for the relative
rates of rotation s*li, ,;lili of body i relative to body j. about a:ies fixed in body y'. It is preferable to name
the body w.r-t. which the inform-aiion is soughC as F. The body rvhce rotation is giren w.r.t- F is called
body l, the body wlrose roation is given relative to body I is named bod-v 2, and so on. Then
4aV -- Clrlr
*
9f-e1e
* uzlr
\
er?lr
t!4r=9lz1r*911r
9s1r
'-r
= Artr * (cr{r + !4rlF x gqr}
tihr = s;qr * ()42* szlr x zstz)
grlr = *-qr * (ri:<s + 94s1r-x_144s)
_,etr=Lh1p*€z1r
ltqr:
;-
-F99slz
\
*9{els
:
t4.rs
g--J
i
llttr
uay= 6rlf +(Oq, * lrrtr x glztr)+(c&f? +
grztF
x ssr:)+(el.rtg+-?t".x !4ts)
I
\:J
,'
u2
an-.
-.....,...--i
,4= itr+ (c.?tr +91x
us = glr * gl:tr *?*42
*tr)*(ri3:+ g*
u41J
+ (it1e+ eb x €r1r)
(
rli = drr + (estr 4 glr x laetr)+ (dtr, + !{r x !4qr)
= 94r * (Or1r + s.r x !&gr)+t=rqz *(4r + qlr) x <t:l
rrhere F has beeu.dropped for the entities rv.r.t- F for convenience- Stud-r; carcfull!'the uudcrlited {irst
ierms in the cross-products. The firs! teuns irr the cross-products in r-- arc the arrgulirr rclocilies relatiae lo
thc basic fiv;mc F and uot relative to the prcvious body- If tlre bodies are natued as described and the
aogular velocities are added up starting $'ith body l. body 2 rv.r-c. l- body J rv.r.t. 2. etc-. then the first
Lerrn, in the cross-product appearing in the derivatir.e of nth ternr, equals Lhe sutn of all the terms to the
lett of tlre ath term in the or expression- Ttris procedure has been follot'ed iu all. exanrples.
The velocity and acceleration of any given point P is found by successively finding the velocity and
4 rv-r.t. 2. including the retative
acceleration of points on tlre a-xes of rotat,ion o[ 2 s'-r.t. l, 3 rv.r.t.
.2,
velocity, relative acceleration and the coriolis acceteration ternrs if [here is relative moti-on o[ such poirtts
rv.r.t- [he previous. body. If P is either a point of body i or its motion retative to bod-r' i.isokno.vn. then up
ar.d ap are determined in ternrs of a poiut (say Q) of body i. rvhose Sa.qO hai,e already becn deternrined:
l+
lp: le + cl' x QP * !-rti.
9p = 9e+ it x QP + eix (st x V) + 2*- x gpli -f gpl;,
rvith gp1; = jplig, Qpl; = Splie. * (i"rlr/ prle^- The last ternr in the cxpression of gi ind the last, tio
terms in the expression of gp being inchlded only
if the point P has rnotion relative to body i.
governor
(a)
(Fig.EI.Sa) is used to couirol the speed of a rotating body. It is
A fgba[
Exaruple 1.5
rotating aboui a vertical axis at aogular velocity qr and angular aceleralion rir. The displacemeat of the
sleeve controls the angular speed by a connection to a feedbacli device- Find the angular acceleration of bar
.l{B and the veloci[y and acceleration of poiut B of the ball at t = al(p, if the angle 0 made by the arms
with the vertical varies as 0 - 0o*01sinpt. (b) A heticopter turirs in a lrorizontat circle (Fig.Eti-5b). Thc
point, O in its body moves in a circle of radius ,l? at speed u, which is increasing ac the rate of i at 1 = r/4p.
The helicopter blades oscillaCe rvith A = 0o * 01 sinl. Firrd.the velocity aud acceleration of lip B of the
blade relative 0o O at, t = rf 4p. (c) A spacecraft, is rerolving about its _aris OD. having a fixed orientatiou.
It
a
I
.r
-t
!
t
tj
tj
u
tj
l-.
L.
l_-,
l--
t;
t--
:.1
:-:r:ir '-
:-'
aL
a rate
(Fig.El'Sc)' Its solai
r.r
l--
u
L.
rl--
ru
IJ
u
u
r_
E
E
u
U
L,
'b-aoets are opeoing
at a programmabte5ale ci'".Tt i1lhg,so.P!:
,t I
.
'<:'
-Find'the
:
."""i*.""j"."rlo..oftl,".ol".panel2justbeforeandJus0aftcr0eQuals0o.-,..'...,
b
oi
.i'
?-
-6
)o
ti\)
u
Lord
- -_-A_s
n
H
-----r^.--;j
*
- '
-'t
5
\-t
rLr
\-
(c)
F,'g.€I-5
<- *.be 6xed to body 1- Body I
problerns are the same kinematically' t*ti 'l' t
r physicat
rrrs 3
l/rr-Y''v'q
Solution
sotutron (a)
\o', The
or-ic ii fixed
6va, to
tn body
h.dv l.
r
-r-.-r about
^L^..t atis
rotaies rv'r't' to bodyl
rotates rv.r.t. grouud about the fixed axis !' Bodl'2
L-
lj
]l-l-f--
--iP-ri.\::i..i,.i..
:,.,
,-;.i${*-i:i}I+
=trh, * =-rt go =0.
-b=
grr x0il
gr
o, A
arcon body
=0, OA=q:
t and points -{. I
!-t=la*.:txQ.A.
lta=!a-f -:zx-E.
(2)
+;!'','
t3)
.::l i
$'e successively cornpute
are otl body-2' Lrsing eqs('t) rc (3)'
gB
qh
* r/, x OA *=r x (*r x QA-l'
x (91: x A-D)'
= a,1 * Q, x .48 + *ir
(4)
= qt
Q,
tt,x -78- -ar6sind!+0bcos0j+06sin0L *
(1)
;{B =6(sin0j-cc0b)
-tik+6'i+-'h.* rii-0i+"'ai
rr+{0i+
The points
go
x(szxAB) =
-u,ibcos1!-(0?+&'2)t'sind
(5)
I
j+O?f cos0
k
(6)
t
= !a=-(o+6siud).i+00(cos0j+sin0!)
as:-12!106cos0*t,(a+6sino)lr+ t-r'a*0silld) -b62sina+&dcosdJj+(602cos0*66sin0]!(7)
(8)
rvhere 0 = 0o* gs sinpt :+ i - fllcospt :+ 6 - -p2l1sinpt
(9)
6=-p'o'l{2
- at / -zrl4p, 0=0s+0r/'/2' i)=patlrt'
4*;jc.o1lin q6
u-ufR,b=itlRand,a,eBreptacedbvt'so,aao-,,
i;;";;;;;;;;;;.';;,',"asinpart(a)rvirtr
... , -r---i-r)itto\ttntrtrl
G'-:$:,1
a
u0ld0)(d0ldt)- G
i =.,. *o.:i obtaiue,r
e:.
rvirh
(a)
0
fi':::ffi;;;,,." as i'
=
";i"g =
j
Stud1. the various tenns in eqs(6) and (?). The ternr
l:l
at 0 =0i,
aOlaO
=,,to/Aa,'
+:
i)
=ti1Os'
!:
is tto! an.obvious oue'
at0=0J. aila=0, + d.:o'
t
rc.l/S
F"i/s''
!
i
Exaruple 1.6 A box rotates about a fixed vertical axis
A
and a yoked artn rotates relative to the box (Fig'81-6)'
Trvo
disk rotates relative to lhe arnr at the tt1o th611'n'
sliders move relaLive to the disk in straight aud circular
have
siors with cosd = 0.8. The Points P and Q on these
relative
speed of 2 cmf s deceasing at the rate of I cm/sz
poiuts
of
acceleratiou
and
velocity
the
to the disk. Eind
P, Q and a Poini r{ of the disk'
a
solution In order of successive rotatiot'ts, starting rvith body roiating
Let
t' be fixedrb-the box l as 5[ewn'
3'
i'
i-'
and
2
l,
rroiio
arm and the disc as
uurr ,i""*oed
we narrre L.e 1;,.,
l!'"tff""rn"
. 6r - ---:- ^a;.-+i^6
"'''.:::: :,:".,.--;:;
rvhich is fi:."d ti thd ground.,The a:<is of totation
k
is
grou[d
to
the
relat'iYe
1
box
of
of
rotation
axis
Th€
OO,l
a-tis of t:tLl1on
of arm 2 rela[ive to box t is along ! which lt i*ud to bJv 1' The
];9i:::.*.
:'*
IL
i
.l
E
-
the arm 2 is 6xed to
tnelri'e. et
tU" glven insl'ant 3.-
l:
\fle successively compose the angular vetocitie"
rad/s, Or = -k tad,lsz
@e= tlt-4L= -4i*5k rad/s
(-4$ =-2L-20j- E radls2
4= tix-zt+e{r x({!)--k-2i+5Lx
9s= 9z-3i= -4i-3i+ir..a7"
, a.\ ,6:
13 I 20 j - $ + i + (-4i+ 5 k) x (-3j) =
Qt i,itt + !* ttzx (-3!) : (-2i go = 0, QE= 40! cnr :
Poirrfs o' B are on body 2 rvith co = 0,
:(-4i+ 5$ x aOi= 2fi)j cm/s
Ba = !2o * 9le x OB
as - as *,'t, x OB + gz x (9i x @)
qr = 5k
,n:
lei +
ll
L radls?'
.]I
cru-ls?
20'2L cuVsi
u*pro"t ,4,gein ter,is of poirrt B of bodf' 3:
=-15i -
'-.LJ
lL
U-J
\
i' *:-L
BQ=-10!+5kcm. sets=2cn1/.s, ;eF:-1cm/s3'
"=
(;ery/5)9* = - i- 0-s! crn/sz
; = SqpQ = 2i cm/s, oQF = islr& +
cm/sz
30j 30 k cm/s, *t x BQ-= -95i- l?si - 190!
x BQ
sj
r.:J
J
I
r
-J
sb x
rve
'::J
!.:J
e=-cos0i*sin0!--0'8i+0'6k' gPl3=2e=-l'6i*1'2kcnr/s' lle1z=-e=0'8i-0'6kcm/s?
q.3x BF= 50j*30Lcm/s, Qrx BP = tlbj+ 190k cm/s"
{!$ x En= -340i* 120i -200L c1/s?
(--l-6i+ 1-2t) - -7'2L-6'4i- 9'6E cnr/s:
2,.,.s, x,els = 2(-4 i - 3j + 5$ x
' llp : sa + ss * EZ! !r1s = -r-6i+ 250j + 3l2k crn/s
x
se : 98+ db x B,P+'e"x (cb x 8.Pl +2er3 {r'1a *8r1s = -1346'{!+r$'6i
,::J
,tJ
lrodl'3: E-Z:1o!crn
pointphastelalivemotionrv.t-t- body3=+$,eqipress1rl'1epirrternrsof PointIof
e has relative .rotio* *,.r-t- body 3 +
il,J
-J
=(_21_20i_k}x40i*(-4i+5k}x200i_-1000!_40jcm/s?.
m:
Points B, A areon body 3 ivith bA=l0k
i\'.-,40j- cur/s
cax BA = ('-si- 3i+ 5LI x t0!= -30i*
,
fu xBA=(l3i-rgj+ll!)x lok:-Tt-130!crn/s3 150i-250h
!k x (ss x BA't:(-{i-3i+5k} x (-30i+,{0i} = -2001+ lle = !!a+ sis x BA :-30i+240j on/s
320j -'*':"1:t'
s,r.= eB+ db x BA+!* r(ek x EA\= -1390i-
point
il
r1
.d
, tl$
i*j
-
l
,.k
1u'
,<l
so = sB + ss x 9Q.+ s4r. - -r3i+ 1-?0i - 30k cnr/s
x (!4 x W * 2,esx !4F * Lets = -856i
e(2 = s s + L3 x BQ-+g
r--
-
390i
-
103'8
k cmfs?'
ions of a space vehicle to study tlre tolerance
sir
t'he ftight condit
to.sitnulate
used to
'ont rifrrac i<
is rrsa<l
A centrifuge
at angular speerd r.r1 (Fig-81-?)- The
astronauts. Its main arm I rotates about, a fixed lertical axis
Example
;rJ;;
i.:
1.?
to
t'rz relat'ive to 1' The pilot is strapped
cockpit 2 rota.tes about an axis fixed to tlre arm 1 al, an$llaq sPeed
pilot is moving a btolk' of mass o-8 kg lreld
a chair 3 which ro0ates.ai angular speed'ar3 relative to 2. The
iai,r" R relative to the ctrair in the plane of
in hls hand, such thac iis centre oi mass Q tio"ribes a circle of
'rind the acceleration o[ point P at the instant shorvn'
symrnetry of the chair. p is a pcfrnt in the pilot's eye.
i.
_
.1A
:
.,\J
:
-
,,,1-
+-
CI
.l<
Lt,-
J*:.I
-
t--
*fiff
iil{;:r":".$f .
*r".,.ry.
*f,,'fi$,-*Iilt'"tffi
\,r/
i\-\2
:------:-
7/(Y;)
t
;
;
t<e't z
;s <.\fBi-cLj=Ji...----
c
','';m:=l;*:5;T"':if:.1i".#":'f ;i*i
sorurion*.0*l,i,Jl.*1*1ffi
t "il';;;i*
=,"?.#;
*
5-:l:*:,:l::
j{'i:l1':rytl$tlll**:t*:'
o:'*
Ji:;:r:r;;-,
:
-
relati*e
Tlr" .o.kpit, 2 rotates
z L .toug
ff Ifl :,,:ffi
--
or
rr" trt" augulat velocities
;,;r:*
\
::
I^- = _cc,itsioo!=;i-0-6i+0-s!
"
g=sio0!*cosoL=0'8i+0'68'
co"1=0.6, sino:0'8'
-
i;=2..:,1;';--'i,'5'-^'.*,::'3
-
The points o,
-lE-
sa
zreou
=::;':il;'i;i:['ja:'ii.
3'
c. p
body t aad the poi*ts
rn' CP =^'oL+
=cj = 10j
,'*6c+-' * t*t' -
OG
ateou body
-02i:'
6s=
0 s L)
tj
t_
t-
+
k'1'r"'.
7 e6
.'-:^,
ib and gi'conrr].*
.
02j + g'6k nr
1(0Si+ 0'6H = 0-8irofl = -a0!+sL*/'3
roiizi.i"-
Ti=;;;x
bodv 3' Hence
knorvir relative to
The morion of Q'is
=':':'
\\'e -ruccessi*err
(gb xe$
g, = gc+ rit x CP + ca x
a'zi-+isza$'+ (-'?'ssi+5-32i 5 $ + (o'032i+
(-40i+
=
03eE */"''
= -r.rnr, -=*r*:-*
L-
've
{i-16$
-
expres5
l
---^*ie*r poinr
co'l'err
c ..t :i
Tl:::..:a
6(03i+0-6u=0,1i+0'1!-0.]E*,,,,,
3err = n
0'8i+0:6b'
*ll"t u* =' *1"' '=,'=
:
:i:'
:
r'-
=.;A,:;#::i:iXX.;:::.i!+,:2!m/s3
i:=;::
* i *|-b*erj
t--
:
(s * 9l +Zwx tQlr-t coF 130i t'u L}
se =
2'152 $ + (-1's{ I i
*
j
+
s
t-'s*'
ro,oi-u"tj-12ji3EY-a0 -* H
=\-=
"ii
rj
l.-.
x
r
+ (14'4i
IJ
Ia,the FBD of the
t-
'
1-.
L
L
L
L
=9,
c
s,=soa
L-*
1-
* ii:
i=?"*;
L
j=0'5!-i*61:.0^:
L=msa :+
0""0,
E1
-
1'6
j - 19'2E
)+(r.6i-26.6?i+1ril=
shorvq
is the
!n Fig'Ell?' Fr
*
-0'8s!-0'8(10'20i-'6?'8j--re'zg$
tt{
J
J
a
:r'ted
force e-te'o=-
ei
'-t
-=-..
equaiion of nroti'xr'
Tlrc
gilotthe
-- '
-- - s3.g2j - 6ai{b
f' = 16'01!-oo'or1
N.
*fr'=xy"-
x ' y't
v\a/
- y."rv ,LX
-.
n-
incline'(Fig'EL'P *-""f"d
Exarnple 1.8 A tank'iihvTrilio-*
*r.z r"tfrt'* the given
of 1g km/h *rri"r, r-J"""i"rating-ai t'c-..r.
--_^r-_
i"l*: : *" :i::
ffiJ:fi,."#"i*,"*.r,
odls rvhich is increasing at the rate o[ 1 rad/s2' The
!
"0".a "f
barrel is rotating,
.l.
tO'
- \.g-rt"
- tt+
:"_1ri,:i,',;:::::'Wffi\L"*,?
r
relative to the Lurret' rvith angular speed of 0'8
a--
-*;:.":tlfr *T:i:.'::"#{*,i!;ll#:{',,i'*:l',$ffi;':;t'*
the
3m'
Find
The poinu.4 is at the'end-of-the-gun-barcl-of lengttr
H6ZE-*"-",."
to the ground'
velocity and acceleratiou of points r{ and P relative
about an a-tis fixed in the ground'
Solutioa [n order of successive rotatio.s. starting rvith body rotating
6xed to the turret
as bodies l ' 2 ancl 3' I*t i' i' E be
we name the tank body. the turret and the gun l"trel
ortrre
o"*", *,.r.r. ro rrre rurrer a.d k arong trre a-tis orrotation the
plale' is a vector in the inclined plane and
tu6et rv-r-t- the tank tlody. L is lornral to t[e ipcliped
.j
accclerati'ons'
conrpose the tngula' velocities' a*gular
I
vertical plane t'rouglr the gun barrel- lte successi'ely
velocitiesarrdaccelerationsasexplained.irltlre.retnark.above:
;:fif:"i:;[":;i'J;;;;;;;;i.
e='sin15'i+cosl5oj--O-2588i+O-9659j. so =(l8x 103/3600)9=5s=
cro - -0.2x 103/3600)e =;2e = -0'51?6!' t'932!ru/s?
=0. :1. = 0The tank body I translates relative ro the ground +
I
/
I
-JJ
I
,,
..
. \r_i
'i
t.-t
l'294i+4'830j m/s
.gt
tt2.= ltr*2h:2k
'rad/s
izt= gr+f k+i4r x2k:8+ k+0x'2k= krad/s?
0-8i* 2 k rad/s
ut = I4+0-8i=
x o'8!= 1.2!+
db = i.z+ 12i+ r^r: x 0-$!' = h+ l'2i+2k
-
-.
l
points O and B aie lixed ou ttre turr€L 2 r*'ith OB = 0.6j nr :
;, --<I
e2xQB=2*xO-6j: -1'2im/s. ' ib'1 OB-- !x 0'6j=-0'6[ur/s?
szx (* x @l = 2\ x (-l'20 = -24j nr/s!
la = b* eh x OB =-0-09{i+4'8!0i m/d
l
rn/s2-
,-"a=go* rizz1@-+.:r: x (,*a-: x gg)=-r-llti-,{3:l2j
r._
-r.!:: . : -1r: _- PointsBandr{arefixedotrbatrel3rvith e'=sin{ltcosdh=tl'Sj+0'6L' &=3,g-=Zfitd+l'8knr:
*tx BA.= (0.8i+ 2$ x (2-qj + l'8U = =4.'8!- 1-44j + l'92! m/s
1'152 k ur/s?
; - e . ;4 = (0-8i+ 2E) x (-a-8i - l'44i+ l'e2u = ?'tsi - rl'r36j ;
Q"x BA= (1.2i+ r-Oi+ \) x (2-'{!+ l'8!} =0'48i-2'l6i+2'88k rn/s3
m/s
1,92L
'
gr = ea + c! x BA=.4i00it3.-!9i+
aa.: eB+ tib x ItA+ !$ x (srt x B/ill =2242i- l?'63i+ t;t8l' m/s?'
,
poiqt p of camnon ball lras known motion r.r.t. ba,rrel 3+
.EE-=
!ZA=2ij+1.8k,
:;:;:;
Notice
that
: ff :
gp =
we cxpress 1gp,gp
eep=600C =480j+300k
m/s,
in ternrs of point
gp
= -e+ZCax S,"l.f +91n'
,.1 ,...,;.1 . '
{5
-
-
I
of uodr
= 120E: - 96i+72!
epls
"=
-,,,,, -,,s?
;::;il','f-.l;:Uf
?T
!A+s.pl3-,
.-
.1
1.6j+ Lrad/s?
!t
6
m'/s
3..
J
,.q
i
.<
.i
.<
,i
t2
:,<
L
.<
.-,
:: -{
.-l
i + e4 r ? r -/"=
.
<
r*
klj-
L.
I
..,
L^
l*.I;:
L-.-,
I
Exarnple
1.9 (") ifr"\gio,b"Isupport o[a gyroscopc rotates
r l?,a1s
l.J
a.5rls
(loiqs-
\
1,
o'
'7c tl s
tspi. r*c)
o
'i1=2,1s
'6;^frq tr:€
ilt
L3
(bdaf,z)
zu
tlt
i
I
t'
- (a) Aff
I
L
L
L
AB at thc rat ss;hown
,ls'
'l'i=rrtrl
gi-bal
l_--'
IJ
]J
abouc a fixed axis
(Fig.Ef .9a). l'he frame m supporting the shaft of the gyroscope rotor rocates rglative to the grnbal at the
rates shown. The gyroscope rotor rchr€s telative to frame m at, the given races rvith 0 = cos-l : O-& Find
at this instant, tlre arrgular vclo<ity and angular acceleration of tlre grrocope ro0or and th€ velocit5r and
acceleration of its point P relative to ttre ground. The kinematically equivalent problems of a spinnint top
and part of a robotic device ate shorn in Figs.El.9b,c. (b) The telescopic arm of a robotic device, shorzr in
Fig.El.9d, is increasing its length at the given rates- Fiud the velocity and acceleration of a 1rcint Q on it
'Jszis'
relative to the cround.
*"ffitI''ffit?fq
tG
e=atY/s, ' o?=
o.5 m
O?= o'5
.--- y
! e=at,/s''
-t-P8:o'i'p..u.e,.9 jf
-Lr
(bJ
tl..
Solution (a) In order of successive rotations, starting with ginrbal
:'"5u
-a"
o!
(dr
tcI
srtpport rotating
';+tls
'r.l-'-
in the ground, rve name the gimbal support, thc franre ar anrd. thc g-vroscope rotor as bodies 1, 2 aad 3.
Choose *-". i; !, E. attached to body t, s'ith k along its fixed a-ris of rotation. and ! along the ads of
1_
lL
L
u
L.
L;
L,
L_i
l
J
J
5
rl
5
er.7
rotation of body 2 relative to body 1- \lte successively compee the anguiar telocities. angular accelerations,
velocities and accelerations as epl&ed in the renrark above, 1'hc spiu of the rotor is abouG the a,\is
e - sin0j * cosd k = 0.6i + 0-8k-
llr:4k taA./s, (2r: -0-5k rad/s3
93= ttr-0.2i= -0.2i+4k rad/s
,Lz=gt_0:Ii+al1x(-0.2[}__05L.-0;ri*4.kx{:0.2,=-0-li-0.sj-0.5hrad/s"
sb = !h * 20e, = (-0.2i+4u +an{0.6i+ 0-8 k} = -02i+.12i +20& radls
.t
,.-- 3 = 4 - 2 e. + u, x 2O e = (-0-1 i - 0.Sj - 0-58) - 2(0-Oj + O.St) + t -0.2i + !) x 20(05j + 0-SU
- -48.1 i+ t.Z5 - 4-5 k rad/s"Points P and O are on bod-r'3 rvith
g -0,
9o =
0,
OP =0.5e = 0-5(0.6j +0.S!)
gsxOP: (-0-2i+ rzj1"2051x (0,3i+03!1,=.,:l2i*0.0Sj-0.06k nr/s ,,
=0-3j+0-4k
.
rr:
I
,:.i{i'
!43x(sb xQZ)=(-0.2i+l2j+20t)x(-1.2i+0.08i-0.06k)=-2,32!-24.012i+14-3&1km/s?
c,t,rxOP = (-48.1i+ 1.2j- 4-5!) x (0.3j+0.4k) = 1.83i+ 19.24j - t.l..t3k rrr/s3
W -- b + !q3 x OP = -L-2!+ 0-08j 0.06\' mA
sp=!b*drsx OP+q3x(r.ls xOP) =-0.49!- 4..772i-Q0.l6k m/s3.
Points P and O are on body 2 also- Eence reworking, rvith g4", gi5 replaced by e42, 1i1s in the above relatlons,
would yield the same values fict g4,,gp- Chect this fact, as an exercise.
(b) Point P, on che rigid e"xtension of arnr 3, has given motiou rv.r-t. bod-v 2 + s'e espress -.;,..gp in tcrnrs
of point O of body 2 with g, = q, !Lo, =0, OP =5$9= 0.5(0.0j +Q.SL) = 0.3i+ O.aE
T:
x OP = (-0.2i+ 4tr) x (O-3j +0.4!) = -1.2i+ 0OSj - 0.06! m/s
t*x (ttz x OPI - (-0-2i+ak) x (-1.2i+ 0.08j -0.06L) = -0-32i - 4.Sl2j - 0.016k mr/*
!5 x OP= (-0.1!-O-Sj-O-SL) x (0.3j+0.4tr) = -O.tzi+0.0{j -0.03k m/s!
t
tr2
r6
i
i
!
:?
:
t
i
l
rq0-45,= 0'4(0'S j'**OSS
024i+
=
=
;;;=o.oe=0.6(0.6i+0.8k)=.o3,6i+0j8Em/c....
ii**p= (-0'2!+a!) x (024j+o'azg = -1'-n'i:o'128i'-o.os6L m/s3
032j +026k m/s
e.p = lo + 92 x OP * Wu- =1'2!+
gPl2 * ePP = -2ALi- 4'2s4i + 0'338k */"''
ee = et +,iz x Q2.+ sz x (t!z x QP) * i'rx
:
Points P and Q are on body 3 with PQ =O'li m
eep
% x fQ= (-0-2!+
12i
0'32!m'/s-
+20$
x 0-'l i =
2!-
12k m/s;
l|-3x(!b*PQ\=(-0.2i+12j-+208-x-(2j--12E)--5{':ti-0'24i-0'4L
k ru/s?
fu x PQ = (-48.1i+ l-2i- 4.5!) x 0'1!: -0'45j - 0'12
03{L m'/s
94 = 9r +.g5 x PQ = -L.Zi+ 2'32i 4'974: - 0'182k nr/sz'
s4 = sp+. tizs x PQ + s.sx (* * PQ', - -56'81i-
:a
't-.t'f
.(
_{
__
:
.-
h
m/s?
t
t,rr,
S
-^i
L
D
fixed.c.l.lirrde'"i:'::::::.(:;':-".1.T*
radius r rolls lr.itlrout
1'o^":.
l- Fiud the.radius.of curr:ature
ott
-.
--^t^-^-.r^... ^r'r.-.^-i poirrts -{,O' C,D r cvlinder
Find tie..,elocity a,,d acceleratiotr of'material
of the
oath of points C, D in this positiorl'
-
l'of
r'rv A
Exarrrple
r'xz.'rlPrs 1.1o
" cylinder
--
-- r -'-rlb\t
(gl
h.
(=oo
U",V6?&c
D
t),.
i---5-'
l't.to
,a
*
A.
.,--_
T, -11t'"q
I'
ia
:-.
-_
:
-q
--
'8
'-\
(eJ
.
F
i3-€l.ro
Solution Let point B of body 2 be in cotrtac[ rvith'po'int
r{ of bo y l'-C-:iroose i, i
il
plane of motion- Let
u6audig=rlgbet,lrespeedarrdrateofcharrgeofspeedofpoirrto.Ilerrce
gB=0, rrg=9, ur=utk,
ar1
='r.l1k,
O':1
=-r!
OD=ri'
OG=-t!'
Po=R+r-
lrodr- I taki[g tlrc geomcirk csefrc as
velocity and acceleratiog relations of trvo poitrts on the tigid
,all points of intsestis the satng 91 x (r^r xg) = -r.,?c- fire
the reference pohrt. since the planeof.urotion of
kinematic condition of no slip,;g^ = 9.8, *
_
W.e use
Da.=b* th x OA=tioi*a'rhx (-f0=
(r,6l
--'tllj-=!:a =g'
(r)
-
_<<I
.q
?
:+
W = (,lrJ'
:&e
-
gO.=
ga1* Q1,4'OA
oO!, + (ub! p.l
-
ofilftlR+ l/'lL
"tiOl1,=
gcl
s,- ;["ir'3 I lR
+ r)l [ + rir
rj'
(2)
*6rb x (-r':il - -";(-'il
(3J
ttrat though the velocities of trvo
The acceleration of point z{ is not zero but is independent of .i,,s. Notc
since diflerent material
points in contact are tlre same at er.ery iustant, theL accelerations are no! equal
is nornal to the tangent
o.f
points_ make contact at different instanrts: se * ga. Horvever. the acceleration
I
B it conlacl haoc lhc same
planeof contac[. I{ence, for thc cose o/rolling w-itbout slip'O1 poirfi '4 4,rl!
retocity'of the con[act point A- is zero
componettt of accclemtiot ia tlrc tal.lgenlial plr." oS corlacL The
poitrt'on thc cllitdcrshodd be
but its accclcrcliott is tlol zcro orL"J rrr = 0. Hence ltc accclcnilion of a
conlact- ' '
c?trcssed in tctms of lhc aconctic ccttrc, rcthct lhan lhc.poirIgl
.tt
p=
s*
gI x O D = utpi *t.,r'( *'r'1 =-'l-i'i.
17
_
..
_
<{
_
-:
._
<
.I
_
tj
It
t
LJ
tj
x (-r!) 1c'11i+.--rri' .
=s, * r=.x OL'>riiiJr'rrk
j'?; i = - iwir(R'+ 2r)/il +')l i 1 ? L
D I atlQD :s, + a, t1 ; i
"
,e.D=**?*?
=p,r-;;"r/(E+rli+pri+-f";1
(-Il
="r'.|,;:t;#;";11'
*
*rirti
' 'i(:'l?
g. et+ eirr x AS'-vio!: sa
ltc
l...
tj
oilco=.=o'oi=eD'(-il='^'fr'(E*2r)/(E*r)' (4)
+ p, =4r(R+r)l(R+2r1,
; -r- iltrt'
;\/Jf.
gn6-= ?i+
'
l- (il itrt
azs =2wlr", sc = vll
r)''+ t" = 2Jir(n+ r)/(E+ 2105)
!\1fr :''l'f*'*"tli(li +
,Z! p" = lb' *rZ*'i-,*
lj
slip:
For tlre case of impeirding
;""iot
of ortact or" oot .,quoi ;' A"
t*
IJ
,. ffiffi::T,:il:,1il:'ii:Trx'nxed
' ":':t':''
.':;,
=
oi=4ulf, *;-;'
u
ll--
ntl !'hc-comPate
cornDottcttls of 9;o
ge - "t'B' afld
'hc ap an''
ie" to = t"tr amd *
nar surface
{rig-er.rib1, rrre'
a.
gr = (l,irlr
gO = t':1t j,
uC = @lrjt
IJ
ii
lhe tatgell'tia'l plate
oad
og
R
oo and e<is(r)-(5)
=
2.
t:']'jl::
rf cllinder I
';
e.p=ugi-, so- lu'!r2/(R'. rIi+nrri'
r[ radius
L
l,-.-
3- If
L
at J{
1- If cylinder turns rvitlrout slip .-:r,3 +acorDer
the fixed axis at A, R= 0 and ep(1){5)
o^
E1'
(Fig-i'l0c]"i${pg- R' ='
L.
I
L
IL
L
L
L
L
L
L
L
g4t=ttgj,
,
:
-R1
1'";E*(rl<.1 *
,,,,
=ti1lt1r- l/Rrli'
' a fixed cyliudtr 2 (Fig'El'l0d)' then r = oo and eq(3)
on
slip
rvithout
rolls
surface
body 1 wiih a ftat
{3}$ For poi,t':D tFig'El'l0d)
q. = ,i*i.,,[p* oe tltisrcsult c6i*it *- "nir"ro"irg
so =eA+ 6-tr
ur1 ! x d!= c'r1di'
v p = 9-e* rar X ;![ =
yields
+
=4r,. Pq=Lfrr'
lrs
iss::e a'6xed.':"t!.:;
rous rvirtrour
ptarte as-that of body.
-sllF
2 is on ttr. s.me=iae of tlie tanger*
as centre of curvature of body
lJ
ii ;'l,l,i, .il1;:l;i;:l.t
]
-a'D
-uig='i1R='d)!+;rdi
(Fig'El'l0e)' then ttre cyli$der
is in-rotation about
G--1
.<t'
ao=Lr1rj, gi =g, . Po--2'' o,=u";.rrrno=fra--.'.ir1i.irrt'
I;;'i; 'i'd l""t""'io"
X';,* *:;: Hi.';;ift::!:i::;':':
t'he *alues o[
or body 2 *ir*';1 ,u1 beirrg
rsoutd
expressions given.ih this exi'.rple
!q *,-r.1.
refere'ce f...r',"
theangularvelocityandangrrlaraccelerationofbodl-lrelativet.ofratrreofbodl-2.
of a lathc
Tlrerollingrvitlroutslipistheprefertedrrrotiorrofvehiclesqnr,rhegls.Forflatroad.tlre.uS"gl5[1avt:
A piniou' i'''ou"tttl ou rhe carriage
trroiion'
tralslat'ory
has
<hassis
t,he
arrd
general plane n'rotion
maclrine,rollsrvitlroutsliporrafixedracktolrovidetrarrslatiorrcotlrecarriagt..
velocity and angr|lar accelerat-iort
shorvn in Fig'El'lla' !.nd:t]re angular
[rechanisrn
the
For
1.11
Example
istlleinstantane:us
ittst'aul'
of rotation of body'3 at the given
of rvheel2 and the insta[taneous ceatre
)]:::t
franre of body 1?
c
*"a.. of.tl6tqtion of wheel 2 rv'r'G- refereace
Vc ti :
c
or_r-$'-.':r,
\tdt; :il-'- _'/
.o,-f:
,,
+ . \\drr r 7':u ;FLe
{,
i<-
f
r
r-
'
5:r.ffr*;ilffi1*,
5
/\
l!(a
^.
rair
(bl
i"
a
foxr,ar
crrain since
ir co'sists
members l' 2
Lhe dth member is the fixed
of
{ ruenrbers-
""u"aolJ;T"I'#;,"T:::*:::::"i:1,ffi
-.^ri^n
::il:"x',tr',X,"1fi
orbodies
*;:5:;.:.1
poi',.c (Fis.E1.11b)
;:::?13:"ffi
ffi$.,fil"l.llT;l"i,lT:f[:ffi;;::;:,:#;;:;";;.,,"*
U.
i-
o,'.1
----'
2 and 3 in two ways in
oi'JtU*
i;;L
.hc same plaae cf,
convenient points of these bodies havi'g
r;i
!4=9-s=9r 4=9e=9t ?:lb
ds:-rt'
=-121' qs="l3k' rirr=tirl' 9{s=o'sB'
(I)
ec=eD+sbx pc-=t uiirx2Dl+!&x_*=5*ezx99'
_,,3 De = so *
(sg * ta1 x E D ui E D| + * x D8
. tLc __ s.o * & x D9-,3D9=
cr-n, ED = -3L' OC-3j cm'
OD
=4 cm. re= -4i+3j
&a x o C
- -3ae' p)
=1Sz-321r/z
to eq(l] and cir2,63 are so*'ed
from trre [rto scarar equations co*espondiug
fro* tfr" trvo scatar equa[ions corresponding to eq(2]' -
The unknowrs
!r3,re,3 are sorved
:+
akx(-30+abk.x(-+i+3j)=o2kx3j
(4)'
--3-si-.\:'?)j=-*'2!
-12-*'ts=0
(3)
i"
-&.,3=-3o1
u'2 :
of eqs(3),(4) is
solution
The
''3: -3 rad/s'
Eq(2}: -12Lx(-3il-4?(_3!)*.i,skx(-4i+3il-(_3)?(-4i+3i)=t^r:kx3i-(_3)2(3i}
(48-3dra+36)i+(36-4t3-Zili=-lut2i-27j
:+
'j : 36- 4r,rs -ziti,=-;-2,' (6)
.,
,.. ,.: \...
(s,:
:_3i,2
: !: L;;r6
t'(6I i'' - sr'''-'.-19'rad/sii' ti'g = 9 *'";
uc (Fig'E1-l lb)'
4
to
Lo gp
vD and
ehc
ar' dre inl'crsection of.eni
'orrrrals
The instantaneous centre / of body 3 is
1:l:::
-
Eq(l):
:+ [:
Fort.tregivenconfiguration..Iisat.o.oszanrd,*;3.catr]alsobe.olrt,aitredusingl:
k"sr:3'
#=#=Frr
W:Tffi:p:l
.u2;u3... d..idid ori t'e tasis
J and of link 2 a'bout
:-t,=W':Y'labout:,r-rr=3;
of rrre direction ofiotariouif rirrk 3
=*
j to !. i.e.. about _ k. This
rotates alrout r iu the directiotr [rotn
link
3
t,hat
inplies
ofgp
direction
The
0.
k- Thercfore
lb. lleuce link 2 rotates fronr j r'o i. i'e', about yierds the direction of gs as srro*,n in Fig-Er.r
Trre sig.s of
and t.r3 are
negative'
of bodl'
c ie.tre Ia1
body I is a[ D, and r]re itrstantaneous
The instanta*eous centre /:r of body 3 rv.r.t.
in-<tamtaneous
gclr alld galr ttu u5 5llontl in Fig'81'l1c' a'he
4 rr-r-t. body l is at E. Hence directions of
D of the tlorutals lo1lclr and-51t'
centre I21 of body 2 w-r.t- body L is at the intirsection
lr'a,rs applies for
acccleratio, of a Poiu[ irt tito dilTereirt
Trre above ptocedure of writiog trre 'elocity a'd
ilr (1):
the mechanismsshorvn in Fig-El-1ld- For mechatristrrs
{7)
r..r2
uB = ge+{,rzk x AB
= (ye *t'trk x o"l)*o:k x AB --vse"
x)E-:;&:'::*t"}lr1l1'fll
ss=et*ti,zL xAB-luiLf =(sa+tirrL *ql-'?QA)t'i:k
andeq(8)
Equatiorr (f) is1lv;d foru2'tts
rvhereg =0,qo =Qandp, =ooif pathof Bisastraiglittine'
vu*"vY
"t'""j,."
-" f : t
as lugl/8, F all tt &no Llrerr sr8lrrs 'rs
the.u obtained J"
*usnitudes of o3.r.6 are trt1"
for it,ita.The
t. .rh" rnugni;;;"or-..;.
ir'toin"d
contact
contact'
\":lt!.'Y"^lt,:.ili::::.':Hffi::#f
o[
point'of
point
"'.'
tlte
the
is
3
body
bod-v
of
ot
cenLre
, 8,4
itlstanl,aneous centre
the insranrarreous
tha[ rhe
knorving that
ug s{, il*;;
Hff'"'lfi
ections of up g,
;J"?ffi;;.il;,;;;,
.A^a
o,s'!L;T,
n/t".
t'oo,
't,
i
!
I
I
I
I
lt
I
,l;
te7
r
{eI
AG r
.
\"
,2\--**3{+^ffist
d !
oa1 - X$l-;ed '!#R\a ,,i*.---€,
trj"-
For mechanisms in (2)
g,ffi^ k17,,,
e-
g,.*r=15;,
urrt,l, ,fri
:
*#Y#
tt.a,*
aB
=
9o
=faa{
*trrk
x AB
t'\1-:
--ttsg'
-:*:-*''t1
t
1
{
I
:
A o..,i€,
.,,ffi:@=.?I]
F'.e.rld
gv
ffi.r-'e1
(2r
+ (ul/p,)g,',J+ 61\x AB
:
-'ie} - uaQ +{u'alp,l*'
a
-(9)
(10)
i
\
i
\
l.
'"1
,^-,1
+
'
.:.
i-"
Equation (9)
line.
and P^ = @ if gatlt ot -,4 is a straight,
= @ ii.p+Aof-{ is a straight line
solved,for<.,1,us*d"oit;;;-igi,*:;n; irl=iatR"t;';l=psllRartd"!:eirsisnsaseobtainedfrom
point of coniact'
where pe
-
is
,
i";iliins
Jt
i;*ot***
centre of bodv 2 is the
ariis
radius R, rvhich rotates about a fuied
A planetary gear train consists of a sun gear S of
r?r'
radius
of
each
P'
*"t"."tion a, (fig'BilZ'1' Three planetarv gears
r, ;;;a;i*
r, *,i,
e.,
u";Lg
thau the
Example 1.12
ar angular velocity
velocity'
g.u Rorritt ir."iau' T[e ri.g Sear rotates 'vith anguiar
mesh wiih the sun gear as rvell as the ring
called the spider' (a)
planets are nrounted on a fraure F
ut2 atdangular acceleration dr2. The beari,gs of the
the acceleration
the planets and the spider' (b) Find
Find the angular velocity anil angular acceleration of
systetlls
transurissiou
irl
used
gear' Planetary gears are
of points A, Q ona planet and point, B on the ring
one or n}\::eats'
jssion has to be altered.
6uev[rv'E"YE by
D accornplished
-' stopping
rlrrs is
an[ereq' This
rchere power ttlgglti:rgq-has
o+. ,zj
fr
:-ffi,'Gr\)w
l'.--'>rR
5.-R
611
4Rr
t \
ijI l!ri \,3$.*_-1 ,.\
.1\%
tz8,
t
V'
-_=q{g
sorrrrio' i;H?;i"raing
o'
r.r r
ot:"
.",t
(b)
coudition implies
tixerl axes i,. ris."-Er-rzb.c. t1e ,o slip
Rt : --oZ R:.
t^r3.R3
(1)
= urlli.{
:
ttedonlltesatnerotatittg6ody,theneq(l)appliesrritlrtbeangularr.elocit.icsbcirrg
relative to that bodYof sun l'
plauets be rd3 aud ara' The axes of rotatio0s
Lct the angular velocities of the spider a$d the
to
relarive
1'2'4
3' The angular velocities of bodies
ring gear 2 and the planets 4 are nroutrted olr spider
at "l and D f ield:
body 3 are &rt - o3,'.)7- t"3. g.l{ - t'r3' The uo slip cooditions
(2)
'
(<.rs
-tr3)R1 = -(*rl -ul)Rl
(.2 --rX& + 2.R3) = (ara -r'r3).fiL3'.
(:l )
r^ra = [r2(It1 +21?a1'tqRil2Rz
u3=[orBr +ura(I?r +2I?3[ lZ(fu* 8zl'
+
{
+
ci3=[ti1R1 *ri'z(8r +24?)] l2(fu
*
*a = li,|la1+ 282)
Rzl'
nrotion, u,
Since all pointsof interes. have the same,plane of
Q.A onbody 4 are obtained in ternrs of poinc 6 of {:
9a
k)=scjr^rrLx 9Q-=,iQ,
l
oiiU CQ
= -Rzi, @=:'83i;;$ince
x (gx
r,)
- ntBi | 2Rz
of points
= -,.'3r' Tlre accelerations
= 9ciroLx
C.:.{
-'iQA,
I
e^= -ri2(r?1 + zRlrL-tW
l--.)
I
iu
lr
-- ff
J,J
i)
r-rl- €
i
-
ori body 2 rvith
@-={fu+
so=9'
2n?)i-- Hence
buc t5e components of q, es'aJ9ng the tangent direction
be, to sacisfy the conditi'on of r'olting rvithout' slip'
L""t"
""-.,.;fU;i"hould
i:
(6)
* lqdtu +2k) - "r:.RJ? I i' r'r
-:,..., sB=qr*orgE*U--"id=-tfi(frr+2fu1i-rilfaizR:)i',
th"t;ir,*-g,,.
tL
'are
(')
g. and G'are poi,*t'sof:bod'v'3"9c is otitairl$J+;,'
wit'h 'W-=lL:aErI!'
k=go+u3k xoC-"'ioc,
+ Eg)li- t'{82 +..'3(EI + n?Ui'
Eq{3) t tO) '+ so=l"fiRz-tr(&
Tbc pcrintso and B
(.1 )
(8)
i at the point of contacr are
.
.':'(-
particvlar ca-.e..:'ThF&illo*iqg results follorv from dre general relations (3).
1. Ifarl =e, i.e.,thes.unisetationar5r,then ur3 -ttz(Rt*2P'z\/2(fu*Ri, c'ta'=u2(P'1*2Rzll2Rz'
2.. lf or2 = A, i.e.; the ring is stal,ionary, then &rs = ur& | 2(h * Rz), ul = -o)1R1 / 2Rz'
3. r43 = Q, i.e., the spider is slationaty if u:(Er * 2R2) = -utBr'
4. .,{ : 0, i.e-, the planets do not rotate but have circular lntslation it uz(fu *2Rz) = r'r181-
jr
pxa'nple 1.13 Qrick rc,hrrr. mcclrcnisn is conr-
monly used in meial cutting'to speed up the idling
part of the return stroke s'hen no cutting is done'
Sudr a mechanism is shorvn in Fig'81'13a- The pin
P is fixed to the wheel 1. At the instant rvlren
f = cc-l(0:8), find (a) tlre angular relocity arld atrgular acceleration of link -48, (b) the velocity arld
acceleration of ram D.
Solution The cenire P cf the pin is comnron to the
slider and the-rvheel. The extretue posit'ions o[ i'ire
link 2 are shorvn in Fig-81-13b. Tlre forrtard stloke
of the tool occurs during rotation of s'heel I througlr
a larger angle d1 cornpared to angle 61 trarerscd b-r' it
during the return siroke of the tool- Ileuce n 3[qrrer
cutting stroke and a rapid returu slroke is oblained'
cosg:0-8, sin0 = 0.6,
--F
-l
I
'-----t
L
'
:)- Z
,"-'1."
fi'
.'0Az
AO
(dr',
d-
:
tb)
cuttingtool
'//
fi:
(OJ
f;g..zt
A
,j
.I
!
AO = ti inr
3 ?-: 1CC.*
ED-- tct+
1L"q11fnosrls
"r"
q*xAP+t,p,r, :+ 2L.x(8i+6i) =,r:lix(8i+2aj)+o(0.3162i+O-9{S7i)
tp:b*u1xOP=ee*
j: . l6=8ta?+0'948?u
Ql
:+ !:
tl)
-12=-24u2*0-3162u
u
=
crn/s- r.t = 0.650 rad/s=9..t * gz* AE---iAE-+2gz x grt:*
11.38
Lep +
9.p=go+ v;xOP --iOp
j)
24i) + 2(0.65 !) x 11.38(0.3163 i+0.9487i) + n(0-3162 i+ 0'e{87i)
(s
-2?(8 r+ 6i) = r.4 ! x l+24 - 0.65'(s i+
({}
-:+ i:
-24=&it-10.14+4-6?9+0-9{87t'
-32=-2*)z-3.38-r4.O4+0.3162' (3) j:
i: -\2-20 ctn/s3, t[ = 0-3150 rad/s?'
The solution of eqs(3), (4) is:
(l)
of rype
Oace a4, &3 are known,'gr, qD are delernriued by the procedure explairred for the mechauisms
in Exl.Ll, since t,he rnechauism ABD is of this type. It is left as att exercise for the'student-
Example 1.14 A
vane pump, shown
in Fig.Dl.14,
is rotaiing at augular velocity
ar and angular acceleration 6 about a fixed vertical axis. A material
watei flows ou0 along a vane with speed o[ u arrd rate of increase of
relatire to the vane. Find the velocity and acceleration of P relative
Neglect the thickness of the \ranes- The problerns of fluid (lorv over
blades of turbines and centrifugal pumps are sirnilar. .
7t
rT
rJ
ttre
u)2,o ateobtained by rvritingg.p ia ts'o rval's and equatiag correspondinS !,iconrponenG' Siroilarl-r-'
j
All
conrpone{rts,nfro;* &2,i arcobtaired.by .rriting gp iu t5,o ways ind equating correpouding !,
pirrr" nrotion. 1{e consider poiuts in the sanre plane of tuotion + g x (- x f} = -n2!'
bodies t
The solutiou oteqs(1), (2) is:
*
AR-- 3(.c- -L
oP =}0(cos0!4'sindi) :8i*6j cnr, .4P = Ao*oP= l8j+oP=8i+24j cm
e= AP llApl=(8i+24i)/(8? + 24')'t" = 0.3162i+0'948?j'
ql=2L, lilr=g, 91=-u4k, .rz=t12k, go:9,r=9, 9o=9t=9.'
Ttrepa.thof Prelativetobody2isthestraight lirre.4B. I{ence 9p11 =ue, Lrp=it9'
rr)
point P of
speed of u
,I
:-,.J
J
,;-J
--J
,,1
.::J
-:J
-:J
--J
.-...l
.:J
to ground.
the curved
F.'r. e i-
lt
J
;-J
':j
J
F
!
Solution With th'e*&rice oferes
'g-riarli'
as shorvn
iu Fig'E1'14' we have
-l
Lt=6t,
9, = sia{!* "gud!'
so
se
=0, =9-, .
go t cosdi-sindj,
QL=
r(1
-cos{)i4(6+rsinf)j'
radiusr. Iletrce !p11 =o9'
Thepathof p.relativetothevane'1is acitcleof
9e1t
= i;'+(a2 lt)g''
gr=9o+qhxOP*Y-e,:.,
ap:9o*r.rr x oP + qr x (grr x QA f 2t'r1 x 9r1r *9r1r
:+ vp: f*u(b+ rsiud) * usindli+ ["1r - cos{) * u cos{Ji
cosd) - 2t'ta cos $+ usin'd + Q? l')cos 9l!
ap = [<ir(t, + rsine) :c]ttisin { + ir cos d - @' /') sin eJ]
+ [tir(l - :.= d) - '"(bt r siu d) + 2r"o
:
-
-"
-
t__
t_
L-
L
L
L-.
ti
LJ
{i
E.-,
Lj
U
L
t-.
L
L
e
i
G
-)
.J.
, .l
I
J
-zL
ilf
,,ffi
',.ri
tr^.-^h^F.-...=;AxIoMSANDFoRcESYSTEMS.,.,,,.1W
2r
or
vecror
..*", q.I :",,.1r, 1: no'o,, **iln,3iw1
J;;ffi)ffi
l?"TI;
a,'d
*,,**" .o.&ce or line-' The distributed over voluc is called fota'
f
I
):.tit rtc,forces
f::'il,:,T**:Y5""ffifii**,?S'::,*".Tt[i1t;i.tr l*iigffi*
-g;',
tr
force
over vorumer
;"L
L
L
L.
L.
"i,i^i*'i^;;;.#;;;;;;;
Mt=1'ptx
L
L
L
raw (axiom) of fonts- (
?" -:*y:\poro,rctosram
io"tpoi*^J:'"rl'
A,forces arc sorctttei! tr
F
:JI*':o
k
j-
= AP-xF:(.ttP)Fsin0n
;i
= Fdn= lr" rv
)
r: l'
lr" iu F,l
,
!
wtereF=F'i*&i+r.'L,LPt:i'i+rvi+r'k*dqistheunitvectornormaltotheplaneofEand''1'
G tf,e relati*e
ottained from
= AQ x Lwhere:{Q
givea by the cross-ptoduct. Note that !{1 ";;
f'e
eq x f' =
oq 4,'sinF/Q x L= {aF"+
position vector of aay c<invenient point Q "" ,i. ft"" "t Y""
r
fr:i":;,,{;..n:y:;,*,o..,,;,et!lite"**unitvectore..lo1git(Fig.2.2)f
is defiired us the "o*porrJ-t
L
L
L
orit= nrornent
/r4e about
cnypoint Aon L
alongg
,! : I
1". O:
f
A{"=tr'lo's=APxF-e=lF'
et! e"ll'
l"r
L.
L.
r-
= er!+
erj
B
momenl Q of the couple'
about every point is the same and is called the
* L=
itr*" r7^'=' Af x F + AQx t-E) 1ap /'
of
(Q.p)F"in o = Q' anaTQP)Fsia0 n is independent
"
rvhose directiol is sat
\4rrench is a system of force F and a couple C
:
L
U
I
I
9
F F;e
e.2)
i
i
* erk' M. is s'ell-defined since for any other point
BAx F'e* rlf" - M., (8alld'
el+ lzl " r'e.l
Its moment
Conple is a se[ of fiorces f aad -f (fig'2'3)'
rvlrere e
il
ii
onL,MB:9= BPxf'9=
-9
1
-6
-F
{4* E: 9t
|
E3
a
ac
A
F'g.2'3
2.2 MASS
-49
Massm(B)ofabodyBistaliena.xiomaticallyasapositiverealscalar,whichremairrsirrtariantrvitlr
the contin$X5tSry'$t*t{ the-mass
By
its parts &'
rinre and equals thesumof the masses nr(P;) of all
'ts
I kg/rn agp"-"|h.g oiffiLttiliution of mass
kg/rn2 or
d ersityat a point exisis and talren ,. p kg/*t, o
nrass
c of. a body (Fig'2'4) is defined by
over a volume, an area or a line- The cenlre of
L
L
L
L
f r-
l* Lrd*=frb, + f.=ffii,
l=Y#'*
dtn
W' or ffil
of
tz'er
mass
drn equals pdt' odA' lds for distribution
rvhere P is a typical point in the nrass element drn and
and curve arc deEhed by
. over volume, surface and line, rcspectively. The ccatroids C' of tiolt"t'"' sudace
"
s
respectively.
c
Jira"
[*dA
- _TT,
!C:
[*d"
lc. = -TT.
c-
coincides with
(2-4)
{d'
orsevet.at patts with
for uniform bodies-'For composite bodies consisting
the ith part, having mass mi aud centre of mass G; (2'3) Vields
r,c
=
[t-
tiY+o o5
being-negative for.a cuiout
rur]l
[t*1,
i.e-, tu!4 =
I*'tr'
(2.5)
llar[-
Aro=fai
I
!iIE
{c't"
*
g
rig-zl.tr
rt62\J
i?. -**t$i t, t+{i'i?:.
'
:{,?:,*:1r"* ,,a *O,T* ipt-ol
lwncrrtzrm
ir= l^*1Fdm, '
\p=mlbltr':
since 4, =GT*1r,=
Errler's Axioms:
([1"
There exists
"'1.'
flap
,:,'L^ir=
!*u".1",
about point
ri w-ri.
frame
f
(Fig.25) are defined by
I^*^-gpaly'*,
= mrcrr.
a frarrrc f such thaf for
dvdrf,
velr
(2.7)
m
.A
trg,r'5
any sgskm
(2-8t.l)
(2.8c)
4r=L'
(2.6)
whete O is a point fixed in .I and F is the sum of all Lb.e cxterzol forces from the surroundings on the s1'slenr
arrd M-o is the sum of the moments about O of at cxlcr:r.al lcads frorn lhe suroundings on the system- The
frame f in which Euler's a:cioms are valid is called an dnertiol ftume.
It can be proved that a franre ? which translites with unifonn velocity relative to an inerLial frame /
is itself an inertial frame (Galilean principle of relativi[y).
The 6 scalar equations correspondirrg to eqs(2.8) are not suficiert lor .determining tlle orotion of a
gcaetzJ syslen. Bolvever, these 6 scatar equations are ,tcccssary ard, suficical for tomplete determiuation
of the motion of a rigid Dod3tsince it has precisely 6 degrees of freedomequatiou of nrotiou of centre of urass C of any systenr is obtained using (2.7), in (2'8a):
,
]he
F
F, : tnac, = rni6,
p. = mo6.- m(ic
i.e-,
-
E*rq.rt, ,
: mic,
tnag,Fu'-
(2-g)
mgs11:
- ""OL), F4 -'ma6o = m(2icic + rcdc),
fi,=niag.=m5c.,$-lIl46o:#Llp",}E*=,noc0=0.(2.l0)
sYsTEMs
F, - ma6, - '?2!cl
F. = ma6, = rrri6:,
Fis.2.6
2-4 EQITTVALENT FoRcE
lfwo force systems are said to be cquiualent if they have sanre total force s-uin E and sanre total nroment.
sum M^ about onc point..A. A sin'rpler force system equivalent to a given forcesyst'enr is callcd its resullatl
1- f,{omeni sum .itfa of trvo equivalent force systems about cay noint B is the same.
Pt@f. Coasider a force systern consisting of n discretc forces sittr the ith fotce I actiug at point i. anrd m
. couples with momenLsfli,i = 1,..-,m (fig.2-6).
:
W:f
i
'
si* grlD e,
*4
B-: -fu.N.qj
+I Qi = U*Lf-+(f
x&+t
e, ) = B Ax L * t{-a
i
Heace the moment sum i4, for eguivalent, force systems is same siuce F and M lor them are cg-ual^ slstenr can be replaced
I{ence. for the.purpose of finding the mornent sum and fiorce sunr. * given fotce
by asimpler equivalent forceiystem, i-e., its resultant.
2- Two-equivalenttforce systenrs cause the same motion of a sizgle n'gid 6odysince it is completely decernrincd
by the total force sum { aud the total momen[ sum Mo, rvhich are'ttre sarne lor the lrvo s.YsterrsTwo equivalent,force systems; in geueral, cause diferent motiorr of a defotinable's1'steur,
4- Rt*Itaat (cguiralcn!) o! o gioen forc,e syslcm ol a giact poinl A consists of a force {4 and an associatgd
couple with moment ep^ @ig.2-?), rvhich are obtained from the tirc conditions of equivalence: i
.i
= t(EA+,q;)
i
. &=De
i
i
(2.rtc)':-l-i,
.
Q.a^=
i
.1r
Ma=!e;x4+Eet
(2-u0)
I
5- frc
-
Ptwf;,
simplcs{ rcsultort {simptest, caaipgkn!).of a given force system is a sirenctr.
Let the.resultant at'pbint ,,t be Fp, en., The *iuttant at, point B i" fa, en, with
(c)
MB=C-nB;en" +BAxER=(qR^h*(Cn^)r *BAxfu.,
,
Flcte ( )U, ( ).r- are the"o*p,io**tf 1 1rtiifei and normat to fs. tt b possibte to choose an appropriate
point A suCb.t-ha,t the last trvo tcrrns
r.hs. of eq(a) cancel each other since !! x fp is .L FB- For
'-,.:
.-E.=J .\
ol
= ,,
.--^ '- *- l.rh '^
--t
.::?:
C.Bire
rt
F.,,kllt
-i ,al1-"r\.^q
--t-^ .q'
:R->
C;
'CotsiJ
.i'.
B
!"
'-d ,Fig:1.7'
'a*
.J
{ij
J
,rJ
,-J
.-l
;J
,"'J
..J
*.J
cJ
G'
I
GJ
cI
GJ
GJ
cJ
*J
L'-I
O'.-|
J
-J
'-J
-l-I
i-t
,--,J
-J
J
J
-
L':-
rri.
t-
l
L.
IJ
f
LJ
u
TJ
tJ
L
L
l_
IJ
r-
'
|
v' Llrt (G.)ll,
isisting of
\sA rrr
ls B vrreusu rLurtD'er'rb
resultiurE k
b
B 9,t9 iesultaot
. , . En,
"-T:&'::l-..1,--t 2-., ingte fotrc/ g sirglc couplc/ a",ll syslem.
j.o
-- -'{QR^'-ER:u-r8c'r"'srrrr'r'sc'
.Fa=
rr(Cn^)U +oij.i,
;;A^il
0,C8^=9ca^ = ernifir .Fa
=.01i:1.,
= e,
a nul syst€m
;'."noti"yrt
::::,:'::ii:T:':'6:
g,
a single coupte
en^ if & = g' c.
g^ ir
!'r b' 1:l_1I
{.*^'*
4-R
4 *t
"' - "*tl---t-t
;, :A;r:
A ,-].--r- f^.aa throu(h the point of concurtence'
,1L;;"#il;;';,6;I_:.::::':::'f:"-*1T*T"1,81ii,
il::-ff :fL:ff
;::$;:;:"::.::
ffi"Y,:fl',-,i:*':::?:::;:ffi
3T:$i',"#.il:H"*
k iompoaent aod &
hi
3'' YRo
auu'
:51;;;laR. aLe=.0 "J* e*"
Plan'e tr'e'r
normal tD thls ],ff
system'
d""E
momeots along
-"",3ITIJ:L',Ji:1;
its simplest
only and components' Hence
L
l,,
L
L
L.
null
ln
cel a.sinrle couple/ nutl
Arnl.single
'"rv
z-axis) and couples.
rn e line (say
s*' = 0 since epo has onrv
Q
n
"
i
::.tt3:-:j::::1,
:T:ffili,;"il*11,:;:::1,I"IT"il;il;;;,rl',r:':"9'::il,lll:*:l';
*' i' singre
l """Jlt"nt
Ii":ti: -:;ffi ;ffi;il':il*i ^t-].,
torcet a
single force/
-"*::['fj.::,' ffl"J
'oir"ni
i::
iTil']']ilil;xH;;;;:;;;a'
ffi";:"rr'*[
couple/ null
!
has
single
sinsre
is a
sYstem'
"''J;rT"T.:"f:::i:;n
The resurtant
:+
{p
cons'stins o'rv or rorces
ofsuch a force s],strm
g'
at ri' i = " "' n such that & #
"::**
Tu-a poini 1;; calted lhe ccttrc of parcllcl forces'
1
irr*ogt
,*"i**"*
E^=r^l{I&}e'
a
i
t" = (Isr')
ir
/(trt)'
* t(trr}rn-(E.
'v
g-
lxe=g'
.:,:.
ve'
(2.121
'
Fordistributedforcethesummati.onin(2.i2)isrePlacedwitlrinrcgrat,i,on.ThecentreofparallelzaiJonn
rtith thc centre of mass
g
graeitatiotto,f:*
dtn o'la body,
lc = LR=(T**)
of
..n"a
"r*io'os
g*n;tg,c,
curve
parallel force norural to a plane
The simples ! rcstllta*lor " aoa.iuo,.a
**'ii*r-ou-iir= ""nt*;arf
thc toodinss*focelif area t' 0) aud
!:'
e-quals
.F
For example, d,he
a triang't*
simplesc.resul[ants of a uniform and
--4ft$#
tltt-&"brtic
atca
J
1l ff;}t:{1"
Ol5|{t{?ffi
laen=an'
_=
"ffj*
taT :
il;; 6;''{
t"=:-T-:-F
.*
"oir..iii
l(te) = f;rar t looo = l *ta*t l
dF=fds-dA, E*=
1
IJ
IJ
l-
'''-''-"tlg
point
uch a p"int
such
1 [-ft
;a*
'-""^L:-t::::t:k:t;
--!
thc arsebraic'ootume
rorce normar co a prane areaequars
?;: *;;X,':*::ff:'ffi,i"*i'o**er
toffirF:f--{t
and ack thrc.,gh thc cctttrvid'ltu'" .,*ti"X'
of tie pttssate spoce(if volume *
0)
d.F=pd*,=dv; El:
lv re
{-
fG.o +
!:,r=
lrav'=t't'. tL;+Wf'
o'spsD a'F'
=
I lzdn + ( E =,&
=.rn *( )!-
glone. creo
NEWTON'S THIRD LA\jl/ OF MOTION
t
t
'''XJ:,::;;#;;;;;t""i''qarts)travee::1"lTi'1"^:::1#:*':*:::';
;'al momel.t rn !{6'" bout o'
;::""H;;; *,,.,r{r,
::'ffil$,ffi :,ff il-'t
then
*. fr*;;i:fiTfi",fJ
aharrt o
O (Fiq2-11)'
Fis,'..}''ihen
ffi;n. --.-- t., - +ou!
f: ::;:':L"j f ?, ;::'J.:i:J;;
a
su
and
(
2
l+ Mtp=-Miz1 o'1.
Mir|= -Mozt,
rnomeat sum tfb, about
of ;;1 and 82 be aforce ium [!'-ind
* surrounorngs
urr. to
u due
pnof,. t*L exterual load
(24) to Ar
axionu ('z4'\
r .-^
,t_ annlv Euler's axio'rs
",rr.o,rnutnr.
^ ^_ ,,- G"
;Yff:"Tr:Il3.:::":"';XIH::.T:H'l4;;&;;il
I
.9
to BtU Bz:'
alone, to Be alone and
3,,!-
f,s
i
t
furers
Y
c*i
t
I
!
I
a
--/- \e-f;,
$,
L
-.-> !-zl
H^ =l
<l;
= MoLz+ &*
i{or='Mcr1* W4
Lrtr:&2+Ei'
Lsr:&r*&'
P^r,:.Qt-r.{-12l
-ilt,u",rr,
= Lrv*l-tt=
Ei+4'
fup,ua,1lt = iI
(c)'
t;
Ftg. z-t[
f\
The paror futdraora
rorce
= Y{'or+aitGl
._<
"+I{-o2
l^!ze1sdm.
point s'ith relocitv 4t) i"
1;'
C-
=F1u1 =F13.
doue bv a rorce
'lrprr
" T':'lT:::'
_
J,,
A Frg,2-lZ
(2.13)
ii; :.;:
Jr,
,__
,, -rr
(2-14e)
(2-lac)
(2.r4d)
-
;-1\r
.._
:1q
v
. <
{2-r5}
\<
{_
Ife(t)actsondiffererrtmaterialpointsatdillerentirrstants,thenr,(,)d,t'dlrvheredlistlrcdisplace.nrett
g$)dt = d1 rvhere d1 is the
ma/'c;,al poh;- fot ail iaoe t' lhcrof a oaterial poiut. I/ E(r) acts on the some
trajectory Cand (2'1a) and (2'f5) yield
displacement of itris material point rvith
rr(rz)
trt = |
L(t)'dr
Jc'4tr1
Fotd6 *
*(
F'dz)=/"1-.,"*
Aq
(2'166)
C (Frg2'12) as:
expressed in ternrs of centre of nrass
conveniently
W
Entities 2, La, T,t !i
-"*rbe
-r
' If^=fu+rc^xm! t'ca
(2'17)
p-nlb
iY = F 'tb
=
E* = JM[^r,^
-"/
'
,:
*
l^nl
+tr'4
(2'19)
1*41 \ [,,tr" d*
Tlreseareproved
''"
{^u'cdrn =ll
-"--; J- :'pcdm= t(cp - gsld*.=bnt- g6rn =Q' =)
x
*^, *lt**'+ !rc^) a', = !;'e x u'6 dnt * tce f^o""
- ;r:;: ='
i,:*.-
+
d* = f,
1,,*
* =iL'u = t&'
i-fi''
<
(2-160)
l.;"t"t (F'dx+ Fv d! * F' d;) = Jct's'z(t'l
[:" :u-:.t(r"a' +
- Jctt.v;'1r.;
i
f,
)-
J*rr.dmxgao*lcn
'
,,
(*c+
o',.
_
r-l
(2-Laal
feY + F';
= F,o, * &r, * Fztr = F.i +
. = F.ts, * F4o5* F'u'= F;i* Fori+ F'i
,re
i:.\
defined by
fjffii;1;;**tu
!
_{
(6)
The result is obtained by forming (a) + (6) IN TER,MS OF CENTRE OE MASS
2.6 EXPRESSIONS OF SOME ENTITIES
?
to frame .F (Fig2'12) is defined by
T-he l'inctic cncrgyof a body relative
rw = L
,1.:--
.-r
.\%l
r;_
t'<
.
d'n
J^
1
T' ""
u\ =tlel' ea *Ie'n"
=
--I
rq
(2'rs)
(2'20)
*%^ { d^=E-c*uaxfn!car;.
-.-'t +^+ii
:, I"{s-i,"+pc),' (grc
.,
F.'!b+T&'uc
AILBITRARY POINT E
2.7 r}uLr;R,s sEcOND AXIOM (iL^ _MA RELATION) IIOIL
*bo x nlboy ='EcV +Ic x n!bl['*
Using P-f8), L4r = EcV +bt x mtlct1t, IIoV = Eay
f-44t: lbV'l,c x mtb1 * be x tnlzt1'
x tttlb
f-at : ibp - ?blt x rnlclr - tc x nlbv * lb 4r x m!4 Alt * b t
^lr
x m9elr = llp' ya x bb lx'qr.lr
M o - rs x E-*b t x m(!b,l r - o ali = ![e - y; x L * u $ L-ul
-{
I
I
rr
<<
<
_v
-q
-r
_
-q
l
.-_
Litr!
a6
'J<
_it
-t
ll
I'
Ii
)
:i
i;
Ii
il:
i
:i
L--
,dl,ro"i1no, =,
bi-
frcae,
W,, mtby = +
b-
""Eilt.= Mt
Er x
>1
rii
rnQag.
[ = Mi Gig2:13):
-i
r
i
provided Point Asatisfiesatl.eastoneofthefollowingthreeconditions:
:
i-e., .A has zero acceletation in [,
:
1- 4,t1r Q,
2- c-.tgftrcr' ie-, acceleration of .'[ is along AC.
-r.e
3-
rc^:\
,
-.
\
e\rt
j
Lr\sS\s\-'-rs
-
Fig.z.\3
A->=*i;.^r+(2-23)
.r
Ip,-feA'
-
il"t, = W'
1
!
,
IfaPu{sclofforceEanditscagzlcrimpulsc4.a^aboutpointAfiortimeinterval\tat2aredefinedby
f"
lrz -
{-.*^(rrltrr= lr,,'*i*'-ro'
r, _L,
,:L .l{, aclilat
if
11 ,:''n"'
poirit
'
tcou angtila'r;*p'it"e toe'(") about
iaslcala-r
tnd
irnTulsc
l(tr)
,;.,.
Iasta.rrtaacoqs
.' t'z
(2-2,t0)
g'
\--- - '
fz
dt
*
L*s^lti = ,l'i,', J,.M,(tl
'r
; ,,' r(rr) = ,lI}, /: a(t) dt # q,
L{trttr.).=
Jr,E(r)dr,
first axiom implies
Alt entities are w.r't' inertial frame- Euler's
r(tr.
t
7=
f"'Eat= l,:,,'r-*= $i,"'o,'
{2.25)
(-a
niiagci '
{t yt2} = 1(t2} - g{rn) = 4Y= mL96 = L
' i-e- r-mpnlic of exter:a al fo;rccf cpals cha*gc it mbmc'rrtun' For instanlcneoas
m;As'- '
I
l(tr) : ag: e(ti) - d{} = m[Yaj I
impulse
(2.26)
in Position'
P
A?, 6rlb,Ag6. are. iastantaneous changes lvithout
momer.htm:
o!
If f(lr.t2) = q]i"o (2'?5) impties io*niotiou
g'hang
nihere
4(lz) =g(lr),
ga:(r:)
**;;;
ef i*;;;
If E(t) =g, then' i=L
If
a paiticular
gi =P(0)'
+
:+
=k(tr)'
zero, then ,[r"
oc=g'
*
zt t1, (2-25)
' f,'uuc'(r') = t':*:(r')'
"o,topondinS
comPonet'.::*""""tum
(2'27)
is conserved'
vt
f,rnreci = Q
+
!runc;(t) = f,:nia'(0)'
=s'(0)'
'.t't
i) = g, L**r0) i,r''
(2'28o)
(2'286)
tz:sol
:g,tt)=r(o} I*'r-'(tl:tj:'10i''
',-l'-.llu'
*uzeroforall]t;then(2.28)'{2.29}holdonlyfortft*i.empoaerrt.
j
RsLATroNs
TMPULSE-MOMENT O: MO.YEN:',*
or e{(t} b
point such t'hat ee(t} = $' or r{ = C'
All cotities.are s't'i':furertial frame' If {-r1 a
' . 2.LANGULAR.
atong
(2.30)
.i
ail-' !4
I
3z
j
-..-l
--:
of l'nr^(t1,t2) alon( a f,xed :-.
a point / nxed in f, if a component
Since the results
, is zero, then the oiirponent 9I.4^ in thit direction is conserved. Similartn if a -omponent .
Los. along a fixed direction of f is zero,itheu the componenL of Hc in that-direction is coascrrcd- Note
direction in
of
':i.an{3 1".-"Ua for'
thatforamass-point rn,'Ho, =(re.+rc,)xm(ie" +r&i+ig,).9.=qll.2dlf an instantan@us.ngui"ri-pirl"" Jo * 11, then wiihout change in position there is an
change in moment of momentum about centre of mass C and about a point O fixed in f:
(2'31)
+-".:i'*1i;.1?;*...-.
C
slavning .. aoo36(.,
a-^.
\MORK-ENERGY RELATTON rOR CENTRE OF MASS
*;i{"';.o?'*c
2.1O
Iiq.(2.9)
=+
F
.
lctt -- ,te,,tr
i.e.,
ic
.t!c11
Q32)
rslrere W' = F.yc= raLe of s,ork done by the forces as i/acting at C [1 fiz in general] (2-33o)
at C fl T in generalJ (2-336)
\= i*u211= kinetic energy as if all mass is concentrated
?6 in configurations I and 2 of the s-vstem and Wi-z b the value of srork done
from configuration I to 2 by the e-xternal forces as if tliey were actitrg at C.
2.11 AXIOM OF COUL.OMB. ERICTiON
(dA
2,r
At a point P1 on.{he,surface"of bod;r'l:rvhich is.in'contact rvitlr point' P1 of
surface force intensity can be resoh'ed into a norrnal force conrponent pN/m2 (prcsszre)
and a tangenL\al (frictioaal) cornponeut rN/m2 (Fig-z-la). According to Coulomb's a:iiom of friction:
Tc,,Tc,
are the values.of
dqt-e<rc&'d fa
bif..ekr'm o^a *-b.at*lo
{ E o<
<{,8?1r.'rs) <$ ' <'tr'r'
(2'34)
T = {tP,onj
i-5 d.ireattd, qp,oosits to aqpoo€ok
*
9.'.9. (t3) r-q 1i\Q- +o'^shtr-\ p?c-re. r Ar = PtP, ood iS dtrectc.d .pi.Eih. to Vi.p,
for no slip betrveen Pr znd Pz:
r 1ltrp,
for impending slip betrveen'P1 and P2 :
for slip betrveen P1 and P2:
the s[atic and kinematic coefticients of friction. These are independent of ?, gl,, p.
and the area of contact. The resultanrt of the contact force s1'stem is, in genera.l. a u'reuch- [f the contact
surface of area ,{ is plane and the direction of r is the same (e-g-, when I translates s'.r-t. 2). thea (234)
implies that the total uormal force iI = f^pdA,and total frictional fiorce F - l^rdA,are related by
n'lrere
*"
I
.CJ
:
GJ
l
t-J
,l- I
J
i:t.-I
yctitr =
!li*'Lt,it,,
{e**,
lUi-2,
Tc,
=+
-Tc, =
= W'
=
-
'
iastaataaeous
e-l
Ft. lrt are called
'- I
(
-thl
J
=-J
*l
,-J
::J
,_.:J
il
LJ
fil
"J
&l
t'--,1
(2-35)
F = ptNand for slip
F = y,'N,
:ql
Eqs(2-35) are also valid rvhen there is a discrete point contact betrveen the bodies (fig.2.15) with .lV =
Nr9,+N3er,/V=(/Vi +N?)tt2 forasmallb".donawireoraslideriuaslot,andF-Frg, *Fzg,,'f =op-,i!P'
I
(Fr= * Fi|rtz for a srnall slider on a surface.
\€\
for no
slip
F < p,N,
slip
for impending
.?t
fa-
oF ELECTR.MAGN;r,"
- l.rrAxroM
"oo.h\+
g iu cledric
Force F on a poinr charge s moving
n"ld A?JJ7
a nragnetic fietd B,i,s
:
ac
an
.
F -- qE +qq x
a.
2.13 AxroM oF GRAvTIATToNAL FoRcE
The total gtavitational force {p, on body 2 of mass m2 due to body
Ezr =
'WJ''ffin,
,.*3
-T ._X
t^'
f6
/ \S.
asr.,s 9b \
1 of mass m1
(2-i6)
3'l
^
'N9,.
i
(Fig.2.l6) is giveu by
i:'
I
I
-J
I
I
.,1
r
'-_J
(237)
- l^,,[l^,c{ar,;}a,,r,
-rs
universal gravitaiional constant. The mass. appearing in (2.37) is calted the gruuitaligacl mcss
wheteas the mass appearing in the Euler's a)iioms (2.8) is called itcrtial rnass. Experimentally, no {ifference
has been observed in the measure of these trvo masses. The total force S1 is called the ucight of lody 2.
Tlre simplcst ttsdtont of the general spatially distributed gravitational forcc.dErz is, in generat, a rrit:clr'cltFollowing results can'be proved from (2.37).
wlrere G
,i.9
I
'
dE'
--
Fi9.2.t6
a8
-_J
l=_J
,-]
\'+-J
E--1
,\.
:
=l
l;
,--:-t
-=
>
]
-I
L?
_
1. For
H,'
c "odiarsrr*iir;agi;r,.,**
2-
mass-pointo:,"u""'-
IU'
;"ffi-qt
?Y:';,;ror
L-
:'u-I-'""
:-'
$'h(
L.-
-Garrn$
rZ =-ctt
i
-,,f
its
L--
L
l^1*o*'
from the centre of *T
In general, the location of G differs
L--
d-rstance from the sphere, theu
r
sp:r*r+"I-I*.;t;1)
approxiinated at C'
4. For troo bodies of arbitrory shapcs thosc sizcs
:^X:l-^6i'
P,
(2'3s)
U*e si*
9 - ]l +
betweeo tbeir
ln can be coocentrated so that
as the uuique point where ihe
T:he a:rrGi of-gntityG is defned
mass tn:
if* *.a"f [o... on it equals that on the actual distributcd
L.
Ll+'
s'a'ita"T"l.r":" o":*^":::-T';:'t'A1J'
o:
'..1
r''*"'ib) is a singre rorce F throush
2
-''--'-a?
the concu""nt
L:,:;:::;;;;;';;;';;;;';-
LE
a
of radius r'
is the mass of tbe'ii* of spheti inside a sphere
of rnasses M and m rfith distance
Ttre force F between tuo radidllyt-*"O;" "rlcrcs
where
h-\.
of mass M and
r4-^.
r
$tg:'l7zu'q;]i
of
thc bodgis small compared to
;"
rcittl,.llff c'ca ue
^*::K;;;";;;
t::':':
=tc'i'e''tL"
,
:i;;;:.:;;;i'o1"'*F21=-G*a""-[1li1"*"
i;,.l"i#::::',!#:Wf&iff
;l
a]
\l
;
.ffiY =
:T;;" :-" or b.dv.is (,.,1( :,neat
: -:the^::!:;,
:-5,i::: :,T,,I;il;:,':T.J
surface of the eartb'
Ri
n'-t
<
distancs
1"
through small
For bodies of small Iir. *oriog
I is the local
b".o*siorritrr t"r"oti" ""a direction: f --'ngg'1hetesf Rrad 0-1"
tlre gravitational force is modelled to
30 km' a =
=
6400 km, even for s = l0 km' 1 =
vertically dorvn direction. .Noie that as-8 =
039r,e'
c M ml R2 = (1 * h| R1.. "'o and F - GM nl (R+ ;t' : (l + n/a)-3
\ri
2-
-h-
2.14 E'[{EE BODY
-,1
I
L-.i
-)
ir.-
-,1
)
L_.
t
L
t,
I
L--
LJ
LI --,
t-
of a seL of n bodies or
of Euler's axioms to a system consisting
For the purpose of propir applicatioa
part of a body' i't is necessary that
particles, a body, a finite or an infinitsimat
jsolotiorr
t:-t::ttundings' and
its sketch U" i*1n in
T"t"
rhe systern should be rocll-identiJtcd and
1-
should be drarvn on ttexerted by the surroundinSs on -the sysiem
bv one part of the
BoD1. ;;Gra;M (FBD). The forces exerted
Such a diagram is caped a FREE
in the FBD.ltJ;o kntal forces and these should not be shorvn
"..
2.L4-L Comluon Supports and Eteactions
to
rvp€ or consrraint provided bv it
by a supporr on.a body depends on rrre
|.],i]lol;;.I;a
constrained by
::j.ITtrj:|i:il:#i;fi *i;:;i:l':::i::::::H:::::,;1,ff :#*"::
ff::::ffi:T.::X.:;;;;;;"r*
^ -r i-^l- rampnt .omDolrents
^
rotatiou comporenrs consrrai.ed':b{dt''R'eactions
r
.,,"J1i,'f
f::f]
for several types of supports are giveo
io
Fig-2'19'
"' i:ii'"1;;:H:::r":'.:ff;;ffi;;.
U.
L
thesection
can-T:::':::.i
I
,t
I
i
I
f
bv its resurtarit
rvith opm€nt
'le
rf-ll i:::::*;1f.:"i:::ffi;:
",.rlT"fr.lll?iIi'j"i*";""1,J;';;.*q"L"r'*'t*"
a:iis:
ilT.E:?::i;:X:fiffi;,X;.:qil;;;r:::'::::H:ilf
orirs cen*oidal H:red'io'!he
*j,'i;;d,":,,X;";.J;";;;;#.-r-*,troruanded
.[n=
u.it
triad
Qa=CtU.+C^gn
(2-a0o)
a:
;
1
LJ,:
LJ.:
Li,,
l .l-
r-
io'* '"'o'"
Lj"
lj.;
'z'G
DIAGRAM
sysiem on anoLher part of thesystem
I
n'1
2. the cztcraauorres
E
-.
!-,-
fo,
,6*ing them ,/O
ate'smattconrnil,{r.1,,:l:T'5;#::fr':t*n./rn*1"
!
o1
rj
U
I
(2-406)
A systein is said to be in efti[brium if its every maCerial poinL P continues to remain at test, in art
inertial frarne f. If a systcm is ra cgur'li,0rirm, lhcn lor its cocry'poti
f=O
Prcof,.
lf
O is a point of
f,
then s.qr
oqr=
and Euler's axioms
-
M,=0.
(2.41)
9, spolr = 0, V t- Hence
rl
J^*11dm=0,
yield F = q, :0-
v t,
J^teolrx!!potr1-=g'
Mo=iIoV =9, :+ Mt= l[8+AO x F =Q-
Ho11=
These are necessary conditions of equilibriunr, but not suflicien0 conditions even for a rigid bodyT,he cquations of motiott of aa incdialcs.s syslem are also.(2.{l), since * = + \, = 0, EaI:oO V l.
2.1s.1 Two Force
'
Mernber
L
T;G..-'-"JjI"r*-:=I;
fl$Ilffl
t-r--=-:A
ts'o.forie ,rr"r,Si1fq8-2-20) is
irvo force nember is a ruernber subjected to only t*ro fo..o]lf
"
ia cguilifrium or a trvo force iacraialess mernber is in motion then the trso forces have cgucl lml.gaifudes,
opposilc dittrtions and act along. the line joiring thcir points of applicatiottProof, M = AEx& = 0 :+ Q acts through ,4 along.AB. Similarly F, acts along ,tB. F = & +& = Q
A
^
= fr--&'
a- Srnooth balt and socketjoiat
L2-
TJ r\'fu,*,
d-s
end (3D
load)
'?ry?ffi:f-fr{
9e
f.z
j.
a-a
"#'F
tnternal force,resultants in a bar k- Coplanar load on coplanar beam l- Coplanar badbn strafult beam
9'\r,z9t
Ai
FBD's of front rvheel assembly (m1), rear wheel assembly (m2), chassis (m3), and the completc rcad-roller
with dtiving tarque M on the reat wheels, assuming no slipro.e giito in 11.r.l9:
.:k;
-{:'Rq -sB
8
tr
x. z'1<jr
30
::
:J
_J
J
J
J
J
U
J
J
J
u
L-I
f-
l. Internal section of a body
^T,Ei.
8i
'J
U
,J
Pin in smooth slo[
b.
U
U
Fr
il
LI
U
U
:J
LJ
J
J
Jil
J
J
il'r
I
J
ed-
*ffifrur*H
;Le?
i
.'..i...,,,,...,.'.".x21"..'.9,,i,,*",".;2.'*,,.,,.
'v.1]5,
tt'
*
FnrcrroN
epif,
pt aod
''";:[,Tr:::::'-iortg,/*J*.lr1',:*1:i5::X***:llffi;'::f
.o"ffici"nts frictior
;: ;;;"rt:
z.ro
are
of
and sratic
i"*i"siv an angre
tens'ron in the bdt
and ,rrP N/m and t'e
;;;;rm
iif.i"tiona
gravity' the FBD of
p". Ar locarion {,, ler *;r".*.,
belt at {' bc7' Neglecting
;;t
centroidJ;
parh coordinares rs:
be T. I*t the radius ***"."*.rtbe
,.,"*r, * anis elcmenr io
ir",t'""'lt
Fig.22t.
in
*
o"
an erement of rength
"n.
"i-*"
conracL over an
tf
*c
AB*a.
get
and sirlrilarly lle catr
pullcv' tlren (1'42} hold 's for slip
the bclt is on a ro{aiiag
for impending sliP
:
lf the inertia of rlre belt
forslip
:
TrlTz=eP'e
Lf"'
:./.0 .
i1-:f,;t = - '
-rs
negl'ected' then
for no sl\r
I
:
rr-}_:
2
" ",.d
fr17
-' :
(2-{3)
-
t'
reduce to
= 0 and (2'a2)'(2':13)
I
fornoslip: TtlT\<e"0'
forinrpendingslip: T1!12=e"e'
,
l.l!
(2-4{)
ll
as in (2"14]slip and.no slio are the 'sanre
irnpe[ding
of
to the
condigions
the
asd
o
l"t ot Ou/' be the force nornrat
Eor a stctioncry bblt, o =
itt
t'
***-t
'
papfot
semi-vertex
with
p
and
of
ropc
"""'
belt or a
N/m instead
f's'
tl
i
l.
llii
!!
i7* -ta iut't
* and q directio"lt"."";#;:
6.'vee'
bett (F.q.2,21). Then
"rJ.
.noiu"rt.tt"n"".q.(z-i:-);;'i;'Y::'.1;l'l*:k*it:jIg;:;"":.ilFT'1
trso pullel= is
is
bel! better fot porver'transmrs
of lap on
t* s-ollct of tt't-*Eio
,rr1/sino. 'Vse'
Ler Tq be ttu *-*iJu'* J,o.."rt"
P lransmitted'
in the expression of the pos'er
i ;" ;;t*'n'i*io"
rve get
i1
Lhe
t-h"'.
sarne fior both
tbis value'
transmission, if p's are the
are used to itlcrease
pulleys
Idler "*"
belt' Suljstituting
to be used in the auo"e c'iott=tntrension
:!
l1
7i
frorn
I
li
it
(2{2)'
;i
:J
ri
I
.
t,
i:
li1i
p-T1.v-T2o=.$1.,:lU2)U(1.-e-,.sc).Fortna.timuttrP.dPldv-_0-.?|.1=3lrr:.
'!
trassmitted when u' = ?i/3iIlence the maximuqo'power'is
" ':
THIIUST BEATuNO
A
AT
TORQUE
2-1? FRrcrroNAL
surfare (co1i1L Aar'
-- axisynrnretric
--,.
*itrr auotlrcr o1,er ao
contact
in
t
body
tb.,,.t'P
consider an axisymmetric
of iodv I under an axia!
coatact
r{
lhe
torque
1"*1'""
of
ring
spherical, etc-)- The ixial
"*1"1
'*;;;;"*ia
ii'Z'zz' Consil^er an elementary
'"t
i
o*r,'it
,t
i.
r
oocmal
body
the
of
Let
FBD
is to be determined- The
*!?:" rvith the axis'
norr.',.r'rr*un!-"o
r frorn
its
*ith
r
distance
radi.s
"t'gtt
and
at a radial
sutface of width ds
t" * itt**tJ'"ntial-dircction
is
force
t
rir.l/*'
frictionat
Thc
force be p N/*r.
yicld
for axial fot"" *d axial monrcnt
the a:tis. Equations "t to"ir:*lrur
,
--:
;
i
'l'1
.
L
L
l*..
l-f*-
t-,
l-
"*.'
u
L.
r
= /tr.i"."Xz"")a"'
.-i[ffi\l
u=
MI
P
f!r,nl(2zrlds'
,9
.-
I
:!!
:
i
- I I r,,'t *lL [ /'n"i"ga{ r']"
*
,t"
R t-.^
i;-Y,t, f .Lflt h V" r.€ffi iiS%
I
I
,
itic
; -'-*
I
I
I
i
u&=fr/(
F
II
,,
:F
.31
,
,.,:
.
::.
.:.,
{l
(a) Conical bearing:
->,i1+
1- for uniformpressure
- drlsinc, eq(l) * Ml P = i !," t,,'odr'l"iool t I f'*a'lel
'(3)
(2) + - Ml | = (2y,l3sinoX"i - fi) l(rl - r!,)-
p=const'
r2)12'
2. for uniform rvear p = Clr (2) :+ M lP = (p,/sin o)(r'r +
to
(b) Flat bearing , u -tllzregd2x{) for various casesr respectively reduce
Mlp.= {2p,fni(rl- "?t/bi - "?t, MIP
Mlp
oa4 | I
= I l,.it
,f
{,'w*1,
r=Rsind,
MIP
ds.=RdL'
s: rl2-d'
and eq(l)
- P,(t +.)12'
(5)
'+
- Rl Jr,o"inz 0 d01t [,fr'u' 0 cosa dll,
(6)
r*lZ
r*12
sin'ceE?odfl :F'R'
cosedlltlJo
*t"Ilr.sintA
MIP=
+'
l. if pqc
'd
"l
2. fcr unifotnr pres5ure p:const:':'t?o," an (6)' *':MlP'.- rP,Rf2'
thenp=.{^Rcosg' and"(6}"
2.ISE.TiICTIoNALToILQUEFoR.ASQUARE.THREADEDScILEw
porver-iransmission-(as in lead-screr*' of
square-threaded scre$'s ate comlxlnly used iu screw-jacks and
the axiat totque Ir' needed to impend
lathe machine) and in tesling machioes and presses- we compute
tt]t:10 of the screiv in
the length
nrotion of a sctew in a fixed nut agaiast an axial tlrrust P' Let
't:
hetlx angle c is giverr
The
{ (Fi5'2'23a}'
contact with the nut be s and it-" rneao udius and lead be r and
distaoce of Zat (Fig'2-23a)- The
by taa c - ll2trsince the hellr a.tiall.v adraaces by I for a circumferential
thread be r l\l/1 al.an angle c s'ith
FBD of rhe screw is shorvn in Fi5223a- Let the normal fotce on the
to the dirction of irnpeadiug
the a-ris- The frictional force is r,p N/m aloog the helix in direction opposite
slip. tfence the- torcc dF ori'an elerueat of length ds ai a dis[auce r fronr the a:cis
1
dL: pd-'(-dncq +ccag€ ) * prpdsl-Goso'gd - sino6.)'
Equatioos of equilibrium for a-xial fotce aod atial moment yi'eld
e=
lt
n,
It
= Jor(ps$o*p,pccc)ds'
Jo-@"*c-;r,psina)ds,.
M / P = r(sin a + p, cos o) / (cos o - p, sin c) = r(tatr o + P') I $ - g, tan c)r;
(r)
The rorque iy'r needed foroimpending motion in the direction'of the a-xia].rnTti."'.!:i62'23b)
lil P,= t(p, is scf-locting if it
fhe
"...*
Mt ) O, i.e., if tanar( P,-
tan c) /
does not advance under
(l *
P in the absen ce
of.
M1- Hence it is self-to<'king if
P}
:1
usPdS
e.+
|
I
...-l
(ol
(2)
P, tan o)-
rqiset- :rE
F;
tt:
:'
since the present directioni of'the'torque and tlre
obtained from (1) by replacing 11/ b-v -Mr and p, by -p,
-frictiou force are opposite to those of t'he previous case:
6iv--'1a
;J
J
J
J
J.,J
r-l
J
;-J
J
J
J
J
J
J
l
U
':J
'*':J
Ir
\\|.-J
-l
j-J
hds
R'3.
3p_
-t
(4)
motncnt evaluated for t'he corresponding
Equarions (2)-(5) imply that the mornent fot conlal- be*,19: :lt"
: p, I sin a' lrence conica'l dutche
flat bearing with r, replaced by an elfect ive coefficient of fric0ion Pefr
pe6 ] p''
arb more efiective for power traasmissiou, since
fc) Spherical ball bearing:
,.
16
r.z
--J
il
:J
:l
I a-
t-
t-
L--
t-
:,. :".s\ -
ExAMPr,Es z
point
the moment of the force system applid on the hinged bar (Fig-E2.la) about
aad the resultant (equirralenl) of this force system at point ,,1-
L-.,
Erampte-2,r 'Fini
tt---
tj
Fr=z-s
roklr
tsi
sB,
L.-
FO=
5
k*
€
:1
tt
r.l.{
j1
I
t<-t
L*_
LJ
tj
The force system is coplanar@mponent.s as shown in Fig.E2-lb:
-So6tioa
h
It is convenient to resolve the forces
and the distaaces
t
into suitable
F3 = lOcos 30o = 8.660 kN,
- 5 cos 60o = 2:5 kN, .F2 = 5 sin 60o = 4-330 kN,
fa= 10sin30o =5kN, ri = [4/(3? +42]rt2l4=3-2kN, F';=[3/(32*4111214=2'4kN
B$- L5cos20o =1.410m,' CE =l-5sin20o =0'5130m' CP=0'5cos20"=0'4698m'
FD=0-Ssin20":0.12i0 m, DG-'EF --CE -CF = 0-0432 m, :{G; 1+ l-410'+0i!?r0 = -z;st1
t1"
The rrapezoidal distributed force s-vsiem is decrmposed into rectanglhl Td trianqllql {!:t1i:::-"":
Sr.pf*a resultant ,r7 of the rectangular distribution equals its ar^ea- L":, t- = !'0 1 Z :-I ? kN' a'n-dacts
Fs of the
tnrough its centroid, at a distance of AH = 0.2 * o-612 = 0-5 rn froar {- The simplest resultant
at a
centroid'
its
through
triangutar distribution equals its area, i-e-, Fe = |(0.0 x 3) = 0-9 k\, and acts
distauceofAl=0.2+(3x0.6)=0-6mfromA.
: ..
product of its rnagaitude rvith
.F1
t-.
L_
t:
lt
l'
\-_
L
L*
L;
l
lt
LJ
LL-
The moment of each forcc component a.bout point r{ is conrpuled as tiie
tends trcr rotate
its perpeudiCular distance from ,4 and is assigned positive or oegatile sign according as it
Ttrus
couples'
the
tle bar in the setrse , (D t V (i) or y (j-) to r-(!)- The sarne applis to the sign of
tti =[2-5 x 0.5+ 4-330 x 2+ 8-660 x 0-5 x I +3'2 x 0-0432
2'4 x 2'581 * L'2 x05+O'9 x 0'6+3 - 2Jk= 0'9938k kN-m
a forie Fn
The lesultant (equivalent) of the given force system at .A is a brce-couple system, consisting of
aad a couple Cn given bY
Ca=I&=(-2.si-4.330j)+(8.660i-5j_)+(3.2r-2.4t)- l.2j-o.sj=e-36i -.:3-83ikN
t
ls
Mi - 0.9938L kN.m.
.-_\
p;ample 2.2 Find the moment of the force shorvn in Fig.E2-2 about 7/\
poiat,{ aod about the axis of the bolc. Find its resultant at r{-JiA.4Li
Solution Lei qand g' be the unit vectors along r?C alo.d AB: a.
i -=-.1k
:
e= (-5j + 12k)/(52 +L22)tt2 = (-5j + 12k)/13
i',
-g' - (-4!+ 3B/{42+32}1fz=,-0.8i+0.6t ,
---J /: .rx--:
,,.'
'
er : (3g* 4ll I @2 q4zlrtz - 0.6e+ 0.8i
:f
o:
.t,
s't?
_
_:?_6q
-_ .!:ZGq=2.6(0.09+0.E!):I-569{2.U6!
./
//1 ,
=2.6(0.6e+0.80 1-56e*2.08i
Q_a=
lt
L_.
tt-
tlr
'
It
Llt
:
t-.-
kN
Ur,l;
lTii'
:soi-ssj * 13ok mm
-"^:.t4C=AE-+pQ1QR+RC=-30i+50i+?0k+65s
I glt^
Ll-
L56(-5i+ 12k)/13 +2.08i = ?.08i-
0-6j + 1-44k
L..HI
^lj :;.1
at':''txu;,lf&
ill I
L-l-
L
=
L-.
L-.
I r
':
t+
L:
Li,
lr
L.
.4-ft*t
. -,
<i-
le
L,.-
31'*
"t
-12i+
198.4j + 84.4k kN-mm
= -1.2!+
33
198.4j + 84-4 L
N-tt
llllf
i,/ f
\-Y
\e,
'
hS-E2.2-
*
{
r-
:':-';i'':
';
'' '
.
,', ''- - ', 1
'
Moment about the 8xis oiih!:boit .*i i-ae'.
N'm'
51'6
x
0'6
(-f2X-tiS1
+Aa'n
=
Mta = Me' - fu ' € =
sVsta,
/B
cons-istinS
The resultairt, of the given force. at 'A is a forcecouple
coupl'e Qa +
of a force & : E = 2.08 i - 0-6 j + l''4{ k kN and a
&' :
\e
198.4i +84'4k N'm'
-1-2i+
its
A rectangular plate of rverght 2 kN is hinged along
ie Gig.E2-3)- Find the total moment of the force system sho*'n
"agu
(eguivalent} of tlre
about po,int.A and aboulline-,{8. tin<tTh-eresultant
given force system at point Ai o 7 Lm",P - 60o'?:
""'',9^:,1."];
list Lhg'fi
Solution Let n be the normal to thc platc as shorvn' We
t*-z-'l
coordinates and poition vec0ors of various points:
Example
2.3
| -r,
t,
{
m,
rs =3!
m, cp - 4!*
2!
m.
s6
-^
i
rf(o' 1'3)
G(4sin30"0"1cos30o) = (2'0'3'464)'
/(4,0,0), E(0,0,3), D(4.2.0)' 'g(0'2,3),
s^-= 4!
F;5.-c l--3
r
\e,
=
2!+3&-T:-.*
= 21+3i648
mr
rtr
- i +3k
;
l
'*2"-j::ln
g=&-lla;1'=,(-qr+rt) l(42 +121r/? = -0-8i+0.6k, alx ADit-<i+3,&)1'6i
+2i+ I-8! m
0'6k)
3(-0-s1+
2i
+
=
3s =,41*
DG = tc - LD - -2L-2i +3-464 b m, !F - Iq +
n: (AB-x AD) tlAg x ADI- (-Gi- s$/$2 + 82)t/? = -0'6!- 0sk' :
t* ,*-=:.""rr* # t,h" firrc forces are given by
Cr - 2s=2(-0:6i'- O.8k) = -lJi- 1,6! kN'm
e:{es=4(cos30o!*sin3oos}:a[0'866i+05{-0'8i+0'6$]=-1'-6::3'46{l+r2h'tN'o
= -4.472i -
+
7'746
k kN
- lagq-ii+2$/(6"
[3=14e2
:
' f,
4-472i
eu" = s[€o6affE: -sin3-oo
bJ
+32
:
+z?f/z: -6i* 12j+{!&N '
8[0i866(cos60o
i+
sino0oil
-
0'5hl =
.
T^-
. --
Ee:5g+=S{cca!*cospj*coszl}=r5(cosI20?i+cos60oi+cos45"}=-251+25j+3536tkN
Thcpcitionvectorsofthepointsofapplicat,ionoftheforcesrv.r.t.l.arc:
m, AE= g.e- La=-ai+2j+3krn''4F =L"-!t=
AD: lo
-Lt=2j
AH:t* -tA=-li+tjf
L*
AC-=-2i+j+15t'
:
+(-28t:
AC.x'.&''+ Qt'+
+(-2i-4j)+(-l'2i- 1'6E)+(-r'6i+3'464j+1'2k)
,..
T.he
moient
w'r'L''{ yi't'h't}e
force:
Q ^ ^.'2j'' 3oh)'+i:18:8i: &365'i-'2L"t, +(-1'96ain-f6;6aii---t:iS
Lrir,e:et.4E:l +'fr'x.\+
M-.q= AD x
(rs.49.i+ES44k)
-2'4i+2j+l$km
m
of its Pcition^r'ector
Moment of each force abouh r,t:is cornputei: by' cross-product
of theforce system about line
4Il
:.
= -40'0?i+0'743i-'56'2eEkN'm '
is given by M.z
:-l'718 kN'mr,' e = (-40-0at-b'i* to'*'ltol * (-'u'ntt0'6) =
* at .A is a ficrcecouple'systern' coosisting of a force &
The resqltant (equivalent) of the given force
M"=
and a couple
en
M.
g,ren bY
"yrt
r* =E& =.(4A721_ 4.472i+?.?468+(-6i+12j+4$+(3,..[|1i+oi-1$ i
d'+(-2'5i+2'5j-+3'530$+(-?H=-e50bi,L1633i+e'282Ekr{
0'?'t3j e* = W= -40'0?i*
j
5629 k kN'm'
,j:--
.
34
I
:-
&=-2LkN
3'4sd1*'6i-ot
I
:
-.
-!-\
Earaple
:i"fi:"$i'ffiT;?.-?Lrr'1."cu:rili"::l::.'-"-,:L'oo'*i:"''
ffi
that
l a' i""*
in
t-.
F-
direction to
"i i",O t'"i"'
outai"J
i"
Ca
Q^a.:Cnk-
t-
I
L.
slampte 2.8
t'L:
Eind the centre or
!
#"t*-i
zl
d
I
Y.
4
\
,!
* *:,o'01::::1j:::t::r1;1ll :|] ;:t"
aa
sl
t
ffi1**",1"inl***"*fi"']'"';;iy;
*;::ff
YJ-'--- 4' a deretion or a
7
aseml-crrcur4r-Dvr's
J'
shell
cylindrical
'vri"d"'
taken L
f#:::"rJ:riffiffiTil;,"-T;x""";;-;ii
semi-circulat
d cuboid 5 is taken-.
Tr-^ *-.o af deleted
u-,11'"-'11llo'"t'
sphere
-.
q"*Y
of size Rlz x Rlzxzi""i "
cubo'ld
',*s
= -(Rl2)(Rl2\(2R\p =
L-..
m5
t_-
IJ
tc4=
-ffp12,
=(4rnl fi)lqlp=tffpll,
i
3
ur='(ZRlr)i+ BL,
..oDr-
m3=(r.R)(2&)o=2*ffc,
*a= (rr2l2l{znle: uf p,
T| )'-
5
antre or mass of each is given bv:
:X";[-;;:;;""d
LJ
talll)-i*:ot'
t,,
\y'ry1y in*'
Ilr-r
i" = ulo'1.':. j:ji'::r'jr:
'l'=,5
t:u = (3818)i+(48+38/8)k'
systems given in-Pigs-E2'6a'b'c'd:following bodies for t'e
the.free body diagrams of the
All coatact surfaces
3 + 4 + s' ia; J' 2' 3'
AB'
o'
'"
/'E: ('}
light' ir{ake
LJ
*r]rrrr" iu*'r.",
i' \12:
i'
{z) ABID, AB/:;E.&);,
''
'l
memb.s' rot 'utti"tt no'*t"t
cables'are
are smooth. The belt "lil it "
'irr'i'-ti" which are trso-force mernbers'
if any, due to the presence of supports
t-
tJ
IJ
tJ
are
t:*:'
Ttre loading on s}'stems
simplincatioas,
LJ
Lj
L
L
tj
relative rotation and
shomn, becatrse thdte is cgmpletc "onrtr-uint-on
tj
t-. )
I
)
-i
a*r-;1":s;;::il-n
X;-;Jr.g.r.!"
it is drawn for
b-ecomes convenient if
iu ris.fz.6s. The FBD of rhe puuev
n"ri",
;
pal
of the belt which overlaps
it'
.
35
:i
.,
,i
I
1
:l
il
'!!
l.
e1'e'bolt' but
in
rotations are allowed at an
are sho*'n in Fig'82'6e' AII relative
exerts tlvo
FBD's
(a)
Solrrtioa
.,The
of the bott are preve4led- Hence, 'l: "',-oo"tt
the.
to
normal
the relative displacements
"*i" to the axis of its eye- The supportinc,,Iembers PQ and
dLections
e-xerts 3 force
componeats of force in,the.two
";;ao;r'"e
Rs o"lrt But the memb..eqfl:
ro.*
.:t
""u
as, u"iog two-force members,
B in theufi.r-s'F'1fBD a4d at the
1 compoo ntsl'iie shotr:n' because';'
a,two-{<irce
components, since it is::not
""J"t]'O''the'6xe!''Su1P"t'
3 i..""-**0.."i* .r,a,s-.ouple
FBD,
secpnd
tbe
in
E
at
force sy'stem ts
ht€raal section
d'splacenrent and the apptied
;il"ttt*
,ot"t"
relative
on
thete is complete coustraint
thtedimensional'
siucr it is not a trvo'
(b)TheFBD,saredepictedinFig.E2.6iThesupportingmembersPQand*'o_::1,.:o-forcemembers,
exerts 2 force components'
?J
*"*ler
*"
U*
only.89
and
cxerts forccs along PQ
i in the first FBD and at the internal
suppor
fixed
the
At
coplanat.
is
' "' one out-of-plane couple component are
force s*mbe" qrd the loa.ding
and
;;**poot'ot"
i"ph"
2
the applied
sectiou at'b ia the second FBD,
rerative d'splacement aud
FigF-E2-6b'c is coPlanar'
L.
1-
TT"
e:z-e z'u.
-
-t:':*HlJT:Tff
*."'"J:ri'*.tlTr:il;:ffi
.
rL:- ---i-rircular ring 2, a t
"":ffi il;;*"'rlu""'""densities\'oa'.dp'
LJ
I
t'
t-LJ
u
l-.
tt*
is' a couple
the-simplesu resuliaat
and thickness dr:
element' of radius r
rR .^
Cn=Mo= JOlrr(Za rldr= Il'',(T)(2rr)dr++ l*,,mlzT)'".=
L-
1*.
force'
o[the tangentiat distributed
,.o F;i1t1*;*n''* "-tte"t
t'he
l
't
.i
,i
,i
i
I
tiI
__:-&,
NOTE: Once a sei.82,of siiplorf, rqictions on body
*t slodd shor the
ai""-t" *" FBD of
Br
due 0o body 82 have O:" :fu"^tl a
reactions but rvith opfoe-i1 sense'
FBD
dBr'
""*" "rrppo.t
joint' as at joint (i' tbco
than 2 members meet at a hinge
moie
If
in
Fig-E2-6h.
(d)
with t&e
preferably, the FBD's of the- members be drawu'
either the FBD of the pin be drawn separately, or
+
le_
T
;".;"" part of one of the rpembers Both these procedure are illustrated'
d!
The FBD'8 are ahown
(o)
,;
;;;
J
J
I
:l
-l
J
6
(c)
,l
I
'-+/
(dt
I
J
n'*,ffgo
"
*'i-? r:ffi!:*,,
ffi
F
i
ol'-o.
o.i,:u
F1
tq3
')i! *'
F7.
-J
\
-
sn e.1f*.'
.
r&g
-
-{
,i
'f .If
i
-
r'r
.J).
zc
s.,Rn
Jncs
8s.".s(i_ffirffi*
m.3
- Iu'ls
*.esl (qg
c \-vl
F3 atl {;
_
Exarnpte 2.7 D.t*-f...-boaI-ailtt-
F.
6-ls
=tL:lolt'
l-:F>.
4n/srz4s'
of bodv
|
I
of mass
-t> >
4 6/3
Snls-
-fiB
lt
. L *i"rtno;]"
Fia V) 7t Tt"
ate 8.d.4'
^--^ ^L^.,.- it" hE.E:j}-ziJ::*:n
,(.],.,-.ts;5{5'tsr ,.1f1
-
ffir
l-rs'L'"
Tt
,sl
':-''
'r
' . , .
:
-q
Y
'-
:h9
If
::i
!*ti
::.
b determined t"ndSotution Lct.A and I be the points in contact. The rJative velocity 9es
pilrt and is dirccted
ytst' 0, then there is aciual slip :+ the friciional force F has magnitude of Fl =
1-
ir
L
::H-#ffiff{;
'
I^!r
6Aea;tl
-t
r-
36
.,_
-r
-
l*
u
I
--i..1
',
a*--l
-
there'is
]il
its . 9t - (Iwhere q is any vector in the tangent pl*l,oJ contact''tlcu
;:';;.,
are
""aforce
directirq'which
arbiirary
E has atbitrary a"gnit,rdu (bounded by f',/V) and
,; "UpTUi"tIoA
cascs'
ao
slip
lhc
zcrc
is trol
fot
detcrmined by solving the equaiions of motion. Most oftitt, ltictional fotce
then there is
contact'
of
plane
the-tangent
tl"to.-in
3: If q^s - Q, anc 9ts - 9f 0 where q is some
has a component which is directed
L
I'-
p,N and
impending slip :+ the irictional force F has magnitude of .F =
ofien, the direction of frictioa forces are
oppositc to the component of gre in the tanglnt plane. Quiie
assumption' For example' for the case of
the same a.s those obtained io .-!.iot'sotuiioul based on ao slip
about a fixed vertical axis with angular
impending slip of a small block on a horizontal platform rotating
radially inward as well as the circumferential
acceleration, there would be frictional fcrce components in thd
directions with the total magnitude of f being p,/Vvalue of gs'A'q as well' are shonrn in Fig'82.7b
For each case, the .,r"to"lf 31^s and if gas"- 0, ,h"o the
of the case' Thus there is slip in cases
and FBD\ completed as described above, depending on the nature
ofk for cases^1,5 and opposite to it in cases 2' 6'
1, 2,5,6 with the friction force acting in the direction
tn"L r impending srip ror cases 4,
r,
k
u
L
L
IJ
IJ
L
;
;#;;'il;.
Bx;mple 2.8
i
"r,a
(a) Chick rvhether the forces: (1) .f' =
2z(-cosz+ y2)k, (2)
'?)vi+
;:?;;;;;n"ra.-'tul
L
r_-
rJ
IJ
8-
8.
.;
lz
+v? + :2 sin r)
?|:
tr'
11'('I A^
i r"l,+ (v2z -' * :^)l*^':::;
frj
Find the.uork done bv these forcesin a close^d ',']
4:
(2x'1,
(c) Find the work done by these forces along the curve U:
,/
(') o-u=l- ^ fr+ zzsinz)
, - \ 2(z+&-,,-,',,-.ftt*rl
2z(-cosr*9
l(-2, f
^,-
+
lJ
f,--
=.\4zs
-
4"v) i
-
(22 sin c
-
z2)s
j*
2zsin z)
(2v
-
2v)
! = Q'
Eence this force is.conservative(b) The work done over the clced path is zero, since F is conservative.
first lind the potential enersi l'- of I '
(c) tn order to find the work done along any path, it is convenieut to
the datum at Io - Q- Using eq(3'28)
The x'ork done equals the negative of the cLange of 7' We choose
IJ
lJ
IJ
IJ
IJ
tr_
Y(d = I p-drJ;
- li" dr=:/ ct',o,ol a'- lo"r"ti'v'0)dv -
lo''''''v'z)dz
'- - f,"tt ,l ar- fr'zrvav- lo'zrt-cose* v2\d'=" -'v' -t'(-n*'+v2)'
C: {r)=2ri-rsj+r2k. :+ rr =dl) -2i- j+rkr.'rz= r{2) =4i- 8i+4I9F
I(cr) = V (2, -t, t) = 22 - 2(-1)' - l2[- cos 2 + (- 1)?l = I * cos2
cos4
V(cz) = v(4,-8,4, - 42 -4(-8)2 - 42[-cos'I + (-8)'J - -1264 * 16
1275 units'
Uf(& *s2) =.:[y(rz) - y(rr)] = V(2,-1.1) -V(4,-8,4) = 1265 +cc2- 16cos4 =
(2) F =(2x-y+yz2)i* !f":2!l;zzli*2tvz\
Fr=2xyz'' ",
F.=22-y*!12, F" :fz-t+xi2,'
u
IJ
u
]*;
(") VxE-
o
v
i
!
,a
.fr
(2r,-g+vzz)
j
kl
& * l=v2u+
(f"-t,*tz2l 2xvzl
I
37
o
..j
',
lt
1
t
'C.
='='
..
Eence this force i" non-iolootir,
l
a;;d;ij;d,r,4;?:iii(i-0)
= (y;o)/(o-oit1 =
1;:g1p-0)
=+ z =
b,y: 0,ir, =,
Jc;o
ahe integral is evaluatecl by replacing y,z in f, in tcrms of - fot poiqts on Cr; z,e in f, ia terrns of y aod
t,y il I| in terms of z. Similar procedure-is adopted for erraluatioa of intcgrals ficr curves Cz ad &.
1Aflfi
wo-t = | (F,dz *-Fydy + F,dz) = I 12"- 0+(0)(2a12laz+'Jo| pqrpllolrldz
Jcr:o
Jo
-
r-
C2: A(1,0,2)* B(0,3,0), (z- 1)/(0- l): (v-0)/(3-0) = (z-2)t@-2) +
z=L-y/3,
i
'
fB
zl2,
y=-3r*3, y:-32/2+3,
.7O
18
-=2t, z=-2y/3+2
-
\.
Fih = I Q, -y+:sz\dz = | 1zr-(-3'+3)+(-32+3)(2e)2ldr
Jc*t
Jt
I
Jcr.A
.
z*-
= 'Jtfor-rrr"*L2,:t5r-3)dz:-e.5
.18tA-13
l_ Frdy-- JCz:.q
I @'"-z+azzrd:g= Jo-lt'|:[-vl3) -(l - cl3l+e-yl3)(-zylT+z)2lds
-'
:
JCz:A
= Jo[l'-ny"pz +7y2rs-
uy/3+3]dy =
3-?5
{' 323l2ldz = -2
+ ttzldz:
Jt'
Jct.rr-'= {henleszn
fis,, {i3r,rr,
trqtl
F,dz
1B
W-a = JICztA@;ar*
-2= 125'
z=0, :+ d:c:O. d'z:Q
Frdv + F.dz)=
(a: 9(0,3"0)-O(0,0'0),- l:0
wa - o
:
fo -J
*
-05+3'75
Frds * F"dz)
",f,o* W^-a * Wa-o :
Wo-a-o-o = Wo-t
:/b.trl - 0 + 0(o)"tdy = 0.
1
* 1.25J0
= 2-25 uoits.
"
The work done for this closed path is non-zero. The work done by
- a non-conservative
is, in generatr;non-zero.
r
.
z-;72, I
(c) C: a(r)=.2ti-r3i:+r?k.-.+', t::':2r,,1=:f,
*
,l-:Q)
= J;"ir{f"dz * F,dy * F,dz)
12
= JII l8r * 8r3 -
1617
12
=
lQF, -
3r2
F,
fo-rce
in a closed path
,.,,
dz=2dr, .dy.='$i2dr, dz:2rdr'
- 6fldr = -10818 units
the clced path shorvn in Fig.E2-9- &) find the work done by this force for the
circular path from F Lo C. (c) Is this force consenrative? C and 6 are constants-
a----:->tF
FiaZz.9
.
38
.-I
l
':l
J
J
.J
*ZrF.)dr
{J
"t
J
J
J
I
':-:--
\,1
tt
r\,r
Solution'
-)
(a)
(1)
-)
l3clslo'*'uoo7
-
For path -r{B:
L
b(dz - 6i
Rzb(62f+' *'uo1=
P^b rt6 - R
1t1/e-c =
l) Yields wB-c=
r=82, dr=0 aadeq(l)yt"ld"
J!"
o'
r0t
wD-::J;,^","::='Br6(dr-dr)
simirarry, wrl,-o: li'*r*\ds=-clll*i-Ll4ll2'
Forparh Bc:
-
7\f
-
a-e-:s-D-t = Wt-a * Wa-C {l
(b) For path
:
fG:
wr -c
-
r=
=
[*
!t-r
dcg{
Kc t
l
and eq(l) yields
+
rD
")dr
dd]
=
li
:
no
to
t'31dr
+
!
u *"
4 de
-c i zB + bd I J''
--
o-
'r- :- --''t f'\ is
t" non-zero' It can also be
q tlffir
'
ua!"
ts* - \-'
l_ l^-^
"'
'_
"
the'\r'c
since
non-con1rvati1":"",:1T:::: ::1'"::;ilY,:::iJ::If]
rhe force is non-conservative
(c) The
F
it tlri. .*"y.
f -.-->1=-
L
Ii
-{
"
"" at
l:i'ff :,j:".i :"::;ffi';:':;:;':1""';:;;
i:*,'sy
s'
P6-X'
t=
I";-Jti,';l'..o,ll'o
(Fis'E2-'10)
":'.n"*i
{ io the horizont'
*"-., .-','-*o Lrre angurar q<I
ffi;;r e'rrF-;ist""':,i:':"f;:T::lj;
" 0';
;51J:3'?:::'"1': ::J:;coe-ot
"
I
+
rt,li -acceleration of the u,heel is u., =
"r'
: j"-:T:ao :,::;.::l:;:t
sriP during the time
= o'
;";;"
Lr-
the work is obtained
oo
t.
+ ! ,.a= us *,;[j"""-"' -,,''l &=
= ,JoJ,
!
a a,
L
= /oi*t*+
'+ t([
L
*he1
cr) dt = co(l
-: 2-ot)'la
"=;i; "=+,
:
^.-:
(1)
..
,
.:
+,crr3]lJ.i' :_:
"*i*u*c]mr+m -'rz)
-aolzla-orIs}cos0*{(Ii'-E)cosO+osrnc}tt'cors{-'+o}"-r'
,'rz=
1',a^o=*,
o,=
!o'n"'o'=Lj#:+-"
";il;;;
:
i
...;
..,
_-,
!^- -rf
L
..
:
to
the distance
equal the nrodult of I and
case of constant
Iltt "= ":l
Norice that the work done for tbis
of application of forcli.*
poiuts
initial
and
final
ihe
.. =.. . ,",
betseen
)
-, -)
:'- .
.'+i-:"- -'
='3,t::.+
.
\
ag
+41!r - -ot1ls - a$
"-otllat
AlthoughVx[=Q,yetthisforceisnoa.con".,*j:i,esinceitbtimedependent.
;e *a o' 6 = o,o = 0'-'cs(1) rieldsi' '
(u) For the case
:
!-.
ts
ra
- !) cosf'+6sin - -
1,12=Ps[{(uo',1-as/a)fr
L
j
"rith
W=L'9r-Fl{os+-(L-A)}cos6+r"6sinfl G
)
. w = /'"ott(**?f-"-*e:?l'-cltedlcosc
-
ba
7"
FiX-a 2'io
* cf
o'
(Ir- 8)jJ: [uo+r'r(h - R]i-1461
x
ttp - !o+ c-*tb-:ooi+t'rk [-6i
bL-
rrr
{ ='0' and
t'o
=
_l
o,lo
5i::::i;';"r:.T;;x;;:*."i"
instauts'
materiar point P at different
*o::Y::"ut
m*: ffirlH::];1i"#::i-"4'i-sind1
rclpect to trnle'
by integrating lar
-
-,
[t'
rt' acts at
0 to 1
tima
interval
time intcrYs'l
a fixed inclination of
l-.
:
""*';-*i
,
''
flj
'/.,
L
L
L
L
|-.-
rj'
L
l_
[-
r{
lJ
L
L
f-.
I.lj
f-
rj
u
IJ,
tu
rJ
l.--'
f
I
I
i
I
'-=;=
-{\.':..'..
. .'. 3.
., DYNAMICS OF RIGID BODY
I)dne
Kronecker's delta 6;1
G;:{;
I :11.},
by
9tP--
,.e" 611'=522 =6s =1,612=66 =631
tg.:J
=5s-6t=6a=0,1q'q'=6rr'(3'r)
of arigid body (Fig.3.l), Wt = ae - 9e = t t x Let. I*t u = @i*t tpt = z.'q,
\ta-- t,Pa' : t2 = 4. + ,tr* zl and let '9= tr"rr * u'2u'2 + ,,3t/i3 -- 'jd,i'
l-.
r
f
For pcints P and A
then
En= J *nxap^dm= J^rrnx (s4x zptldtu= J\Ar^'tpt).-(tp^'s)tpold'n,
:+
rici) dnc,t;
(z;ut;pildrn =
f ,{r'6r, l:r"r, t;r;)drn,
Defining t6 =
l*{r'A;; go, Ilwi = Iirr+ Ilq+ IAq,
=
i-e-,,Ifa. = I{rur * If2u2]- f*.r,
7{tz = IAq +1*uz+I$aa. +
=
Ifil]
[i*
I"43u3,'
I{zu2+
Ittq
*
.H*, "A
[-; = g-9; = {u;c;} -g -
H,t. =
t.,;6t-;}
(3.2)
(3.3)
it] tIl
'+
i!{at=u'r&q}'
rfrl
[-'l
,=
rhete Lo,r= lflll
,
rr
^,=
Ln,;J Lrrii I:A r*.| L;.| .
l-rr'
4'
(34)
(3"5)
I* = li. The real syrnmefric matrix [IaJ is catled incrria'.molriz at A relatite to the ares
rr(q) at A- In glneral, fi.f b not in the direction of 1g' If ';l is not a point of the rigid body, then
fu: flc*bt x mgs^ rvith |fla.]= tI"lklNotc thar
t
AT ,{
", ^=to*-r"*too
of the mass element drn from a1-axis at .i4- The elemetts of [/Al are given by (3-2):
-rua,dbtance
- F;=i;;-;;=
in;l,z*,sr- x!td,,,= i:'i+z!)dn= l{rr;a^
4'^= f d2rd,rr1r2)dra : t',, =
l rrr, d*.,
tlr'013 -
r;:
loi+x!)dm,
Ifr: I!r=-
trrrza*,
$, = t"1"2+ ri)dm,
€r= l,!,i+21)dm,
tS: tlr=-
rlr-r{"--1,",,au,,
f ,rr"d*,
t, = !(o'+ 22)dnt= *1rj12. ,*: I:,'+,\in--,,,(tf
r!:ii=- f ,va*,
(3.6)
)2.
':
=
J-!"
+ v2ldnt= rn(tf )?'
I$,IA,I{3, 1A,1"n" are called Lhe produck of idirtia
of the body w.r.t. axes ri ar A. k!,k{,&f are called IIrc ndii of gymtion about axes t,y,z aL,{- Fo'r the
case of scss dis{fiDa liott ott a pldnc arca ilt e,y planc with point A in Lhis plane, (3.6) yields I} = t!" +.lf;u '
I{L,I*,1!3
x;
are called the momeats
ol
incdia and I(!,
Lcr-us'estabtish the trrrslortnatioa rclalionsbetween the elenreats of the inertia matrix [-[^] rv-r-t. aies
at, A.aad the elements o[ the inertia matrix [I'r] w.r.t. axes a.l (d') at ,'1 (Fig-3-2), where
rJ
d :"t.it {oiz9z*aisgs =Qp$. =+ 4 = oirsr, + 6ri :4'di = orr9r'aiq\ = a;oo;$s,(3'7)
(3-8)
- a+ - component of 9! atong % = 9.1 . g = cos[Z(e!,$)J = d.c- of g!' w-r.t- E,
*1r^-=ee=a.lg!,,
(")
:+,?1 =zet -15.t = apgp..9lr=o;prp, :+ z! =airxr.
llr9, t
r-:sing (3J), (3.?) and (a), Ill
'
= f^(r25i; - {xl)dm = aipaiq {dr"6o, - trtr)dm, i."., S\
,-9u
d-.o\
,'Y5)-\
E
l;
Xrr9,
.t-
L.
I
l''-
=
ar(
.lv*
Fi-.
-2
.1
- r.L
r
Dj=' *' "'io;'I*'
summiti""'*tt";iiaiilni; *"t ti" t'n"' "i*iq 9) i" th" sumr / of\, 9 terms:
,- - ^r:-.^1.,
(
respectively'
il;"; the elements of an-cotity w-r.t. bases 9i snd 4 by ( ) and )"
zcw o.rdcr kasot'
l- A scalar, like temperature ?, sltU So(f) rilement', su& that T =T' is called t otdet
ico"o''
such that rli = a;e,,p,is called a first
2. A vector, [r. gp^, ;Jii itiil
"L"r"";:
otdcr tctsor'
t:"::d
calS{
is
3- An entity, tite fii1, with 3? (9) elements' such that lll = a;ra;rff
"
tcasofotdcr
callel
is
4. An entity D. with 33 (2?) eletoenrs, such that Di;r :i.rfir":.!of
i'l'l
:
tt":
atP
n'
l'e''
using (3'9) with 9't = otpge =
The
--
Ifl
'
9: ne* at ,tl, is obtained
If" = r{f - avau!* = nunll{r,
121{2np2 + 2l{p2n3+ 2Iil E3!,i. = ,;r* : r1;; + I{ati+
'{!n3
about an a:iis
i.e.,
"
If, about othogonal axes n and s I* = \t = olpozqllo - t o"r{'
sre{ at :4, is obtaiued using (3'9) rvith gl
-9zs\ = g' i-e-' c3q :
(3'10)
s?:
tj"--tlorrro=/frn1s1 *I$r.2s2*I$n3s3+Ifr(41s2*nzsr)+I$(42s3*n3s2)+I;{r(n3s.+41s3}-(3'11)
3.2.1 Principal Axes of Inertia at z{
n' The corresponding I*' i" called
g
An a-ds a is:called a pitcipal atis of ittertia at' A lt I:i :0 V lpincipal
mor.cr,,Ii',.of iner-tia-
i-e-,
lrct u"
:
$rno = I{rne in (3'11):
4f;"=/flarq::tss{-t,'+=0 VgaE, * 1=^a,. + u6=}ni;
It*r- lnr, * {fi., + I*nr+ Il1nr = }rro
rfra1*rfrn,+rfin3=Ia1', r4t rr,
["'r =^[:il ,
14 I I
t* {i"r
i.e., I{n1 * I*nr+ I*"zrs: lnz, *- |
"' | ["'J
'li
L";J
liil
=
r*,,r +r$o,
't
-;;:=;":
+
tr^]tar:r[3],
(3.12)
(3.r3)
'?')
of [IA]' since z a8d 'l' are defined
rvhere [eJ = [r, nu nalr. Eence n and I ate cigca- lucilot ond eige,.':oal.'e
(3J0) and (3-12] yield
as the eigen-r.ector and.eigen-ratue pair of malrix B if 8u = lt- Eguations
I*!*
:
If;rnrr.e
=
Iosr.q
= l'
oJ ,;ntettia at A
=
l4'
(3.14)
g
arc' *spcclite'ly''ti'c cigce--wctors
llence, tlrc prirrcipcl axes of iaaaia aad pincipal momcttls
least 3 ort'hogooal cigenottd eigea-,alucs o{ *c;ocrt;o trrertri:-l'i'Al at.4. tr' reatsymnretric matrix has at
principal azcs of iacfliogi
vectors g1 rvith 3 eigen-values l;- Eence, there aluays crisi st least 3 orthogonal
ot A uith rzspect to ubich lhc i?;c lia matix lal'es a diagono.l fonn|
[), o ol [l;f o ol
rr'^k:13
Let
s1 : r.ri{.
Using (3.4)'
g.^ in terms
t
f,l =13
"f',,'fl
is
of principallx&of inertia at' 'A
(3.15)
,',
."i:,
If ut isirthedircctiotof agir<cipalatisof LrterlcioatA,fhenLoi"ir.thcdirectionoflv9,e'g''if<^r:<'''9!,
then cri =u, ui=o! = 0, and Ha =Iifrgi = Ii.lg.
In Fig.3.3, ry-plane is a plane olmass"yr,t*etry, "+ P(x,y,zl = p(x.y,-z\ :+ I*- f*zz'dm'=
0, I:! = j^"yd*=0, =+ z-a-xisisaprincipalaxisof inertiaat.A. Hence alir.cpcrltcttdiczlartoapla,'e
of *ar" igi^ct ! is a pincipal ar:r of il.cr'ria at a poinl lohcl il':':*::?
yz and zt ale pl'aocs of rn11
(since!::_n_'::":
A, \e"'vv
at 'ri
principal o+a
a1ces qe
ate PtrrsrPq
t
and y .,{c
(Fig3-a),
rg-,J-.rr' z anq
revolution tt
of revoluElon
Dooy ol
For
tOr a body
aprincipalaxisof inertia
z-axis
isalso
symmetry) + I*= I!, -0, rf-: i; =0, + I!r=Ilr:0,i.e.,
at,a. H#e ori'poiotonthca:crsof symmerrgof abodgof reooldiott, lhcaxisof rctolaliotctilottyluo
I _r
-,
, by symmctiY f* = Ifyr.or1/rtalJo it oaslittrle a sct of orthogonal ptitcipol ates.
axes
orlhogonal
-
._17
:
.J
./\
t-r'1tZ)
1
crrgr-?)
Fi 9.3.j
'
4r
J
T
J
J
l
J
r
(3.16)
-:r
,t
,!
\>j
I
I
tt
'"*
'g&
't_!i
J
.-l
-J
.-.J
7ai
-r-
i
Ti'b\tt-"'iro Teram of C ;
Consider rwo set of parallel oo]r, at i and i;
3-2.2 Prrrallel Axes
:'i:rr1r!)drn+
t Lr*,'il f
t$ =-t^rrrra* =-l @.+ ,crXiz
:-j'rrrro*-,:or,*
a*.+z'c,
*
L
"1T*j I
f *a'o+z'c", f
---+
c(-c-J
(Fi&3'5)
ad'',-=:
7,",
!ar 7.r9rFie-35
f;'+*(""'*'Lzl'
xsr)&n
l:*-'", f 'ra'n-'c, I
'
Ifi = lfi + ,o(r'"r+ "3"),
$, = \9r+o.("Z"arLr\.
/S=€+-('3, +rZr),
-
L.
at centre
lye+r!)dm = fito"+,,c2}2*(te+
t,=
-
'
tcig' then z; - i: * tci and J*;;dm = 0'
xs.l\dm
I*-L 71,*: aigir Lpc = i;gi,4t,:
L
!e. *,
*tE,
z'a*= I?'-mzctzc?'
I?2-,rltc.zc2,
tr=
tfi= t$-
(3-17)
ntz.cze,ca,
tt = I$-Dtz,Q3:,Qlt
inertia matrix at A due ts + discrete nass rn at C'
the bodies and.adding;the individual
Inertia matris ro. "o-o*ir" uoai* ate obtained by decomposing
tul"tio"" for rotation' if d&aeacontributions using parallel axes theorerrs and the transformaiioo
/I
+
[I^J = tlc]
+
3.2.3 Inertia Elements of a Cuboid
"d
a Cylinder
-L-.
L--
:
!-t.L-.
i
t,
I
L-.
i-
ti
=
L
*tP,
l_ii,t{
R2 L?
'nR2
t'
'
I:, = -7.
.
* ;),
- t?v = rn(|
=
A RTGID BODY
OF
3.s rvror{ENT OF MOMENTUM EQUATIONS OF MOTION
'
3.3.1 Motion of a Rigid Body with q = corts0ant' i'e', or = a'e"
(3'{) =+
If A ,s a poittt of fhe rigid b_odyor.its 3-dimensional massless extension, then
tc
ti"
ru^t=l',X IX
ffi]
Lrf' 's""jfil
tl]
=
.
[iA;]
i-e-, E^= Ifrr'e1 +rtzuzz+rs'e"'
to t*e
*l fr
ei ro
nvenience, *'e
ttzgi
Ia.feneral, I/.e is not parallel-to tr, even for plane motion. For convenie"Tl
'l',j "i';:
isid
th;i.1:0.iiiaad,i!itionAcoitlcidcsu;thc/g^|r=9./eN&alorr9/C,then
= iLttgi="'o,,t.+a, x H,+=Ii56e, + I*6e2:+ Ifual%+u'3 x (I$t'rg, *
= (I*t - I$rf)e, + (ISa + rfiar?)e2 + /$ri'e.,
Ma':
i-e-,
ilJlt
1u{or=I{rA-It.laz,
Mti= tt* + Ifr-',
=+
Mt. - I:,& - If,.',
tr{tt = Ii,,;t + I!,r'.,
Ma. =
tA
'.
lvl A3 = l33u I
L-.
Lt'
!-4t *0" =
19,:r.C#-
I
L:
(t2
d;i*
T*
"1"^T'":"'v z?dm,"t5"oj':-=,"('y:11'.::,*ii*":":l':i
I!. = R2drn/2. Hence for the cvlinder: f,
:
R2dm/4+
rf.
n-,."";;;i";rt.'1J". For the disc
*,'tio= =,n(+ * Lrr),
,f. =
I
t.- G
-
:n," .r'nna", (r'ii.s.ou) is divided
L.
L
l_r:rY
tf"'=rref!,
I
\-
"l'^,(blz)' +, z"bl2l+d,:1fr1",
-,
*t"pt +otl?t'
'{rr*
I
+g'
I!&-
t!3u9:-+
(3:]s)
bods
*
ri!4es)'
.i
(3'19)
r
-.T
J
In thissection,a witi tidt.poiat-of,body and in addition A=C / selr =91e,t4r is along 'AGIf'aris 3 is a pritcipal aris of i,;c/.ia at,4, then I$= I$ = 0 and (3'19) +
M1" = I$b,
i-e. M,q, = I{2,o. I If z-a:<is is principal axis a[ i., then M,*. = I!,q-l- Equation (3'20) is
centre of mass C and any point;{ on 93 through C for the follorving bodies (Fig-3.2): f
1- Body having two plones of mass sym11.elry rvith s3 along their line of intersection'
2.
3-
M-.*
=
*
-1f",,t291
iJ-J
L
l9z
Its-292-
Fii.3.7
(")
Mo b needed for lrotion ai constant t r, since 91 , 92 rotate rvith the body- flos'ever,
,/q * a principol axis o! inertio a! A, then (a) inrplies M.a= Q". /A - Itr= 0, i.e., no rno,tr?.ent M^ is
g;1,
needed to mointain motiott at constotttar. Thus if es is principal asis at
? :"0 ?"f81.9*!::Eq.:
A body ro.raring abour a ftxed a-xis (Fig.3.S) is said to be bala,nced if thc 9#;"7E-*
beoritg reacliotts al O at& B arc zcro,:urhaa it rololes at co:nslanl aiigular ulocitlll
:+
+ rc -0o
tngc = -mr,frsg.= f :0
-I?"n'Cr+ I?"r'*= Mc =9
i-e-;'C is on the axis of rctation and the a:cis of rotaiion is a principal a-tis at C. + If:. = If"+ azrzcr2c3 =
O, I*: I=9"+mtco1,c, = 0, :+ axis of rotation is a principal atis at, everl- point on it- I{ence, thc necessary
Hence, a variable moment
:
and suficield caulilioas oJ balancing are:
I-.Centrc of moss C lics oa atis of ';1tatior-- 2- Aris af mtatioa is
For angular impulse about axis qj thrcugh -'{:
o pri*cipal
axis
at
A of iticf-
or.e point
1
(3.2r )
fonr."(tr.ru) = I$[a.'(t2) -,r(tr)]3-3.2 Euler's Equations for Three-dirnensional Motiou of a Rigid Body
.I*r A bc o poitr.t of lhe igid body and eitbcr .4 Z C or ga1 = 9. or gs1t is alottg,{C, then tr'-r-iprincipal a:res of inertia g zt A:
.
r,^t=
['f + [] [I] tl3[]
Chocse g. fixed to the
Ma = E-^v
.
= [fi6r
=
tigid body so that i$ =
E^g=*o, * * *
il-e
:
O.
= Ii'&*1
+ I!r6-er+ I$n3e.
*
ez
ltr
l
s3
r[l, ri:-r r#".
-$Nr-l*)-+""]er *-..
Ifr6, - Q$MA" = I*zbz - (IS It{a, =
i-e-,
+ rrr,=rnr"1s1 *Ig1u2e'+r*-s%
=
1.43),t2-3,
If1)t,3r.r1,
a,
16^r=I*,i, - (If1- l{2lop2.
Th€se are Euler's equations for ptincipdl axes of inertia at .l{.
3+
3-3.3 Direct Equations of Momeut of Momentum if I{1= I{2
q b6ing the principal axes at 4- Every axis n(- cos0g,*sin0g3)
31
I g
is a principal a:iis
Id = t{r, siuce for !.(= - sin 0er * ccdg) I n &
/f, = rfin1s t * I*r.zlc* I$n3s3 = (-/fi * /$) cos0sin 0 = 0'
J- 9r, eqs(3.10), (3-11) yield:
Ifr = If, rr,.0 + ISa2.0 + I$n3.1 = 0,
I d.c.'s of g
If;^ = Ifynl + I{szi+ I$nl = rf, cos? d + I$sinz 0 = If1.
43.
JJ
ffiffi
J
L:t-l I
JJ
JJ
Bodg of rcuoldion with e, along a:ris of rerolutionSlab like cylizdncal body rviuh g" parallel to generator.
If angular velocity is constant, then a' = 0 and (3-19) yields
.
(3.20)
are 0,0,
l]
J
J
J
J
J
J
J
J
J
J
j
J
J
J
JJ
J
J
J
J
J
J
J
-
kr
Lil
L.,
l-.,
LjLJ
L_'
LJ
L.
L:
L.
LJ
LJ
LJ
L-.
L_
f
LJ
L-
LJ
LJ
L_-
L.
LJ
LJ
L
t_-
L-.
LI
L-- -'
Li,-,
L:-
LL.
Hence the a:<esg are.q[rcetsg t-hat9" is aligned with the principal a:cis 3 at,{ (Fig3.9) rb6eas 91,'9
could rotate relative.to the rigi(LMy about q3, stilt i7, = itr = i$ = 0. The al.grlar oeldtg {l o! thc
fmme 9i dtfcrs fmm thc a;egzler oetociej io:of lhcTgid bodg in its 93 componcnt:
Q.=Orer+Oz9z*Qag,
+
", =
Ht = Itrurgr* I-42,,t29r+/*atgj = ffl(trrer *utzgz)* I$ar39",
M,c'= Lev = f.,11g.=.-,.-*o x tt^= I{rQ)r9r*t:29)+ I$ti39"
!.r*
r
glo.cedu-.1g
u2g2+ a4g3,
urrgl
revolurion.
can definitcly_be-used for a body of
Or =
*ax
ulr {lz = @2, Q: # -s,
H^-
(3.23)
&c-
..
5'.i:#T".#'*of;"ai't;"?il;;"tff":r:*."-;;=';;e,*;.e-*i.s,+&^"r.
'6;>
-1
'
.e )
A translating system behaves as a rigid body- Equation (3.22) + M-c = Q, s!4_c,e.g1_;';Q. The equivalent
force system at C is a force F and a coupte of nioment M c. Herce Mt = M-c + rct x F : aa, x ,ngcv.
Hence for a translalirg sysletn, Mc
,: Q o:ad M a = 0 iJ AC is along 9cg. The moment about cocrg point
is ao{ zero. For the translating system shorvp, Ma = Ma : Mo = Q, but I{s t' Q-
3.4 CONSERVATryE FORCES
A force F is called .conscrualire if the *'ork done by i[ fron'r time f1 to [2, during l,he motion of its
nraterial point of application from location !r b L2, is i*dcpend.ent of the poth Ct colllectfiig, thcsc. locations-,
wr-z:
lj,'r-ool=
tJ,r:-rr,
vc1 andvr,rr.
(3.24)
I{ence conservative force acls oa thc same malcrial point of the body and, is iadependeint of ik oelocity
and time, i.e., F = &r). Equation (3-2{) implies that the integrand is a perGcL differential of a position
dqpendenL scalar function l/(d. called the potettial cncryl!, such thar
:
dW
(325c)
-,F - dr = -dV,
rI'(t")
W\-z = -. I
(3.256)
dV -lV(a2) - t,(rr)I,
=
Jue,)
i-e-, urorl' dora from L1 lo r, egaals ncgatiac of the change ir potential cncrgy from y, !o 1r- Eeoce roortdone by a cotserttaliue force in aay closcd path (rz = rr) is zerc. Let 7e be the dalum for V, i-e- y(ro) : O.
Equation (3.25a)
V(r)=-
"+
i-e-, potential energy at a given
!\'riting F ' d1= -dV , ia
terrns-
r!-
I r.ar,
Jb
(3-25c)
positionI equals negativeof the rvork done fronr the datum to this positionof the componenk of { and dr \r..r.t . various coordinates yiel&:
f,dz* Frdy * F,dz = -#* -
ff*
- Ynr, =+ ," =
-#,
1
-X,
,, = -#,
F.d,r q F6r dg * F,dz = -{*
ort Er av
F- - -ol" ^
- #ot - #or, 34 ,."-Ar, fo=_fi, L=_*,
AV '
:+ P, = -Y,
F,ds= -7;*'
Os
+
av. av'
F=-#L-ffi-ffx--v%
where
F, =
( ) =i#*i#*t#
=Gradient (
)-
I
)
(3.26)
r*-*t*
".u=l+
lr" e
Fe +F
el
++
=r-#.ffii,*
=Q
ii
tI
?
1b.zzy
1':
Jt in direction g is obtained as
.
-: ;;
I'. = F 'g= -gf'-e- -(directional derinaiive ot.V in thedirectionof 9).
.
.2i,
Thus'if.F iscoasentotiocthca F--YU- Itscoaoer:sc knolhzc,e.g.,forT=fgrccrrrl, F:-gY-is,
.I'-,,t
notconservalive. IIF rs conscractio.th"oVx r.
=g.
The force corhponent
t,j
I
A
c
E: Q itt o stmpty'cottt'cc'cs "s'e"' a clced path C (Fig'3'10)'
w
10), W
from Stoke's theorem: since for
foll,ows
Proof
iiv
x
gJl zf,-dl!- (xrrl,z)
tcr 'oL= [tYx
- i"E.dt=
I
r(x
Conservative Forces
3.4.1 Potential Energies of Sorne
the piecewise linear curve C1 (Fig3.1r) Yiel&
Using (3-25c) and integrating along
or -edmk (Fis'3'12) is
"'o^::!t,:f*:":!:=*:"'
;:,;;";;jffi"H,
";3;;;',:";;;
-E = F(r)s'- (Fis'3'13): h€.
l,u r,ou L, a,'l o2, totith Er="':
r.
-(Ft'drr * L,'dxz) = -& 'd(r, - r) = -Lr'dt
-P(r)dr'
: -F(r)!-dr=
-F(r)qP
2r = -4..4,*=
"r
v=For datum at r = r'o :
l,'r1rya,.
t
fc,
1.
Fig.3.t).
L
dV =
F.g.3rl
gr
(3.2s)
F.3.3.i3
-
:
-GMmlr'
datum
-GMmlr3 *'ith
For rnutual sravitatiou force f(r)
1-"1-=rTI'r^'"tf:SJ
an extension'e = r - Lo'
for
be
f(e)
tlre'spring
pirll
on
(lr). For a pair of sprirlg fo..e" (Fig-3-ralJ.t:trn"
(a).
rr.lrereIoisitsunstre[.t,".at".,gtLH",,."f(r)=-J{e},d,e=drand(3.29}yields
Z..WP;#%
eiq' 5'i.+
.-r=!;l-€i
v=!o"rkta",
":
(330)
(3.30)
.
r:-.--- -6F
sp;.,,g of sti{in'{ss f' /(") = &e, and
a- li*ccr
witlr datum at unstreic5ed conliguration 16 = f,,. For
extension
ke2 12- k62.'2 rvith 6 (=e) being the
the
-,-ields V j"l
Mr = !@\ for telatite axif lvist,of 0 bet$'een
couple
(c). Siaitatl y,tor lorsional spnag."itl
vith
(
k,,02
and
12
&r0
=
torsionalstiffness kt' l*=
ends, V : I:t(0)d0. For a ;iicar,o"siooollsPringof
untrvisted configuration as the daturn'
3.4.2'Workless Forces
pait of forces rshich together are wotkless though
lVe lls*, sonoe forces s'hich .are workless and some
individually each may perfQrnr trcrk' .
i- ft"go"ti" force F = 9gx B since W =E-'9--- qvx B'o= 0
slip' since li = A, 'fle-,.; &1;0 = 0'
2- Rr&tioa ft, at conf,acr ,P1 ivith a 6xed U"aV *itt-n"
slip' siace W = N\' g' = IVu* : 0 as
3. R.eacti,on at smooth contact rvith a fixed body rvith or rvit'hout'
Idsa
no separation'
eirher rrr :'0 for impending separation or ?o':0 for
4. Reactioo R 4L Aat smooth ball and socket or smooth
joinr with afixed !ody, sincelil'=
hinge
gr^-
-, __=_.2
.{
H:i'tr j:";-,fi:'i5#:i.:::f i*i:'i^:Y;"'*:' qej
!i:!t*=,*t,0: l_, -^^,.^.
i.
fY
,-vv
g_
N
*_
?,
' Y?r Nr
P2 with noslipsince
*itft
n
-\9-----&r
w = &r-gpr +(-&r)' w, = &t'(qe. -!rp,) = 8t'0= 9-"" !!.r, =.!e'' u' (gr-.zj<{
T.Pairofreactionsatasmoothcontactwithorrvithoutslip,since14/=
0
8'
/VrE,'9rr+(-Nra) '!p,7/rrr(,,'!rP. -4'cpr) = Nr('a)" -(up')'l =
no'separatton'
for
(up.)o =
o,'('o1^
1i::)"
0 for impending separation or
- (opr)^
as either nrr = o.lt*ll"oi.t*JL."
. r*
l'no'seRarati1".',, a$'
,tt
g- pair of tcnsions in light inextensible cable since since W =T9' cr * (-7 g) 'ge Y
-g-;rffi."llU,
inr
for impending slacluess or 9 ' 9t = 9' W for tight
= T(e - 9r - e - gz) = 0 as either T = 0
e. pair of teacrions {Be,t:,lt$u':L4gl* ".rt #rt_ball and *Y i"]'::
.
x:":x'::
s x ( A a ) s = 0'
N. = F e -:::1ff::x1
]; rfi L"i"*"Xi;"1#ffi;-i--"6 . = r r. @ -*.J=', *
6. Pairof reactions Rr-and
aicontaco"f
ffi;#;,:
@
]
t
I
I
I
-.,,b-
-)
v'E
.iJ
--Eg. Z.rS
,<
iru
I
I
I
l
-
S
I
lr
I
{
!:'
siis&€ix EeTPssIoN"or A RrGrD :oDY
rccanexPressthekineticcnergyf ofarigidbodyas
O"iO6 1r.19)and lpc:gifsa.''
3.5
t rcrNETrc
x rt7'6dtt
rPc)d 6 = \ma/+ *gr' l:'"
i:?.?: **o71i l^*t(crx
gs ' s= I{ciui = I;c;"';"';l (3-31o)
y = lmul + +; s: **rZ + lt$.r.;
fusing (3'3)
(3-316)
2lf'46r"1!'
i.e., ; =i;; *itt?r.?+ $"$+ €-3 *2lf1't1,t2*21$wt'tt+
(33rc)
r-
lmtfq* i
.
L-
where
ri,";'1,
imi,;' i;i';'i
T
-- lnrf6*
r
reGrs to principal axes of inertia at
C' tf pohrt r{ of the'rigid
body has 9't.= Q' then gp
: 9P, =
ll$-i-;'
(3'31d)
ia^'s-=
r = i !,'?,ai,= i ! *^'1,ax=,,^|d*= fo-' l:r^*
:^o:=
r -= ;iijrri + I*lrtr+ I$"'f * 2l{+'tr'tz + 2l!*uzut * 2llvl-tl'
r = lfli{ ui2 + t;! w;2 + r#-i'J'
-
i.e-.
-
can be expressed in ternrs
wbere + refers to principal'axes of inertia aL A'' T
rigid'body through A:
of th6 instantaneous axis of rotation.(Fig-3-[6) otthe
L
,
Q
(3-319)
L
L
-
ra ar,\
(3.31e)
For the case of
9' may or- nray aot be a pdncipal
7 = !rno26 + ll$"tz '
g = -g", ;;
axis ac C' (3'31b) becomes
(3.31n)
RELATION F.OR A RIGID BODY
3.6
- - - WORK-ENEILGY
ener&v T
The rate of ruork I4l of the e-xternal forces and the kinetic
'ot a rigid body can be
expressd
as
'F.
-l
t
L
-L-E-
+'i = tnec,b i7 rr"' e,"=dtn:
= F'ec
+r'
''a
1
1
*'
l rr"
E-' o,
r'.a'ol
*
! *"'u
='L' vc *'''it-
lfence rve ob[ain the following uork caergy rclalio*
'i : iv,
Lr
x
it
apgdn = F
-
w +z'
l^r"
-- F' vq + Mc'q' [rncc
mtc Jon:n
a:l.d'
in
x
gesdta
- E- Ec : Mc]
integraled fornt
Tz-Tr - I'{/rr:'
(3-33c)
tnd 2' DenotinS. the rvork done by the
where fi, T2 *ethe values of kinetic energy in configuration"-rl
+ I'iz-",, 1-3''33a1 reduces to
consei'ative and nonconservative forces by i, *dwn", iI/.,=l'v"+li,r- -'it
(3.336)
(72+V)= (T1+ t'i; = Wn,r-|,
I and 2' If 3ll forces are con1lati5'tfren
where y,, Vz are the values of pot-ential energy in configurations
ener''r):
.oo"".-..io.i"rmecu;'cal ener'y (sum or kinetic and potential
#". =;'",'i A;;;) ;;;;
i+v
=i{n.,
Tz.*Vz=71*Vt'
t+ir-o,,
F.
t-
L-..
System of Rigid Bodies
3.6.1
- - Work Energy Relation for Interconnected aad added up to yield relations for the oholc sgstcm
Borr"rl"". tt.iri are applied to iudividual bodies
the system since some of the' forces-t"'.tltn
which involve work done by the intcrnal and the external forces ou
the s'trole sysiem' De[o[ing the cootrilut'ion
arc external for an individu-al body are ir fact internal fiorces for
relations corresponding to (333)
of internal and external forces by the s.bscript int*ext, the work-elergy
teduce to the following form fot various cases:
*
L_-;,
L--;
L_
L.
(3'33c)
44
F:3
"/lc siig
no stip
'"\
-
i = i\lint+cst,
T + Vnt+cEt =
(77
Wac!n1+l"=i,
*
Vat+e"rzl
'i-*Var+e,,=0,
4 -Tt = Wiar+c:rt-2r
- (Tt * Via**t1) = W?t,,.+crtr-r'
T2* Vat+a2=Tt*Vntgutl'
(334o)
(3'340)
(33ac)
q'ork of internal
internal forces of interaction are togeiher r,r'orkless (Fig-3-18), then in eqs(3'34) the
conveoient to
are
form
integtated
forces is eliminated and these reduce to eqs(3.33). The rare form and
provided the
obtain acceleration (or angular acceleration) and velocity (or angular velocity) respective\',
a priorill'- frr general,
s],=tem of connected rigid bodies has one degree of freedom and. W can be obtained
and
rsork energr relations are conveuient to obtain velocity as a function of position. Impulse-momentum
velocity
obtaining
and
for
problems,
angular-impulse-momeut of riromentum relations are useful for in'rpact
as a function of time provided the force is a given function of tinte.
If all
3-z NECESSARY CONDITIONS OF EQ1}ILIB;11UM 01. A RIGID BODY
TIre rrecesscry artd safficicttt cordiliors of cquilibrirm of a rigid bodg are.: l. E(t) = 0, 2' i{-r(') =
and 3- the body is itritiallg in equilibriutn al t = 0-
Q
Proo-f Let q. be the principal:.a-\es of inertia at C-
.
Ma=A, + 'i:w=F'r'c+Mc'u-=O V t
T -- lnu[ + +I?t-?+ Lrrg*i + iI&-3 = coustant -.(0) = 0:
F=0, Me=O, +
-
- g,
:+
u;;(t) = Q,
i-e-. the rigid body remains in equitibrium for alt time.
e{t) =,9a-6
!ac(t) = q'
T-Agd
3.8 CENTRE OE PEBCUSSION
e
-P
'WY=\a
rrr[gc(O+)
-
Frg,=.tSCt
,J
jcy'G'
- -(0- ]!co = t= I.s. + (Ie + Iar"4,
I' = 0,
t6 + Iq = nrlr[<.r(0+) - -(O-D-
si(0-)] =
rn&!.,(0+]
(o)
(6)
(o*) -.(0-)J = Iaaeo3- (&+d)rqThe solution of (a), (b) yield: A'*r = (Ir + dllel i?s and,
(")
16 = {h(h+ d)/(&&)3 - rlra- .
Thus, in general there is a1. inrpulsive reactio-n at,O. The locotiott of poitrt of applicelioa'Q of e 'roTsucrse
;ropulsipi force for ultich there is no impulsiutEaction al the beoring al O * callcd,.lherccn1.;v o{ percassion'
€t
It
-rs
obtained [rorn eq(c)
as
[(h +d) =
(rg)'=
1t$12 +
liz, :+
(3.35)
ad = (&$)2-
For a thin uniform rod of leng[h I, hinged at the end, eq(3.35] yields d - L/6;and for a thin uniform circulir
disc of radius R with hinge a:iis normal td its plane at the periphery', eq(3.35) yields d = Ri2'
2- Consider afree igid body at restsubjected to instantaneous impulse Ieed L CQ al Q (Fig3'f9b) with
principal axis g" at C. The impulse-monrentum and angular impulse-nronrent of momenium rclations yield
c,(0+) = dlol&,
Iansc = d Ig = Iaca[u'(O+) -'(0- )] = /$ r'r(0+),
+
a
L:
4
ix
t
n
h
4
e
1
t
:
i
Ieea
-
*[gc(O+)
- !c(0-)] - mpc(O+), :+
i
= Igeatn'
Eence the distance r of the instantaneous centre of rotation 01 from C, at t = 0+ is grven by
comparison of eqs(3.35) and (d) imply
t$/md +
rd=(t$)2'
YctoJ
that ot:- o. Positions of 0 and Q are rccdrrocclt
r=u6:(0+)/.(0+)-
:
l
I
sc(0+)
+'r
vy
,//\
Fi
xe,
9.:.rsb
J
.J
j
]J
J
-l
"-l
illPulse
Consider a rigid body with fixed a-.iis of roLatiou at O along 5 (Fig-3'l9a)' Let an instantanebus
ti"n
the
impulsi"Sffi
and
let
0
ati
ta
OCQ
normal
directiol
=
(impulsive force) /a(0) = Iegoact at Q in a
relations:
of
momentum
impul'+momeni
ar O be -IO(0) = t.g. * IO9+. Impulse-momentum and angular
l-
J
rJ
(o)
positive terms is zero:
Each of the 4 terms on the l.h-s- of eq(a) should be zero since the suur of these
u6(t)
i=
i-.I
.J
,:- I
(d)
14
IO9,p
J
J
J
U
J
j_l
rJ
'--I
-J
J
J
-J
J
il-J
-J
-
jJ
J
J
k,
U
'..r I
U,
'lllhile working wur J[aitmci or hitting with a cric&ct bat' if the imp11 point is a centre of percussion
w-r-t. the point wiere ih" h"rrdt" ilnlta, thi there b oo impubive reaction'(characteris[ic'stin8') oa hand'
3-s TMPACT OE RIGID IODIES
3-9.1 C'enerat Smooth lmpact of two {Jnconstrained Rigid Bodies at a Point
point of body I having
Consider smooth impact of [wo unconstrained riSid bodiej (Fig-3-20a) so that
'4'
plane at
is called
mass ml makes impaci rvith poiot B of body 2 having-mass ryz-The common taugen[
"t to the
g,
normal
aL
A
direction
The
planegla.ae of .impacl. I*t.g, e1 be orthogonal urrii vecLors in this
impagl b called ccatral impacl if lhc ccntt's ol mass G lie on
ltrrr" of impact is ciled line of impact- The
'rol-cent6l
impact.. The impac0 is modelled to occur in zero time'
the liae of impact, othervrise it is called
L,eL O be the point fixed in space which is coincident rvith / at impact'
i just
Let el and 96- = u;oe.n+uisq+rri'gr be the angular velocity aad velocity of cenlre of mass of bodv
ui,S, * u.:.e5- \4re
before the impact- Their cbrresponding values just after the impacr be ari' and li. = ui.g." +
in direciions
body
of
each
momenta
The
set up 12 equations for the 12 scalar unknorvns irr: ,Al,C2,vb1,!.1,9, 9l are conserved, since the external impulse ou each ii only along e,", *
TJ
tJ
u
u
u
u
t-.
tJ
LJ
LJ
LJ
There is no external impulse
I{o
1-.-
*m2rt!2o
= rnlutn *
along q;, is conserved:
tn?u?n.
;
(3-366)
ofeach body is conserved, since the external angular inrpulse on each about fi-ted point O is zero:
v'on
t-
-
a!^o
(3-36d)
= -e(o p, - t:*)'
ttAa:9a -9, = &c', *g, xte6r)'ea,
(3-36e)
iu*n = tB:gq
+9zx
taCr)'9*={Slc:
If body2isamassivebody.thenrn2=cp + 4=-;t, s'c.=9gr,t"go-ularand(3'36b)forthe
sysiem of bodies and (3.36c) for body 2 do not yield aontrivial equatious. The slt scalar unkno*ns erl, 4r
]
are determined frorn (3.36a), (3.36c) for body i and (3-36d):
(g'gZ)
u'1,
= u1,, oir = ure , ELr+yrs,xwtv';;9- = qrr*!crox',r,t".:11 -:" -::" =-",(.q.ea -"x^)If body 2 is a massive body,,at' rest; 'then' the last of the'equations iu (3.3?) reduces tor,ufi,l.'= -eurn '
3.9.2 Surooth lmpact of two IJnconstrained Rigid Bodies in plaue urotion at a Point
rr'en=de-9^=$br+sal xL{cr}-g.,
lubo = ts - % = (!L2+dz x rac2) -9o,
:
rvith
u
u
L.
L.
LJ
LJ
The six postimpact unknowns (Fig-3-20b) ,u'ro,llr,rtln,{2r,'-t'r,tt', are obtained using (3'36a,b):- ,i
u'rr
= ur,,
,'rr7tt2rr
mro'r..t*rr!.
=tlI1u1.
(338o)
*mzx?a
and (3.36c,d) for the configuraLion ot'Ct,Czshorvn in Fig.3.20b:
L-
(3.386)
+ mlurnrl,
€* rL - rn2v!2nr2 = &' rr- rnzu2n rt'
obn-t7o=-e(oao-ue.), i.e., (ul, *u'2t2)-("t -arirr): -e(o2o *ur3r3)-(u1. -&r1rrl'(3'38c)
since for the given configuration u,tn = uh -(rtrl, aBn=ttzo*ttzt'2.
If body 2 is a massive body, then rr2 = oo * dz = t tz, *z: \2, utrn = !gn, and the t 1st e9ua!i11
in (338a) for the momentum of system of bodies aud, second equation in (3.38b) for body 2 do not yield
nontrivial equatious. The 3 scalar unknowos ,1, l/1o, u'1, for the configuration of Cr, Cz shorvn in Fi8'3'20b'
.io^\ /e 2aAr /a aa^\r-.^--:-^J r* ro
t mZVz+
(3.38c)1,(3.386)1,(3.38c):
--^ determined
from
are
iSul, +m1u'1rr1
LJ
L:
LJ
I
L..
L.
(3-36c)
oir =
impact and e is an experimentally deternriued constarrt, called the coe$cicnt of rc'stitution' Hence
t--
L
,
ELr+rcrox ,ar/cr=E-cr*tcroX n1u6, + I{Lr*lcpx 'nlu'rng! =E-c'+bPx '}rtulnci' (3-36c)
ttt?!'-c2+ EL,*lczox ''2''2,.e'=Ec,*lczox
EL"+!c.or.*r*r=Ec2*lctox
l"u"t''
where Wr)=[rc Jb,], Wrl-- tfctltgl'r], etc- Equations (3'36a.b,c) constitute 11 scalar equatious' Atr
of
crnpiricat ilalioais added to conrplete the set, viz-,e - trfto..vhere:u, is.the't'elocity'of separation
just
g.
before
along
B
g from .r{ along E just after impact, uo is the 'velocity' of approach of A tot'ards
t-.-
L:
= ur,
mrui.
Lj
L-.
0rr, 4. vr, , u'lo = v?a'
=
on the system of ts,o bodies, so their total monrentum
u'r,
tll
- I$-.
,i'
1
7
ii
i
j'z
LJI
/be
-
-n
F;3.3-zot)
+8
--J{vza
'n.
V-
l
I
l
r-I
,^-J
i r..'
(3.39)
$u'r+m'o'vnr1 -- Su1+'rtlulnrtr
trt')J'
*uzrzl
orirl)
-,(".
(u2- *cr2r2) - (oi, = -c(u2.
to u'1. - atlr1 = -e(r,L -4111):
reduces
(3'3?)
in
equation
last
the
If body 2 is a massive body at Eest, then
Bodies at a Point
3.9.3 Smooth Central Impact of two Uncoustrained Rigid
the angular velocities are unaltered and usiug
Since the impulsive normal force passes through G,
,'r,
-{3-36a-e), the postimpact rralucs are obtained from
rr'r.
ntoio *m2u'2o = mlula
= or, u',, = uga, oL, = pzr. lie = Qt,
*'rR21'3n'
a'2o
- ttln - -e{oz- -
ur, )'
(3-40)
,l.he velocity of the centre of mass c of the system of two bodies remaini constant since the orternal impulse
system is only due to changr: in g' component
on the system is zero. Ilence change in kinetic energf of the
nr:) = (m1u'1, 1m2oiol(mr * rnl) l:
.rf velocities of G relative to C I rca =(*rrro + rrf,-lrz)llU +
Body
1:
A?r = |nr1[ri;
]*,[',, - I!]+*#41'
- Wl'-
-1
:.}.#[,i._oi.},'.-(o1;.-,,.),]=,ffi(e2..tl)G'.._.t.),
'-
arr + alt:
iffi(c2
3.:o<
isGatled gcrfecttgclas{icimpactfiorurhichthereisoo
=+ e3-1<O:+ O(c!f.T\case€=f
pciccdlg ?lastic impact fot *Lich +he
lcs of kinetic energlr (A?r - O) a$d ?, = ua- Tlre case,c = 0 is called
loss of kinetic energy is ma-xirrrum and u' = ['
and the consenratioa ofrmraentum
For impact of body t with a massive body 2 (Fig320c)' *, = k,
-\f <0
to
in s,, does not yield a nontrivial equation' Uence (3'40) reduce
: oz, u.L, = 'l'r tLn = u2,,' 92" to
In particular if body 2 is fi:ied. then {4 =?.tr = O and {3'41) reduce
ui, = ur." ,i, = ,ar, ,1, 0, l2o :'0, 4. = 0'
u'r, = ur,
u'tt
: ut6,
o!1,
t)'rn
=
o"n
=
-e(vzo
- q.r-
(3-41)
(3.42)
-e7t1u'
=
3.9.4 Other Cases
impact A has zero velodtY just after
If body 1 impacts a perfectl-v rough fixed body 2, then the point of
of nronrentum about o:
irnpact. The postinrpac0 angular velocity. gr! is decermined fronr cotrs,:rnation
(3'43)
rvhere l7bl = [ro][4]'
Hb-,= E-c, + rsp X rnlucr 1
::
ttre rigid body I'
:r:
since just aflsiimpact !r+ = 9o: 0, i-e', O becomes a point of
of each body shoul{tbe draws shorving
FBD',s
point,
then
lf one or fhe oiher body is constrained at some
points' The postimpact data is obtained
all the impulsive forces at the point of impact and the constrained
of momentum eguations a'boilt centre of
by writing impulse-momentum equations, angular impulse-moment
and the coefficient of restitution equation'
mass (or about a fixed point, if any) foo
3.1O GYROSCOPIC
COUPLE
"Jbody,
inertia at o equal *' = 8''
1. consider a rigid body with a fixed point o with two principal monrcnts of
g.
itself precess at coastant Y" fg
and
L,eC the body spin at constaat ratc sc3 about body-fixed a:<is 3
about O reeuirld, to maintain this
about an orthogonal axis E fired in f (Fig.3.21a). The mome nL e-o
acceleralion qif of the body ane
motion is called ggttscopic o11plc. The angulapeio"ity 91 and angular
@
i-e- orl
- pE+
:
= p, u2= 0, u3 = s, t.rl
:.r3
se3
=
Pg.s
* se3,
tir
=
P9t
x
sgs
=
-psg?
/i
cquations of motion:
= 0' r'r2 = -ps' Applying Euler's
3
+fl
rt<t
''J
';-l
,-J
'.J
I
,-J
,i'.J
,J
.J
,J
"
- t)(u1" - t':o)"'
:-J
99r
>p
J
J
L-J
,::J
+
t3J
et_-J
ri--J
il
-il
'=:J
ilr
.J
.:J
J
J
:J
il;J
iJ
J
- )'
o
:t
- l'
\i
E
o
directions'
y = pE- s =sg. The gyroscopic couple is' surprisingly; normal to both precession and spin
Hence
2- Co;ds the case of precession axis I at ao angle 0 l'o the spin axis 93 (Fig'3'21b)'
= -pssia9s'
ut = pS+ sg- p(sindq *cosaq3) * sse, ,L= pEx s*, = p(siuast *-:*':]
1"*
eqgations:
g--tt=tt*a$t-pssind- Applying Euler's
i.e. <.r1 = psia0, on=0,@3 = s-+-p.cos
tr,[o, = fr-t
Gr- E)t :rs = 0,
Mo2-- Ir9r.r--(€ - Ifl),.l3.^r1 = fflp2sin0cosd - 4(u+p"*O)psin6'
7[ot= Ea" .. (.[f, - I$)urp2 = a,
(3.44)
i-e-, Co = Mo: [Iflp?sin0cc0- /&(t*pcosd)psin0]g:'
rni[r'q1z:-":l:' oI
3. Ler a Uody with ifl = I?ztoLale,abour fixed poinr O (Fig.3.22a)- We s''rsh to find the
cos &in (3'44) , "+
r^r3
+
g
and
=
<,4 so . that preccssioa al cottslottl . is possi6la :Usiug Mo = -rrrgh sin ds2
l
-s
rvhere
-
Lr-
L
L.-
flflrp?ccd- $r,,4p+mgtlsind - 0 +
iL-
>
tj
I
L.
tj
LJ
Ir
t---
L-.
LJ
L-.
L..
Pt;
tz =
.llllrngrlc()ss\l/zl
?r*Sh cos 0 ', r7:
:--fr:-€'t
ff".*t,t(r-:W
tIg**:a
It
LJ
Ll-
F.8
Eg.=.z2cr-
(")
t
FortIrecaseof-!>lt|,,nghcos0|(I$)2'tlresquarerooLternrisoipandedby.binomialtlreorerrr
s9S
retaining the predominant leading terrns, the precession rates are obtained as:
?E * A a-,'4
pt= n.sh/(f;33-il! '
p1= I$u4/(.Ifl1cos0),
,-NfY^ ^
-
(,
'' ./i,"";g
The relations of tlr'rs sectioD are valid for a spizrrin g projcctile*itr, o "ra
nrgl replaced by C and F.L, tvhere F is the aerodynanric force acling.opposite
ut ='
1,
-'i.r.zb
:tr;:ll'xff:ffi;"j."ffiH"Jj.:"$;l'#i'.,o"
:, r il3n+.rrse
axis I
The spinaing top shown in Fig3.23, spins at rate s about its axis es, its
precess€s a! rate p abour the fixed vertical
axis g.r ia the vettical plane through &, gs. The angular velocity r.r and trlo ate grve[
* &:i,1Y:::.1::Td;*:.,o..
The moment about O is Mo
- -mghsin8qz.
?+ v =
Since
Ifl - I?r*d
\\bAe*O
kv tgr-'
i!i.,
by
z
i
I
!
i
!
]
OC'h
j*gg.
-"'{:3
t
Ee3.r-3
t
Mechanical energy is conserved since forces..".or15gF.ative:
rll?L@2sinz0+d'?)+ Er3l *mglrcos0= constant=
Mo3- 0, the third Euler's equaiion yields o3 8s constant:
74o"=I&,i,r-(IP, - $r)u,'.o2-_ 0 :+
cir3
LL-.
l4nsht$lQrlccsl'
_1
(3.46) that the sleepiag top
Tire condition (3-a6) is delinitely sat.isfied if -3 > llnghlfr/IlrJ. It follorvs from
conslant 0, is follo$rd
(d = 0) is stable it t t! > l4rr.ghlfllf$!1. For a spinning top, the initial precession at
at some staqe condition
by rvobbling rnotion with nutation because as r.r3 decreases dire to inevitable friction,
/f--\
rc i
(3-46) get-. violated and precession at constany 0 is not possible'
/",
If condition (3-45) is'satisfied then the trvo possible precessioa rates, p1
(fast precession) and P: (slow pfecession), are given by the roots of (3'a5):
l--
t-j
(3'45)
0; or fflp?cos0-I$qp*mg'['-0-
of the quadratic equation
Thus tsoo r.alues (rools) p1, p2 of uniform precession.exist provided the discrinrinant
(3-45) is positit'e, i.e- if
z
(3.46)
u! >
i.e., if
4(I?rcos0)msh
0
U|*i' -
-
sind = 0, i.e-, d =
l
=0 +
qa3
Eo' t'
'
=constant=os(0)' -
(1)
(2)
t1
t
a
50
I
i;
,il.
'-r
0' Eence che
a:ds f4 is zero' sinle {4. fo = -rrrghi*e*'Es=
The moment about tie fixed vertical
0 is constant'
.
abour rhe fi; verticlt axis E: through
momenr of momenru*lg, E"l
,+ Iffpsin20*I&r.rgcos, = /fo.(o} {3)
= consL.
ffi@sind9'-tiq)+se9.l.(sin0q1tcos09")
Eo.b=
OF .A' RTGID BODY
3.12 TORQUE FR.EE MOTION
priacipal axes at c'
H (o)
moment N-c =q. Let g'be the
consider a rigid body rvith external
l.,LioLh
gc(o) = const'
{i.e
*
E-c= .f1-1g1 * I&rrgr+ rfg!'atu =
iavori<ibrcThe constant
W =g
vector [6
provides an izacn'a6le lite in
:-q=F'2s=ig
,.!-l;rig,Il,:
+
f (Fig'3'2a)'
T17=T-fc=const' +
=;"; * w*=
;*;;A;
\Hcucap = coust'
pt.na
*
C
..-^.
ct
ucospA - cN
: co''st'
vector r'r drarrn from C lies in
is-const'ant and the tip of the
as rvell as directiou'
lU' i"-a"t*"f ' g is variabte in magniiude
invaiobleplane norma I to H.through
i.e., the.rotational'kinetic.energr [1
3.13
ToRQUE PREE MoTIoN OF A *:-
!.
3rd Euler's
(1) +
eq:
BoDY \I/ITH I?' = $'f
'&.-
+ Ec:/fiitte'at'e){r$"e"=consi'=[r,(a\
Y=g
*--"'*=ttf'-('4i:id)+r$'al1=const'
Ta =Mc-!*:o'"
' it
+
-3 =' co4st=0
l{ca: I}r-'-1'fi :
'|)tt:'-=^'
G) * 'i+'l=cor'xtii:Wftd+-3)itlg1.(,3=consr.
the
(l)
Q\
(3)
(4)
(3).(4)1-2-=-?+-3+-i:GotEt.(5}o=arclan(ai+u$1ll?1-"1:const...(6)
:corrst.+HgL)cos|=cotls!.(l),(5)+f=coost.i.l
(2):+ZTa=fi.C--
(1)'+'t']tt=(J'rrsr*..'eesl*-'l -';--';'*"+*'3q
=ff*ffi**'
*
:
s: Per{* sea'
Hence
o
rvith l )
u"t
t?"
:
t':"t
:':':!l.3'
(6)
iaclination
trre body'fixed a-'ris g" and
*rhose i*crinatioo a
is a vector of constant rnagnitude.
-rvith
Equat'ion (8) shorss t-hat
Irc nxed in i'ettial frame f are constant'
o
'
p with the invariable li*e es along
boriS--fxed
isintheplaoeofglyandg"andtheiirctraation0ofthebody-fixeda-tisesrsiththeinertialspace-fixedaxis
;:l,f-g"."" trtt motion is as if the
ffi;;;
shorro
case
rhe
for
constaut
a'-xis e,.with
{
:inertial
ss rernaias
cone 2 of angle f
on.*
*ti,
."*tott
**,
a-,cis,qj,
*ith
a
gtlRace-fixed
cone I of angle
about atis
p
* along s'itlr spin s of the bodl'
The nrotioa consists of preeession of -ri= ", "ootlt
initial dat'a for ..r as
the
determined abinitio from
f...
totque
for
-Jion "*1t
The various e$tities
rdb;:";;;il
ffi
**i*:"ljJi;:
Ifi-,g,
'E-c
+
=
I[6rsin0 = Ar'r.r1
+l
+ I$t.ts$ = Il6(sin 091 *
H'scosl:=
o,c"
de3)'
I{'ql$l"ua)'
= 5d:Pcos0 = $ccosilI{s
t,c - t{JIIccc0lI!"
s=Ufi-$slnct*"':"
regtlotprccessiott(Fig'3'26)'
the body is said to have rctrosrcitcr''-""i;::;'';i;
-t4,4,
.r, 19
p..!
tand =
ar3
.r=<^r1/sin0=Hslll1,
L]
cos
I?P*
= Psing = Hc:ilnlll?r'
,
ForIfi >I83,s>0andthebodyissaidtohave
+ (s +
= ig'
u,fis,/'o
Pto".
er\ Wu
5t
("}
- (6}
ta tcoaod
-^ <Is'
-n--,
ForIfi
'i if : ,:::.,":::,:,:t:"i:::,-1ff-n"
=
,W,/
":lli..
"f
P- --.
j
);
{.
ti
r
:r
-.I
--l
J
JJ
JJ
,J
': I
J
JJ
J
J
J
J
J
J
J
..J
I
!
J
-J
-J
-J
-J
-J
-J
-J
I
-.J
S.J
)
1.;"'1-,:
r
)
f-),J
-,.:..:
btirr:'ivnonr.r norerroN' or A RIGrD BoDY
, ,,1i
of insrtia r-t'-;'d'
*"
g
where
P::":Ptl,axes
,uo?,
,t=
*.*,angular velocity
;rra o"art."a*a srndt distur-baa1 is giv6n :'
warG
to
in
cstigit€";hether *.u,,."io" bounded if
We
at C and Mc.= O
tothebody.I,ettheingulatvelocitybegr(t)=.,,o9r+e;(0gidrrdng:,Y"*motion,thelalrreberng
;J;-i'6i;;;;;. the dbturbace r" 'ppii"a Jii " <r"o' I.ence s= €is:ur=trg*er' -@2:c2, &15=€3' til=i1' d?=i2t r!'s=€'
.
3.14
-'-;" srABrjrri
,)
I.
L
l,
_--
-_t-lr-:..
Neglectingtermsofsecondordetofsmaltness'Euler,sequationsofnrotionatCyieId:
''.
Mcz- 1,9$2-(rS-S.}15r,6=+ --$rir-(6-frlt"(t'te*er)=0'
L.
L
rL
Mcs-I{"ar-(Ilr-$.),tp2,
' r$'"-(fr-
*' €'a-(€'- t?**zq *0' {2)
8rl<'1+'1)1'=0'
$'a-tf'-r3pe2:0'(3)
'"'r:'
'
(1)+e13e1(0).Equaiions(2)and(3}arecoupleddifferentialequationsfore2,e3.lAreeliminate€3by
forming (7I *d using (3):
(4)
* tqi', rg:Xr{,
8z€z- (rS - rf,),oa"
iu
_'g':'..'?'i'j:' eq(3) implies
is harnrc rnic (bounded) [ *d hence
:0. +
i
L
G) If, * €r, $r * I&r,
that
ea
The"solution of eq(4)'forez
coe$cient of ezin'eq(a) is Posiiive' i'e"
the
is also boundedt if
$r> I?" aad If, ,
ts
IIence, tlre rotalion of a rigid body o6out
rS,
a
: o-
-
or
if
rfi < r$ and
:
t,
i::
< I&'
'
(3.48)
of itcrtia is {rtc lcrgesl or
prittcipol azis fot ulrich lhe ttomellll
slc6le(2) +
(3(u)' Equation
to:":'-::1.L:* e3
c3 = e3(0)'
I$: Equati'on (3) + es = 0 :+
tr9,
If,
{
&)
.h^,,r. ,,
=
arcis et ls unscable'
rotat'ron aboul
e2 builds up with time and the
e2 = e2(0) + (Is - Ifi)r.roes(0)t/s- rhus
Equatious(1)-(3)* i;=0 + e6=e;(0) +thetotationaboutanva-xisisstable'
(.) f;, =Ilr=I$:
TLYING CIGAILS
thc smallest'' is
TO
3.15. I.LYING SAUCER.S AR.E STABLE COMPARED
a-'ris 3 (Fig'3'2?a) ended uP b-v rotating
An early cylindrical satcllite iaitie,lly spinning about lonSitudiual
The
the s'ell-established theory of Sec-3'14'
about transverse axis i 6g.f-eZU1, "ont."ai"tinl a."*"ti"aly
is
body
The
rigid bodgliag. thc objcct wrongly as a
e-xplanation of the actual obsertation lies in ,noacl
the'rotatio[al'kinetic energy ?a is noc conserved
deformable and though E-c iscooserved "irr"" ub = Q-,
because of dissipation of energr rithin the body:
(1)
- E lz = flZ t2I&
E-c :€".e : Ifru'e,
"n(0)
"
constant, tlic body ends up turning about 1" 1*it
?.6 keeps decreasing due to d-rssipation aud since ffc is
the dehomiuator of gxeressiol (l) for rrt -tl'"t
for which ?p is minimum fot giveo I[6:, Hence it follows from
ubich the pritcipat m:o:i: is flre marimurn'
eventually the body would c'|d eV rctalittg obovt an n'i" 1o'
By the sanle
transverse atis I since lfiEence a cylindrical satellite ended up rotaiing about a
-19
'
in a-tial rotation rvhereas siucer-shap4''bodi"t
;
reasonr cigar-shqped bodies with I$ < ffi aru not stable
with I$ > /fi are stable in axial rolation'
p
i-
Tau'
Sa.rCc< - Sr'.a?eJ
I
r'\
c
()
5A
i
t
o(ga
*r* Ar4(n ;},rff
-1
tr's'32s
'::
:A.i'['ii*
o)V'
,r
i fixed
rr.-r ilF' pu"sti
.-o"iihroutr
throug! 8nx"a
that
.
MorroN ,ND"R .ENTR.aL
3.16
Eo,.cE
tr T
-.-L
cetttrcl !o'ni [*"h
Consider a body (Fig-3.2S) moving .rrra", "'*t".n al
and 3-17' for
of
fot,,g' ln Secs' 3'16
p"i"t
the,ccilrz'
o i.*"0
point o ia ireriial frame r and F tt r".
frame f'
ire.'ity, rve write (')1y * ( ) and !L a are rv'r-t' inertial
+ Lcxtttgc=lcxii*=6@=9
(1)
:+
bxgc=llbl*=b=constant'
+
is its value pet unit
the mass is concentrate d * c and to
all
as
if
mornentum
of
monrent
irthe
rvhere E-b
thc t*iectory
O wlrich o::*"t *
t-.-^1:"*
Equation (1) + tc-L fu, i.e-, s6 is in a ptane through
FIlcc :+ 1"xF=0
lcxnlc=constamt =Eb,
with
Choose tty-axes in this ptaae
iu-oplane passiag tll,roug-h lhe ce-alre of fotceO'
"u*"
fu = lrse"
origin at O- Equation (r) + rc x gc = rg. x (i9" +rC+) =
nrass.
of C is
a
plane
"!e'=
(3-49)
: constant (3.4S) i t hol'"' v4 = hslr'
0 :* O =coustaut', i'e'' the trajectorY of C is aradial
Ilence 6 retains the same sign. If ho:0, then e=
r'ector of C fronr o is given by
by the
line- Thl rate at which creo ;i is srocpl tris'i'28)
' :*ttt::
(3'50)
;:,i$1t = h6l2d't - r"di,lz,' :+
,"6 =roo =
*-
rouco
=
h6
rrnotion't,,
is Kepler's secoud larf of Planetary
Ilence the area is srvept at a constalrt rzrt'e:'This
3'ITMoTIoNUNDERIN\/ERSESQUAR3GRAVITATIoNALFoTicE
7- of F as
${ n I r2)g'" the potential srefSSI
F:
if it
-(G
For the- inverse square gravitati,onat central force
telation for
rsork-euergy
the
I{eace
is
at
c
is acting
-cttmfr-
c
'
moving at speed o'becorrs:
f"=lv._-r.,:+Tc+V-=const.+n,,i'{2-Gi{m|r=,aifi_cu,a1rg:Q-.
e)
+ZGM/r -ZGM/rc'
body escapis to infinity if speed Do at
occurs at r^6 and u-i. occurs at r-"r. The
u? ='uE
i ,orro*" that u-*
1=
oo is
ftence
)
0, i-e.,
if
,L :4 +2Gn{(L!a- U'al= ol-zlt{/'b > 0' i'e" n "oi {2Gltlrdrtz'
giten bli
inilependent of its dfuection''is
the rnr'aimu m csca?c spccd t.to escape at rs. which is
1,=(2GMhiltt2'
For circufaror6i{of radiusrs, i=0. f
,n(i- rods) = F, =
- Equation of the Atajectory
"3.17.1
of the -trajectory' in
;;;ion
equation of motion using S= hol'?"
'
moti'on
=0' u=r{gd =uegoandcheequati'onof
1,. = 16| = {GM/4ltl2 '
-Glfmlri +
(3.52)
C
r',
f
by solvirg the
coordinates, for tlre ge*eral case' is'ottained,
:='' =':.;r:f'- [ilt3 +GMl"'=o: ''
- t#) = F = -GMtnlr?'
Llr and using 6 = no1*:
u
The nonlinear equation (3) "* be sirnplified by the Crapsformation =
m(f
r=*(ij
Substituting
i
from
=-"+ =-)##-=-oo*,+
-lrfrus
+
GI*{uz
=o
d?u
:+
G'M
@+"=a?-
Thecomplementary,particuIarandgeneralsolutionsll3,ltptuofeq(5}aregivenby
u. =
Acc{+Bsi'o= Dce(6-l?),
vo=CMlhi,
giverr
where A, B. D,p are constants' Ilence the trajectory is
-$3
(3)
Lgt?:
i=-ho##=-u""#'
(l) in (3) yields a linear equation in u:
-trer'#
t3sr)
by'
tr
= u.*up, =G^tltfl+
(4)
(5)
aco{d-fl'
,l
,J
.T
JJ
-T
JJ
rI
L/
-j
$, q)
-r*
t-
LJ
I
\w cent.o'a.l h /b a .\-.
Af *
:
LJ
L.
f
LJ
L
l-
tJ
lj
(6)
tw'
L GM +Dccl(4-01'
. n-*rt-al
L=\E
rh6
from the radial
.o TIo ,r o! =0. Ilence d' is-the polar angte measured
p,then eq(6)-+
positiooatr-;.(rig.3.29a).Forsimplicitnwedenotec.bydintheseguel.Thusuith$mco,s$'Elfmmlhe
LeL6- -_ 6 _
;;';;r;;"i "*,.,'*I
IJ
l-,j
l--
IJ
IJ
L
L
t'
L
L.
u
L
U,
L,L
rL
"qrotioo
of
tLc
ttaiu"f:ryj;"*r.
,
(353)
ht,
(3'53) w'r't" time an'f usrns
"::::':
The rodial speedlsobtained trv differentiat*:I !:'o'*-t
f $ = ha:.
(3.54)
v'=i=lrsDsin{'
+
-il.=--DsinCi
consf,.
D from
f-^. O)
6\ //frlist.ance
(distance tof P from line .[) = haD|GM:
(dbtance of P
D
fllll
-rccsal
=+
of fotce'direct'rix at distauce ll D trom
(Fi8'3'29b) witlr /ocus-at thc ccttrc
Hence the orbig is a conicsc(:,t('n
:'
(3'55)
it and eccenlilcidg egiven [l5
e = ttiolGt{ -
Eq(3-53)
This is Kepler's first lErv of-plarrt.tarl" 1uotion'
of .xl'its
The
on the value of
Es nrui
-
e--trf,o1t;tl l
oa > {2GMlro)rl2
e)l
os= (2GMlrs)Ll2
e=l
q <{2Gt{l1ttl2
e< I
tao
t-
L
l_
)u
--
--
L-
Ix
os
;;t ---l
rs
Eo>0
escap€s
Eo=0
Just escaPes
6o<0'
Bo<o'
= (GMlhlLP
-GMnf
Es
= -Gt'lm/2o
Eo= -G!!m,l!
I
L
closed
orbit
closed
orbit
I
i
;
of senri-major a:cis a:
ffi..ltip,r.icorbi[
;:::;;:::'".r *, = *. :: ,'= o 1 ,:'=;"i" :;l
T::::?:".;lx;;11;:;: n"llz
flUg=r2U2:Ir9(o)
Gl[ntr2'
tnvtt - G l{ n lt1 =
= ntfr/Z
" nl'z + ul =
"-:.:"'.:;;.:,--.r,ri,, =
Pur u2 from (b) in (a): rnti/": -GMmf4 =n(tp1fr212/z-Gl't'',
^
fu= -GMm,/(rr + "z) = -GtIm/2o' D
Put ul from (c) in (a) :
E'o
E6
(0)
2Gt{r2l;1(" +'z)'(c)
/---fft7]>-
vru'e
*<-re gilru-rT
t-;ttrs<u Orbit
tbr Cltrs<d
s.L7-2
3-L7-2 Time Period for
i"
\ -r --_: ...-i^i rnrl qer
. .,,,
Thetimeperiod?forelliptic.,rlrit(Fig.3.30),ofsemi-nrajorandsemi-minora-\esband6'
I(3.56)
'l' - (Area of ellipse) / (Areal velocitl') = dabl(hs/2\
(r)
r)c.sd = olffi*;ll
i=#*
;e*o=
rou
+
t*i" =
(3.s6) & (2)
i.e.,*
3.1?.3
=+
=
Irfr'
'
*
ZiiI;'
|('-"* *
rmin)
=
e
D1f.:1
(2)
.'-' (3-5?)
2;.a3lz
-4,a)t/'=
- Lo 'D,
-T
T =Zroslzl(GMl'l',
o3, which G Kepler's
t
'a
[-a(l- "'l'l',
c?ltlz
ttrid l'arv of planetary motion'
of Initial Data
Parameters of tlrtr Ortrit in Terms
oc
r=r(,;-,o)'
- 2r;ab= 2rq2(L'l'=
T
Ftg-3.10
16' us'
hs
hiD
,
,ttz_
\GMD,
(3.58)
os
I
TbeparameLercGM,ft6.I)arecomnrlledfromtheinitialvelocityugal,radiusroatanaag[ec6rvith
position from tic minimunr
(F€'3'31)' The inclination co of the initial radial
due to gravity at the s*rface of
isdetermined from the acceleration I
radial poeition is an urrt no..'i- CM
: nt!! * GM = 9-8?" Eeuation (3'a9) =+
earth having .rdiu. a, d Mtnl lt
the circumferential directiorr so
5+
!
116
=
F4uations (3-53) and (3'54) for rs are solved
=
rouCo
-
l/rs = GMlhe+ Dcosdo,
+
uro
= uositlco = IroDsiaC"'
"=[(* -W)'*(H)']""
D=lu"o-GM/hal'
(3'60)
,r#M'
{s=ar*a.
(3-5e)
= rouocGco'
simultatreously for D and $6:
rP6
* uro=0' (3'60) *
Eorlaunchingparalleltothesurfaceof theearth,o3=0i
using previous results:
once GM, Irs, D are ott4irred. then all required-entitiesarc-computed
ho
116 -'
Gi{ n
I
GM
:,,1---- = t '
t
rmin ha
'io'o
oo3(l - e21u2 11no1z1'
and if e < l' then ?' = 'obf A=
'-t'*' * t-'')/2'
e: ttf,OlGMt o= (r*"'
cosd- ftlt-GMlhlllD'
wittr
u'=heDsin{'
u4=h6lt,
Atgivenr:
fr(,m ,Irvz 12 - Gl{nfr = mufr - GM*lrs'
alternatively, u! = * - ui, .r,here u2 is obtained- \.
i
earth iu a
of:tbe
velocitl: ui relative to the surface
la:U:,tchirrgposifiorr is at the equator $'ith
cr is the angular velocity of the'6arth'
direction West to iEast so that,.us = ,,* a.,tol, *.,,lvhere
is
The &csl
'*"';:j';';ili;;,
.
*".,.,".
.'.to"itf{ior
the area
(Fig.3.30); the tim.-fes
rraverse from B to D, since
tle
areal
from
t1a
:: "i:1::thb'time'?5pO:
ss'ept b;--the eosltion vector w't't- 'point
.-1
: $reaOABil) = laobl4 - b(oellzl!(h612'l = lt - ze:!r!12;abl(l t!h:12\:."!t Similarly,Tso=T{t+%!ella'TealTao--(a-2el1l{*+2e)<r'
T.te
to
%l*i/a'
GeostationarY satellite
b'a circlc a[ou{ :\'-5
relative to the eart'h- llence iUs orUit
Geoeta.tionary sateltitc has a &ted position
tic cct,rl- of forcc
trajectory of, a saGellite is coplatar uith
azis ttt'uc*cl ot the ralcL.r of [he earth- But the
cqtatoial ploac
lhc
i*
r'
r ortit ol ndias
o at the eartb's centre. Eeqce geoetationary satellite has a circzla
and moves ftom West to East at the rat'e r'r:
km'
r" J (Gl[lutz1rls - kfr luztl3:'{1000
+
ac:,trtfc- (GMlr.)U?
3.17
4
3-1?-S alansfer Orbit
should be
requires change in Eo' The transfer
Transfer from one type oforbit to another type oforbit
for
achieved
*hLre ttre greatest increment in energt' AE'-is
executed at the optinrum location in the orbit
(i.e., for the same elxpenditure of fuel):
rhe same change lAgl in the magnitude of velocity
5f,_(vatueof81inthefinal.orbit}...(laIueofEointhe|nitia|orbit)
As)2.- 3.?1 ='-ft:4, $? + za g r&1
mu],G M m / ) - lm(fi -uf ) =, !rn(31' +
r],
i
LI
G
(lmrfi
l
!
-(
=
a location u'tcrt 15 is maa-inzm' i'c:' ot the
should be large, i.e.. Au shottld be alotg 4 and'al
Au.gi
Hence
localiort of the minimum radius r*;t'
a Satellite
3.17.6 Effect of Atmospheric Drag on the Orbit of
1\'e consider trvo cases.
!
---L -: rs- Hence
a verl'short time $rhen r- 3 fmin
1. The sateuite is in clliptical or0it and is in the atmosphere for
lsc applied ct r ='ro $'hich causes instantaneous
the atmospheric drag is modelled as an i[slcn la7lcots imptt
in t6e next orbiO (Auo is negative)' i
change iu speed from us i' the initial orbig to uo *iAuo
Es=\flir2ol2-GMn/rs-
-GMnl2c+
<0'
AEo=muoAos= Gttm6l(,l2o2 + 6ia'=2o2rta6oolcM
tends to a
by the above amount and tlie trajectory
Hence the'semi-major axis of successive orbits decreases
circulat orbit.
5q_
JJ
J
J
JJ
JJ
J
J
J
:.-I
J
J
JJ
..!-'-- :,7.,;= <-;+-:
: : :' ;,":ri:
.
:1
:l
,--i
l'-
tLT
L_-
L*
LJ
-
:-:
-ra.
'
-€-i:!r:*'
11j--
foict F"i1
atmosphere undei-drag
cir<rla.t otbit of radius r-in
nibtly
moves.in.a
2. a\esatellite
ilr= -ZFlmo'/ -(t)
o{*ll.'r2 =-F,-'
;;'-t-+;:
speed
tu=
Eer'ce rdecreaS€s and
p-Yiewa *lr - -2clm'
cu'
t'l
potcatial
drag,
viscous
the
""u
For the case of linear
ttis occurs-sine the decrease in
atthoulh thete is ;;g';;t;'
increases,
(GMlirl2
a=
by f'
mot" thu" the negative work done
*
-GMm/2r,'4:r''+'ri' :
;;
TO NON-INERTIAL FRAME
M
3.18 MOTION RELATfVE
of the centre of mass C of
Coosideranoninertialframe(m)havingangularvelocitvurandangularacceleralion{w.r.t.iDertial
TLe uot'Jo" oimotioo
g,nu'('';'s.t2)'
has accelera.tioa
frame r and its point A
a bodvot
I',r[r^r, itx rct+'- x (sr x tce) +24'x ecw + ec-.l=
tltg.clm =
+
Thus
F
-mgrl1t
u:
f-
*
T;
- trt9_x'la^ -
t,i+'-x
[; ;:l
gct*:
{gx lr,i) -Ztw-x
pseudo fforces'
physical e-tternal fiotce
rngclm equals the sum of actual
orrrame, angurar accer.erarioa
F and four pseudo
(3.61)
'forces' rvhich are due to
g-?i:-:t:Xil:::ffi:;l,JiTt:I":":ffi:JI
accererarion
.
'
frame I attac.hed.:,9i.:the-ccntre
to tlre:earth and 1n inertial
attaclred
(,o}
ftame
angular
ihe
at
non-inertial
frale m rot"t*
Corrsider a
stars €ig.3i3"iat
under gravitational force
"n"
onry and with its axes to*rards
of the earth
to frarne
velocity sg of the earth relative
*d other external forces'l' Let r:
;
'
"rn="a'g' Considei
o' ooa'
"
a=
";;
(3'61) becomes
"-i
g,;ln'.4m-- gcla- Equatiou
'uo'
*=
(U
x
m4^= L + z-- rnst;,ll - nry'x (el r) -2ruex9lm' gravitational fotce
r = O'g1* = g' q' = 0' the
(FiS'333b)'
O
o:Yo
the
at
1. For a stationary plumb.bob
t2)
eolrl= -ri9' + g<= !' bl'
-mlc--=
:+
L=
nsal!
-lE<o-'rrsold
g= ho +'L-itnrd dLection
general differs iu magnitude
acoedration .1 g, *nhi.h in
&
where
4
is the effective gravitational
from g-.At
2
Forsmallvaluesotl,
w_e
-"R)e':'
x24)41 =1 ar!(<'r'
& -t,'ott=d;a'1*i:T-A:itcetn=2rl(3600
obtain
tl",rt" c"tJl"[s d;m in (1] and
tt" "qu"to'] L-=
-'n- 1--'.nt'f
-- -@
compa."d
neglect rhe centrifugal term
f,.En = ?-+ mg--211'!4x
then
as given by the plump line,
, - ..---^,
only the coridis pseudo'force'
of -Zttlt't x 9l-lo Ecst according to [Ut dit*tion
ia North3lrcmisglerc and to
we have{o iodude
Eence, if rve use a
tleoioles
l- A. bodg ia free oerlicat fall ot the c*zator,
arc dc!c*!-d t.o
;;;
4- The sinds btorving on the surface tf
tbe lcftin t;c
so,4bcnt hemiiohae:*
tt"-;gu
@
3.33o.
Hencet'he '
E-'tr.1+*3si',i
r..o.a.o"u'*itt tr.",aitat ';''-2ry"* *q*lttd^itrii'-tcip
do*n' &e
rcsiolorl.o;-ftuout"'
of the wini;;;;"
cyctoacs formed due to motion
ottttctclockuisc itt Notlhent bemisphctc-oo| "*"**c-ia'sordhcra
I
(3-62)
4il'
a
X / fr'
=..=u trt-{r"
,
hcnisphcrc(Fig'3'35b)'
i
'
,rrf\,
51?3-'ffi.
\"i.4,^
3'
t3"
3
\
w
f #r
\
\}
z' X&F
3'3so.
:=:.{
i
l;
:
Yn
'%
L
I
3.3sb
',/
+
irI
!rJ
.
:-r_
5.
Approximatesolutiorr"fo{Jree fatl: E =
:
0' Equation (3'62)
reduces
Lo
(3)
g=g--fu-xuon the r'h-s' o[ (3)'
lt is solved b) successi*'e approximatiols, usiu! g frorn the previous approximation
u- yt*la
lstapproxinration: usingg-0orl r'h's' oi(3) :+ g= g- +
:+ a = 9.-?4'x (g"t +go;
2nd approximation: (4).is used on the r'h's' of (3)
g: go + yt-,!x (g.t2 +29r,t)
r: ro + 9ol * lyt2-t' x (*g,t3 + !j2)
(4)
(5)
(6)
(7)
to t he usual results for unilorm acceleration'
The undcrlincd terms are the correcLions. due to earth's rotation.
J
-T
J
J
=I
J
j
J
J
J
J
-J
iJ
j
J
.J
{
J
J
jJ
j
:J
J
97
*J
-J
J
}-l
F-
;i
|
,',
- i ('
U c-
lq
- .
-
';-'t -'
.
''
r
t.i^!ze\a"e
-
Bodv 1:
Cr +
l-
for point .t{ *'-r-t- a, y' z axes-for-
th: systcm sholvn
gia.tftu'd
rtu'd
&-ia.t
1yF$a
E's'
E's'
'
Axes 1,2,3 are the principal a-tesof
z-a-tis is principal anis at cr *'ith
pt"ne
ingi'^tt
I;r;'rii,'$2^
rinllProne
*t,:'#:,'-*=
*.-:--' b- '.-----+'t
F-*il.83' ,b
o'"
or mass f-" * t;;;;i"l;""'
"**
rllass synlmetrl'through
inertiaatcl . Plane
y-- is a planeof
"
:;,==0,'|u =o,'"'
r,ul',;:;' *u':i,,=,,".;{:=:i"'0"'"'
")i
sr=si,o' s?=-coso' sr=0'
""''uli""
,1 =coSo' r12=:sillct' rl3=s'
*
:+
.i=g=sitrogl-cosoQ2'
k=!!=cosagl-:-siuaq"
-
a" cos2 a)f 12
$; = tl"': Ifr'"1 * 8;ri+r$"3 = rn1(62sin2o*
$; =t!;=lfi.r,i +t,g;ni+t"g;n"=rrr1(0?cos2o.+a3sinro)/12
rn1(6? - o?)si.ctcoso/12
8: - r'co' :,f,'"',.' + Ig"sznz* r$s3rr3 =
rnas s-r'tnrnctr]'
aL c2- Plarte r: is a glarre o[
Body 2: Axes I-.2-.3- are the pri'cipal a-xes of iuertia
';
with
through Cz + Y-a-\rs x PrinciPal a-tis at C2
l-IE--I
l)3
.
1-..
l_..
f
'arious
A'B'F'G'H and bodv 4 aL
3 aL A,B,F'G'H
Kt
i
f]
I
*.t.i*
l
' '*_ o'-.:r-'tri
_T o, _ ,4
r
E=-=,"'nr,'ft1,.,.11;tt;;".8,i;l;*L;*'"
l--
I
!-
inertia
llrff3;T" ilffi:U;
^
H
f
tl.:.e
'/
3
i is pi Nes,ecr:++:i:a:*"*:"f
l-#*t'.,,':::'"lli,"jio*,
1 at
;::;'li.i:::T::;
bodv
''l''J'x. bodv
bodies at ttre rollorving poin*:
iasfition'ror
*:' n,D,P,Q,bodv
;':;;,D,P,Q,bodv
"i'u
i:
E*AMPLES
meoti of
r,#:,
HT
I
+
L:
I
,=
rr.2='piTR2L,
,
1;'r
-
r===';
26.=csiuO,
:l:-r.1t.;:'::'"
yc?=0'
*l'"",'n=r,tt:;l":i'f .-
k=a'=sigggl+'ccoei, +
t
t:
E t
I
= I?.',--
tg'=b"+ecos0'
n!=0" ai=siud'
ri;"i' + ri;"!' + rfi':? =
n2l{R? f 4+ L"
r' =':': ='ii"i":-f1:u:::::::
Body
3:
!,
j, k are the principal
a:ces
Ll-
t
R2 sin'2 0121
I!
:u:,,',,nrvith
of inerLia ag f,3
z6'=b3
:;==':"::,:i;i-1;;';-,;:;"u"i^l'",,,
tf; - p3a161c1(ci + oil1tz -Za.'t2c1l(r" /t'+ cilr2l +
I?; = pso$ (ol + b"L) / L2 - 2,...,r2 cllt'' lz* p'l
,
tr
l--lL=I
lt2lcos'0 +
L21'lz1sine'."0
tr
L-
rt]=cos0'
1cz- I!3^-=rfJri'+?ni'+i.?4'=n2l(R?/4+t21tz1sirr30+n2cos2o1z1
t
fL=..
1"1'
rfi =',.2n=12
''c1
'
J
s8
P2l
'lt;
=
'lr"
=
/'!'=o'
!:'
Body a: I i r
ptiniipal axes of inertia
are
;'he
ac
Cr witl::,
,,^
8 = tf, = I!! =a,
fi - zm+nils
t!; = t!,- nu(3i?r/8)2 - nalzlils - (3nr/8)2J, If"' = E;'
ma=
pa(2etlfi\, ic. = yc
=
0;
_
3c,=.b4+3ftt18'
values of nr-o'i' abouL
since for the hemisphere the
a-tes
i.e.,I!,= If;u= rfl = f[i(z'no]EiJ= im,&l'
at .A for *,J
The.elements of t#i,".ii. ,..,ir.i*
"y"t",,
;;;;.,
of
arc half o{ the values for a sphere
x'!' z at E
are finatly
obtaled
using parallel a-xes theorenrs:
.4-*
E.=Dti+rrt(y},+,3,)J,,r,=ir,i;+rn;(zi+'3,I..I*=EtL"..+",'.(4,+v},}l'
t'='
.
i=l
4'"
- n;rs,vc,l, + = it'i'
=flfi
r=l
i=l
-
r,;v6':6-J
I*=
'
*
ftf;
- 'n;'c""c'-l'
Theprirrcipala..iesategir'errpoirrcofabodycanrsotrtetirne-<Ireobr.airredbytoolrirrgforaplaneofnras-.
s1'rnmei;yoft1lebodypu.,inselrrouglrtlratpoirri.Thea-tisnoranal.l.osuclraplaueisaprirrcipala.lisat.
trre above Procedure''ihen the third
thar point. If trvo pri'cipal axes have been J".".*r1r"i ": i'Tt::a-.ies'-r' Trre ptirrcipal'a-'ces'a*'1'ariousip'f,i*ts
of trre principar
principal atis follorvs from *tutual: orthogoualiry
parenthesis:
rirass synrrnetry git-e* i* che
of rhe bodies are listed rvith the planes of
l- ,{:
Bod-v 2- ,{ :
Bodr-3-6:
Bodl- 4- S:
goas
!(v:). J :
j (z:), p :
j(:::),n,
i (c:), g :
9r,9:' ec (t2'23)'
ei(?-3'}.
i.i.!(:::.9.:),
Ii: i'(v:)'
Dt i:5'S' Ca (1'3"?'3-)" Q'.'4 t1'3-}'
g= i,!,k(r:'v:)'ff'
r', !,j.,k(r:.1[zv].
b(llrvl'
i'J-'k (r:'Y':)'
about a no[-principal atis
mass C' is rotating
Exatnple 3.2 A rigid. body, of nlass nl Nit'h ceutte 1f
tlre bod-v has eleureu[s
fior axes c'y': attached rc
poiut O
of inertia (Fig.E3.2a). The inertia nratrix at
prof
position' The-bearing ai B doeS not
bearing reactions f", th";;
the
rind
-..
*.Ign,.lr,t.r.inr. Consider i* particular tl::?: "i'oi"" tn't'* 'rj32b'f9z ,t
'
'pr&*
J'ffi3o'' Lt{ks:,q,
;:-''-""fi,'
,ry#, 1"*.-<3o''lffir
(G)
:;
(1'
ide an-s axial
*,9,
--rtl*t;:l
(bl
Rrk, 8a = &!+ Esj', be the angular
= ark, Ra ='R1i+'R?i+
The FBD Ot
p*itio &a does not have axial comPonent'
acceleration and bearing reactio*s at the gir.o
''is obtaiued fromgp Q using
=
the body is shown in Fig.E3-2c. The acceleration of C
Solution
ut
= u\., oc
= ei. I*t
or
sc = so+
dr
x aC
-t-' ^:-^^
^
This can also be obiained directly since C
;;;,;&,8s
+ s-x (sx @)
ntoves
=
in a circle of
radius e ag rates
point
are determined using moment about ftxed
M o =-@ E x (R, i
+
Rz
i + Es b) *
6
L
x (& i + n'si) +
- (I?,6- rf;.r.r?)i+ U3,6 + t!'-aii + t!,ay
q
(l)
-'f ei+i;ei
i{
o
L
and
E=
r"'r'i' The 6 unknorvns
tng6:
+ c ! x nrg(-
sin 0 !
-
cos
0j)
I
|
-"-
I
--.)O
F.',,
L-- ,-l *
L
L
l'
'- ''
,
.
[:
j:
'-"
i>Rt-
'/
t?12-
t?,' 6'R4-onl=jrf;,a+r!''z
D[s*=
(2)
i,
+ ;[;:a;#(Eri+;.;,.'f;r;;:"*:'1-"* "'
,=*j
*
i:
:-l
(5)
Er*fie-nrgsino =-mtt2e
L
(6)
.,;',
f
(?)
:::I-mscosB=m'be
.-a. .o 2 _..?onl.nna
7, ao=gfla+ tg,"-m:''r?ea+ntsasind)/r
eorts),(;)'*
R5=(I!'.,+I?"' +'t1,,ca4rngacos,)/L
^r=-Tr?,o+I!.,,,2+tnu?eb-nrs6si.0)/
R2:(I!,6-$,-.+mieb+*Ou*"rtlt't'
*
Eqs(2),(6)
f
I
1
i8)
(9)
!
o[c'? term and change their directions
rotors ,lue to the presencespeed
rrigrr
foi
rarge
very
are
in grou.d refereuce
These reactions
respect to the fixed directions
their
consequently,
--0"";-::;ith *'ear' The rotors nlus! be balanced to avoid this'
as the axes rotaLevibratior,.
forced
^r,.t
r,ary rvith time and cause hea'y
r
LIL
*
i
LJ
-
We consider some parLicular casesi'e'' lhe ceatre o[ mass C r' on the a-tis
e
=0'
)''
erf
1
:,
I
i
,
I
t.
rotat'ton'
i
'tt
I
t
i
,i
2- I!" =
L
tf;"= 0'
R7
L
I
l'
i'e''
ac
z-a-xis is a principal a'xis of irrert'ia
o'
L'
= -(nu't2eb - ntsbsitt.)f
fu -
{-rw't"ea
+
ntqasin
:
/i3 = (,r-'e,
R5
0)/L'
=
l
'
ntgbcos,)/ L
(rrtr'ea + ')rgo cos?)/ L
R5=ntgocos,tL
t-r.3.a=o,I!"=I?,__0,i.e-'tlrea.dsofrotationis:rlrrincipalaxiso[irierriaatCrvhiclrliesonit.
Rv=ngbsin!/L' E2=ngbco,\fL ',l,=rrrgosi.0/L'
L
l-
;:':Y;',:;,T:"lTi::*1.,:"j'i':';ff:.:l;ffi:iii"i'i:;
L]
;
L
F
L--
,)
)
,
Ul-r
Lic
l.=-l :
lJi
r-.
lI
'
cosoel
lsinae2'
"'= : - -'.:::"
-ccSoQr'
';;,=i::i3"*;='f'"'"'*"rf,"":ir$s3rr3 ={I? -rfl)siiro'co
]
ltj
ljl
* ;:*:;':;,=tu;'i;*1;:;:;;I."::;;1*'i=
L= s=
'
l- '
u
[
"
?::;.;ti*"::"? :i**;;;;*i**:::|"'i=
L
er' e: aL o r+i'lrr s' a'l
',i.,e*ia
j ispri.cipar axis a'L o ie
Hence
Eqs(8)-(9)
*.here
rg
=+
R2
(I!'i l
ntebcos|)l
(10)
L'
=
= -(I!,at2 - msbsinl)lL'
:
(-r.P + mgocos9)f L'
n's = (-/fln
0)/L' E'
fu = (8,-' * mgo sin t0)lL'
R1
is given by eq(10). For rrre
bodies,'a:r;,::1:,
p2)/r2isinccoso
_r,n(o2_ p21/12]sinoroso
(l
plare 1!.:7rns?r,z- nry2lr2)sinacoscr = [rn(q? (0
\:, ,"";r"gr;ar
-q cos a
lflsinocosa
-1nR2 l+lsin
r:-- 1!,
to -:' r*eltt---- Misaligaed
mff'12)sin c cos o = -pnP(2
--.
thin disc
1'lrrz14 - rrrRzl2)sinccoso
' tL:(ii)
(lQ Misatigped thin rod. E :(t- mtr2l12)sinccosc = -{r"h2112"""::,:.^ D..
1!,_lm(R214+h2rr2')-,l,n2izisi,iacoso =m(h2!r2-ftzl4)sinocosc
a
;;;;;;;*ru".
.. :
6o
,
I)
,i
t r\-l
-''
note thiiihe
>
platc and if Ir : tfrR
di'a'amic reactions ate :.r,ro it p = q for a rectangular
lc ls interestingto
a piinc-ipal axis of incrtiaO and the axis of rot'aLion is
for a misaligned cylinder since for tke cas€s lf,, =
(Fig'83'3a)- The
several pulleys aad rotors
Exaaple 3.3 (a) A shaft is rctatin' about i-axis carryi*g
z;)' Find the reactious from
the centre of nrass has coordinates (e;' y;'
mass of a typical element is mi and
torque is ?' Bearing
at anSular velocity c'r and the driving
the bearings at .,{ and B when the shaft rotates
for A w'r't' body
matrix
The elen'rents of [5e inertia
B does not provide any a-xial constraint to the shaftpllkular find
Neslect sravi!.v'
fixed rotating axes e, Y,z are
!U) It
P,r*er1+es|- 11= r,
1*
. rF
shaft in Fig.E3-3b'
I:",C'---
t,
f
=lji3'lrf'ai;):
j,mi
;,;
--,
i il
-r
i.
1
"?::i::Lt:XI""X;i: i'e,i
'ru,ru2
t
ri
o'
a
0-l^
_- iE
& tsJ
"{
.i{
tb)
(o)
Di,r -:-p-r.,n)ThgFBDissholvn
Solurign (a)Letthebearingreactrionsbe&,=:Eri*&j-r.,fe;nsi+&!+AoE
*o*unt' eqlati-o'p:for:lbe 6xed
in Fig-b3.3c. The
6 unknowns
&-&r;fia,'r,fr, A, ;;a"r..i*'ifi
-r
__-
!i
.3,
-- -.,
--.
u-,
lat.rU
e{
-
from
t'lre
-
pointr4andf:frl.;gc;-Forptane*otlo,rnuitt4.'in,t'lrediregtionofuon-principalarrl"'E'at'-'
./
M-"
-
trk x (Br i+
R:i_)
+r1= u:"a -Iid:)i+(Iin+ I:,u2],i+-tine'
Rz--$*u'tf;'u,1lt
{l)
El = (/fl'r}+ t!'u21/t
(2)
_q
; =T/I!,
+
k: T = I!,n
a:i
t= Eri* 8zl+8r = )l*'i4g, + &,^. =Inu*.-'!'i.
where tb,=t,kx (t lJurit--'ttri+vi!) = -(dryi+a'"'ili*(d':ri -'"vil!
(3)
,.=
+
+
:+
-i*r--i:,;-t-".,''''
LRt i rf;"; + t!.'r2
i:
j:
{{}
{5}
bearing point can also be talen in Iis3J'
This appioach of taliug tbe moanent equation fot thd&'red
O, :1 -O.lm, ml =f-5ks, r3=0' T-a2N-rir'y2=0$6rer'
kg,
(b) nq
=2
rn3
= 4 kg,
,, I-0-O4mr 93:O, zs=O-{rn, z2=03rn L=O'5ln' 14:l00tad'fs'
kgrp?
I!, = -2x 0.05 x O-L - 0-5 x 0 x 0'3 -4x (-0-04) x 0'{ = 0'(}$4
lf;,= -2x 0 x 0-l -O-5 x 0:06 x O'3 -4 x 0 x 0'4: -0'009 kg'mi
tg-ott
r!, = 2x 0.1213 * 0-5 x 0.u3l3 + 4( oj€12 + 09421 = 6'o3s*62
,, , o.
:fi, - 0.2/c035467 = 5-639r rid/s? '
r
ar- =.r?(0.04) i
-
6(0'04)i ='t00 i
-
-r
-r
-'-
,.
rc)
(7)
R2 r?^ :
Rr
..
\
:
>-ir
_{
(8]
0'2256j m/s3
<<
I
_{
Usingdatafromeqs(6)-(8),eqs(r)'(2).(a)yield
l'
-q
\'
.-I
!b, -' -..'?(0-05)i'+6t0:05)i --:1t^:::'::i::'::'
(rr- 2-.?1o.oo15 tr;(0:06)[= -0'3383i-600!m/s?'
+!
:'
x 1m2)/0.5 = 1080 N'
* rm')/o.s = -oo'u'n,' '
-(0.054 x 5.639 + 0.009
2(-500 i + 0.2820 j) + 0.5(-0-3383 i-600j) + 4(400i-0'2256i)-1080
(-0.009 x 5.639 + 0-054
i+
180'6
j = -{E0'2!
-
1r9'?
j
N
pass m is
carocs' An autorrcbile of
Baarnple 3.4 Bal'lking or srtperclcoation of highway au! milrcod
proper rralue of 0 it its centre of mqss C
travelling over a circular road 'banked' at ar1- angle 0. Figd the
-
<{
u't
.i<
6t
t
I
.,<
-=-
ilL-
'i-
J-j
* "o*,*iinH
travers
L:
IJ
1,
fr-a
: -tt'
tL
sorution
;-*ffi-T
(a)
"'
o",'*"' ll"ni
tb>
tty5:1,;T"i:"ffi'X3::ffi*:::':t;';:i':
lixj::;
sc
- lo'l nls' - ("/Rx-
cos 0
sin 0
cornpo.enrirof che sum of rrre
trre
represerr
Fr
Nr,
Fig.E3.4b.
in
lormal,aul-1r^:r,.",
The FBD is srro*'n
f:,
lJ .
fif,ff,':i':l:*m;l:*iffi.::'"*:f;*:',i"":T,$::n:'L:i:J't:'::::ffi"fiilliliT;
L
r-.
?'f:["t':1"ffii[:::"'::':$:::[":"':;:::;:T:
of inertia at C'
;i';':it::H:.:;i;':*il':::il
is principal
0' as ! =
t-L'
=
no'-principal
w.r't'
C
equation for
re3
u.e momerrt
l_
L
*
L
L
tr
'-
-Ft -
ir,
"-,rilJ
axis
a
''"'
jol';l;l';'
,l=;;'*f:-:i;'"
ufi
s"3"" '.,i:t,:::,
tf, =-i' = 4i'"'= If'n1s.
.4c')sin0cos0
-*
=
vietd "'**'+
+
Fz -mgsin 0 = m(-uzcos,/E)
F
using eq(l)' rhe components of
tr
L
:'=::;:,
= itsc
*t""'.'r*' :+
cos0/R-9sin0)
''v' + 1;2 = m(u2sin'/R+ecos,)
Fr +
F2
= rn(u2
(2)
(3)
(4)
-$'"i
Mc--gg,a-I3,"li-+(f,;!E'")i+11;t=inet*,(al.l2
(N2-/vr)6-L(F +F)=-1f,''2=u?'-i;;f
1 . .^r,
U,1",;'i"f,l=,.il;,.X;;*;;:'i;:;i'::'ll^fi,r'!iil:?i::if);J"*'^'"'
o?l R)2;!2b (7)
- rfi1sin
I+
l{
o'cos
(ricz
"il
)
f
1. #;'J;::Ti,'fr::[:t:fffi;rion
;
For no oduard slip, Fr *
E
2-
f
eqs(3),(4)
r-
Y
;H:'i*ilil,:I"j# ;lx:1';q:;j:**:xtlxrf;:*:;''::'";:"':::
:,:::,H?J.':llJ;:::'::';'
:;= i*
:jrfli'tl{;t'",,",j,s':
us
(l)
9")
e' +
l-
L
,
i"j333.,i'a1*:l:i:,:.9ff;"I"'.*r:Y$i,;:f#"iff.iffli
-i 7ru,-
f
L-,
,:-,
;
+
Fornoinucrds,ip,
Fz
rhe optimum cnelc ror ba*kins:
force shourd be zero. Eq(3) vierds
s p(Nr *
N2) where p
-
taro is the static
coeffrcient of friction:
-(" ca.lR- esin,) s P*(o2sin0/R'+ecos0)
r;;::',1lf$l=ar='Rilll"l"'-o'
tlll
i
II
-
4-'
'
=-"-
'For r,o'tippittc
:+
ftocflii&,g),th:e snialler noraral
+ hsiallb)-(o'ln'i{hcq,0lb-sina +Uf" frlsinocosaTmDE}} }
u2 Se.l?ltao o+b!h)ltl-(blhrtan0+U?z- f,1sine7lant'1
iv1= |m[e(cos
'..J
reaction /Vr must rcmaT positive, i'e', usiag eq(6)
0
JJ
J
J
o
(11)
and for larger rralues
Equation (1$ rerrcals that the car is more likely to overturn for smaller values of B'&.d
of h,(Il2- Ift), r. lf (Ig - tfrl/r',,Al ( 1, then (11) simplilies to
tl" S e8(tan e + b/hl
lft - (blhltanil
(12]
p
comparison of eqs(9) and (12) reveals that in this case tipping $'ould occur befote slippingif > blhyield
(6)'(?)
S---F-or-no 6ct*ing (d = 0)- The aris of rodation is a principal a-'ds- Equations
I
/vr,/v2
= lms(ll tw2.sba).
J
J
JJ
(13)
Equations (9) and (11) yield the foll6rsing speeds u, and ur for the initiation of slipping and tipping:
t! = psL-,
(1't)
* =(blhlsB
are srnaller
Hence tipping occurs before slippin-e, if r, > blh. The values.of r'! aud-urfor the unbanked road
than {hose for thd:'bariked toad6- For a stmighl'Ian'ked rcad (R= co)- Equations (6) and {?} 1-ield
Nr,iI? = |mg[cosf *(h/6)siag]
than
i.e-. in contrast to the circular case, the inner rvheels are subjected to greater normal reactiotr
;:*J:I'T:::::*f:::X*"
vertical aris
rrr
,reigrrr h
a'd ,.id,h
t
,^[*ll
-J
[er
^
a.'1i"8,L
,-r'i&t
(Fig.E3.Sa). It.is rorating at. lT .T I 9'' lis hinged aL A and B abour a
.,\.
angular velocity a,o when it bangs against a stop D at the ground level'*fi-J. 1.t-- I
-IThe coefficient, of testi[ution is 0-2- Find the irnpulsive forces on the t i':i I D
i
trt
;4
door from the hinges and the stop D during the instantaneous inrpact. <F4P6|
Sotutioi The 1rBD for impulsive forces is shosm in Fig-Ilii5b- T'lre line of impact is along;i The aqgular
velocity,beforeirnpscyisgr-:r.o\- {.ettbeirngularvelocityjustaftcrimpactbeC=o'L,andtheirnpulsive
reactions from ttre bearings
be
.
, &=Eri+ari+4"L, 8a=&!+a"i+nof
The velocities of pointi 8, D just before and just after impacc are
u'p'=0'
tzr=-l.i,sb, o}"=0, ol=-''b'
lLE=-uob\ gD=0, !f,=-a'$i'{a=0,
= -0'2'o (1)
Coefrcient.of restitution e'; o'g=-'v'4=-c(ue, -oo.l + -u'b=-0.2(-ra6!) *'
,,-!
The fixed aris of.iotation is,not a principaL a:<is'of,inertia,at'tlre,fixed point r{ of the,d9o5*
-m(Iix-dlz) = o; If. = -m\bt2l(-dl2l = mbdl4, I!, =
Et =U!,L+ ti,i+ r*H-'q
6'H^= (rf,i+ i7;-+rLhx-'--; = -r:(*'aodj-+
1=
+'
'rrllb?
fi2"+ l6ti)"1=
'rr.b" 13
|moj\)uo
The angrlar impulsemoment of morneotum relation for the 6xed point
.A:
fang^ = A
ff,
q
:
t.I
I
I
63
.-J
iJ
J
JJ
J
I
rJ
-J
(2)
\
(3)
..-J
yields
t-(id+c)k+pjl x ri-dkx (R.i+R6i+8.$ = -l-2(frnodj+ trn&?k),"6
:+ . [l:
+
Rs=$
dfr5 =0
:+
i' = 0.4mbzuolp
k
-g,F =-0.4m62r.rs
i - (id + c)f - dfo :-o.3mDd.rE :+ [* = rn0[0.3 - 03(0.5 + / d)b/pl,.,q
J
(4)
(5)
(6)
-J
-J
J
:J
J
4- ' Fo, or'tippittg (ooct'tiidag),tb:e smaller normal
Jvr
=
+
reaction rltr musi remain pcitive, i.e', using eq(6)
|m[e(cos 0 + hsinllb, -$,2/R'fihcc,llb- sin0 + (Ifzu? (e.R(tau o+blh)lll-(blh)tane+u?z-
If.)slngcosa/mDE]l > 0
ff|rine1*nn1
(11)
Equation (11) reveals that the car is mote likely to overturn for smaller value of &e .d and for larger values
of h,(If2- Ifr),r. lt U&- t?rrk"ru ( t, then (11) simplilies to
u2
< el(tan
e
+
blh) IIL
- (blh)Lar.ol
(12)
Comparison of eqs(9) and (12) reveals that in this case tipping rtould occut before slippin g,if P
5.--Eor-r,o.ba;*ing (0:0). The aris of rodation is a principal a:ds- Equations (6)'(7) vield
.
/vr,/v2
(13)
= |m9(1 1twz/gbill
Equations (9) and (11) yield the folliwing speeds'u, and
,t -
> b/h-
u1
for the initiation of slipping and tippiug:
ti
pgR-
(14)
= $/hlsR
Hence tipping occurs beforeslipping,if f > blh, Tlre values of, p' aud-u1for the unbanked road are smaller
than ihose for the banked road6- For a stmight.banked rcad (R= co)- Equations (6) and (7) -vield
!
Nr,ils = |mg[cosf +(tr/6]sin0l
i.e-. in contrast to the circular ca-se, the inner g,heels are subjected to Sreater normal reacLiotr than
rvheels foi the case of straiglrt banked road.
;:ffi. r::T::ilX'HT.;;*. 'r.
t
rreigrrt h a.d o'idtrr b6{
outer ^
fai.'tt-"&{
-,--irl&t
is hinged at r{. and B abouc a vertical aris (Fig.E3-5a). It is rocating at
angular velocity a,o when it bangs against a stop D ar the ground level-*l
The coefficieni of testitution is 0-2- Find the irnpulsive forces ou rhe
door from the hinges and the stop D during the instantaneous impact- <fqfa
Solutioi The FBD for impulsive forces is shorvn in Eig-til5b- The line of impact is along,!- The '-gular
velocityteforeimparcy is gf-={.rok- tet the angular velocity justaftcrimpact be +f = r.r'\., and the impulsive
I
&
= E, i+ azi+
a"l
0, v',= -.a'bi 'gb = 0, og, = -usS, tfi" =0, t/6, = -u'b, o'D, = A,
:+ -'a'b--0-2(-rob) +. 'ut' = -0'L*1-o (1)
restitution ei u!a-vb,:-e(upr-rp.)
CoeffEcient.of
gD
=
The fixed anis of.rotation is,not a.principal anil oCinertia,at*the.fixed point A of the door*
4, = -qa11-itrz) = 0;' t{.
tst= U!,1+ tf,i+ r* $q
z+
:
-tr(bl2')(-dt2) - mbdlt, I!, = m[b2 fi2 + l6/il21
:+
i:
k:
j:
'-
-0.4m62aro .
-(id+ dF -dR.- -0.3m&d.rs
PF =
:
:*,
(3)
d: Iary^ = A ffa yields
t-(+d+c)k+pjJ x .ri-dkx (Ieri+R6i+8"U = . I.2(frnodj+
d&=0
+
Es=0
JJ
J
I
--J
J
\
-J
JJ
|rn6?k),.,6
F = 0.4nb2wslp
f,a = m0[0.3-0-4(0.5 +qldlblpl.no
(4)
(5)
(6)
-J
-J
I
rr
J
rJ
-J
:J
i
t-I
I
I
- ,rrb"13 (2')
' a'Ht=(rf,i+ td,i+r*hx-'--o)=-1J(im0di+|rn0j$:uo
The aagula,r impulsemoment of momeotum relation for the fixed point,
iJ
J
&,=ni+a.i+nt
The velocities of points E, D just befqe and just after impacc are
ls - -uobl
JJ
n
J
J
J
J
The velocity of
u
ul
i:
=*
L ' lL :4'&.160 i =
iaqla*-*"t*ttt*
f.i+ erl+nuk+ n i+ iri+ nuf + F! = m(o'r + 05)&zoi
El = rn6[0'3 - 0'4(0'5 - cld'lbfgl"
A, + il+ F: O-6m&ao +
O-f6GroL Using eqs(4!(6), ah."
)
is
gb = -0'Saro6
r"t"tioo ; f=,mLgr,' yields . ' .
bcfore and just after the impact
centE:kESi}io=t
itz =o
*
j:
.Rz*Es=0
itr - -8"
+
ftu+8"=o
k:
,?3 : E6 = 0'
provide a-tial constrail''
63 and & cannot be e,raluated individually. If one hinge does not
:n""and a thin disc
o[ axial m-o'i' 11
Exarnple 3.6 The system shoru in Fig-E3-6a consists of an impeller
mo hits tbe impeller'
L_-
tJ
has an5ula velociiy ,.rqtust-beforc-a^-bird.of man
The suppori at B does not provide any
Model the bird as a mass-point and the impacg as instantaneous'
0
(a) if the coefficieut' of restitutio"^"
:
axial thrust to the shaft. Fiod the impulsive reactions of the support
with a slight misalignment of 0- It
LJ
ffd;d;;;;;;il.#%"
L--
r.tt-;5ef:=i n:,_
L_-
LJ
LJ
LJ
:
- lfhe
-(o) -^^
EBD for impusirc
Solution
i. ,W"
sh6wn in Fig-Ei!'6b rvith
forces on the systern of the bird a nd the ,otor is
the impulsive reactions at A and -B being
L*
&a
: &!+ E:i+ tratt'
ftr = fQi+ 8si
at d rvith angulat t'eloci6- c : :'16 ! just
The rotor rotates aboul a non-ptincipal asis of inertia.of tire disc
1.2.3 are Lhe principal a''ces of inertia of the
before impact and with a/ : rz[ k (say) just after impacg. A-tes
[.
LL-
discarCwirh t1r=1]o6q,,'E:'r3= irorf1.I$=0si'ce!isaprincipaldirectionarCasi=-car!3.=-sirg, sl =sino.- s?:cos9k: n_cospe1 _singez i-"nosgr*.J"eg-.r,,=.*0,
fleace for che rotor
L
I!, = If, -
rn2(0)6
: O.
..
4,= tf;r-m2(0)b :€,=-[,co ='f1s1n1 *f2s3n3+I$s3't3 = lrn38isin0ccg'
i1 = tr+ fr'? + f;,'{=+ rgnS = 11 * f rz2Rllsins 0 +2cos? al'
H a = {I!, i+ rf i+ 11$t.l
\I
lI
t-
(.)
The line of impact is .lo.,S
The velocity of the bird just before impact: CE : :o(
(l)
(2)
sil ct!)'
:-ai=Henceitsvelocityjustafterimpact:i/e.=-usccoi*u'j-- The?unkaorvllsu"r^r{'8.r'l?2'83'8{'8sare
o{ molltum telation for
determined from the,'coe6cient of rcstitution relation, *gul". impulse'momenl
and:bii.a: lte sdocity of D
of
the fuied poin[ d and the.impube-momenium relation for the system rotor
,-,
i
i.t
I
A.
!
:
justbeioreandjustafterirnpacti*gp=-uroBri,cb--u,Rt!.Ttlecoefficientof',.restitutioare}ation
,'r,-rb"--.c(tq-?oyl
-"
:
Eence
E--
{e = -vscosa!.
o'-O=-0(-uosino-0)
A{.e for rotor*bird yields
i+ I* k)('' - tro) * AE x nrs(gi -
*
o'.:O'
(3)
Using eq(2}, I+ng^ =
zbk x (A+ i +
)
+
*ri) :
(If. i + ti
gB)
!
J
L
<-
r.l
'^fI
-l
L'-
!
I
--
J
c
O
f
+i:
j:
k:
,
'-"r
t
:
c.a!t,l
a
.ii
l'
".-:;---
.'
h
I
The irgulse-rnomentum rq!.atioa L:
it, i + a, j +
:+
A?for rotor*bird
E3 k
+
Fa t
+
yields
a"i =''o(!! -
!:
.Er+&=0
l:
k:
A,+At=rlouosinc
E"=0
:
q'El
=
rnouo sin
cj
(7)
(8)
(e)
il"=-0'5msussinc' R1 =h"= &:0'
Thesolutionof eqs(a)-(9)yieldsa:r-rro, 1is=1-5m6u6sino,
Ians, =afl..t and r=AP forrotor*bird'
il;;;;;;;;;,ii;',ir,ir",q.E","udeterminedusins
di{Iering from that of the rocor alone
After impact rotor*bird form one rigid body with its inertia -"it
Afu for rotor*bird vields
conrriburio, o[mass-poinr rno ar'AD--Rrj+34k. !-ang^=
;;;;"
zoL
*
x (n"i+ Euj) =[{ri,-mo(0)36}i+{ti,-rnol?r(30)}i+{I:,+ms(o?*E?)}\l''
(Er j + 36 !) x m6ue(-t(fi, i + ir1 i * I!, L)no+
cos
di
- siu ct i)]
(i0)
- I!.(u' -,o) - 3mouobsinc
2bR4 - lf;.{r,r' :yd- 3moEl0o" - 3.rn6usbcosc
j:
g I!fti
-.o) * 'os&!"t' - m6uq81 cose
E:
yields
Tbe iragulse;mometgum relation I = Lpfor rot'or*bird
j)i
R1 i - uot- cos o i - sin a
E, i+ ar!+ Ito f + f-o i + Ar i =,ro(g! - ge ) = rns{-u'
i
Ar+4.-rne(uecosa-t'r''81)
:+
!:
j:
ea+At=lttouosilro
r?"=0
k:
!:
-2bfr!.s
(ll)
,
(
12)
{13}
(
t4)
{ l5)
-r,E .j?s, Rr,FL",ff3 ".e computed successively usingeqs(12),(11),(t0),(13)'(14)'(15)is rotatirrg at, the rates shotrn relarive
e,.'.6. a.z A body 2 consisting of a shafr B D zrtd a rcd G E(Fig'FII'?a)'
The rod.is of length 'L and
to platform l totaiiog about a fixed t'ertical atis at the given rates
the
a:iial testraiut-
'4't'
lrut' prof ide an'r'
ma.s ra- Neglect the inertia of the shaft- The bea.ing ot B does
(b)
trvisting
the
SD'
shaft
*r"rifinil (a) the bearing re4ctions at B and D and the couple applied to
;i"",
;;;;;;;;;;;r""t
t;
-a
force and shear force at section :lu"r tt'" slender rod'
'orSnal
;[i!,:
ort,t,,
atI
iro! l,lr'ldWt-*
ik*,,,
7FE
e,f
.
--l+
{t
La.J**,rrit-u*{;
$:,
(at
Solutioa G) f*t t!r,4tand sh. Q2 be the angular velocities and acceleraLions
4r :
gfr =.i'r i
L
93 : t4r f cr3i =rr1 i+-ri
,izz = ri;.r+.i'z i + tlr x t tz L=tirr i *
q;r
'z
I
J
f,
,L
leJ-
sEa ! ;.
!
lg- - *il*
k-Lle
-*i
(()
of bodies l 2 t'r'r'!- ground:
'
(u
rile
j*
rer1o2
k'
(2)
the momeo'
of body 2 is sborvn in Fig-83.7b, where C is its ce-nme of mass and lf is
Tllil '-"
of motion of body 2:
equations
the
6
fronr
31re
determined
M
Fz,
F3,
E:,
Fr,
81,
unknowns
The
6
the s[a[t-
Tte FBD
6s
_..
JJ
JJ
J
J
''l-J
.-l
J
J
]J
:J
:J
U
r*J
U
r'l
,tl
-J
J
-lJ
J
-J
-'J
J
J
k:
E= trru"oa a eo;pS\tiatlo-qp
[-
f
. lb = so *
-'
L-
lJ
[J'
E'
H
H'
H
L
L
L
1-.
L
l-
tJ
lJ
t.
.
l
]JJ
L)
r
u
lj,-,
l-
2 remains fixed'
foa C-'-foi-* O on the dgid cxtension o-!!oar
cz-x 9C
+
* " !*x gC)
c,,i(D + dlli + &ar(6 + d) - i'{;'"I L
Mc=
- (-683 + 6r.3) i+
The principal
x (&i+B3H+ ?iLl+ei) x (rri+rzi+rsE)
(M + iLr.g+ !z&)!+ (6& - it& - 6rr) k
ii+eir'i-Di)
axes of body 2
at
c
are parallel tb
z,g,z
axes
l;:'
ll,'.
(4)
witlr I!, = o. tc = Ig' = mL2 lt2'
Forbody2eqs(1)-(2):+t')s:Qt'tdv-Lt2''tu'=O'&t=i1'j"=it2'"t)'=t't1<'t2
body 2 for C yietd:
The Euler's equations of motion f-br
+
Mc, = 19,a" - Uf" - I!.\,,,r,,t.
Mcy - Suau -99"- €,\,.-'
Iy[c,=t,6,-(C,- $"\-'uv.
=
+
(Fa-8")A=O
(5)
u + \t(Rs+ Fa) = nL2&2f L2
6(Er - Fi - iLF? = mL2sYu2t6
(6)
(7)
F'--nec+(fta+rr-,r,e1ii&!+(Ra+ra)E='n[-]r'!f,!+{"tp*-410+d)}i+{t'rr(6+d)-}62'}LI
Er * .Fr = m(g'Vtit'Itt '
B',+Fr-m(g'Pt|tlzl'
+
:
'
R"
+ Fs :
m[dr1(6
+ dl
{')
..
Ft-ttt-tllttzL-u1(b+d)l
*;lLlzl
t'O'
.
- tilr(6 + dll
d;lzbl' 8s = f3 = ]rn[ti1(0 + d) - ''"Llzl
R1= \m[g -.!r1z l 2',,p2l.2lcb -ctiL(b +
dlfztJ; f' = molp:f, -:'11(6+ d)|
Ft= i*lc -.trLlz-:,"'tru2Lz/ra+tillo *
The solution of eqs(5)-(10) vields M = lmLl?fuf,l3
when ri2 = 0'
Notc that a torque M is needed even
rvith centre of orlss
of mass
rod
of.the
(b) The FBD of part AE
'o(L -r)/tr
(d 6)i + !(l + "1i
i-ointr'O, f are on bodv 2 with aC' -- +
C'
is shor*rn i1 p;g-Ii3'7c'
4 t OC' +r'z x (g& x OC'l
+ il luz(L+')]k
=l\.trtr * .li+ lrrrr(t+') -'i1o +d)lj + ['".r(6 W.= CR-rG -.')!xEn
gc. =
="
go+
Thegrincipalaxesofbody AEatc.".:p1.ll"ltot,r,:-oowith
r'rt'dl""
The Euler's equations of motion for body A'E fo' C.';ilh
l;c"
(11)
(12)
t
=0'Ifr' ='nfi'=m(L'-tl3lt?,L'
as forsodf"2- yield:
"t-t
- t.a, - ({, - Ili)Pt""t, =o
rfa"llzL
Mc.y - €, a" - g9, - lllY',', = m(L M"., = ry; a, - t*; - $")n,', = n{L - rlsuYo2l6L
Mc.,
+ W. = rrr(L -.)"(d'zi+
r-[m(I-"]lLlsc'
:+
?*tPzL),ll2L
Ll9x.'
-[*(, - t)l LlgL+4* = Pn(L - 'll
l--,1'
lj.-1
(3)
- - ir; r, i + l,-t1tt2L -
G
(13)
I
t
-;=
(r5)
(tuapt(2L-,)-&'?(6+d))LI/6r
Mc.+i@-dix& rn(f,-r)2ttr?(2.t;;)-&,1(ii;)$
is the normal
e the ! comPonent of !P
is in the -idirection' Henc
at
cross-section
the
io
Ient beins .he i
'{'
The normal
tht ihear force- The twisting mot
together-consiitt'to
j
components
and
!
of ea together
force and the
sirice tlrc i ana'b ilponents
n,ment
b;;di'u
tht
is
z,eto.
Qailself
componeni of Qp is
moment'
bending
the
.""",o"r"
vertical shafr, at O (Fig-E:t-Sa).
mass at C is hitr; a a
axis at a
pxa-rple 3.8 'A body of mass rn *ith centre of
roaft is- rlated about the fixed'verticat
ti",-i?r,.i*
,r"
o
ilr,".-,h
*o*"rrr"ii
tts principal
".
"rrig to tlte vertical' Find
oc has a consta"i il'.ri""aun
J..uvv
such that
consranr rate c.r os-"
=
"-
i.,
,K,rry-=iffi;:"
"""'ffii;;;;
+=I'
N"L(-l .,,.n "-"LNi:
';,{o'L
(b)
. ,' to)
L
^
.
rls equ
lt
-:
ti
tio*s for
Solutionsincethe.bodyhasplanemotion'thesolutio"-t:t::O*tnedusingl'Euler
moiion fcr non-principal a-'ies'
equation for plane
tuomenf, in the direc{'ion
rrirrs"joir* does nor,r-tert any
Fig.ffi;;--,*,.
i.
sfterrn
is
body
the
of
tlre fixed point O'
FBD
Trre
rfrom 3 Euler's Jquations ior
att"'*iita
L"
Cr,Cs'g
3-unkno.'os
The
axisof the hinge
*"utut"i-"i-J'ut'o"at tt'
The angurar velocity r,r and angular
principai
axes or
Z'
-o*""t
trJ3=-L)cosA"av=:-'sin0'
tt=t9=-{^rcos0i+r'rsirroj' u:g+
Mo =Gti+ Crl-nrghsindk
d:=0' ut:it":rit:O(t)
(2)
TlreEuler,sequat,ionsofmotionforthebodyforo.usingeq{l).(2}.r,ield:
Mo, = I?,6,
-
(19,
- $rl,,-o
0,
,0=0, or
sin0 =
i-e..,
or
0=t,
.
Cz=O
- rrrg& si t0 = {€, coso
or.
{3}
ft.=S:
r+
+
*
Mo,=19i,-(8-*.Yu-.
l{s, = I?uau - 1t!, - f,Yo'-'
iff }-:
sin 0 cos 0
#)
,a
\j
t5)
-L'
= *st41;?r- {?,1'"
0 =cos-![rrrgt71
€r' 8)t"!
a'?
Tlre third solution exists provided -L' <
> WgnlT?v
- 19"|l'
!
imshlt'fi.- !?1"-l: 1]: 'if oierr *r m"-*,"1r in .be directiirn
r,ils.ioinr does not
z- The FBD of the body is showu io Fig.Hi"-aretu"
equations for the 6-xed po'int
determined from 3 moment
Mt,Ms,e
unkno*ns
3
The
a:ris.
hinge
the
of
O- The axis of rotation is the noa-principal
.
oc'rs
9"=g=-cc9i+singi
= g= -sin0!- c1s0!
8r=O
9,: L +
sc
-
ry^
=
'nt=-cosd' a'=sin0'^ nt=0
0
s' = -sin0' s' = -cosO; s' =
since k is principal a'tis at O'
+
*
t!"-*
,i'. = *rq, -
+ I?r""n" + f;'s'n'
mglrsin o4* Megs
67
=le..-
-l
rfl)sinOcose
'
I
t
E-
I.:,
Mt=:
-+
Mor=f;r&-$rz- 'r3*
1y{oz = tSa + 8*"
i. *
'+
Mo"= t&'
MoL
)
--)
)
Equarion (?) ror 0 is
-
,-,, -r',, -row2r*g.*e
$yf sir-e"s'e
(?)
(7)
raghsino ==(e,-
(s)
Ms=A
rheT-".::(:l-.y';::::;::":;:f::;,3;H';r:l*-issupporred
* Td teagth tr' (a) Find
uniro,m srenire*oi oi-*
force"shear
the norrnal
1u1 ri*a
on
ff;3,',::r.
i"*t
"tfi$;.lffiilffi;"
ex.rtea;;;t
load
au"
|oom'
-theof thJb.om-dt-d'
rhe rension i' tt " c.biJ "oa
t+rytion
f
*"-""'';[i
beading
ana
momenr
twisring
,2t4t
. lef
l_(J20,'T
(
force,
, i offi,*,
j\eft;i.il"
_ffio,
;W
t
o$r-.
'.4:8N7"
'.-=Q+t
I
'-rt-rr
to ;
.Q,E
C\'
'"
with the a-tis
so rhat a-.ris y is coplanar
aL o are choseu
z'y,z
inertil
of
axes
(
its centre of
is
principal
T
rvhere
'rs
Solution (a) Tlre
boom shorrn in Fig'83'9b'
au,
The
axis"iri" ,,, trre directio* of trre hinge axis- Tbe 6 unknorvns
of the rod and the vertical
*"*.",
any
exert
uot
and 3 Euler's
mass. The hinge joint does
i-notio. of the-boom: L=',.gc
n"r",n"'l]j";;of
of the boom are
R1,R2,R3.,C1,Cz,Tareietcrmined
augular acceleraLion r':
*;"0
*ili,
TLe
o.
para
noa
the
equations for
""*,*
L
l-.
L.
L.
L.
r--
L
L
L
L
L
.rtbl
I tal
(1)
u' =osin0' t'tt =0'
<" --!^'e:c"cc0i*trsin0j + "t'=t'tcos0-'
-'' =arsiu0' t'r' =0'
t't:i9=ricosg!+cisin0j * i':ocosl'
I
)
ly 2
sit'h
gI.=|li'
tb:
k) +
(2)
usine eqs(L) aqd (2)'
,o-x
&+ ex {gx QC)
'
''
+ r"2 cos0i- - !)
= llsin o(-"'! sin0 i
rng(lrsin0)lL
b= G i*cai +ffLcosl -
(r)
(4)
C
:
qithl;o'=0,lf;,=I!;:nlz|3and.ec(4},tlreEulecsequatiorrrofrrrotlotlfiortlreb:omforoyield:
I
11i[or=8",b"-(g-$1-'t''
8t)",
Mo. = I+'., -{*-
+
+
'g;'=''!ml1;"itta
TLasa- lmgr'sin0
ro
[:'mgs +
-
lmf,lt^'zsro
e*el
i
(6)'
(?)
'
(8)
j;nrlsip0(+rzsin0!+r12c0!-t'r!)
;,tiiri* R8k+r(-si.0i+cm0i-)*rng(-cos0i-"']'aq:
0+31Di-lmf,sino&!
sirio} !+! m sin 01c,! f, cm
+ & = Rr i+a,i+Rs k : !n9(1+cm2 o)lz.oe 0-}m t*12
the tringe joint on the boour'
where B is the totat reactiolr force from
6g
E
I
(b)
The FBD of pait'-r{8 of the-rod of mass m(L
- {l L *lith centre of inass D is shown io Fig-E}$c.
i"i"o o.D ateoothu'EooH;ith O2.= +(t+r)i,
-i.,,. .,. b,:9o,1 ex'OD* otx(gxgP)
''i':
M^o
The principat axes of body
:
(s)
*U,*a)sind(-ru2sin0!+r.,2cos0j-&!i''
An + \fg,- a)coork- *tZ -z)1 x {a
AB aLD are parallet to c,y,:
aries
(10)
n'ith I!. = O, 1f,, = I!,: m(L - zl3f t2L.
F=rn(l -dL)eD
:
= ln(L-z)?sind(-,.,?sin li+uzcali-Au
The Euler's equations of motion for body AB tor D, with
Mo,
..;,s,..;.z,. . . same as
(11)
for the b6om, yield:
- t?*" - U?" - I!),*t"u, =a
l{o, = t"Drio" -(e. - I!'b,-. = n(L- r)3asinp/tzt
0' ca
M o, =T!.b, - (I!. - tfll-,., - tu(L - ,c)3,.r3
| t2L
"in
M-o=.i'n(f, -zfsiao(<irj +o2 cos? \)/\2L= ea * lqt -r)cosd!- j(tr -,a)i x fo (12)
F'n is obtained frori eq(f l) and thea C* is obtained from eq(12). Ttre rrormal to the cross-secti6o at .e is
in the - ! direction. Hence the ! component of {p is the normal force aod thp i *d k components together
'
+
0
.
constitute the shear force. The tsisting nromeut being the ! conrponent of AR is zero. Eence Ga itsetf is
the bending monentAs in Example 3-8, this problem cal also be solved using the nroment gquations for plane nrotion-
'Blairrple 3.10 L gytoscogic gtir,dcr (Fig.E:I.10a) consists of a'uruller'or grinder in the form of, a disc
of radius r and mass zn, which is rnade to rotate abouc a fixed vertical a-xis ai a constaot rate f)- The disc
rolls wit^hout slip. Neglect frictiooal force on it iu the radial directiou. The disc can rotate freely on the rod
about its a:ds. Find ihe force exerted by the grinder on the pan- Find also the reaction of the pirr aL O on
the l;shi rcd-OA-
d in
tF y't,i
s qfu.zEg
;'
*K1rYK''-*K#
d;ffi-r.r\'.;
i.toVr. -t*W6,!
c, Y,
\]_
d/
.
\\J
f
-'-.
<h"'
s .f,-
-tgt //,> !
7T
t-i"sg
Inl f3 (dr
.
i\A
-
Solution
The axis e of the disc is a principal axis at point O on its rigid eiGnsion] O remains fixed iri
the ground reference. At the instant shorvn (Fig.E:|.10b), the principal a:iis y is cliosen in the vertical plarie
through e-axis. The FBD's of the disc, the rod and the rod+d'rsc, are shown in Figs.ES-10c-e- The radiat
friction force F1 - O. Cn has no component along the axis of'the 4;rsc.- Ma has no component aloag the
axis of tlre pin. The 12 unknorvns Mr,ll[z,Ct,Cz,rV,r'3,f'e(3),8(3) ate determined from 12 equatioas of
motion of the rod (6) and tod+disc (6). The angular velocity of ttre disc 2 relative to body I is aloog i say
r.r1 !- Let ur, % and rir1, ri2 be the angular velocities ald accelerations of bodiei 1, -2 rv.r.t. gro,'nS- \trIith
O B = 5i - ri, and g,a - 0 (uo slip),
ir
9r = Og: Q(-cosdi+sin0i),
(1)
cnil
t-J
'aU
'at
':I
J
j
J
JJ
J
J
-I
J
J
J
JJ
JJ
J
Jrl
..-l
JJ
J
JJ
,J
JJ
J
E
L.
....,'j-'*
t"---..
1 -1 "..-, '
lj,
lj,
l-'.
L
IJ
TJ
L
t_
IJ
IJ
L
L
L
1_
Il.L
IJ
u
tIJ
L
L
L
l-
L
I :.
'= (.rr
,..o, -
Eqs(2)'(3)
':+
:+
,t:''ro+
r.rr
gtz
-
x@'= -(arl-oi#"blr*6Osiu{l!=
+
=A(-ug -bsiallr)
L
1-
(3)
=0
;:e:;*;l;;:;,-:fi::;;l;T""ff:-eanolr)!
- m*/2, 4 : *, = m(?14+62) and eqs(4), (5) vield
htc - -69sing/r, r.r, - Osin6, d. = 0, 6, = 0. 6" =0, r;r, -
For disc
For light rcdOA:
-O2sin6(cosp
- 5sing/r)
(5)
(6)
.
=+ M.a=qE+0ix(,tr!*Ezi+ae!i)
W=Ma-en+&ix(-$-9
:+ Mi* Mzi.=(Cz - 6Es)i+ (Cs * eBz)k
:+ Itfi = O; Mz -- Cz- 6Ea, 6s + 6Rz - 0
.
(4)
Z, I!,
(7')
For the rod*disc, Euler's equations foi trted'poini O are applicatle dnce rod is inertialeS:
' Wl{6"
x fiEitr'
-:rce0)Irtk +'M:i+'([i- "j]
- -rfsi+ (ll:-[fali+ [N(6sing -'rcos0) - mgDsind]!
-n96sin0'k'+
: f,.u, -(r?r- I?,furur2 +
(OsinA
-r.F3 =
:+
0
&=0
(8)
i
(e)
+
!t6,.=I!,b,-(*-(;l-;,-.t,
+
JV
ir'(6sino-rcoe p)-mg0sin0 = mO?sind[]6rsin0-(62 +
= fm96sin0+mfl2si!0{}&rsin8-(63+
Irct the normal reactioa
:+
r1I
for thestatic
JYlnrsr
:
case
(A:
f plwO'117(0sin0-rcosO}
0) be IVs1.
r2/4)cf]
(10)
Eg(10):+ I{"g: mgDsinA/(6sin0-rcos0).
I + [sind -$l2b+2bltlex0$gr!2s
(ll)
.'.
For 0 = g0r,cq(11) =+ IVflrI"g- 1+ Qz{2g,Ttrusa'largevalueof Oimpiesaverlrtarge valEcof AI aud
the crushing action of tbe SdDder b better.
,t mo.,"s in a circle of tadius esiaA 1 rcte fl. Ilence q1 = -fF6sia0l.'Fot t'he rcrd*disc,
F=
mqa +
&+
(lV -mglJ =-O2&sin,!
+. .&
= -O26sitr0f +(m9-Ar) J
N is given by e,q(U).: Equatirins (?) qnd (9) imply It{ p = l{1i+ ff3i = 9'
p,1a-ple 3.11 (a): A uaiform bloctr of rnss m - 120 kg is supportcd at D b: a b:rll qrd soctet joint
(Fig-1p.lf a). It is subjected to aa instantaneous inipulse with P = 2'10 Ns. Find the, impulsive reaction at
iUe-support Eind the velocity of C and the aagular velocity of t\,block ju1 after-t-he applicatlduof the
n'here
,
instantaneous impulse. (b) Rcwork part a if the bQc\.is lupportea ii D by aierticat incxtcnsible cable for
-(i) ncw-oik
nail b if the block is supported at D by a spring of
t*,o cases l. p = 240 N.s, 2- F = -2{0 Ns.
jost before
stiffness 120 kN/m. (d) The bloc! is not supprjitid'biiU b trEuslating vertiiilty{orru at 2 m/s
just
impact with a corner (rcqr ncar it) of a emooth horizontal table-at point .8. -Find its lagular vetocitl'
poiot
t
tras
is
rough.a^od
p*J3,if
the
after impact. The coefEcbat of rcstitution c = O.6. (e) Rcrrork
,1a"!t9
joint
is
hit
at
B
by
a
zero velocity just after impact. (O The blodi support€d At D b, ; 6all and'socket
L1
L',
L."
ri1
Q
It
€*5
rf'
1.>
in
-6q
,
ur
o'
'.:g-
+rtr'
'*._,
i.r99''
-s
'.5
31,/.*P* I
&l*,n)
o'rj
/+_2t
GI
t9
!-he
*:
TrW.,ffi-,ffi;'G"
t1'u t
{' t*} (t)
(z)
.lo*o
J
Sot.rtioa (a)Let&betteimpulsivereactionatD(Fig.E3.llb)and u=u'ilu.vj+u,kbetheangular
t
,n,
velocityjustafterapplicationof theinstantaneousimpulse. LHo=Ia,.E, yields ar, llc=oD*oxDC
yields uc aud mAg6 = f yiel& !. r. y, t a.e the principal aries of inettia at fixed point D with
= 120(0.52+0-3?)/12+O-252]= 10.9 kg.*r, 1f;, =120(0.12+t?11t2=
rf, = 120(12 + o-52tll2+ 0.2s2] - 20 kg.m?.
larg,o= DAx (-60i- 240j + 120\) = (0-5i+0.15$ x {-60i:240i + 120E)
= 36!-69!- 12OL Nm-s,
Il,
10.9 kg-m?,
(1)
(2)
The angular impulse-moment of momeutum relation for fixed point D.yields
/.g.o: f.ogo +. I!,.,L+ Iflutr!+ I!,tr' L= 10.9.2'!* l0.g,y j + 2&', E = 36i- 69j - 12t)&
+ .,. - 36/103 = 3'303 rad/s. ov = -69/10'9 = -6330 rad/s' ot' = -120i20: -6 rad/s
=+ r.r - 3.303!-6.330j - 6k rad/s
+ oc: b +u x DC= 1a-SOai- 6.330j :6!) x (-O.i6j) = -1.5i- 0-8258k m/s
-!
+
lfrya
Je -OOi-240j+ 120k
:+
Ij,,= 120(0.52 +032)/12=3.4kg.m,,
I!, = 1fr(12 + o -52\I 12 = l2-5 kg-*'.
"ogUl"t
If,, = 120(0-e3 +f2;7tZ= 103 kg-nf,
(-60i-l'i*r20$ - (30+0.15P)i-69i+(15-05P)L,
impulse-moment of mornentum
-J
:-l
--l
:.l
.-J
--J
J
.J
the instaotaoeous impulse applied by the cable be ' i (!I. 2 O).The FBD is shown in Fig-E3-llc'
The principal axes at C are e,y, s axes with C,{ = 0-5i+ 0.25j + 0.15! m,
ite
-
-.1
A= -r20i+ 24Oi- 219-1k N-s
-(b) kt
Larrs.-'GAx
-'l
-l
The impulse-momentum relation yields
n&pa,
!
relation A fl+
= Irrrr.
rJ
-/
Lr
(3)
(4)
for centre.of mass Cryields
- €.-,L+ Ifru"!+ r9,.,k - 3-4r, i+ 10.9r.r, j * 12.&r, k - (30 + 0-15P) i - 69i + (15 - 0-5P) k -'
:+ o. - (30 + 0.18P)/g.4 rad/s, o, = -69/10.9 = -6.330 rad/s, u,, = (15 - O.5F\l:r2-5 rad/s
(5)
=+ r^, = (30 * 0.15P)/3.41i - 6-330 j + (15 - 0.5P)/12.5lk radls
(6)
lb = I!m= [-60i+ (f - Pli+ 120\]/120 m/s
(7)
uDy = acy = (f - F'11LZO
9-e =yr+ glx CD = tb + sx 0.25j =)
: I0 and eq(7) yields f = F - 240 N-s > 0u5r, =
then il^
taut, than
thai +1,the ^-Lrcable ---.i-o
remains iarrl
l. F =240 Ns. r{ssumc rl.-i
m/"'
Eeuce the assumption is v'alid and eqs(S), (6) yield gr = 19.41 [- 6.330i- 8-4 [. rad/s, g" = -0'5 i-t- L
2. F =-240N.s.r{orr-.thatthecableremainstaut,thenup, - 0 andeq(?)yields f = F'= -24$ Ns /
0. Eerice the assumption is not valid. Ttus the cable becomes slacL. with T = 0. Equations (5)' (6) yield
g=_ -1-?65i- 6.330i+ 10.8k rad/s, sc = -0.5!* 2i+ t m/s.
7t
-l
l
il
r
-J
-J
-J
J
J
J
J
-_-J
r
-J
sJ
I
since its stiffncss is.8'mte- Henee
l.j
t.
IJ
L:,
l:
t_.
IJ
IJ
L
(.),,.Tb9.:PIl$.G,:.*-qbl:.r*rO;;#'Provide anv impulsive reaction
#ti""'.ui-!qs(s),191 *ltu f = 0'
ii'.1;J;ir;;
"H:T":
**1ol lrorizinfr .Atil ::',t"": t1 .,''T:l'"J"
er 'rt'Po
(d, l're [n€ :ffi
ff
j::1t':', being gt,cb, gq1. The FBDf is sLorvn in
(,- .. f,ranaa
l,o r,t, be .the values just begre 1:-yr*1
T"
Y:
m, oc 05 * 0'25 j 0' 15 L m' Hence
;;';;fu ;;
'l
*-;::
!
=
?;:;r: .i;,- @= -o.si- 0'2sj + 0'15 k
e i m/s,
m /s,
ar = 0;
0. * = o!"i' d =' ,LL+ ursj + ,1 g
t^'
gc = !E= -2j
{p = * + d x @=(o-r*rl +0.25e,)ia (oi -O't*n,-0'5ur1}i+(O'5-i -0'2&.,1)k
conservation of nroment of momentum
The 4 unknorr ns u',,u!r,4' u| are obtained using
coefficienl.of resritution *::-t_
abour n*"1ryr"i d""iraaorr wirh o, and the
m5)
+
+
llio
(-
(8)
(9)
Ha*OC x
_
OC x rr.elt =0 *OC x rQc
Eb = Itc + €.t'.i+ tfirt'ri+ IgyiL+
(0.5!+0.25j - 0.15h} x 120(-2j)
3.4u,,i+l0.9oii+ 12.5a,: k + (0.5i+ oj'i] 0.15$ x 120uij =
(10)
ruf = -10'59 - 529au!"
:+
!: i-*'.* o"; : -36
u|=o
':+
10-9.ri=O
i:
(11)
a'o--o'Lfu''-0'5tI=-0'6(-2X13)
obr=-0'6ue, +
(12) + o'' .stibsrituting q,,,d,'fromeqe(10), (12) in eq(13) vields ui = -1-2X7 "tlt-a1^::"(q'
3'662k tad/s'radls, tl,= -i-UU, ."alt' U"o* lc-= -L'237i */t' i' = -4'0'tli-4-041
point
0 is consen'ed s'ith 0
about fised
(") The FBD is shown ia FigB3-lre- The momenL of ulomentunr
(0'5'0'25'
jt'"t dter impact' since g! = g = sb- Usiog EC =
being a poiqt of the rigid Utai
.l oa*'
]t'' .
13.6 kg-,.u, Ifl = A- 120(0'5X025) = -15
+0-1521
;
e. uL-0='e(ue.-i)
=
Ijt, = 12d(0:S2+0J?)/12+O2sa
l2(0'?5)l
It=0ks'*''
rfl = 120[(0 .32 +Lzllr2+Gls"+0-52] ={3'6
:ill':4'5'kssr2
9 &gro"
If; = lz}f(f+ 0.52)/12+0-52 +0,5?l = 50 kg'nrz' t?' =0 - 120(-0'15X05) =:
120k,
[r€}[!4J
:
ilo =9.+ QC y mgc: (0.5i+ 0.25j - o.15$ x 120{_2i1 -36i-t+l
- [i: *j * ][;il =[-il] +
gr= -3'6't3i+0'3?63j
=+
Note thae this
(f)
=[riff]
2-I3h md/s
does not comespond to the case of e = 0 for
thesysteaof masspointand btocl(m*m1) isslacrsg in
case
TheFBDof
-
lfi]
ofg! - I
whi& only the-cornpone:t
"'-""^: 9'
Fig'E3'llf' DB = 0'5i-0'3i+0'1L'
* = +uif+'i-tl _,'.
=Q, e8 =-zi+i-: L m./s,
s't
- !8 - 6 * ! * ry:(-l i+'i'i+ oi !) x'(osi - 0'3i+ $'
uif
c8
I
'':10:.;r;'+o-&al)i+(0'5r.t'}-'0:la';li-(0'34+0'&ailE
,
The mogrcnt of momentum
g'o=
gD of the systcm of mass point and
'
(14)
-(15):
the block is conserved:
ED + t*4i+1*r'r!+I!,u',L+Dg x nir* = 0+LExmttb
=+ro.erli.+r0.cr;L+2erl&+10S1-03j+0'1Llx2(uli+ui!+ui$
!:
1
f.0.9ar!
:
!:
.,
-
O2tti
-
0-6o!
:
0-'{
,::ro.*.r! +o2o'r- "1 =
tu" + 0.6u! + d, = 4-2
LL
;...r:;1
ixi.: i tj
.Lil;.'.]
l+I,i -'
=+
=+
+
x 2(:2!+ i- k)
= (0-5i- o'3j +0'1k)
u{ = 0.03669? + 0.018349ui + O'OSSOapu!
. -'"
u!,
.\.r,''
1'-'
=0-9-t5046
-
0'0183't9u!
*
= -0-01 - 0.03"1- 0'05u'
0'091?a3o!
(16)
(1?)
(18)
,
=...
1. Thelindof impactisidong!e.,---[.
Eence
ot,=t,'g','='-1.i-u..gt=uli+i-k
ttecoefficieotof
restitution relation Yields
:+ a'k't'r+0'&'1 -l/'--0'6(0+2) (19)
ob,-'6. = -e(aar-'*)
(20)
eqs(16)-(rS) .+ uL=O, @, =-0.036697-0-01gea9uf, o! =-0-06-0-03ui
.using eq(20) in (19) : i, - 1.165? m/s, eq(20) =+ ari = -0.05809 rad/s, ar! = -0-09497 czd/s
+ y' = -O:05809i-0.09497! radls
2-
Eor sticliing together
v',
*
=
=
O-Lt
{s-
Eguation (15}
+
t'o* o,*":, io =o-*'t',- 0-tari'
Substituting from eq(21) in first part ofeqs(16)-(18) yields
+
l-rt.t 0.3 -0.r1 [r.,'-l I-o-sl
rdtil LrlJ=
L_tl
*r
Lj;il-
y' --0-03455i+0.05168j
1"'
(21)
= -0'3''t -A'lt',
-
l.',1
=
[0'03455
I
l:iJ L;nu:.1
0'00965{f, rad/s
(g) , The FBD's of the block, rna.ss-point, and mass-poirrt*block ..g 5[esn in Fig-E3-1I5' ,Bet P be the
fi.ted point, in space coincident rrrith B-
'
CB=0.5i-0.05i+0.1!m,rc.=-0.5i*0.05j-0.l!nr
gc=98 =3!-2j m/s, ga=O lb= -2i+ !- ! m/s' 'i ='',L+t'ri+t'ri k
Forblock Ep- I{p +
l-
The liue of imPact g'
=i
8".!i+tfr-'ri+/9"-'.1+ PC-xmls-g+PC xrnk
{22)
e3)
:+
$=o.!-2jm/s,
Q4l
*=u',i+j-km/s-
The b unknowrrs or,o'ri*t'.rtt'r,ul ate deterrnined f,rom eq(23), cou*rrration 'of notuentum of
a-dfuection and the coefrcient of rcstitution relation. Usingeqs{22} and {24) in eq(23! yields
tr + r'zr ifl
3.Ut t+lo.qrik+12.er1h+(-o-si+0-05j-0.1t')x120(u"i-2i)=(-0.5i+0.05i-0-tux120(3i-2i)
.
!:
j:
'
k;
p,for m*mr:
3.aw!-24=-24
=
!")'_
10.gr;
:+
:+
:+
*:'r:
-12a,--36
12.*ri+120-6u"=102
.L20a,+2o',- 120(3) +2(-2)
+ d. x cB =.o,i- 2i+ (-li+.i,i+-lH x (0-5iu" * 0.1tri, + 0-05(,1 = t-13409.u, - 030228
'oBz =
F
(25)
=0
-3.3028
+
t;', = -L.44
*
uf =
60u,
178
-
l-1009u,
0-48u;
0-05j + 0-1k)
la:*
:+
:+ (1.13409o, -0.'10228) - (I78-60u") - -0'6(3+2)
ob.-oL, - -c(as, -re,)
e:
:+ 'tr'
a- = 2.8691 m/s and eqs(26),(2?) yield u| = -0.L442 cadls, u!,= -0'06283 rad/s
: -O-1442j - 0.06283k rad/s
+
2- t€t gL - uri+lryi+urL. The 6 unknowns ufrru'r,tt'r,us,uyru:
conservration of momentum g for m * nr1. Equation (23) yields
{27)
(28)
(2e)'
are determioed from eq(23) and
?.*,f.L+LA-9uiL+tZ.&,t' k+(-O.Si+0-05j-0.1$x 120(u" !+u, j-+u. k) = (-0.5i+0.05
-I3
(26)
j-0-l $x120(3i-2j)
J
.JJ
,l
J
J
J
J
J
J
rJ
J
JJ
J
J
JJ
I
k,
-
LJ
'
:
lr. ,n ';*turr* 12'"---24 +
--t'
,--_l
j
L
:
B:
*
U
u
L
t-
=
= (tr,
*
0.1c.{ + 0'0&.,:)
"
i+
{o'
- 0'tr'{ + o'st"')i+
of momentunr g 1
Usi:rg eq(33), the conservation
t* '
. li:
-
(32)
o's'"i) L
+
1
+
L
tJ
L
l--l--.
L
L.
for rn
*
120u'
k:
120o.* 2(2-Ow3-0'55045o'+0'l?645u',+2'8405u')=.i
;*,
o'352s
p'i+
mr yretds
u' B +
*
2(4'?859
,li,]'l
[Il
125'68 J lu'
:=:ffi."l:::,t?l:[i'
3.12
*m15
j:
[[-i.roos
Exampte
rny4
i'
,i,#:^
-
m4 + rnl* =
24 = 120(3i- 2j) + 2(-2i+ i- $ 356
* o'24rtu - 0'55046u') =
l20u' *'to'*,li'J*1'13409o'
120(u'i*
L
k
+ 0-240"
=
0'1764?u') = -238
+ 3il5Z9o'+
tr_ilfkl
rad/s'r'r' =o'r881
,,*::^ l
[[l= [;li:;:,J
rad/s'' = ;o'"t1*::t:t
n"
ope,,.
oPsl'-l1i.ig.Ea.rz.) *o
na,.
one face lralf
ace' has oue
A cuboidal bo:t, out. in space.
J;
*"^'i-"t""t intothe c'on$guration i[
.,o-rhereisnoexternarmoment"::lI'*i*ft=i*:il::1,"",$i::*;::"
;tffi':i::;::ii:T::T":'T;i:il;l"J*veroci'lva'clhisi:mii
IJ
l'
l"
L
IJ
L
rJ
Fug
:a-O
e
conhgurar*'ra in the: .oloig,r."aior
Ut the centre of mass
0
tEcJ = trn![el = trc[o
r*t o
"olr +
surface atea of ttre
bethe mass per unit
box
z
(cl
zo' (e\
Zo-
Solution fr"t'?
,-
rrc':
7
&
l_-
:
Es
uox'-1 = *:::*nea
lOo?c :*lu.t . c"'
= lz(ir+q(u')Jo =
mr
(r)
F . - tG - : r rG,.rl
='i:""::]::*1.*f,Lt
u1
=
aao11;ms
9
:
i.
j
"
deletion)
2-
:
;
eatdAEKF
'?t3
=o26
-
:1Ut
.T',:"i-"E
"
*
planeyzisaplaneofmassWlmltrratc ant iirapriacipalasisofincrtiaitG
lo*:1o'
nara{H
l*-
)
)
Ir--i
)
L-'
o
'J
Ir.\
l-
(31)
=(0.07772*r-r340eo'*a-24a,-o-s5o@;r$:r*l!;i,,Xi}iir.f;T:11"(33)
L_'
L-
(30)
-l!1647o';
"'d''=8-16+o'48u'+43ur
L2.5u!.-60or-6u'=102 *'" ('Lii
(0'5i- 0'05i+ 0'1E)
-;i+'t'^u)x
r+
o'i+
x
vL + d. CE=o'!+
ls
=
'3'5294u'
,
+., l, . !1,,=-',*'111'1009u'-5'5046u'
-39
10sc.r!+60ur-tfi.:
o::-i::::
;ffiHJ{r,*.;;"i*
GCt=..(i- $o/zo '
lf;, =-mr(ec0 (Gct)'
u
= - 10o2o(-
o
| zo\(a |
cc'=u'l*'719(iJ
=9(i-!)a/20'
fl
decornposition
Gcz=bz-'rc
-
*'(@"@'-
2ol- (
+
*s(Gc")'
";(6t-*l'ol
.r+
-
I!'=0'
:
tG c"l'
o" o(tg
oI
nl { -
1s4/20}
Sol2o'
= z,"l c I to
:'
-i4
rf
ilf
;Y
-\rlA:
t.:-d
jv
,;Full box:
paoel
ABDE:
panel
ADKF:
_!{
t-
aa
oPen box:
ec(1) +
;; t- L- il,- o-a.tr. vr
just after closur' eof
Let t t: ar" i* u, !* urk be the angular velocity
rvith
G
mass
of
centre
principal a-tes of inertia at its
$, --
{''
+ A, :3o4o * l0o2o[02 + (al2}l"] = l2laaat4l
f;. - -azalr.zllz+(9al2Ql2l= -343aaof l2A0
S. : a2 ol(az + 01 ! 12 + (tsalror?l= t283o{a/1200
E; : (l2lI 40 - 343! l2O0 + r2$/ 1200)oro = 457 o / L20
Hc = (8?j * 457\'loaousll20
lid.
For aclosed box,
Ql
z,y,:
-,. : 0,
tny --'8t"tol740,
$' -
.
:3n'o
(3)
-\
i
a{
J-\
_.r_
eqs(2) and {3) Sield
*'. = 45?/360vo, :? t=
. <
-.-.t
gr=Il"ut,1i--f;lrrj+I.c.r.,.k=I7aac(u,i+-rj)/6*laaou'k
II.c = fl;' llelce
are the
I
If,o =2l2a2o{a2+(2a)2}ll2l+2[2o2o{(2o1?ltz+(alz)?}}+z[axa{a21tz+o271=37oaa/6'
Since tlre'eriternal momenf, is zero,
qq
iEToto/i40)i + ({57@s1360)
!
-c
\
,'
,r}j;xc,
A spherical sateltitc of radius 8.translates rvitir
\
i
velociry uoi in a rarefied atmosphere having rr molecules per uuit' i.;f.
f;q
I
- ;1>/) l{!(Jg3\(fi1'=u/g
volume, each of mass m- Find the drag force on the satellite. \r-=
'\------l
th+g
restitutron e.
o[ restitution
1:
T- - - -- -fY€ Assume frictionless collisions rvith coefficient of
sphere aucl
tlte
[o
frame
i[ertial
an
solution r4re neglect chauge in speed o[ sate.llite due to irnpact. Atiach
satellite in
the
by
srrept
zrR2rts
volutne
consider velocities rv-r.i. this frame- The motecules in the cylindrical
of the collisions Pet unit trme'
unit tirne make impact with it (Fi8-E:!-13). The drag fotce equals the impulse
The impulse l, on it is expressed in
L{olecule 1 moving with to maties impact rsith line of irnpact along'g,.
terrns of its velqcity componeots just before ap-d just after irnpact:
Blarnple 3.13
u'r. = -eugcc9. t'1. = rt1, = euosin0
uoccr!
f1 rn(ul- -urn)e, = -(I*c)rnuscos0e,.
=
ula.=
t'_'a-
_r-<l
'.::.-,'__._
1.
,|
.",--r
l-.ti '
of these impulses ori the
The opposite impulse !2on the sphere is Iz = (1+c)mr5cos0g,"- The componeuts
in the ditection i add
sphere in directions oorrrrol to ! due to all impacts cancels out. leheri'as€oulPo[ents
up. Tlre component I, of !2 in direction i'is
I" = (I*c)rnue.*AJ(-cos0)= -(l
Let
JV be
+rlnruecos?d
the number of molecules hitting_ the sphere per unit titne ar angle 0,
,lV
betrveen angles
0 and' 0 + d0'
in unit time)
= a x (volume swept b1r the arc between 0 atd 0 * d0
cos 0(ftd0) and length us)
= n x (volume of cylindtical ih"ll of .udius Esin d, thicliuess
a R2 o,sin I es 0 d0
n[2r(E sin 0) (coe 0 A dlr]ool =
1
.!-
!1
c€ 0d0'
The impulse per pnit time due to the i*p""t oi'rV *"f"1ks- is .:\'I" = -2r(l + clmnffvisin?
by
git'eu
rl2,
is
from 0 Lo
Hence the drag force D, equal to the total impulie per uuit tinre for 0 ranging
*12
'
:' .',. '
'
D.:-Zr(t+c)1a1n113l"'"tr.*. iA;,-2"q1+c)mnp?r,i1-|cci{g}fl/?=-}:r(1+e)ma.Ezu!
-
:
' +*r
rD
:!
-
I
Ll.
tj
l;,
"t''1'; 6i.fo
,i,Y ht
slarnpre 3.14iDisc-rof iass
and its centre has " ""t;"'L!il1;nt-*:"Yn
z attached to a sPrins
Ij
lj
l-,
rJ
IJ
I*L
The line oi
tttz=05
L
rj
L
L.
L
L
I--
olr^,rttzo are
i*n"o
is along
kg, .u!,r=t'?r
eq"trr,(ir-+,:; =;;
-5€060
.
tal
and conservation of'fo-
rnornentum:
-
,
r/-
;r"'1;1_:;,j.-1;,1',
on disc 2 is Jv'q'
lnstantaneous impulse
The mor.ion of disc 2 aftcr lnpact
lltt" r
,.
.., . ..
talies place under a
t"
iffi; - ;;:':1;;"
f6lss
3,s
rvell
--,:" force which is a couservativeenergv of disc
"ott"n";:;;;
* ianical
sr.:*:ff.;;;**"n'ls'Ius'ia*erimpac'l''Lrre
0s
r:*::l*"**:;,[T.*TIffi
1
*. ..u'rJilffi ;-;;; = aa+
l-o.z *
tq
".a
spring extcnsion,6,
0'2 =
-l'l:h:
- -1'129' *E'16go
-si*og)
=4(-cos0g
nr/s'
*t*'*"
st2:'12g,,.49.'=?'2(-sinde'
'
+ ,i= -t'izm/s' u! = 8-16 m/s'
a r1
r = 0'{ - 0't = 0-3 m a,.nd spring extensio*
- e" = B'
spring'
tlrc
of
p*el
' t'ol'
In the maxirnum compressed
t'aoalv' i" this pcition be
*t""i"'j'"t'*ro""ti"t
rn'
51 = -0.1
* t*
ur.
=
' 'Q'=
;)
u-l
u,,
t'--\
'T+v
m/s
I'6'32 rn1
= lo'rz
O'6(8'16)103
0'6(8'16)/0J-=
:fr+V1=f*V,:.1*,1o,i4,1+ite1=!::,,:+.,i1+}t8,
..;,,.
.'.,..
}(0.6X1632f1}t(0.r)"=.}(06)tq.ro?.+r..rzzJ+}[(0.?)?
. &=39?0 N/m:3'9?0kN/m
,r,"l-onr
exteosioa
u" b -rtt "
'-
t'elocitv uc:
.1.= 0 and circumfe*ntial
=.q'j}[oit-t
+ 6)
6(8'16)/(0'{ + 6) = 4s96/(0'4
"':::1
-"1t0-ulto:+esg6'?/io'4+5H+
Ho,=@*o=,.*'{dn
r
,
??" -::
*
uto)
,L^-r'ro=
-e{u2*
.', mroio*m*Ln=rlr1o1o.**'-:''+l2u"o+0'6r'!*=l'2(6)+0-6(--3):5'a(2)
=+
IJ
0't
determrt
*
!
t':
and' cos
' .'-'
*+
IIo' =ttr2rluct = ta2tlo!4
i
9*B
* d;;';*)
are gi'en by
titu i""t t* t-'-:t
just
U'*"
veiocities
of centrq *a 'ogt'lo
or' j l0sing=6 m/s' 'dl="t1 =2taAfs
L0co0=*nll"'
u'r.
=
=urr
ttL=1'2kg,
=-3 m/s' u\ =t'4 =0
rtt
m/s;
=,-5sin0
:
= -rl
Solutioa
L
f-
tZry -trt'*1
{(;:;)j:-o
\-1ff1;
)"1^
i 1*;Tf:#: V
\|{ B GFIC-it
Hfji"i'3i"n"oir'iiilillii;il*i.*
i
ill*":: lHilj;II:'lJ:"J': .":[
r"-t ''JY"nu"-"lrrl'
and b transr,tioe *ith'Jffi;
'il
is
Gv'€r
'
*t:'ffi*'.xi'i'::5'lu.";fii:i:-':ii:iff";il::
T'"n'l;,.
I\y - /'"t
surface. rhe impact
I
;of
1[":J:::'";;;;;
i,t,,
;t Y"H:t
compression
=
?U
0'?' 1'hema-'rimum
'.
smooth with coefficient "it*"*tt""
*-'t.dY.,:
-fr lult, i1^e"- /%
thespringin subsequen' *o":n :t' :::I},'#jfl:tjilff*:
and the
"""rsrduringirnpacr,instantaneous
;5T*-#r"*1r::H:X.*"*t
qrnL
o",tiilri;
,
ou disc ";;i;;;ngjnrubsequeor
impurse on
-:t:..':y;i;1ff.'P'ins
motionvelocities
-;"::"",,
:+ sino" =
-::6- The
of-tbe-spnng----]--*r-r* o" r4b) and cos60;a/5
= 0'8
=
--rnaximum extension
aogular velocitv
=f +v *
i(3{0}61='
=lionltt,iu'+t't221+}lrszoyo:)2(4)
66J +nsa-ze: - 1s9.9, -.--::-;
+ 39?05{+317
* (6+0-1)[39?A5t +r,tg52+ 15?'86 --tz,s..fl =-o
1?5i,9--=,8'"
=+ 39?06!'+ 2ng62+ 15?'86-- "= "ifil t'i17.538
=o
(5)
,
-
''i' r'Jg "i
:
;
:
'":83')::
']16"+i
-'=:t .''
'
al
I -r
ll
fi'<
compressed position 5 = -0'1 is a toot
where (5 + 0.1) has bei& fgtofed out from eq(4) since the ext_:me
Cardau's method,' is 6 = 0'2005' which is
of eq(a)- The only real rqot of the cubic *"oiio" (5), obtained by
,-<
g' (b) ti'ewotk part a for the
[he plate at angle
coefficient of restitution e = 0'4 and 2' if the rod imbeds in
the motion of the rod just afLer impact
plate rvith
''---- the same motion just before irnpact- (c) Find
uvv r'e'v
vr free
L@E
case of
fs
;nrooih rvith c = 0'4'
()Jt
in part a if the plate is massivc, i-e., m2 ) mr and ll1: i*O1rct-F.t
\-+v^
-
{
<
2-tls
Ai"
r-il;.-
, ,i L lJ*a i"l
oo-:-ri'es,
f-Y1-o{
:'
(cl
Solution (") t
*
t*
0'24 m'
u'rr: it. - -5sind = *3 m/s. t'la = -icos0 : -{ m/s' ri = 0'3cc's0 =
:O-38 m, ''' = 0-4 rn'
oL = -2 radfs, o.,: = 10 radr/s, dr'= 0.16 m, @
0-2? + 0-05?J - 0'016 ks'nr?
r9; = 0.1(0.6)3/12 = o.oo3 kg-rn" 19- = 02[0.6? + 0-f ]/12 +
[{o' for rod L' ttb' - Ho' for rod+plate and
The 3 unknos'ns ut'1,t,\,u". are deternrined using f/i- =
grouud coirrcident. q{th '{'
coefficient of restitution relation, r,ihere D is a fixed point of
tr":ri- rnlu!,r11 = f9't, -trl1tr1'11'
HD,' :+
:+ a.,i = 8ui* +30
0.003o', - 0.lui.(0-2a) = 0.003(-2) - 0'l(-4X0'24)
:+
I!,r;'2 + r9.'"ri + rrrlr:'1.d1 - *ur+ I?:-, + r7r1u1'd1
For rod+platc . IIb, - Ho,,
,
:+
For rod
IIb,
==
:+
0.016a,1+o.OOi1tu',*
+
.u',
=
*
30) +
(t)
0.1ui-0.r6) = oI)16(10) + 0'003{-2) + 0'r(-4xo'16)
{2}
-2-btt'1n
U'Bn-u'en=
+
ui. =
-e(uso-tA;)
-0.9?143
m/s,
and
:+
-!'ro -
eq(Z) +
l2-92oi^
*72) =
::
-
{3)
Jt + (-2X0-24) = -4'48 rn/s
= ola + (-lrr a'tn =u'r* +r.ri11 = .r'ro * (Sui" +30)(O.il;: 2'92u'ri+7'2
t)'8,, ='-tt t'z = 0-a(-25u!* ) =,-,oti
'n
P. = .n2rr':0i4 x'10"-'4 m,/S.
tu/.l
ei
:
14\
<
-0'4(4:+ 4'48)
.a!r=2-429 rzd'ls.
O for rod+plate
from Hb,: Hs. for rod+plate' Let m'o'i' about z-axis at
kg'*''
Ifl+[I9,. +m1(di +a{f --0.016+0.003+0.t(0-163 +0'38?)=0'036
2. -i (fig.Sf.rsc) is obtained
be
ro- P =
:+
lo,.t'o = I?r-z+ I?.'., + m1o1od1 Hb, = Ho.
+
0.03&ri = 0-016(10) + 0'003(-2) +o'l(-{XO'16) - o'l(-3X038)
=+
(b) 1- The FBD's are shown in Fig'Eil'l5d'
For rod+plate:
+
m1u1rd2
ca!
= 5.667 rad/s
e.:, = ao +.r2 k x OCz = l0 L x (0'2 i + 0'OSi) = -O'5 i + li ^1"
,i = orr='-0.5 m/s, u2, = 2 m/s. I9,' = 0'2(0'61 + o'l211tz = 0'00?5 k8l-"
71"
t
*:
:
:
f^,,
f.:
r_
L
L
L
L
L
LL
IJ
;'
fc tod*plate *a
For plate
2,
+
:+
tri :
+
:,.t1:
::,':l
'-l
=8ui' + 30
t::yi-rn2u!.(0'2) = *-'
(5)
-
m2o2'(02)
(6)
:+ rnlolrr + m2t/2n = ntlvto * rn2t'3o
p!. = -0.5u,1'
0.lo!o *o2l/,, = o.l,(-4} + 0.2(2) +
tL - 'S^^ - -0-4(u3o - urn)
p.'r: p,
(7)
- 0-13334) - (2.9,2ol' +7'21 ---0'4({+4'48}
2-066?of. - 2.92oi,: 3-9413
* ,i" = -o.oozo *7"
[2-066?(-0.5) - 2-92]u'1" - 3.9413
,4:0-{985 mfs,' . qrl'= 1'992'rad/s
(2-06674
usingeq(7)
,Eqs(?),(6) Yield
The 3 unkuorvns u!,d2.,Q Gig,E3-l5e) are determined using P', =
rod+plate, where .B is a point fixed in Sround and coincident, rvith c:-
2-
+
.:---
0-00?*.,1-o-2ui.(0.2) =0.00?5(10)-0.2(2X0'2)
5.3333uin
-0-6666? +
ollo : ?2o +{.r?(0-2) =2 + 10(0.2) = 4 rnfs,
e
+
:+
u
(8)
P" l'" = pr| H'8. - E2'
uc. = -b.i - 4j m/s, !c, = -0-5i+ 2!ni/s, {c"= 'i'i+ 4vi
(ui"
*, = *,+-lk x CzCr:4-i+ u1, i+ -lL x (-0-04i + 0.33j) = (u!, - 0'33r{)i4
u1r: -3 m/s, ul, -4 m/s, p2. = -0'5 m/s. tr1, = 2 m/s'
o'r"=4,-0.33o!, *: n -o-oa<.r!
-
tot
O-O{:ri)j
(e)
"
pL=?, +
:+
plv = ps +
l--_
IJ
]J
l-
L-.'i''
Ll)
0-1(4= -O-33&{} +O2aL,=0-l(-3}*02t-0'5)
itlTtlr* tIr=4, = tnrrrr; + n!3u2e
b.r("r, 0-04-i) + 0.2ui, = 0.1(-4) + 0.2(2)
+
u!r-zS36
L_--l ,
+
uj,..
=-
1.3331i
+ O-ir-i1f
= 0.01333&-.ri
rn2sc.
::
0'22',i)l
O;
(tl)
l
rad/s.
its 1nglrlar velocity temains unaltered- Ileuce ot' = o9" 7 O '4(10) -'A;m/s'
relatiirn' where
The 2 unknowns ,1,{to are determined ftom H'9, = HD, and the coellicient of restitution
D is a point fixed ia grouid ard coiocident uith ,{. .
(c)
Since the platc is massive
Usine eqs(3),(4)
13
r,',".
0.0071,i + 0-00iu'l + 0.r1'g 0a,-t'02666?t^r!) - 0'33(-l-3333'0-33(-3)l
= 0.0075(10) + 0.003(-2) + 0'lt-0'04(-4) -
As in eq(l), forrod
Lj-i :
Lj"l ;
+
*, : (-1-3333 - O-22oili - 0.02666?q'i j
I?: riL + tf,',ri k + czc r x *&,1 = A:.; k + tr,? " t f
]$
+
Hb.= Ee, *
Eqs(e)-([l)
t'[J:-
mrui-+roa{, --riirur: *lztzttzt
-
+
IJ
;
Eb.
::
:+
:+
: ED,
E'p.: Ee.
1
as in eq(f), for rod
Using eqs(3)da) and
l{
IJ
L-
,
-..1.i.:!
':
For rod+plate
L
I-
f
platr-2.t'o= po
CiaetAmined uiing f6.-= Eo. Yrod l. rr;' = HD. fot
with :{'
co'incident
groirnd
of
point
*"fiA"i:&o=t6otloo'retat1cn, whlie D'.o afixed
ate4ualruow*ri:*li}fi
1
IIb. -
If
o,
+
= 8ui' *
urn)
r.r',
*1 "
_l*,:]1..= -0.4(us, ' +- .i - { -.'(2.92ut;'+?.2) = -0.4(4 +4-48)
Eq(12)
=+
r./.=
30-526 rad/s
,..
-[g
(12)
30
u'1,
= 0.06575 m.fs
<
to a thin disc
of
mass
lringed
B1-mple 3.16 A'iqd. Q4 pf y.nass 0-6 kg is hinged aL O t:
ptates at 40 rad/s' the
sijiona*,.lhldisc
hetd
is
rod
position,
the
given
the
In
.I.t'" hinge at is frictionless
0-f kg at / (Fig.E3.16a).
asupport and
.'j
'A
spring is unstretcbJ'
spring has a compression of 40 mnr and the rotational
Find (a)
released'
is
rod
The
the rod from hinge at oon
N-m
of
0.1
torque
frictional
constant
is
there
and
velocity of the
from o just after ,.1"".., (b) the.angular
the angular acceleration o[the rod and the reactio.
t"'
from o:'n:i_':::::i,:.t:::::o;,'$":to
rod, the angular accelerarion of rhe rod and the reaction
\'
R, ioi,,_
lot o{r=4d
h9v/-s
-l:Tl
Ad#,rF:t++T_-{f
e,
Y
,ph"..ffi
,,,{T>€i, 33i
;!'rtn
e+l
;(,{=o*6,
(b)
tbl
V6=
V^
o'Bwr
= o.BCr
and the disc o.l"Lo*uo in
Solution The FBD's of the roi+disc
;ffiH+, :T
:-q(.-N
,,;
/ t'tt
ttll
- T:t-"3, t', :'
Fig'83-l6b'
Tt Let
,1,0;1,It,"ii;,i,
the
rn?
: ;, *r, ,*. = 1:1.(u1),/2 =
i;i;;i";;i;l ;.;l;,,;
L')r'.l,2'i'o'tttug"1"*f poriitionofthecenlreofnrass'4ofdisc2'
a,,gurar
0.0005 kg
t'elocities'oftherodandthediscbe
t'
Ma=!
+
t-'2=0 +
I,o.u'='1{a-=0 *
(li
u3=co8scant-40rad/s
the ntoutettt'
nr and g^ - 0-&'rt go' Fo: the rod+disc'
The e-xtension of the spring is 0 : 0.2 siu 0 - 0.O4
L" [or tlre
ai o, kiDetic *"'g'-1'and potential elrerg}'
of mornentu m Ho,abotrt the fi-xed axis of rotation
general position 0, are given b1'
EoT
:
r!,,,t1+ gf;u2*
0.8nr2ua)
0'o2
O'OSe'rf + 0'4
+(0.128)ri + lto.r(0.s,.'r)" + oioom(o1:1 =
- 0-128*rr + 0-0005(40) + o-sto'l)(o'o.11":
: l€,4 + l!m2vi+ !41-;1 =
o':11'
+
(2)
(3)
y - lkraz+|r(o-zsina-0.041?-n.r9(0.4sin0)-*zg(0.8sin0) =92+50i0-2sin0-0'0{)?-0'32ssin0(a}
ro' = No, gields ri1 ' ur = 0 arrd rRr ', R:
(a) The FBD of rod+disc is shorvrr in Fig.E3.l6c. using "q1Z;'
are obtairred using f - I*iec- and kinematics'
I{6'=!t,fg,+0.192&t=_0.1+{(0.2)+0.6g(0.a}+o.1g(0.8}:}4.,1=20$0rad/s?(5)
t6! m/sz'
* = -ul1a.a)i+6r(0-a)i= Sitnfs?, ee = -"'i1o't)i+i1(0'8)i=
+ Eri+ Eri+ ('l+0'6g+0'1e)!= o'6(8i)+0'l{l6j}
F= Imicc'
+ ltr = 0, '.R2 = -4'467 N'
yields:'r1'
configrrra*ion 2 f9r 0 =3}9'='irf6
(b) Work-enerpr relation from the configuration f ,for.0 ] 0-P
putl on
The
and kinernatics'
,"i"g l:imtg6,
Using.q(2), H6. =':Ms,yields dr1. &,Roare obtai*ed
dre spring for 0 = 30'is 100(0'2sin30o - 0'04) = 6 N'
+ Vr) - Wn"r-2 +
50(0'2sin 30"
t(o-09&/? + 0.4) + {(r/o)'? *
(Tz+vz)
*
r.rr
-("r
-
0'04)2
- 032esin30"ll -
(0.{} + 50(-0'04)?l =
:3.451 rad/s
frs.
Ms,
-O-192ir =
-0.1-
+
100(0.2sin30" :0.04X02cos30o) *0'69(0'4cos3f )+0'1g(0'8cas30o)
+ &1:2.772 rad/sz
a = -uzr(O.a)e. +6r(0-41e4=
-4'?638g.
*
1"10889 m/s?'
?1
-0'lr/6
(6)
'2(z.16) (7)
(8)
:-
C
yo_
',,,.,...,..
='', ' " ':
'" :--; r'-;
:
t'
--' - j05(:4.?638)+o.l(-g.ozis)
E=Err,.g"r. :+
- ^:
4:'L45N
-'(9)
fr'+
&+(0.6g+0-1g-6)sin30o
:
* (10)
s.
+
6) cos30" = OO1'''0"1 O'1(2J1?6)
O'ls'(0-6g
*
+
R+
:.0't*
:
er-*
general configurati'on' ri1 can also
t"
coovealent oipression
with
system
d.oJ..1,"
one
{f
ia
f.
For this
.
o
1
-)
@A=-r.r?CI.g)e''+€i(o-s)+-+-9'52?5p;+2,21?6qrtils2'
-
1:
"
0.096(2trrrirr)+[2'+50x2(0'2sin00'0a)(0'2cos0]-0329cos0\'t1=-0'1tr1
(ll)
cos0] = -0'r
+ so x 2(0'2sin0 - o'oaX0'21T0) - 0'32s
+
0-096(2;t)
[zo
+
eq(7) obtained earlier from
Equation (11) for f, = 30o -is: identicat to
o1factor
cornmoD
write the quations of
on caocelling
for rod+disc' $'e catl also
*"*"oilequation
th.
writing
of
lto..,Instead
t!.
ilo, =
to obtain
e[minat,e the r:11,ioFs 8s"fi4
t.he disc separatery
and
rod
tlre
of
",,a
motions
is shown in Fig-El}'t6d- N'
(c) The FBD of part og of the rod having miss rns = 03 kg
.rjng
i
f' itf, are obtained
aB:
3,ri(o-21 :+ N - s-osg'N
8'+JV-+(0'3g-6)sih30o
+
. f." =m3aDi
Fe=m3aso + E +S+(03g-6)cos$!r=::t(o'l ^=.,',i:=2'814N
equutions of motion
of
/t{p,=I!,61:+ilf-0.1-2('/6}+(03g-6)(0.2cos30.}+S10.+1,=.tcg(o.al./31-t
= 0'5954 N-m
N,F'Mcatral.obeobtainedusiagequatiotrsofrrrotiorrof(partBr{)+uisc.
+
,1I
3-1? A 4-bar rinkase -*"Y ': ::::::1 ::i" ::T:i:T::'i.:':;:::':J
r"Hji'x'I'l
the glr'en instant
'::.t_i:j", i;;:iT:""'.":J;
; i;;;.;;.ir.). Find rhcsupporr reacLions Iat
Exampre
i
1)
B
,in
Solutiorr \\t
first exPres
&r
t'tz,t">3,A2'@3
-{Y
?,1t Bia*-
9'
the connectcd tinkaggs'
iu ternrs of arl'&l usiug kinenralics o[
Pointso.r{onbodyl;.r{,B,Coabody2andD,Eonbody3havetlresanreplaneofnrotioa.Heuce
rlB=0'4jnr' 9B= 1-2!m
!r3=r,3k' !4=!'p=9''
m' OA: tn
95 =t"'3k, go=uD=-ql. -t4e =0'2j
gz:-rL,
ir =.rrL,
0-4j -'''t'(t'2)i
g8 = ee +;'i x AB = os(DB)! +
a^ =u1(OA)j-- 2 * lj = 2j'
"*.1:':.:
t1&-+ a"! x'48 = -n{(pB)i+a{DB)i
,^ = -u!(o.A)i+6r(oA)i: -4!+ alj-, ea = etj
+ -4i+ar j +ti2 k x 0.4j = -1'666??(l'2)i+'3(l'2)
+!:-4_o.4uz=_3-3333'j1w1=|.2t;4+til2=-1.666?rad/sz.til3_o.&83&.lr(2)
(3)
sc = sA-uiec+-zkx Ac= -4i+ti'tj+ti'zLx0-2j=-3'6667i+d'rj
9 (3 x 3)
g,r=2krad/s, gta:-2k,
The 9 unkuowns 8r'
The FBD,s of 3 links are shown io Fig-E3.17b.
il
;l"TI"" il::r,"",' "f'
,J
I
I
I
I
I
U
U
_)
24
-
a1e
t
i
t
I
:_
;
I
determined'using
+ 0-3333&;"
[1(1)2/3]ar + & = -te',ss
R6=-10.886-o.*ilt
=l|.2(L.2)2|3]ri3.:+
o.$s+ F4 =
link3: Ittp,=IDt,t3 + -6_r.2g(0.6)-1.2R6
& - Et -O'1,
Fy - m2aai +
link 2 :
"''R5'61
1
(4)
(5)
= 0'4cir
8o
t
... :
:
- o.&r + , 6t = 3-ZS1 Bd/s2
tis = 3.151 rzdfez, fufi5 = -12.40 N
E€(2),(4),(5) :+,
-17-83 N,
:+
:+
Bs +.8" : Q-A4444
Iink 2 1 Mc, = 19.&,
-0.2(r?" t ng) = ll.4(0.4r2 /121(-1-6667)
+
-10.886 +o..E.,
:
(-19.095 + 0.3333ur)
-0.4c
.F, = n2aca +
Es(8) and (9)
liukl:
+
-&-
0.4(-3.6667)
8s :0-7555 N
= -0-7111 N,
l?r * Es = t[-&,i(o-S)] = -2?(0.5)
Rz * R<- s = l[.ir(0.5)] = 3-781(0.5)
=+
Iiak 3 --
r.
= m1agr" 1
= tn3acai +
-
Fy
= rn3aca- :+
ft: J?5
=-35
-
Rs
=
1.2[-&13(0.5]]
L-2g
=
1-2[d,3(0.6)]
=
:
f
(7)
(8)
:-
(e)
= -l-4667
Es
Fr=mracrz
Fy
R5
(6)
r-2[-1-666??(0-6)J
-2{3-l5l(0-6)J
-
{'i
(10)
:+ -.Er = -2-756 N
+ E2 = 29-53 N
:+ l?r - -2.711 N
+ Es: t-641 N
I
-r
'\_
g1:mFle 3.18
Each of the trvo rvheels in ttre mechanism of Fig.E3.16a has mass m1 and a-dal radius of
gyration h- Each link OB has mass rn2 and is modelled as a thin rod of lenglh f,. The collar of mass ,n'3
slides on the fixed vertical shaft rvith i constant friction force .F- The spring has stiffness t and is contacted
by the collars rvhen the links rcach the horizon[al position- The s.vstem is released from rest at position
0 : 01- Assume that the friction is suflicient, to prevent the rvheels from slipping. Find (a) the acceleration
of the tolt". the.instant of release, (b) the velocity arrd acceteration of the collar *'tren d - d2 > 0, (c)
"t
the r-elocity and acceleration of the collar rvhen 0 = 0 and (d) The maximum compression of the spring. (e)
\l:ork ouL parts a to d if the rollers are smoolll.
T.
:
._
:'._r
l
._q
,s
b
J\
:-\
\a/
i-_j-
v
_i9
L
:.fl+r---"
>
:
l*r
dotqm
(di
it
tbi
Solution
The work<nergy pdnciple is conveuient to use for the conrpletc s]'stem of I bars. 2 rvheels and a
collar, since this s1'stem has only one degree of freedom and the expressions of.T,Y and tft6 can be s,ritteu
conveniently for the general position of the system.
B moves verticatly down and O moves horizontaily. Eence the inst,antaaeous centre of rotatircn of link OB
is at the point of interseciion 12 of [he norruat to uo - -to t e O and the normal to og = -u j at B- llence
the angular velocity r.r2 of link 2 aod os,uo can be related to u using IzB -- LcqO,IzC = Ll2, I-tO = f,sin 0:
f-ul = ol(I2B)
':
__rf
:i
-j
,l
:
a
.!_
L
os
.
trr=uo/f =utatilr. ?and'[/ are expressed iU ternrs:'of,trand 0:
7.- lrm3tf +2flmpf;+ ]msefrolJ +2{lm2y1+ +(m"Lzl0)tl,3l
(3)
= i[*. * 2rn1(l + k]o/':')tarr2 0 +lm2*c2 elvz
(41'
V = tn3gLsir.fl +2nzs( |f sinOl + +&42 =(rr.1--l- nr3)glsin e * l}d,2
the t+rm |ta2 Ueing included only if thespring compresses by A- Wac= -Fu. Work-energr relation is
."_
,a
1
F-
.,|
Body 1 rolls withsut slip =+
applied in iategrated fiorm to get u and in rate fiorm to
t+'t = Wnc
:+
Tt
ri in the configuration of iuterest; ,Using eqs(l)-(a)
ii
't
/(I2Ol : vc/{IzCl, :+
lal=o*.€{L.. ry = }usec0, uo = uean0 (l}
(21
$:-tsec0lL., .'
Ya = Lsift?i {;l,:r' ii=,Lce00..-.:-a . +
-
_.,=r
lF
,i
;'-r
-i._E
' r\s-{l
i<
i !!;
r--
:!
;:
_<
r
.<
',' E
[-"+2m1(1 +k|lrzltzrnze+!rn2seczrJlon+llhrnr(l+t3/r2)randsec2d+$m2secz
0tart0lv20
*(m2 +m3)grcos 06
,,81
= -Fo
(5)
,---
-'iii6.ri=;1qt
Substiiuting 0 from eq
2
"*a
ll'!'
."*.ayl,i iyyu,".F** ti1:," r:""
iiigl*lo"
[m3+2mr(1+r3t")t"n'zai3"Iirf;.itt
: ::, :
^',,,,,.,
,s<;c,'*$mziec'dt"n'lo2.*otL"'.
(a)Attheins0aniofrelease,0=0larrdu=C.I{eoceeq(6}yields^e.r,.
0t1
gr
rl / [''e + 2*'(.L+ *3/t') ttf + ]m3sei
1r = (ra2 +
'o3)9
0 :82 and using
zt 0 =01 to configurat'iou 2 aL
c< >nfiguration !
from
relaiion
Applying rvork-energy
(b)
eqs(3).(a):
1zi + va)
-
(]I'r
+
V1l
=
Wsa'-' *
-sin82)
orlo'+(mz*mslgL('sin92-singr) - -Ff'(sin-gi
k,k.rt,"..'2e.+$*r*";
(7}
0'}l'/'
+
it*r+2rnr(t+
{ms+2tnr(1+e3lf )tt"t0e }"'2sec?
+.
o
=12{(*z*
rns)9
- F}L(sin,, -"r"ili I
03 for d and u fromeq(7)'
from eq(6) oir substitucing
obtained
then
is
The acceleration'u
i
*i:"tttt*
o: ,Pt(*r'imr)g
(.) For 02 =
n- [(mz+'ta)s - F]/(ttt3* 5tn2;
Fop'6,- 0, eq(6) vields
rrheels have zero vltocitl'
Iz-is at () and hence tlre
configuratioa'
this
ia
since
*-hen the
m1 does not appear in eq(8)
3 of the syst'em (Fi€1'E3'f8e)
configuration
in
;;tt*
of
at0"ction
co.figurat'ion I and Lhe
{d} t et 5 be the t-*i*t'*
."Liioo u.t-*gr;rhe initial
verocitn
4ppryi*g';;.;;;.gy
zero
wrrore slstem.has
yields
.."fi*.ran 3 and using eqs(3) and (4)
(T3+Ve)-(fi+%)-Wnct-s
0l= -F{Lsindt +6)
rneo6
][62J-[0+(m:+ma]glsin
(e)
*ti;t::'ff
[0-2ra,2sf6l2)+
+
rlLsin,1 = 0
(*' +
(}, eqfT)
1;;
*
(E)
j'Fl'f,ntn"'/t*
:
Yields
-
-
"r3)e
-
'16
of the spring'
is the nraximum de0ection
;1"'f; :";;;:1",":':#itH*"il'H.:#':'3;*1""..x.*:::,T;*:ll:*""=""t'1"
l-€', tJ1 = ur' rrErrvv
*'flo;l'
of mass is zero,
enrain unaltered
arrd (9) r'.fi:"::L:::
external forces about its centre
to o1' Inciderrtally eqs(S)
at'":.:'.:"::,ti::,Xl:l:;ir11::i;;
"*ta.r."l
since tlr-rs"t".'n
terms involving t6 sec to zero.
pcitions
because u.r1 = 0 in these ttuo
"ti*t
t*"" fot it'" case of rolling rvithout slip'
^ ffi n *-------I
r"'
;;:-,;;;";'';".'::'
]iilfffi
"i"T:;':i;;.';.1*' ;n"-*o*'o'k
:Y'::ll'":lili;'Jllii
-{[-:',-- lFil
tle:::::1.:l::: lffie,'*a'&[
-P
lfQ{-i'"
;"";;;;*i'ir'"i' cen*es'r
t-)
e
i*:' Tn',*:::-:t *Y"'l:::
svstem 1L3
'fP
'cthe --ir orrorarion is 16rnrfr. The
ffi:,ffiJffi:;:;""tsrops
\o)
\o)
!.r^.1-<
:.
,. ,. - -c,r.^
brocris*-v--Hq^i
-.tr' rt= v-*Q-,
s1.s2 preventing sudinsor rhe
inertia or
f::[[fJ:ii. i,i;;r.
tt*ou"a' 6)-ttt io =-1'7'n1t"
ind angulat acceleratioir of Utt CD' velocity . :'tl
The stops $1,s2 it€ simultaneousl'
find angular velociry
-
CD' the ttnsion in the clle-an:d,
-l-t'd
-fHL*l^
otthe block"a when
so- thal th1,1a"timu1
r'rs
be
should
What
(b)
|
the axis of rotation'
uro so ".*'fr ;t
l*01*"*"*'oi uu"r A is rs? (c) What should'be
;;;
.,{
rq? Ilst9f
i|]fil Jieiust b""o-o slack when r{ has moved.out by
bar
and acceleration of .A relative to
il:;ffi;;;;-cD
TJq; ' l'u+.'
.l::'" '- --9 ;T'?-t'")l"a
^r fi
it: "t'",:::
R,C
ibl
8a
^
1
Hl:
':**,': "1:::.3.":::::fJ"T: ;::i':"ti;::'":
T;'n:::i*:k;*;::;T:',ri::Tri!;
:,::?":JI"T:1Ti;i*il.ff
":i'ill.ff Hi:.,T*f
sol.tiorr
The momeirlofdheextern'
-tl'-'
:::::.JH"fi
ff;i,ff
ffi :*1;tt;f"';;i::"'",:X::"*
leb)
f SmT#;'TlllT:T;T;:"J#'tr:{;"%;;::,iil:-"P'Qrravemoveddown
f"i:$;i ilT:lffi:::;;;;;;""i1' 'l';|2=: i|2(rig
n{52ti +t}y" =
E3
--
Ho. =16rnrfro +2tntlc't+
m(52r1+2rz),'t =
rn(52rfr a 2rfr)r"s
^ 1r.
. a -.-.-2r-.-
2(2m)(3rs\2t't
coostant
(r)
54mr3';o
=
(2)
. ,--.^:..-rl
=+ Ho,=m(52rfi+2r21-+4rnrir^r=0
(ii)')i + 2[]r(r -'o)'i -4mg$.-ro)12
2
+ ;'11-*'41(zm)fl&ardt +
(3)
+21,1'ml$'t')'
T +v= !(romrfr)c,
2m9(r- ''o)
2-54ry(r'
+
1'5mi?
'ol2l'a
- rn(26rfr + ,'),''f
radial velocit'y of A be ur'
for r'= 216 be srr'g,r and
acceleration
Ho' = 0 [eq(2l'
(a) r-et the angular velocitr--and -"Y
y =*n.o.]'i 1*i"t "*''.
;ili;,
f,lo,
from
=consramr
arl,11,61 are obtained
_.0
(4]
0'9r';E - 0'9(g/re)ti'
54mrfrc^rs
:- , :.:=
Eq(l) ? m[s2rfr +2(2ro)i]-r =
i =:n(26rfr+rl)t.,fr
Usiogeq(3),f+,Il=constant+,,4zo,fr+1z,.o1?1..,f+r:6rrruf+2.5,agnfr|tg-2ttt.gti
*
ffi'lifi'ir-4nr(2r.s)u1&,1 =0
l*r, *
p for-trris position "ru "r.or.,r.
The FBD,s of btocks ,{ a,,d.
btock P ;
block -.t :
Using
'
{G}
:a' = -8,1ro1l60rs = -0;r'*53(c/ro)
block P
of
in Fig.H}-lgb- The equation 'rotio. 'of
2T *2rrrg
:+
+
F,
=2*up'
{8}
T -2futttg = 'r{i-r'i(2rs}j
ri'l
mG
f" =
F6-m,,r$+z;61+At1=rn(2r6i'1-l2r;;-1)(9)
(tO)
1"-rng"j0
+
F-:mi
rz?=0Sirng' Atl=f i39mg' N2:1tr,!'
i=0'089'
))vields
eqs(?)-(lt
of
eqs(4)-(6), thesolution
i 8-O8g ia tlrc
arerrl = L211(grs)U3 a*d =
CD
tod
to
relati*e
.,{
blocli
and acceleraiion of
The veloci*
outward radial direction'
*!.1"':: ,*:y:::.:::"::1,.f,:
ilY.Tl'ilH"ffi,".
eq(3)]'
::;[j, :."."**i*tl)l a'd r +v =coustart [usins
Eq(l) + *L52rt+2(2toi21.,n :!4tnilato
using.eq(3)
, T* v=
(.)
e
con-staru.i;:
''n
nor which
yietds
t^r6
=
0
J2,t s af€ obtainCd
a
..+,,, ,n[2ffi + (2r:d;?Ic3 +i.5rIg1fr/ro -
Irom eq(11) in eq(12)
i
(11)
2m9re = nr( 2orfr
+.ipl
(12)
= 0'4303(cl"l'1" -
r- FBD's of blocks
the
ror
r*Lu3be
19'J:'::LH:.1"ff?r"."=;""."'t ["q(r)],
l:'Y""
",,go1".,"locG
* in Fig'E3"19b with ? =!'.4r3'are'r al
and P are the
""*"
ra\for block A and F' ='Zmap" for block P: '
Substituting
A
t?)
=zrn(-ilzl
=
ca2
:='*
rhe
-.it -
Eq(l) =+
(13)
m[52r[ + 2(2ro)2]t'a = S4mrfrars
; F, =2tnop,
btock/ : F.=*(i-tC2)
block P
o3 = 0'9rs
+
(14)
:+ i =2g
+ -2mg =2n{-il2l
(15)
+ -2-5mg=m[i-r'f(2rs)] * uo=l'666?(9/r3]t/z
Exalople3.20Acylinderofmassraisstation..y.,ll:.a.brllkofnrassmllringedtoafixedsupportai
is 0'2 and for contact of'the
of brock rvirrr trre cylinder
o
(Fig.E3.20a). The coefficient of friction
i.,
"".r*a
83
-.
}..
tr
U
u
klj
rJ
) ..\:'a :tr:
'. .! f'-' r '
L
tJ
L
lJ
L
1_IJ
l.j
l--tJ
u
IJ
lJ
t-"
u
L
tl-.
u[.,
[-=,,
[j]
1--
L.
.t:
-il
'i
_j.ri
crylinder.rrith tlre gtglit+5p-+ -A constaat forcc P i1.1qnti9{i"?:.th9 c1!inf1 at a cggstant height of L2R
-'ra1"9
-6f
tf,i'cyUnaer fot trvo cases: 1- ra1 : 3m,
of P for impendift'uiiiii6,ii
aboye the gound. (a), Find tUi
2- tal: 9.6m. (b) For mr = 9.6m, find the instantineoG'iititei"tio"rof C and the angulat acceleration
of tbe cylinder if the applied forcc P is greater than the .v{u9, gbtained in part a- (c) If P - 2-3rng for
at bottr
rrl1 :9-6mg, 6nd thc velocit| and position of C w[en ,tip
S--.-^
;+-ti +r
r ,-
lj
IJ
L-
/- llrl
ii
.":-.::'-
'r
"i*i"
Y8'3 .se ff'8
ilffii-t.,,t'o#1
',;ffh:
3.(ffi1,,T-{ h)x tt'.(.ril}*
Nr
-*e$
f"1-,?,
"PPE
-.1
^
ur
roltiuf s'ithout:slip at either -A or B- The FBD's
(a) The cylinder o,iit-tlve ir"piiaiiig;niriiJn
"f
of the block and the cylinder are shorvn in Fig.E3-20b. The $ un[llqrvns P,/{r,Fr,N2,f2 are deterrnined'
usiag M6, = 0 for block, 3 equations of equilibrium for the cylinder and the condition of slip at B or A.
Sottrtion
Mo. = rVr($.66)'- m1g(O.it) =
:+
O
/r! = 5m1gl,6
(1)
ltr6asslracimpendingtollingrvithoutslipatSandwitlrslipat,riwithfr=O:2/Vroppositetothe
directiron of impending slip (Fig.E3.20c).
Mg.=0
.Fr=$
.+
&=o
1-
F.or m1
-
-
:+
P-A.ZNL-Fz-o
:+
A'2-rng-/Vr=0
3m, eqs(1)-(E) yield rVr
whether the assumption is correct:
#zNz:0-1(3.5nr9)
-
+
2-5ng, P = 0.8333nrg,
- 0.35m9 +
P=NJ3
=i
0-2N1(2.e)-L.2RP=0
f
f? =
F2
F2l = O.3333m9
S
0.1333Nt
=
0-3333mg,
(2)
(3)
At = 3'Irng' \\'e clecl'
PzNz.
Eeoce the assumption is correct- =+ lmpending rolling without slip at B occurs for P = 0-8333mg- [t should
be noted. that slip does npt necessatily occur at [he surface where the coefficient of friction is smaller2- Fot m1 = 9.6m, eqs(l)-(a) yietd N1 - 8rn9, P = 2.667mg, Fz i 1.06?m9, JtI2 = 9rn9' \1'e clcct
whether the assumption is correct
pzNi:0.1(9rn9) =
0.9m9
:+
lFrl=
1.067rn9
f.PzNz
Eence the assumption is not correct. :+ Assume that the impending moiion is rolling witt-igut slip at
.A and
wit.hslipat8withFz-0.1/Vzoppositetotheditectionofimpendingslip(Fig.E:l.2M).
fr=O
M* =S
.. Fr=O
:+
_, Nr-*g:-.9*C:.Q
:+
0.84P-0.lJVr(28) =0,.,.' :
:r::' 'ii'.ri
P:0.1rV2 -'Fr = 0
=+
,
. :.
Ifih cfcd:whether:the assumption is corect:
PrNr=0.2(8m9)-l.6rn9
Eeoe the assumption is correct.
+
:
=)
i::
:+
*
'r:+
JVz
P
'
f1
=
=225mg
!'t'35mg
:::.:
.. . -
|Frl=.1.35mg S PrJVr
Impending rolling without slip qt .A occurs for P
8!+,
,
9m9
=
2-25m9-
:
(5)
(6)
(?)
'I
';'.
")
.t.
.
:,
:
.,1'.
:
i
I
,]-.
'i t'
'
'
.,
,,'.,t,'
^.-- .,'
rolls'---lvrthout slip at A with ri =
ft)
\-, For P > 225mg,'tf,E cflintter
i, , R o are oLtained using 3 gqua'tio9L:. of motion of the cylinderl
,
F,=0 *
,A'r':tb:8mg=0
t,'l
(8)
.t-
"=Y^^
Ft=P-0'9rr.g-ma
(9)
*
o=t)R=(1-6^P- 3'6ng)l3n (1O)
F.rE-O'1r1Iz R-O2PR= lrrr&z(alRl +
F1 = 0-3rng +O'4667P
:. o or*2-7861119'
O'2(S'ng) i'e''7f 2'25ng < P S
is valid if lprl ( ,.rgr, ,i.e-. if O.3rng + 0'4667P S
F,=ma *
t{6.: I!,,) +
Eqs(9) and (10) +
This soluiion
*
,
alL(Fig-E3.20e)' The 3 unliaovns
P-O'1JV?-F1 =ntz '
(Fig'E3Jof)' Ttre 3 unknos'ns
,Fof P >2-Ts6mg,ttre cylinderrolls with slip at A and B
A'2'6'c
cl4inder:
obtained using 3 equations of motion of tlrc
+
Fr=O
F, = qa +
!{ct = I!,'b +
are
(l t)
=+ Atz-9mg
(r2)
=+ o=(P -2'5mg)/n
* ' it= (1'4mgE -}'4P)lnR" {13}
N3-nr9-8nry=9
P - 0'1N? - 1-6n:9
t'6m9.R-O-lJt!:R- OSPR=
'"'"t''--
(c)Since2.25mg<P=2.3m9.<2.?86ryc'tl,uirri+ialrnotionisrolliugrvithoutsliparS{EigrE3.2og}.
For block
?
I{o,-:Y1r-9-6rnd0'50] = 0
:
geuerat position
The equations of motioa <[ the c-5rlinder for
+
&:o
0-f iVt
Tlrus slip s.arts at both contac* fior z
,--,!:=
+u?
+
(t-0e333
-
=
o-722gb.
( 1-{}
= 4.8m90/r
':
1\':
= (1*
4.80/rlm9
*ma
=+ Fr =(2-2-}-4ilblfimg
(1-09333 - 0-64612)s
g't = iilR"(al Rl a, a =uR=
0,160/zlrng < 0'2(4'8rn96/r) i'e''
\tlocity
u1
o[c
(r5)
{16)
(1?)
if z ( a'7n9b'
iu rh'rs position is obtained using eq(l?)"
0:41')'g d'
l:,'?"e333 I+
u1 = 0'1738(sb)'tt'
-0'6461n(0-7229/A'(|;ls
ossl)g *
= [1-09333(0'72296 -0-66)
1
- P, = ina
F"=ma :+
t{s, = f,a * & B: o-lNzE - 0-2R(2'3rn
.Fi = (1'106? * 0'16b/c)rne
Eqs(16) and (i?) :+
(l:106? *
Tlris solurion is valid if lfllf ar1^r1, i-e-, if
+
r yield
At2-rng-I[s:O
2.3tng
:\
fo""'d'=
the rvheels is rn and tlte centrc
] massof aself-propelled vehicle (Fig.E3.2la) irrctuding
radius of g1'ratio. of t1 ' l-3 '
wheel assemblies have uIESS rr1 , rn2 and a-tal
of mass is at C. The rear and front
driven car'
ttrs,rvheels:.arerp, ild pt' (1) A rear'-rvheel
The static and dynamic coefficients of friction 1si
u is f,he
torque M1 ' The rvind resistauce is cu2:'where
starting from rest, m-oves without slip uader a drivi.g
the
t a*d after travelling distance z' (b) Discuss
velocity of the car- Find its acceleration and speed .i ri*"
marcimum
the
(c)
Find
slip of a rear-rvlteel driveo car'
basic method of analysrs' for motion with possible
achie=e
and the driving corque required to
possible acceleration for a rear-wheel dri,ven car without slipping
driven car x'ithour
the maximum possible acceleration for a fiour'rvheel
it- Neglect rvind resistance. (d) Find
'Mr.Mzrequired on [he rear and front wbeels to-ltrie;'e it' Neglect wind
slipping and the driving torques
driving
*" car and the angular accelerat'ious of its u''eels if the
resistance. (e) Find ;"";":i;J;;
"i
wind resistance'
torques are Mt,l{z and the wheels slip' Neglect
8s
,i
:
:
J
JJ
J
JJ
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
iJ
J
J
J
J
J
J
J
J
.+v
re$
-vrQ
t-1 T*. ql
lNz
i,tq*,
F,
rPe-
.-.-o
of rouitg uirhort sfip. Irence *ork-eaergy
'o' ,.r*t".?0"."". esuation'
ir" mot or the car' rront *'heers
gou"oios
rorce
.TlT;?. ':? ,]*;:;":"il:tlt:ffJt'-'t'"* oo
rriction force
rt.^ r*ant ..hels and the friction
iaput torque to- rrre rront rvheers and-the
f::::til*t::ffi;'J;ffi;;"fr:"i"*
of t5e car'
In ttre FBD
che desir-ed angular acceleration'
on these is in the backward directio' * u.o''i"
provide forrvard accereration- The rsork
trrc forward direction to
in
is
wheers
the
rear
on
force
frictiouar
the
the verocities of poirrcs o[
zero (such as frictiou fiorces since
citrrer
is
'
forces
internar
and
external
doae by the
and the driving
tU" wtlf.aone by the rvind fotc* ctt2
Lri
..
(such
zero
pair-rvise
contact are zero) or
a,.d angular
".""0i
of the car' and the aogular velocitl'
torque M1 onthe rear whee\. I*t, o,t t,itu" 'r'" """ulrati'on
car and rate of
& = alR- The kinetic energy of the'
*
,i*
.
implies
slip
No
=
wheels.
its
of
acceleration
given by
by all internal and external forcqs are
;;;a;.
+ tnr2&!(u/R)?]
+
1 = |(m - m1 - r",r)r'+ []r,1u1 + !nr'*!(ola]"t []'zau?
(u
m2killttllu2
(mr&?
+
+
''
(2)
. = if*
g{ - Mp+cuz(-u) =(M1lR-c'z)o
qnd' cancetrut.
cancelling rcomr1roa factor n:
t{ QII.'
T= tiz'
in ratc foun ?=
-- -r- --..-'d
'
ahe acceletation is obtained using worh-cner'r.retatioi
(Mtf R- cu?)u
[* * (*.ei + 'r'r3]/E2l oit =
'a=(M1lR-"'?)1@*(*tt-? +n+ki)l?2l
:+
tb
obtain u(t) [o. coEstant Mr' iategratc
;;;;4ri
lr"
ro.'"o*tant Mr.integralc
A#=ry
.- I
t
lm
+(-"t?
+
(3)
wariables'
the
cq(3) w.r.t- t by writiag a = dvldlaad separating
auyanaad sepaqatini t&e tariables.
cq(3) rv'r'L z bv wtiting oI - u
"r?ti)/E'zl'
l r' ffidv
=t
/ [m + (m1!]
+ m,Li) I
R21'
.:
obtained usiqS'the'work'e'aergy
For eoastaot Ml, o(t)caq also be directly
being Mt(xlB}.
work done by Mrin-distance e (rotatioa z/E)
.,
tz-Tt=Wt-z .3 |t',"+{1rri
For.Mr(t), o(t)
is obtatuied by intcgrating eq(4)
,=
[/
Ho, -(m1tlo1
*
ir1uh1)
=MplB'
rv"r't' timc
u,t t] dtl I
(b) Tte FBD's are showu in'Fig'Eii.2lc'
+r'f.2k1.llR2lu2
relition in'inteigated form' the
RVn+ (m1ef + $"k11I Rzl'
coincident with A.
Let O be a futed point of the grcund
f (mzti rz + *rYlrr] +#,1X13 =
86
mplr + mrti w1
*
mzkloz-
(5)
lr
,ll
,<-
-,1
.t
.
1'q'
be deterrematic viriables a'61'&2' can
O:'
foidei;\il1;IViifr,Pz''&
8
unkaowns,
11
The
"iU.tbodn front and rear wheel assemblies of car)
'Rz'fu'\
i"di* (main
the 4
rEised usi.g I (3 x 3) eqiraiions of motioa "i'g
we shali writc ? equations iavolving-ouly
and 2 conditions of slip or uo slip tt e "rra i.-g;;et*,
For the whole
For rear
- Ho,
Fc=o
car:
rrheels:
nrgb2-
Mu-
. Fr=ma
Msr. = $,'itv
euzd=lrtoh+m*lua+mzkZ6'
'vl6me-ilr-Nz-t
+
p,
_._
1--
_
1:l
-
trr
- &R = m1klbl
.I]
ContactcoaditionatB:ifnoslipthenul:olI-'ifslipthenFl=peNv(11)
or if stip thin
if no slip then u2 = a! R'
Contact coudition at z{:
For tLe case of impending slip
.,..<
::i
(8)
&- F2-ctt2=ttto
Mt
<
i,,)..
F2
= pvN2-
-{
\
(12}
assumption is made regarding condition
replaces p1 iu eqs(1I) a,nd (12)' An
ofsliporuoslip.at"orit,i.t"landB--/\rr,Nz,fr,Fsaree.teyledinterrns'ofo,6l,b2usingsuccessively
equations of (11) andldTz) La 5'ield
-
<r
3
appro'lxiate
ir
eqs(6),(?),(g),(10). These are substirured
made 'about the
"-tii""o
are computed' A thecl is finatly
forces
and'the
solved
are
These
o,riliria.
for
of the
equations
slip and rvhether the value of the mignitudes
' p.op". directions of fiiction for-ces at the contact(s) rvith
frictioaforcesatcontact(s)withnosliparelesst'hanorequalto[lrerespective.O.t"dl"Oofsiaticcoefficien[
rsit'h other
satisfied then the problem is rervorke<i
these ;;;"
If
reaction.
normal
and
friction
of
"ot
por
is gi'en b.v 'q(a)'
i* Fig.E3.21o *,y:l_lrl",:-'i* force. no:l*,."
the front rvheels rrill lose
amount since at some st'age either
florvener, a cannot be increased to unlimited
can be determined b1r Grst
Thu *"r.i*om.pcsible aceeleration
stipping.l
start
wheels
rear
the
or
contact
1olR u1 = i2 - al R'
i,ll =
findiug tbe forces as in part b. For no slip,
"'2 +'rl2k3)/R|v (13)
Ho, =(m1l;fr.l1*m1uJr1} *$n2k!u2}n2llh2) }m3uh3 = [,nn+(m1[i
-_
<
r!
il"H:"rril,:15;::*":
Forthe6uaknown'qJVr'IVa,fr,fz'd.'M1'rveusethefollowing6equattons:
NiO - mg0r = {mlr +{rar*? +'rn2*?2ttlRlo
M6. : iIs.
For the .rhole car:
ra9-lVr-A'u=0
Fr=0
:+
F, =
Msr, = I!,"t1
tna
For rear wheels:
tu)
(15)
(16)
-F2-no
Mr - FrR= *1*11alL
F1
aA
(18)
-
wheels:
i;;;g
,u" *o1r.-"nergv retatien in rate form' i'e''
eq(a)J:
.
+rr.zkl\lR]rollb'
N1:lrnsb2*{mlr+(mlef +n2ktr')/Rla)lb, N2=[msD1-{*h+(m1&!
,/r.2hlllrzlaa
f'r': (m + tuzk:'/Rz)o, M1 = fm+ (mlei +
Ft = n2+],alY\ ,
Substituting these in eqs(19) yields
no'slip
"i-.reil
wheels
if
and no E$-off of hont wheels
:+
I-
The ma:cimum
o
if
1
o1= p,9bz/6[1 + m2kl/rr.}}
a S az =
s\flh+
(m1ef
acceleration': a = min(c1'42)'
-
1
^-{
,-
i-
i!
lt
(17)
FzR = r.2klol R
M6r, = I!,'b2
nr, > g- (tg)
w\ee[s:
and condition of uo lift.off of frogt
1I4,Nr'
Ft
aL
B:
slip
no
of
condition
and
in terms of o using successively eqs(14)'(15)'(18)'(16)
The forces lvr,Nz,fz,Fr and M1 areexpressed
t
For front
i!
(20)
(21)
P,{h+ (m1ef + m2L1lll'rl.Bll'
-
._
=
+ n2*lltmBl'
<
g'r
:
q
'r!''
of
wheel
gom eq(21)' -The liftpff c- --';*"*
'tto(t'
is obfned
tot'
ac&lerarion
this
g3ea
.t Atrf :"* T::..;:;;aed ii due
being stry
a beias
of c
* the vaue or
:":
ao' to
*qd;;;;";i*
*ohuo ti"
teo-sheel
a
9$-'-+i;
'"
"*t".
thi direction of frictioo force on
analvsisfor;i;"!*5r-**"i:"l;:fff
ris-E3-Jld.. No'ice
u
= w viaa". ":":,'::' "'
-'
i)
requir"d
Mr
of
Tbe value
.
""'i',"
lll,:f #:; :::il:;::**:"T"'T'T:;;
(mrt? +
{i[1 + Mz)/t*+
tR
LL.r
[rn
(M.1 + Mz)ol^ *
+ (m1tf + mzkzz\l?2]uu =
a
=
r/.zk:2)l
P.1P. (22)
-
,R:. ,...
h,E
-
;3
*#-ry
d=#
-E--
'
it-t
LJ
LJ
rvheels:
For ftont wheels ,
For rear
L-.
LJ
u
*
IJ
-i"-i--ii)"r",
]
(23).
'
(24)
[:l
*[z- FtR= m2klolB
=(rr+fz) l,m<P'{NttiI:}=
jol"- '
?hema:crmumacceteration "
l{t
eqs(20)
from eqs(26} arrd (2?} ujing
. Mzarre obiained
Mt = $.el&R{b
rcl
Ho:
The FBD's are shor'vn in Fig-E3'21t'-
car:
\)
Fq front
.and (2s} for IrIr , J{z , Fr , F2,.o|
Jqb
F, =ma -.
a rtr'PtNz=mo
ttllrlr+
IrErrI'
wheels: o"r;
t''ioz
+
:
t2..Mr-ltNrJl= mrti&r
m*?6t
\'-- ^**'R=
:)
j
a1s
(31)
(32)
t33)
\oo'
i
(35)
Snbetituting&y&2,oftomeq9(33).to-.(35)ineq(30)arrdusingNr*JV:.=rngfromed3t)yietds
'
tv'L---Mzllb- (36)
E))
tttl-- Mt
Pt(tr
JV2
'r
[rng{61-Ft(h
=
8]] + Mt* Mzllb'
fV1=tmgt6z+pt([-
rl
f
(29)
.
- atg
mg-ltlr-N:=0
i{o'
Fr=o-,
!
_)
",- ^^.tit**t
' F'=o'N,-
5
is given by eq(t')' The -unkno,sns"I{r'JV3'a'ur1'<'r2
Mo' =
.
y'{rr"dim=pts
ri/E}
+
* v,lrrruit+ (*tei + mzti)}/l '"i
:'^:
For the whgle
l.: J
IJ
1-,
IV'D-nrgbz='
'''
mg-Nr-J\r'=0
Ft* Ft =trt?
Mt - Ftf ='rni-{c/R
+
f. = t'td
Mct.= I3-"i"
h{cr, : €:a,'
'-.
l-
u
L
':
brirrnno*r,, Ai..lv,;Fr,F3,o,Id1'It[2:
EguationsCI3)and{2a)yi,eldlVr.Hzgivenbyeq(20).Itfollort-sfromeqs(25),(28}and(24)that
t_-
l-r.-
_*;.*.,,,,..#
q'h$';ii""f;tt*::.i,::,,it;
frovidedthereil'"m*e "r'ttie
' 1t6,, = trur
For the shole car:
lt
u
tu
1"
u
L
f---
"##*,t,iT-,\
or-Tt,
.. (d).
aa
[t
^'--l
,j-J
from eq(36)
eqs(33) and (34) usin5 lv1'il3
,,,t.rizare theu
all the wheels
"o*p.aiii;m
the rear *i*r" tral;i'r.and,with
Tbe analyses of the retardation oi. ,"r,i"u with only
reveals that a
(35)
and
equations (29)
":J
f:,J
Compirisoa of
braked, are sinilar to those presented for accilerttion'
slip on
in such *f1jt that there is nq
"
gr€t€r retardaiion of lr,gis obtained if all the wheels are brakedthere is actual
jlip on the ground yielding
are locked, then
the ground. On the other hand if all the rvheels
para*er
km rvith a verocirv orsoboo km/h
is raunched ar an arrirude
aod its minimum and
'to the earth,s surface. Find 1- the minirnum and maximum altitudes of the satetlite
tlles of travel from the
otUit and its time period' 3' the
maximum velocities, 2. the eccentricity oi itt"
velocity components
and 4'.-theradialanrl cirlumferential
perigee-and apogee to the end of the minor a-tis
and (ii) after traversing
altitude of 1600 km for the first time'
and the speed when (i) the satellite reaches an
to a malfunction
o""* ?*i,--"' (b) Work out part a il 1ue the satellite in
a polar angle of 60o ar earth's centre rro* ti*
(c) tlom
s" * iL" circumfere*tial directio''
the vdocity at launch is at an outrvard "rrgi. or
the optimunr location in
Find
of-eccenLricity 1-2'
trajectory
a
llur"
to
launch"d
be
is
to
probe
part b, a space
the objective' (d)
in its *elocity at launch to'achieve
the orbit where i[ should be launched and the "rr"r,5.
be launched from it so t'hat
U'-aA oUlivation module is to
At the maximum altitude of the sarcllite in Part
;Xj:T:;t'i.T;::;erire
"ry.
in its subsequentorbit its minimum ai"'"""eno*theearth'siu:-.::i::i:,
taunchiirg'
viug cases: 1- circumfettntial i:"""*il?,
rockets) &r'the-'follot
required at launch:(imparted by appropriate
.f'z
t.
of the earth as 6400
2- radial launching frorl tlre satgltite. Take tlre radius,E
|^,I,
,<u nn;.:
ilit'*itlffi
il;'ffi
determine the
sofntion 3'*"1*,
i
\-
ro
-
=
6400 +650 =?050
cMlh|=
rmin
=
km)?
sR2 =9.s1+{6400
1-16416
km,
. . -,
= 19245 km/h,
E,:,650 hr.n,.
omin = ftolrr,mrx
,.rrffi(64o0
x'10-4 km-r
?050
1fi<fLl!"|
,-7
t,r
km), =520?56 x l0r2 km3/lf
km/h' L6 =16o6o = 211-5 x 106 km3r/h
=
omax =
- ft=aigg t*'
:*1
Irglrmin 30000 km/h'- e:=':y':*:^0.::*'
km
=
*:'
=':ly,lz):
[ = c(l -_'?l't" =
(rmin +rim.,g)/Z = 9020
: 1t' 1
1
the mindr alris (Fig'E3 '22a) a.d
of
end
zt'the
to
r*in
Q
at
perigee
P
Let ?rbe the time to travel from
be tle time to travei from.apogee '4 at rm'a;g to 8'
!i = (area OPQ\ I $ol2) = 0'5078 h'
area'O PQ = tobf 4 - 6(oe)/2'
?i (area OAB\ | (hol2) = 0'6?17'h'
b(oe)12'
J
.c
:
',: I
T
'J
J
J
J
8802
OAB
81---.
"--f
J
,lJ
I
U
-T2
rabl4+
=
T-he launching pcitiou turus out to be at rn in'
Llr'= GMlhf,+ DcosC :+ e =70'27o '
At r = 6400 + 1600 = 8000 km :
tim./h' + e = (ui + o?)rl2 =ZbStZ tmTl'
v6 = lrsDsind - 5062 km/h. 04 = hglr=261137
At {:600 :
km'/h' + u = (uf + v1)u2 = tl7o5 km/lt'
v, = hgDsind = 4658 km/h, vo = l:,ab = 2?311
area
J
Sl*t1 Br..
oo:ooo:30&}0
km-!
D = lllts - GM ft:|il - o'25428x l0-{
imax = $Mfte- D)-r = 10990 km'
Lma:r
-J
*,' \
the given dita for ihis porallel }
basic constauts Glr{,ho,D from
o
km'
Kl|zth"+ D)-r =
[*ir, = rmin -
=
1
1b;
laraching.
GM
'-
(ffi"9*K# J-I
sY.
J
;q
k-r;=--+{YF:"r
-t
r
..-J
J
J
J
"rJ
,J
-J
JJ
J
!': \
--/
L l-S
f--.
'-'t ''
The speeds
-w" u could'il;*
first determine
i"
"ro
6400
=
ui,=
t- :
I -
uscc3o
cM/h
t
-
=
116,
299589
=1.16236
using energr conserva[ion:
km/h' -'
x t0-{
km-l
;j*'
E
tr
=.roud'o
(Fig.H]'22b)
=2LL'2LX
106 km2/h
D=l(tl\-GM/hil2+(r'"/ho)?lt/2 -
-*
o'zot8ss
x lo-{ km-r
**'
;:
=
umax
= l'o/"min =
"'J^:;'='*'-;,'j:*
km/h' e=^io1cu =o,243r
6-a(r-"')'t=6?90-6km' T=rob/(hs/2)=2j59h
30186
il':'":,:'"',::::[::;';,:]#::$.",:,11,';]ili']';1i:l''ii"iH:[HffiS:::T::i
r',=..gi#jqx"1;=-.
At 0 =
"; =';'"1"; :,ffi:}f]
l-_
The fact that the values of u at
IJ
il,:r:::::L'ffi:y;e
f
,1$:;,,'_:?km,h
;, = o"',-, |-ffir,?'-:'';='liij,i ),r -2a{ss km/,r
r = 6000 km are the same in parts a and b follorvs directly from
the equation
be launched ia the circumferobjective with the minimum Au, trre probe should
:if"*:l':X#i:l[*in*;i:;i:l':]",',1'u,;i'.1.H"15:,H;S,I;'*"*
D1/r'2
=)
=L2.
+
l/rmin = GMlhl+ DL
+ D1 = 0.77g555 x 10{ km-r'
+ ur = ucr = h1/r*11-401164 km/h,
er:h2rDr..Gi't
,
L]
I -'
Gr{lh?=
l/69e?
- (I + tlt'2\Dl
h1= (L.2Ct{
Au = ur
:+
tr,
L
I
:':,';::xlT;:.,
...,
km/h, +
:+
lJ
ur
r-
263'129
x
106
km2/h
I'02?8
km/h'
1/11044
-=:::'":;,'o.*''^
[r, = ul -
19124
=
-154
,'
km&'
1i:::il111[*'.]i11ilm:':II[
Lffi';*,:lt,'r;*mlm'*T1;.Tffi:
is giveo bv.
12
=
6700 km of the module
Conservation of energy
l-,
L'
fj'
Dr\'t'=
themoduleare l1044kmand67fr) km'
= ucr = h6lLL044= rgszo
o2
t-
l
-r.,1.*=40464-30186-
(d) l.-r"rnt"...",ttu--u*imumandminimum,radiigr!.treorbitof
Li r,o, D refer to the trajectory of the probe (Fig'83'22d)"
t
L
= io.4 /cllnltgi
^::Z:o,=::';*i!,,",',1,i,', ;=[X Zi^o,Y,',i',!,;]r:]::::
16'49o
bndo : (""//'o) I $/'o -GMth?)'=0'2960?' {s =
lJ
L"
L
'ro
o-(tr,',ir,*rmax)12=9020'5km
f
tr
L
L
I
F
.
= no/'*o*- 19124 km/h.
";;
\ma2.-GMmf t
'- '-*-- ;'
*ft'
uo = 30000 k,.r/h' uro = uo sin 3o = 15?0'08
,-,
"urained
D foFttris ao,.pamllclloonchiog
kg1,
zo50
==:::';::;,
;::
lJ
I '
+ 060
ui;
Y
:
-- ooz = holrz =
11044
lm(ul+rsrzl2l-G
x
19124/6700
MmlLL*a4 =
q,hich'ts very large compared to laulobtained in part 1'
ea
=
31523 km/h'
\'tttti-GMmf tz
+
u'
:
4(F6 km/h
**--)
r,-*l
;;-I
aJ
exc-luding the fiork' is mr with ceotre of mass
I}xarnple 3.23 The mass-pf -a fork-lift tru& (FigE333a),
at Q' The fork is supported by a
U
at Ct. The mass of the fork "od t5" o"te i" -, ii "*:"."f .-4* and vertical reactions' The inertia
Ugttt nofzontal
smooth roller at A and a connection tt .B *hfu-;upport-s
just su6cient
at A, B , E,F if the fork is given
of the wheels of the truck is neglegible- (a) Find the'reaction!
(b) Find the ma:rimum deceleratioa the
one of the ground reactions tP *to'
..'iJ
t.'iJ
t:J
upward acceleration to reduce
slip nor
nei[her slips nor tips and the rvheels neither
truck can have with four-wheel brakes so ttrat th" "."t"
the centre of its
centre of mass is at height h3 above
lift-off the ground. The mass of the crate is m3 and ils
the fcrk is f:- (t)
the ground is p1 aud for the craie and
base- The coefficient of friction for the wheels and
at tlre isstant the fork has uprvard acceleration
Find the ground reactions for a rear-r,rheer driven fork-iructi
/t/r' A;ume
o1 relative to the truck and th-e.-drivt'r t"'o'l t'
tn::' **""FffTfjpfiI*
,,J
-J
:1i#-i
u
l
fffffiFffi*#,ffi"ffi
'ffiffi"H$qf*4"Hffi#j
to,
;f 1
JOo,
,;a.i.qffi
'z
J
*-fN,
f:*S'{'h_-_Wffi tf
'i{r. Na +*/
,J
,
^a:
:"F.?('ft8'i.. - e. Nr
_!-,}oJ'
}#, i#,:t, -}_u" solved
N1
iv'l'n.."tr'oa''iit
s.r.,tio;'dlhis probrem,t:,i,:", ="...3,J,:i,,#ra;.1.",'
for individ"'i'ot* '"oich invoh'et:T;:"'r:ltl;l"ll1r";:::"J.,lll=
Mc==00forindrvroua
for dre
rttsc, Ma
Using F =
lixed points t=r:lil
1.
1- usins
=rttg<.t
= io, forconvenient
.
ir":1i.:_=*5::*xT:HTl"ii:-Ti:J:rternar-ro,ces
'
follos'ed io 8x321'
on trrc svsreor appear
ia
'lrre
"
tt i. "pp'o"tt' was
o.vsram cao be ren'tit as F-ngcattanslatingsystemcaobere$'titten
3. Theequationsof motion,F:tng6, rn-=nof
act'ing at c constitute a null
g, i.e., the external-forces *a",,il. ruirh * 'itertia lin'-**"
g-, Mc:
using F = Q' [
- O for
this augmented force systenr-and
with
rvorking
^
in
consist's
E thac
method
systern- The
adt'antage'over the first trvo methods
ttt"
poiit'
to"'""itttt
any
is
A
where
this null force system,
is zero' Method 2 rvould require Proper
;;'ii11i::i.:::*
sv"t'm."b;;;
force
augmented
the
of
3.
momeut
equations-
computation of H6,,though
be the same as in me|lrod
the resulting moment equation rr,ould
acting on the
Weillustrat"*u,hod"',and3inthesolution.\1,efirs[.aPpl1,metlrod3.
er<ternal and inertia fcirces -rn;gc:
The
c'
be
fiork
the
of
(a) t et the upwaril acceleration
sysLems' Hence
in Fi5'83'23b' These are null force
t'"
wheels
(l)
and.the
fork+crate
truck,
"ho*o
Fr=O
Pront$rheels: MP,=O
Me.. = 0
B.ear wheels 'Whclettuck:
MF =0
; =;
Fork+crate:. MB,=g
F'=0
Fr=O
(2)
F'=.9
'
:+
JVr=[m1963+m2(9+a)(61+62*6r)l/(bz+6:)
;; *; .-'*'(o+ a)Drl / (62 + re)
''
N =-.z(g+a)bqld
Er=[
=
ft2=m2(g*c)
Jj
J
'J
J
,!
i
ir
J
-J
,J
!
::
i
-_.1
,-:{
(3)
:-
l/,\
(4)
,(\
(5)
<
(6)
-q
(7)
-i:t'--
si
.v-.<
+r
S
-!
ttI
,..
'
t*
.
-:i.i.1;
!.;;
.; ^:f Ii r.-i.-:::-r'=.':-:
'r,ar!{-i
' r.' i.
:.
-1,
tot a '=
(a1 tu.t /v1 increas€ P,9 trr lecrlases with a' 4rz becomes ryxo
(?}to
eqs(3)
usirg
fvi;vr1n Ef;e;"- be determined
{mg7/m2b1- 1)s. lyith this-viluc-ot i,
**o1. *d iuertia foi:ccs -m"rc- a :ting on the
(b) Lct the backward accelera6on of the. trudkll"'t,-fr"
lThese are null force systems' Eence :
'
truck, crate and the wheels ate sihown in i'ig-ea-23c.
(8)
N2=liig6,2-mz9br-(m1alr1*m2olr2! /$z+b) -'
Ms,=A
Fortruck :
lr is observed oo*aug6-iq{
l^
-.[.
t-_
L-t
(s)
- FY=0
t_
'"E --0"-t"
(10)
o=(rt +Fz)l\qt*mz)
ord N1 increases by the same arrcunt'
"
:
(11)
o1o1=(m162-*rtrl!!-Vrlrr*rnah:)
Fornolift-offofrearwheels,usiugeq(g): JVz20 =
(Lz)
pr$ir+N:r)l'ftar+"tz) noslip of wheels, using eq"(rg),(9i,o - (f'r+ Fzll(mr*mz) S
"Ls
LJ
Equations (S) and (9) imply
l
that
'L-r JVz
^,^ decreases
^":-'' with in
ease
For
L*
t-.
For
fr = Q
cra[e:
fv=0
:+
M9, =A
L=
l.l-
(13)
II3 ='fi13a
= m3!
0
Ar3D - nr3sd7l2*n3efu-
(i4)
JV3
=+ b =
d1l2-
ahslg
(15)
msa.f l.?@3g :+ ' a-S g,zg
: Fs ( ,raNs
!i:l
(17)
a{:gd1l2hs
+
b--'|1./2'-"'A"ry',0
F6r'notippingof,crate, usingeq(15):'
c:miu(o1' Prl' lt29' gd1/Zfu\'
Eence the maximum retardatiou :
wheels are shosrn in F(-83'23d'
(.) The external and inertia forces -m;lLc;acting ori the truck and lhe
Fot no slip of crate, using eqs(13).(14)
LJ
t,
L_-
These are null force systems' Eeace
L-.
L
LL-.
L-
LJ
lt
L__
u
t-t.
L.
LJ
L
LJ
L*.
L-._
_i
Frontrvheels
:
Mp.
=0
Rear wheels
:
Mq,
=0
(18)
Fr:0
r;- LhlR
'Il9l
t20)
:+ a=MJR(m1 +mz}wholetrucli: {'=0 + (mr +&z)a=fr.*Fz
mzahzll (bz+fu] (21)
nrt = [mrgts*mz(g+atX0r*6a*&3)- rzlolrl MF. -0
(ba *03)
Q2\
JV! = [m1s6 z- rrlz(9*ar)6r * mralrr +m2oh2ll
frr
(22) is given bv eq(20)' '
where o iu eqs(21) and"!.
(3)'(5)dB)'(15) aad (21)'
we illustrate method 2 for obtaining rroment equations
Fig'El]'23e' consider points o and
(") lrt u be the upward vetocity of the [crk- The FBO'" are showtr in
the given instant'
S n*"a to ground and coincideat with .F. and B at
:+ ec(3)
Ho,=-m2u(61+02+h) + Mo'=U:'-m2o(61*6:*6s)
Ttucti:
:+ ' eq(5):
+': Mi;'='H5t = -'n2o6{ '
:'''l :
Hst J'-r,..2u6l'
fork*crate ,
Consider
(b) Leruberheforwardvelocityof theforkwith.i='-a-TheFBD:ar:.Yn,rnBie:EXlt?3ft
'
given
instant*
ai the
:
points G and K fixed to ground a.od coincident ncith E,and,D
Ttu&:HG,=m1tlIr1*tn,thz:+Mc,=iIn,=-m1aIr1-m2ah2eqr6}.:
tc(15]":'
+
'
Crate: U*"=fisah3Y*r=i{x'=-rn3o[3
velocity of the fork- Thc EBp' i''
(") lrt u be the forward. velocity of the tructi and q be the uprsardwiih F' Iet' the location of Ci from
and coincident
shown iu Fig83,23g. Cor,"iao P*t'g fixed to grotind
pr.and
iih".g,r.* instant c = 01 *62*0s:9 = lr2. For.the t14',
0 be giveu by e,y, whe re i:,,,i,:=
i
Ho,Jmsulrl{t.r.zoy-rr.2v1, :+
- -rr.
Mo'=Eo,=m1oh1}m2cr!*nzvil--r/r2a1a-m2v1i=m1clr1+m2ay+?,/l210u|-t,12.,|Z-T,.:I..:i '
':+ eq(2t)
i: .
- nt1ah1 *mzohz - rr.2r.v(b1* 62 * De)
.
L-.)
.;
.- -- ::r=g:ri
:
€-|t+''::
ii
l:.
..::,
"
-' - - -::l:i":"t':':-; -. -::;=".r,1::
'
L_-
LJ
L
L-
ga
l:IJ
Qr
I
Ct
.:
lj.
lj
.
.....
a. tdrrarroNAr, MEcrrANrcs aND EQUTLTBRIUM
lj
tj
tj
t
q
is called uirlual displacentent 6q(t\:
6q(t) = q-(i)
pathofmoiion.Notation'6O'differsfromthenotation'dO''LetaforceF;(f)actatthematerialpoint
the virtual
1*O]"t-":lt"t.":
i. The positiou .,r".to. 1;(tiof point i cau be expressed in [erms 9f S(t)' Let
in thG
rvork done by F
The
'-(t)
poinr i due to virtual displacement 6q(t) of the sistem be 5r;'
;;";r*;
F'(t)'tirt'
t'
is
time
value
at
at its
virtual displacement iu zero time, keeping the value of the force conslan[
external and i*ternal forces acti'g
the
all
b-v
done
rvork
uirlzal urortdone by F;- The virtual
i;;;ilahe
denoted b1'- 6l'{r(t)'
oo th" system in virtual displacemenc 6q(r) of the system is
the path q(t) is catled [he tarialiarr
The diflerence in tl're value of an entity for tlre path q'(t) and lor
r-alues of the entity is called the f rsl
of the entity and is denoted bv 6( )- The first order diflerence in the
Hanriltoo's Pn-nciple:
uaiationof the entity and is denoted by 6(r)1 ). \'Ve norv state tlte
.A geometrically admissible motion c(r) of a syslenr betrveen prescribed confiSuratiorls {1 at tl and ri2
indicator (V-I') vanishes:
at t2 is the actual moiion of the system if the follorving variat'ional
1-.
u
[J
t:
tl-.
l-
v.r.
t-.
l-
= J['i
r,'
;.rr, * 6w)dt =.o
in tlre srnall
/,rbitt1rry varied nrotion of the q=i.r,. s'(r) = c(t) + 5q(t)
frame'
inertial
called
valid
is
is
refereace frame in rvhich Eamilton's Principle
neighbourhood
42 LAGrtANGElq EQUATIONS FOIL DISCRETE SYSTEM
coordinaies
Consider a discrete .system o[ a degrees of freedom rvith geueralised
u
l_
91 ,
"
6w= I8i6c;
and sumrnation
L
l-
: Q;6q;, where q, =T L,'H'
(4-3)
,.
generalised coordirrate
is called the generalis cd forccfior the ith
*rr"ntior. nl" u""r. used- e;
? is obtained as follorvs- Let 6g(t) = e4(t)' ie"
rvith {rr) = r1(t3) = 0, and e (
c'(t)=q(t)+ea(t),
1'
(")
a function of e, and expanded by Taylor's
T.he kinetic enerst function ?' for the vatied path is expressed as
series about e = 0, i.e-, about the original path q(t):
* #1,=o{ i#l.=rt? + ':''
6rtrtd'(r), q'(t),4-"[.i(t], c(r),4 -"(.) -rl.=o = #1.=o' * i#1,=;2+-"'
?\.'
:
i
I
a-.
g;. The first variation of
1*
t
a
i
tj
lj
t
' ' !In, rvhich are all
in terms of 59:
lJ
1
r-l
of g(t)'- l'he
independentofeachother. Leiq=lgr gt ...- {n-l tld f bethethecolumnvectorofthegeaemlised
actjns zlr-i =I;({r'"''9'}' can be
coordina[es. The virtual work do[e lry internal and external [orce-'Fi
ercpressed
(-;
(4.21
for
t-
cg
(4.1)
-o(i),
,virtual, displacement since ar actual infinitesinral displacement dq(t) occurs in infinitesinral
This is called a
in zero time and is not along an-r actual
time along a paih of rnotion, ivhereas virlual displacement 6q(t) is
t--
L
t-
::
t t7
4.1 EAMTLToN's PRTNCTPLE '".' .:
FiA'('l
move
system
a
qLet
be
Let the ge'eralised coordinatrs to, d""".ibu the configuration of a system
t2- Consider a geometrically admissible
frc:n the configuration g-1 at time t1 to the configuration f2 at time
tir, C({z) = 'i2 (Fig'4'1)' Consider
motion q(f) of the system between these configurations such that S(tr) =
these confiSuratious such that g- (' I ) =
another geometrically admissibte motion g'(t) ;f the system bets'een
q(t)' Tlre iafnitesimal
qr, q-{lz\- f2, which is an aroi{raryratied rnotion in the smallneighbourhoodof
tinre I
,displacement, from ar.r admissible configuration q(t) to a neighbouring configuration q'(f) at the sanre
t-:
t_
tjt_j
tj
Q.
:
- ?!i'({), c'(t),fi} = r[d +
€q, q+er;, {]
= ?(e} ='[.=o
E3
,".i'
'*'-''
chiio,rokj of airjr"T
i f1'1]sfrief ';"li;'$9i€i.--.. , ;
o,ui"-- #1ffir'#Tr' ; Yrpil} = i# *'+
a\e 6rst o.a"..L,,sos:.te-:l-.tlt-
tj
it,
SuLir'*1-r.g(43);(4'4)in(4'2)andusing6s;=er;;vields'-'r'--"'.'1
*ffo'*Q;t;l,tte
u
tj
zf,e srsr
t_
LJ
(
tj
t-.
tj
]j
-ide
4f,r
by parts
term in the integrand or "q(i) is integrated
g\r€'f,ogrcag c's egtalidrs for discrctesystcnr Y'!U-rU
Qi=-ast
strbat'iunigtr
-
lJ
t'I
,'
t-.
lj --,
I
H
H
L-
a factor of r1 as in the
othir two ternrs:
trtence' the inCegrand in
t::
zero as
:
:
{
i =r,.. ., n.
O':*U
aud
AV
l-].
get
(6)
(4.s)
aotdinates itilepettent of each
and nonconservativq par ts Qr'
other'
a
Qi"
3 #i:r*t'"'"u"*o*ve
T:ffi###";ill
:H"ff
.j6w=Qi5qt:(Qi+Q?.}6qt=j?+Qt.6ci=?x+Qi.)6c;
rrie &hoe
u
tj
tn
(a)'
ni(tl
0T _ n.
aldi,) - 6fi= v;
lJ
]j
:
tfitffi-#,-Qllrdt=8' ' .':"]'., r ,, (c) should be
-ts
zero since 6(t1) = 6(t2) = 0 by
O"<rjniegral term
o,o be valid for arbitrarv
TlreSe
':
=o'
l,l't#*
tffia,(t)l[ - /:'
'ii"ihn
ff'ti'
t_:
LJ
tj
.
Qi rromeq:ti;:
Q; = Q?'
_'av
-:i':':i,
i=
_ u; ''
+
;(ad,) - aq, 6*= ^..
the Logra*gian fittctiott tr o[ rhe
"y"i"*
*=#_X woRK
*If'u
+.2
a*d(46)
r....,n.
bv ,(,i, g' t) = ?(4' c' t)
*
(d)
-
(4.6)
y(c)' llence
*rkt_#-_o,., i=,,..,n
I.TLINCIPLE oF \/IRTUAL
in equilibriutn, ie-,
t-trnsider a system rvhich remains
c(t) = {-r.= iu:
* d
o
*
T = 0-
(4i)
For rhe
^
of the first order aud hence the
generalised velociiy i'(t) = 6GI
the
6q(r).
+
Vqriedparh C.(r) = C(t)
" (4'2) reduces a [1"6W at =
order, * OtUf = 0. Hamilto"titin"ipt"
is of ilr"
?*,
e
knetr?energy
"".o,..1
6w
."'f q(,r ro(,,) = 6q(t3) is initiatly in equilibrium and 6l{'=0 for all virtual displacements 5g' thetl
to the
t\>nr.ersely itthesystem
generalised coordinates q(t) corresponding
are zero. Choosing t'he origin of the
forces
of the
generalised
solution
rlr
0' Hence the
initiat .oJiriont are g(0) = 0' d(0) =
ttre
configuration,
equilibrium
:o:ttal
and' the sy:tem remains in equilibrium'
of motion for this case is c(t} = 0
uork
lben the
hsms{eneorS.L"g...rg"{"quations
'.;.16o{
of virtual work: If a sistc* is ia cguili6ritm'
principle
the
proved
have
s.e
gcoilticolly
a
fhos
in orbittary uirlual dkploccmcnt b zctt' Conrterselg'
clone. fC thc intcmolatd. exlental lotqcs
j
- 0 +.
=0'
uJrta<sieleconfiguvtiott,ofosyslenlls.cllequiliDriunrcottfgalrrtiott.!!"*,:,::"::o,""bythcexlental
is iai{iolly i* eqrilibrium:..
loirtlrat displacemed' ptooided fi]e systea
.rr.A inlc*o I lorces is zero for arbitmry
(4'8)
.
6W=0
v 6q'
Foradiscretead.o.f.(degreesoffreedonr)systemrvithiidependentgi,eq({.8)+1
:+
Qi-O'
V 6q;
6l4r=Q;6qi=0
lhe generuEsed [orlr,s ate zc,to'
ip.rforc discrele system in eguilibium' all
94
i=l'"''a
(4.e)
t
-
,HE .
ll'Ki'
ilD.J
\,
)'--
--
H-i
A
porENTrAL ENERGy
srATroNany
4.4 pRrNcrpLE ot'
-. J.{.
-.-- -:.--' - ---l-ret 6W. and 6[7," be the virtud ivorF doae by the conservative and nonconservative forces, then eq(&8)
6W = 6W. * 6Wn. = -iV + tVn. - A,
=)
If alt internal and external ficrces.are conservative, then eq(4.8)
6w
=
-fi :0,
+
V 5q-
6Wn": 5V
.+
v 6q.
til = o
I
j']-
e
=+
(4f0)
lq
11
r'
({-11)
-
i.e-, cbe first variation of potentiat energl is z,ero. Thus the potential energl/ has a stationary value (an
extremum) at an eguilibrium configuration. Thus priuciple of stationary potential cnerry follows I/ a
systcm srbjected.onlylocotsert;alioeforccs rsincguiliDnunt,thet6V=OY6q. Conuersely,if 6V -OV5q
and if lhe system is iniliatty in cqailibiam, lhen it remains in equilibriumFor an n d.o.f- conservative system, eq(4-tl) .+
:+
V
v 5qi,
5q,, :+
ilr=Y&,=o
=fr;6qi=0
i= I,---'n-
Y=o
U*=O
6V
;-t
r<
G-rz)
-
ir
Equations (4.i2) are Lhe equalions of eqailibrium of on n d.o.f. conserttaliue system.
4.5 PELINCTPLE
:
Or RATE OF \rfRfUAt .wortK
:-
Consider a sei of adnrissible virtual velocities ri .at an adnrissible equilibrium configuration- The rate
rt. rateo[sork done s,ith these virtual velocities- Since ri.an as re{l be
of virtuat'work 6liz i, a"nn"a
".
v i
,r,ir-o
(4
-
=-
\
13)
i.e., for a syslem in equilibriurt, lhe tole of llrrtual work dol.e is zero for orbittory viriual oelocitiesNOTE: If the system is constrained and the forces of constraint (internal or exCernal) together do zero *ork
in virtual displacement consistent rsith the constrain6, then in eqs(a.8)-(4.13) the virtual rvork is only due to .
the active internal and external forces. If all the internal forces together are *'orkless, then in eqs(4-8Ha-f3)
the virtual work is only due to the e-xternal forces on the s-vs!el'rlDircct Proof ol Prittciple of Vrtua! firo* for a rigid body. l{ for a rigid body can be ecpressed as
_
lA = F -ya + W.u, + d.W = F -gcdt+ Mc.g_dt - F.drc + Mc.d4where d0 :,"tdt- Hence,..*
(4-14) !
6w- = F -6u + y"
A rigid body is in equilibrium iff f, =!,M.c- Q, provided it is initially iu equilibrium. Ilence eq(4J$=+a ,{h
rigid body is in equilibriuro iff5t4z = 0, V virtual displacements 5t, 54. provided it is initially inequilibtiu.m - 44
-
4.6
srABrLrry oF EeurLrBruuM
.=.-!
-E
:
lj,' :-
!s.
coNFrcuILATroN
W
1Ji r;
-
An equilibrium configuration of a system is said to be stable if ony smcll initial disraficncc (displace*ar/li \ of,f,o
and / r,elocity) results in a motion rvhich b timired b a small teighbotrhood of the equilibrium configuration :&
W
The equilibrium position o[ rod I in Fig-42 is stable but those of rods 2 and 3 are utrstable. Rod 3 is unstable U,rl
since a small iuitial angular velbcity causes large displacement from the equilibrium position-.
F,3.q.z
Cotditioos of stobitity o/equilibrium configuration of a corserualiue system of connectcil'igid bodi* arc:.
(r)- If the potential,energy of a conservative system of counected rigid bodies has o lbcol''iniaimua at a
position of equilibriurn, then that equilibrium configuration is stable.
\
Proof,. Choose the datlrm of V aL the equilibrium configuration 96, i.e-, 7(96) = 0. Since V has .
minimum at qo, for posiiion g in the neighbourhood of gs,
Y(c) > 7(q6) = 0.
Let the iaitial displacement and velociiy imparted to the sysi.em be q' and ri'. Then
'
=,n
(1)
(1)
r(,i)= E(c)-V(cl
Q)
+V(c)=?(,i') +V(c')-E(c')>0
=)
"(,i)
E(q') has a small positive vatue since
> 0 and tz(q') > 0 by eq(l). Equatiou (l) + there exbt aset
"(,i')
ofg,int,heneighbourhoodofg9sottrat7(q,)=a(g.)andeqs(2);(3)+
T(i') =.8(s')
-
V(s') =
0,
9s
+
i' =A.
\^
.
,
:
.{
\< -
i
-'l
-t
>
__
-t
_J
.,-]
-I
l
-J
S/
I
t-
Ll-
l.
L
L
a-
ffiIt*Xy":I;1":#rI"1["#;;H.ffi;E
kt-_
l-
equiribriirm
.""rir,i.."rti* {s and th't6anm"m
t
gs' IIence V has a local minimum d go'
where g' is in the ngighbou+;d of
is given by
with coordinate g, the equilibrium configuration 96
For one d-o-f. systern
LJ
dvl
dq lq=qo
L--
tJ
L:
orr
dq' lc-co
(4-15)
=0F-tli
.t2v #v
*'o
ir ffro,
#:#='^"=';#10,."'q=qo
aY
^.r.._.-bY
ci^ rs grr€I
Civenr b-t
.
---€-.-alinn
-. -^-a.
dt - the equilibrium configuration {r3'92s
For trvo d.o.f. system rvith coordinates 91, 92.
0t' 0. AV
* qr =916, {? = 920.
-
LL.
ac, --n"
The equilibrium confiEurt
tJU
L
l_
IJ
L
1-
0q,
i'"" i. stable if
1z tras
('t-ro)
(4'17)
derivat ,ives at 96 satisfy:
a local minimum at {o' i'e'' if the
4.T,,NUMBERoFINDEPENDENTEQUATIoNSoE.EQUILIBRIUMoEAftIGIDUoD:
g. Altlrouglr Mg = g,it does rroc -.-ield
bod.v are: f-= o L{.:=
a
rigld
o[
equilibriunr
of
Tlre equatiotrs
.-\
anJzmoreequationsindependentoftlrep,"'io,'ones.T-trecumberofequationsofequitibriunr.rr'ritterrin
.olporr".rt form, for different force s1'stenrs
are as
follows:
:
1.General,4=9,1'11=0(6)-Parallet'll:-a-xistli=0'Mt'--O'M^v-0(3)'
0 (3)'
4'
Coplanarineyplane "F'=0'F"=O"{f;' =
(3)'
3- Concurrenl;F=0
F' =0 (2)'
6- dpl"n".concurrentinr-yplane:I."=o'
5- collinear ll r-axis:& =0(1)O (2)'
?- Coplanar parallel, ll z-a'<is in zy plane : I1 = 0' I6a' =
0 for '4 on lg- Alt forces intersecring the same line r (d , F = 9, M t:Q (5) since M-^'9=
SYSTEM
INDETERMINATE
,"O,ICALLV DETEI,JVIINATE AND STATICALLY
XN."
".^'='"
I
reaction forcbs and,couples atthesupports
,----
'
U
u
IJ
IJ
l-.
I ,,
t-.
l-
-
.
if the
aud the
A system is said to be statically dete -rinate
9'' Llt(f ) =
using only the equations of equilibrium [E(P) =
internal force resultants can be determined b-v
fiorcc''dcforrnation
case'
p; otherwise it is ca.Ued .ror,.oit, itilclcnninate' For the laiter
Ql for all its parts
relations are needed for compete soluLion'
be dererr.ined lry
the teacrtions if supports cannoc
if
itdeteminatc
is'cxlentally.slatically
A system
intcrual force
system is iateraally stoticollg ;il,c1e|{.inale ifi|the
using only the equaiions.of eqtlilibrium- A
systens
tqt'"tions o.!'g'uilibriunr' Stiticalll|.tinf.:,t:1"*e
resuliints cannot be detetminea UV ,r.iog oniv'ittt
timperature'
during assembly and due to rise in
are subjected to forces due to initial mismatch
in
in Pig'a'3 is preseated the fotlorving'l'al>lc: The information for statical determiT'Jc1i;";";:*t
Remarks
:
force resultants No' of Eqs'
No- o[ gq' f
Rractions
1 Rr, Rz,R: (3)
Rt, Rz,Rr' R< (a)
2
beam
arch3 Rr-&(,%,86)(6)
beam
beam
4
Rt, Rz,c,
(3)
5 Rt, Rz, r?e, Cr (4)
r---^ A *hf''3!t]
P- P- p' 13\
. 3
3
3+3
-
I3
3
N'
"t' n4' s (3)
3
N'M'S(3)
3
ili
a'r' s
'v:
/v' M' S'T {4}
staticallv deterrrrinate
externally
stat' indet'
staticallvdet'emrinate
3
::*'SLi".;=J:;tr;
r-1"".l{a
]J Jrtrft*iu
{
i
i
NHH{L
+=#
t ",fr'
ft-,\
f?u*t{F
:TiTk
^"a""'si
"#
IJ
"*
l-.. '
".,rX":P.**
r-- I
lj
LL.
beam
I
i
r\grr'\Yr--
L-_
u
l.J
*"
n
-,,
96
3
-
int'ernally
srat' indet'
(.)€ Ll
-S''.
wwi{,e
<
. )€
-)f:: <
,,.
Alinternalcross-sectionofabeamwhmeoutrvard(fromthebeam)ao-rmalisdirectedril-1:::::::: is
is directed io tf. -y-.*'-o-':,::,1'rection
same
morneDt resulLants are assigoed the
and
force
called a -ve face. The cornponenG of the internal
in
directed
and
face
components actiag on a +ve
siga if their effect on the beam is similar- Thus ttle
in
direcLed
and
-ve
and those acting on a -ve face
+se coordinate direction are assigaed a positive sign
and
corrrponents acLirrg on a +ve face
.*:J[:":';fi":;;;;.;;;;';;;;;'
ditecad in-ve<oordinate-direction
"irr
sign'
in +re coordindtc direction ate also assiSned a negative
,-<
e-axis along irs
discretc and distributed loads' cl"9*
C.oosider a straighr;; JjJ;+t*.
;
of tlre
components
plane of toading (Eig-{-'lb)' Let ihe
ceuuoidat a,xis and y-*ds transrffse'to i[ in rhe
distributed
the
and
q(a)
N/m'
r be n(z) N/m and
distributed force in the e and y directions at location
N(r)'
oiequiliblium for lhe axial normal force
equations
tl,e
momcat be m(r) N.m/m- ll'e.raot to establi.L
i
from
gf
bear'
a
element
ru(r)- ihe FBD of an
rhe ransverse strear force S(:) and ihe bendiog;;;*
are
eouilibriul
of
in riq.:a'sa- Its equat'iqns
ta t*.Lx$,hich.does ror carry any discreb load, is slrown
.=
'
:.
.i{
f" = S(:r+32)-S('}+g"*Ac =r'0'
F,= JY{e* az) - N(r}+n-At = 0'.
ta}
(1"Ar}atr'e=0'
A[t; = it{$ +Ar}- Jr/(r)* rnry&*S(t:+A'r)Ae+
.
l' Dir'iding
load goAz from '{ and thetefiore rsLete 1Ae is tlre distance of tbe resultant of the transverse
"
inFolvitg
tcrm
the
r'
at'
values
average vdlues becorrr the
cqs{a} by Az and taking the timit as Ar - 0' the
q drops out and rt'e get
,
dY
nt
{4-19)
dA'
,=-.rr,
E- -;$-nrE=-C,
-
t''
:=
.
d2.u
ds dn
E=-E-E
o!
-r
<!q
(4-20)
E=c-6-
Eqaerions (4.19) are Lhe diffcrc.*tiol cqvalions
f-ir!
coplanat load- For
cqnilibrivmof a straight beam under
f =-r.
'#--,,a
dtn
#= -r,
#--''
__
(:r'2t)
--
CoasidcrthefB{}daoirrtrnitesimalelement(Eig.4SbJofthebeanatorrgd'o'$'he.eadbcreteforce
of eguililrrium, [ = o' & =
f-.g+ Frj_ aad a discele couple ifok acts *j;:;,rr" b"..*-'Tlr.the.qo.tio*s
ol quilibittttt "t
jump
coaditians
0, lf;, = 0, of this elernerri io tt Emit as .,
- o, "i"ra
il __.
"
---<
i
N(rf,}=N(af,)=-F..'s("f)_5(16-)=-Fyo,if(:f,}-l{(rf,}_-l{o.(i1.21)
'
/
. ''
Tteiumps in ir-, S. M at a6.equal negative of f'o' Fvo' Ms' tespective$'- ,+
Geosctrical'Itcz,Vltctiot of ncrlt.Its.' E{uationsi (4'19)
-!{
fl(,,}.=rv(e1).i_|1,ou..-.-(areautrder-a.ediagram)
l8z
-
S(41) =
:
<
g.r diagram)
-(area undl:t
- J,.'oo *=
M (rz) -rlf(21) =' j:: t * !,,' ^
S(az)
i<
5-z and ra-e diagrams)
-(sum of areas under
-
[=]Sif.nr=ol.
(slqcof&r-edurve)=-n, (slopeofs-rcurve}=-g. (slopeofi/-l.curve)=-(S+m), a.part
ort'e bearn
considering
bv
obtaincd
r". aiil-.. roaded seg;en* "iu".- are
si;';i;
with rf' s' tf
-:TA
a = 0 to z in Fig'(4'4b) is considgr-ed
to *. side of the sectioo at, r. If the b""", *;;;;f-.
* Irc.e.
ri -
!e
g
-(ortrer rorces in a dir-I, =
-(other.**:::,ri.-r'
1;
-,::1':-::f"::H
-r
.v
:-{
ii:t*''
i{
i
,,1)
*r4,
tx+
&>(
_{
_{
The
cootdinate direc[ion are atso '..i8;ned a.pcitive "ign lEig'a'a1'
thce acting on a -ve face and direct-ed
and
i.g",ii"
.
are assigned
d,
*M
_{
I
oE STRATGHT BEAM
EQUIiBRIUM equertoNs
t;
li
tol
4-s
,18 .J
)-'
l
-{
-{
-q
:
9't
:.
/jt
il
'i
i
i
i
a'
t
i
L-.-
*gn*"t are computcd and the N-t, S-z' M-z
The end values and the extrgmal values if any within "".h
results pt""""t"a eatlier is helpful for this'
diagrams are drarvn- The geometrical interpretation of the
and the constant's of intcgration
If q(r) is not a simple function, theo eqs(4-19) are directly intcgrate<i
de[ermined from the end conditions'
L
.,
L
L
t
L.
L.-
-L,
)
L-
-
Lt,
L.--
,
iiqfliry:ts;
I
,
:
\--
t-
.,
)
[,
t-..q
t-
_)
)
L..t-
)
t---
L
L
)
_)
tI *l
Li
lr
_)
LJ. )
L_
4.10 EQUTLTBRIUM OE TRUSS
so that their centroidal a'ies are
A framervork of tiangnlarsf,apes formed by joining bars'(members)
lruss' The joints ate idealised
concilt:rcnt at the joiis a'rd the apptied looils ect otly-aurlcseToirls-is-calleda
joints for space truss' The members are idealised as
as lringed joints for plane truss and as balt aul soctet
joink- Thus alt nrembers of the truss are two force nrembers
wcighllesscompared * 1L. t6rads applied at the
o[ connection to the joints' t'et Fi be
ttith the forces necessarily acting along the tin" 5oining their points
of a joint' the forces due to members are shown
the tensile force on the ith member at its ends. i" " fiO
r:alue for Il''
equations of tire joints yields a negative
as pulls on the joints. lf the solution of the eguilibrium
then the actual force in the ith member is compressive' '
j joinG alld having c -roun{atLn consttaints' 'Tn"
c,onsider a frame made of, 6 rigid bars, connected at
j joiuts is 3j (2j for a plane fran're)' The tcrms appearing
nuqrber ofcoordinaies needed to specify position of
t.t'""' The number of constrain[ equations specifying
iu ihe parenthesis in this section refer to a plane f.*r,',u 7
z^- ,,B')" = L2al'and the number of geometrical
the fixed length of 6 bars is 6, e'g'' (ca - 16)? *(y^ -y'f +(
O".tn: d'o'f- o{the
c' e'g',
constraiat equations duc to c foundatiou constraints is
'-e =,0'-UC:0'^Tt:
n=3j -b-c(n=2j -6-")' [ngeneral' lhe
frame- lfallthecousfraintequatio*s are independent, then
Hence n Z 3i -b-c(n>2j -6-")numberof independent constraint equarions is S(0f c)a mecicaism in which some individual member(s)
is
it
then
If a - 0, then it is a rigidJrcme and if a > o,
still
of sonre nrenrber or some founda0ion coustraint
catr move as rigid body. A franre is over-rigid if renror"al
frame is
foundation constraint is calted rcilvndattL A
malies it a rigid frame rvith a : O- Such a member or
constrainr nrakest' " "t:tlTl:T
fisr-igidif removal of any member or any foundation
of
forces at j joints is 3j (2j) and the number
concurrent
The number of equations of equilibrium for
unknown fcrces,in 0 members amd from "
?
condition for a' rigid frame is that n : 0' i'r
,er of unknown forces is less than or equal
numb
the
taat
lly
dctctminc{c'is
condition for frame to be statico
"
equatiotls are arranged in a matrlr
to tlre nunrber of equilibrium equations, i-e., 6+c <3j. If the equilibrium
the unknotl'n force vector [flte+"1 and
form: [n][Ft = [P], where [A]1e;]xtc+c) is the coeffiiient matrix of
nccessary
rank [A, P] :=16 * c, nhich is therefore the
i"f,"r, I ii" U.ir""to.. Irs solution exists, if rank [A] =
ooi snffcicot condition forlhc frome to bc stat.iullg dctcnninatc'
and su-ccessive addition of
A simplc spaliol (ptator) lruss is formed Ly a basic tetrahedron (triangli) so formed is just-rigid'
joinG- The framework'
3 non-coplanar (2 non-collinear) members to form other
are independent in this ese' tf the
The constraint equations provided by the fixed length of members
motion o[the trussas ":i9:j:e91t:j:
fotrndati,on constraints are also independent, then [hese Prevent
3j:6+6(2j:6+3)'
aodc=6(c=3). Eence,thed-o-f--ofsuch.asimpletrussarea=3j-6-6-0 =+
of ttre whole ttuss'
equilibrium
of
The 6 (3) foundation reactions are dltcrmined from the 6 (3) equations
considered nerd,' The forces
Tte jcrint which is the last to be formed during the fotmation of the "lls:-:
(2) equations of equilibrium for
in ohly 3,(2) members meeti'g-at this joint Je'determined frbm the 3
and solne its 3 (2)
fi" next clnsider- the last but oni joint that was formedjoint'
the concurrerrt for""
"yrt"**!ltT:*d that
ite procedure
equations of eqtrilibrium fior the unknorvn forcesin tie3(z]membe*
which qeated that
is repeated for the last but two joinL ttiat was fiormed and the fiorces.in Q-*"*O:rs
'
all memberc' Eence'
joint are determined. This sequential procedure leads to the determina:io1 o[ forces.in
yr.l rT:_i?TIr.
asimplctntss-roi{[6(3}ilrdcpcndcltfovndatio*co*sltlirttsisstoticallgdclctmilatc...'.
@
-ro
fi:::*j:I
i
1
:
1
I
i
!
i
t
l
I
i
I
I
i
i
i
1
I
itre
Es
\t,
ry' . -\
.r-{r.
,-F*
'...r f
op-y
nn
Fq
u!
:5'
- - sls.q.b
--Cig.k.6
F; is negative'
if *: *l:T": of
Ia the jo,:nhwise'deresrinarion of meniber forces descr'rbed .{qve,joints in whic'h it appears
-anv subsequently'
of equilibriumoi
then this negitiru rrdu.l"l"Urri,r,.a in all equations
plane
tbe special case of joints 'A and I of a
consider
pnlts
Ft.
oaly
The FBD's of all joints should shorv
joint r{ yields Fz - 0 and
applied lload' f' = 0 for
truss, shorvn in Fig.4.6, rvhich are not subjecteJto
4 7' 8 are zcrc-totxe members
F! = 0, & = 0 for joint B yield Ft = 0, F.l = 0' Thus members ,r,cmbers
of a plottc {rzss' For this purpose'
o
Sometimes we are interested in littding forces irt iusl lcro
it into tro parts by cutting a member of interest
a section is taken through the truss so that it divides
aodgnfenblyoilyluoothermemberswhichott,nolconctrt./rlltuithil'TheFBD.ofthetrusstooneside
sectioaed' using the equations of equilibriunr
of this sge.lion is dra*'n shorving pulls on all the meiubers
often convenient to select A at the point of
Ml:0, F" = 0, .F, = 0, lve can n,.a ro..o i,. l **b.o. lt is
above FBD' then these
ficundation reactioos aPPear in the
concurrence of some unknorvn membe.r forces. [f
l1
l-
...:-
t'i
--.
,
p,lr directed oppcite to the impending r
I; ( prJv- For impending slipping rvithout tipping of a bod1" '=' .---.^^4
impending 1..
simultaneous imoendinq
For -i*rrtranonrrs
othe area or conract- D^".,"r.o.
tipping aod slipping, F = y,N aud .tr[ acts at a corner'
load is int:teased, tne assume
2- Ia ordet to decide rvhether tipping or slipping occurs first as 1he applied
is
'that it tips 6rst and find.4 N and .r,".t ,ui.tt* tat 'S P,/V- If the c'hecli holds' then the assumption
of slipfing-takingPlace first'
cortect- Else the assumption. is *.rong and rve ret'ork on the-.basis
a prescribed load P' we find the values P1
order to decide $'hether the systen'r is in equilibrium undet
-1* I*
tso directions' If P lies outside the range of
and Pz of the load for s,hich there is impending motion in the
ittremains in equilibrium'
Pz. then the systern does not remain in equitibriurn, btherrcisi
Pr
"nd
and no slip at another'
one
-A:"*:
4- C,onsider a body'rvith trvo cont-acts $'iih the po+sibility of slip at
opposite to impending'stii: direction] and
impending slip at contact 1 (say) [i-e., /V1 and Fr = P' r Nr directed
check wlether'the a-ssumption
equilibri"no slip ar iontact 2 [i-e., /Vi and Fe]- Solve th" "qrriions of
"n9 a-ssumption i'S correct and the
then the
of no slip holds, i.e., check whether l&l I FrrN2. lf Lhe check holds,
with the assumption of no
olution b over. Otherrvise, the assumption i *.ong and the problem is tervorked
'
pe,N2 directed opposite to impendiirg
slip at I [i.e., /Vr and F1] and impending slip at 2 [i-e'' N: and F2 =
slip ditectionl, and finally check that lftl S P,1Nl'
including atl the fric[ion
5- For a.system of bbdies with multiple contacts, count, the number of uuknorvnsslip.equals the difference
' fotces as'the unknorvns- The number of contacts at wtrich there is impending
of all the bodies'
rhe number o[ independent equilibrium equations
of the number:of ;;;J;;*;;;;;
stip, and employ at these contacts & = r'jffi
Assume the specific:1oJ",., .f these contacts rvith impending
.; t*n"nutn, slip as obtained from assumed kinematics' Solve the f,quations
dtuect€d opposite a il;;;;
at the contacts rvhere
of equilibtiu,m and these kinetic conditions ani thun check whether lql< t'jtntY:
has beun obtained, else r.ercork wiih
no slip is assumed at,the beginning- If the check holds then the solution
;;;;;;";;.
--<
'...,,<
:'r<
!\
.-
I
Thus1/rcteaclionPoothejottntalisla*gctttloacitc,leofrcdilsr=.Rsinc.
4'7 il
INYvrOf,\/rING FRICTION
4.12 rIrNTS FOR EQI'ILIBRTUM PROBLEMS
- t r--^^
-thenormalreactionNisatacornerandt,hefrictional'::""\.[A
'{:r- For tipping rvithout slipping of a body'
i,i#I;,iJ'#T',:[
I
-1
4.11 N.EACTION AT A JOUILNAL BEARING
journal beli-ng of the same radius (Fig'4'7)' The inevitable
consider a shafitof radius R at rlst in a short
(radial) foice,v and atangent'iatfrictilonal f11ce 1'N
clearance results in a li,e contact *'ith tle no.rrr",
to Nn *'hcre
Luthe bearing' The tot{ reaction P is at an angler'c
relative to
slip rera,Ye
about to sllp
is abouf,
rrhen the journal ts
'Tlre
.^m
tha
..'is Tlre
a-xisthe
r
tr
from
distance
,ut at sorne
some dBlance
1P
p
act radially but
o _ t*_ip, is theranSle of friction- Thus does not
j-'
b1./mourerrt M of the reaction about the a-xis is given
" 6 :- ^
;t {,;Itf I
r=(P'N/P)'R=Bsina -i+)l;
\
-i.
-l
1
of the whole truss' If more than 3 members
als-o used in the solutio''
=
:
,lI
equilibrium
should be determined beforehand b1r considering the
of equiribrium are
equations
irr"i.
made.nd
are
are cut then additional sections
;;=;.^;:;'.o
1<
:\
1-1
.:
i
+
i.
i
1-rrr
I
.-r
:
;
,i1-
l
?-)'
-..<
.,<
.jt
<
:
:-a
.a
99
\l
,
r'' .;:--''
- "'1
...
'
l ,.
lr :'i'"
Exaraple 4-1 .A .0iuck with e-ear wheel drivte:;'- i'";
(Fig.Bt-l) has a'winch-mounted al its back: 'Thii's';
winchi!: 't-'i."':'
driving torques on the rear wheels and on the
ti-"1 :'
are ?1 and ?2. Assume no slip for the rvheels"itte'
drag force
coefiEcient of friction for block 5 is p- The
c1u4 and
wind'
are
on che truck and block 5, due to
ct'c2 are
c2us where r{'u5 are their velocities and
system
constants. Derive equations of motion of this
l_-
rj
rJ
{
using c aud
l{
l-..I{
t{
IJ
Solution
f:. (+r*q
E)(AMPLES.4
,',.1-
.ri,
l.-.
i<;\:
as generalised coordinates
3,a>f)
G.vll
F;3- e q-l
rotationd-R of-.he-r*'heels
of the truck a,d tt u;.ot"ti*7 or the rvincrr cause
ilR and the velocity of block 5
of block 5- The angular velocity of tlrervheel s k
The di"ptac"nr.nt
r
and displacement (z * r{)
is (i + ra)- l{re take the initial configuration
and potettizl energy are given b-v
o?
tnt =ytt"*
tt tlre datu.r for pote.tial
energv
Y- The ki'etic
l.d]=
6d2]+ ]mo;' *.l*'tt +
+ I2(i/l?)?J+ jt"rii
+
r1(i/a]2]
][,nzi2
![mfi|+
,*+''ut16'+ msri'i'',;"'
m?:+:m3 ] ma'*rn5J {t' i lepa"li?'+}(rr
*
=![rnr
0
T=
/
-v'
' '::i'l
: (ti1 'r ,n2 + .f,3'1'' nta)g'z iin O 'l- fisgr(z * rd) sitr
(1)
(2)
'..'
5 moves up by (6r -i- 166]- The
rvheels rotate b-v 6tlr-and block
rhe
5r;6o,
displaceme[t
virtual
In a
do virtual rt'ork on the rear
The driving torques T1 and 12
:
prn5gcos,.
pN
is
s
block
on
frictional force
do no I'irtual rt'ork sinpe it'
torques acting on chassis 4
opposite
Tlre
respectively-rvheels and the wincrr
frictio'ar forces on the
trlo.r 5 are c1i a.d c2(i + rp)- Trre
tra'srates- The drag forces on trre truck "rrd
Hence
of points of con0act is zero for no slip'
rc-heels do no *,ork since the velocitl.riXor + t tdl.
es 0 (62 * t 66) ; c1i6c' "z{i+
. -, 6iryo. = Tt(6r l. Rl * Tz66 - pmsg
1611166
Pmsgr cos 0 - t:(i +
+
ri)l6z
[?r
ct(i+
ci
cGd
- = W I R -pm5r
s
Qi"
*
Ut + I)lR?li + 'ns"d'
ff =yl.r*
rn:
* m3 + ',,{ *
ms
= (rnr{
rn2
*
rn5)g sin d'
dv
0z
=lT2 -
=vrl R- pra59co6a : c;i:'c2(i+td['.'ar
,rr3
+ "r.{ +
.#
fi
pm3'orcosf,
=(ra *msr3)d
L
L
[rn1
{ rn5ri'
=,r,sgrsino,
*(#\-T|-X=QZ''
*rn2*me*m4*ms+(Ir d h)lr1li*msr6,
^
* (ror * arz * rns * rq * ms)g sin 0 = T l-R'
Prr,sgcos 0
T:o.
ctz
T-s.
A.E,
':
The Lagrange's equations' of motion are:
*ff)-**T=e2",
-c3(i *te)tl t3)
-
i
.:.::
c1i
-
c2{i + -d)'
(Ia+m5r2)6+m5ri+m59rsin0_Tz.pm5!!tcos0.""(i+'d)"
U
U
L
L.
r,
L
:
-
A
m1 with a thin bar of mass m2
consists of a thin disc of mass
Example 4.2 The System shorvn ia Fig.E4.2
by a linear sring of,s;tiffnes
a horizontal track and is restrained
on
slip
rvithout
rotls
disc
The
it.
to
pinni:d
the d'sc' The springs arc uodeforned
-rs
connected bet*reen the rod a$d
E- A torsional spring of stiffness !r
motion'
rvhen 0 = 0, a = 0' Derive the equations of
foo
l,
The frictionat fegge -on ihe rotor does no rvork since the I'elocity of
the point of contact is zerq. The displacement z of the <iisc cenf,1e aud rotation
Solution
d of the bar, imply rotation 6 - z/f-of thedisc. The angular'velociiyul of
the disc and aagular velocity ar2 of the bar are given by u1 = if R, u2 = 0
and zA = u* f,sind. The telative twist of the torsional spring'ts $ + x/R)'
The virtual displacement 6:c, 60, + 6xa = 6x * Lccsl 60, and
6Wn'
:
P 6z-a : P[62 + Lcos|
ei" = P,
qt'=
601
=
Q'fu
5).
I
+ Qi"60
JJ
J
+
J
J
J
Eis.€tr.z
PLcosA-
The velocity of C, the kinetic energ,' and the potential energy rvith datum at O
are given by
ec=lLa .dk * Be=;!-akx (I/2)(sindi+cosdi) - (i+ ILitcosl)t-*Lisinei
T=llllrrrgzltilRl' +rrr1izl+ |nr2[(i +*Li)cosllz +(*Lisin0]'1+*(*rtz1tz'1P
= l(3m1/21- n2\i2 + {.n2l2l6)it2 + lntzLii} cos?
V - +k* + +tt(d * rl R)2 - f,m2sLcoso
= 6*rt2rrr.z\i+
ff
lrr.:Idcosd,
{ =
ff
X=lr1k{o+x/R)/R
rnrLzlft+ l,n2Liccso,
= t ,te + x/ R)+ *rnssl
The Lagrange's equations of motiorr are:
d r?Tt 0T aU
d
OT
AT
o0
AV'
a
J
:
= -f,m2liflsinfl.
(l}
0
= PLcs0'
i
AI
(2)
4
,Exarnple 4.3 The system shown in Fig-D4.3 consists of a rotor n'ith axial nroment
of inertia / and two bars.pinned to it. Each bar has mass ,,1 aud the mornenLs of
inertia about principal a-xes'g.at the centre of rnass C ate'. I1.,lzz,Ias. The bars
>i
\
i\
I/
'
vA+ s2'x Ag --
(u a u, x o A) * ttz x &-= 6 tx
= -(B + csin 0)d & + aa cosdi+
o2s = (E + osino)z$z + o202.
o0sin
gi
I
tol
Ri
* (Oi + o g x o(sin d [ - cc0!)
9L
d
are restrained by two torsional and one linear spring. The springs are uudelorrned for
0 = 0. The torque applied to the rotor is fi and an internat mechanistn ipplies lorque
7i to each bar to raise.them. (a) Derive the equations o[nrotion and shorv thaL these
are directly related to the.p.rinciples.,of, moment of momentum and sork-energy. (b)
Obtain the first integrals o[ these equations. (c) Find the angular velocity d of t\1
rotor for which the bars make constant angle d6 to [[e vertical for T1 = ?i = 0- (d)
iConsider the case of.free roiation of the rotor, i'e', Tr = 0' Initially q = uf il = 0'
way
thai
a
such
iu
bar
to
each
torque
applies
t/2.
mechanism
The internal
rvhen 0 the bars are lorvered ro t,1r" ,"stical configuration and 0 = 0 at 0 = 0. Find the angular
iD
velocity of.the rotor rvhen 0 = 0. and the work done by the internal mechanism to
change the configuration ofthe bars from horizontal to vertical- .
Solution Tle angrlar vetocities of rofir 1 and bar 2 are g1=, jj- and gz = 6i+0L- Hence gc' is tiven by
gi =
J
J
+
{3m112*mr)i+'}rn3l18ce 0-i2$n?l**a* k.(0 +z/R)lR= P
,n2L2612+ rlrn2tr(i cosd -. irisin g) + !r*2Li0 sin 0 * I;r(f + zl Rl * ]rnssDsin
n t26p*ImzLiced+ k40 +zlR)*|m2gtrsin0 - PLcqA'
F.quatlons (1) and (2) are the equations of motion of the system-
JJ
-0,
;lril-*+fu=Q';"'
a\7*)-fr+*:Qi"
:+
r?Tt AT
sin 0,
AT
J
.J
{
(
a-.
.\
JJ
J
J
JJ
J
J
J
J
J
J
JJ
J
J
--.1
J
{
r
1
The exteasion
"q*i*p!ne*,j-!$0:
The ki19!ic.+ergv and potcntial
' ''
energr
:
G;G;ilr.,iaa"i"il;i.'d':*..9^n,'i1x...,.:;.-.i1:..,.'.'...
t = iI i'+ 2(i)ft{(r? + ;i'* ai.i'+ a2021 + '.!' .f" o+ rzed? sin2 e + tr.itz1
='Vli)+rrrt'o+I:zsiq?0-+m(R+asin8l16'+lls+'rar.2102
acoso * 2,L'sin? o *
+z(+kcazl=
-2mg
= -2tngacosO* |qzrsinal?
f
(1)
ktoz-
(2)
reversea gotqYo
by the torques ?i on the 01" O.:t tlrl
In a vittual displacemen t 6l,6l,r'irtual :*.1 i" done
virtuat rotation in direction
1i actidg on the rotor do no virtuil work since the rotor has no
.
6Wn'
:Tt6i
1
Q{'
46dilSt6O-+Q!:6e-+--
l-
Hence
Qt' =zTz'
=Tl
l
:
qa0
=
2lI 12* rrr cc?
I
{ae =ilfu+
* I32sif d + rn(E + osin 0)?]<i
*OQ
A - 2*orsin0*
a0
--0,
5=0.
o0
2kL"sio?i +2:k.0
=o'"' +
*(#)-#-#=Qi"' *(#)-'**#
-
$avn*
r:,
*', * ,-y1-
Zlle*tt*210 -Kt"r-
siuof
)il
(3)
=A
11".::,": :rr:::::^-a,,1r
t
r11)$in,cos 0 +Znlft*asin8)acosa]t'3+?rnsasin
a
+2kL2sin2o azb,o:trz(41
the **T* about the fixed a'tis
Equation (3) is the moment of nomentum equation fot
at 'r1'
Equation (4) is ttre moment of, rnomentum equation'of:::t'ty::':'axb
fo.* ih"
6
fottowing equation ftom
I t2* 11 1 cosz 0 +
*, tt r-'- r, [Le
12{
'-
1
*
,i
The Lagrange's equations of motion:-'
(")
-
mazli
Izzsinz
.in
o
+
D0' i{st = i{6,''
:
cqs(3],({i: eq(3)C+eq(l)0
0+ m(E + csin0)2}d
2',.(R +a sin
0
)a cos
alit - 1U* +,,nailii
+2k':1!0=T'6+tTzi
-(Ir"-Ir1{sinpcos.0'+2n(R+osin0lgccd}tid?*[2mgcsind+2&f,?sinzd
trtrlr* rr; i";r,r +,r;,sins? +daii"ioon+6qg6,-.1;;fcisosinaiz-€ru#tsin8)acodltii?
i"'i' si"? + 2;u 6z'1'26;1 zff iil6' = t'Q + ?ui
+2[t$ *
i'l i
:+ fiUrn+frrcos2 0*I22si,:0*rn(8+asir,efl6'
'
rhich is the
--
-1to
+'ro91162
-
work-energy relatiou for the whole systcm
have been writtcn
utr*ti ;;
; *"-"otoirro-*t
=T'i
ECuat-roT^(3)
rm equati'on and wort-energl
syst€$r. lntegrating eqs(3] a,od (5]'oace, we get .. '':: '. ' ;-:.
t
I
i
zmgo ccO +2&f,2 sins 0 + *ta1
in the rate form'
t
'
adt{5)
t"luitT for
i
+tFzit
1
fould
the shole
,
a(r'tV) =Wo'
l
i
(6)
i
L,-
)
L.-)
L.h:+
L-
r)
'l
i
I
I
t
a
.lo5l:-,
f
L--
i,'
t)
_(c) Tr=Tz=O,0=dq'"]\
Jf,
-
i
.:
i
= 0,
lzmsasina +2bL2 sin20
;-r
t--.d
t
= a, and eqs(3),(i) yield d - constanl = fs (say] and
--
*2k.01/ IUzz- r11;fi,racosa +2nr(E+asind)aerdl
(d)
LeL u2 be the angular velocity of the
displacement from 0 = r/2 b e
-0 to get
I
rotor rvhen the bars are vertical. We apply eCs(6) for the
-.<
i
1
+ Iu + m,B2)utt - Z{I 12 * Izz * m(R + a\2)u1
t t2
= II /2 * 122 *,n(.R + a)!q | (I /2+ f,, + mE2)
t['"c - (T + Vb _ (,l+ rz1,
J^Ut lz* r- + m(E + o),|-i + zkL2 + k.(r
/2)?l +
2(I /2
=+
(7)
/z*
lU
r11
*
ma\,.ti
-znzsal
-
(8)
The rvork done by the internal mecha.ism is given by eq(g), *,lrere a,3 is given by eq(7).
a
:
Exarnple 4-4 Atl instrument to monitor conditions iu hazardou,
(t-S ,r,2
B l*
"ntironmentispositioncdb1,alazy.tongtypeder,iceslrorr,ninFig.E4.{a,lr-vappl.ving"M.).jlE1
torque.tly' to the handte of the bar zlB. The mass of ihe instument i",r. und
mass of rods is
I
*:::,^:.j::Y
";;.* *;, *;;;;;t;;.
kg/rn. The pirch o;,h"
t'::'"",::t:,Y"*'*T:tion-,Sor.tilrheslfl
ihu*t
one
tlrey-coordinatesofthesepointsofapplicationare.<
y3
=
(@
+
2b
+c)
=Zac-rx,l
:+ 6(A8r=
Applyiug the principle of virl,ual
c.r./:\
c,*/'..
'
rvork
c ' i\
;1Zi
sin
-2asiad60
6W
-
b3,
considering
- p6gl2r :+
6d
-t'.\
/S:--\
-\m
*k#."
}t.'
= _4rcsinl60lp-
=f,Frr6y;* l{6ti =0 +
+ 4\bg 6y2 * 4\cg 6% + 4\dg 5y4 + rtug 6ys + M6O O. .
=
[Ig{o' + 4b(o+6) + 4i(a +2h + cl+4d(a+26+ 2c+ d.ll
+ory(o * 2b * 2c -t- 2d)] cos 0 - Qtralp)Atsir
2\ag
+
i. :,
\:H
$ir'a'{,,:li{-u
"::.:::::i'!;^
-48
i.
X\,
1
0,
\Zzt
e+(o* 2b+2c*2d)sinr?. 7\
or u oo; j:" =^," *"1
:
:': ::"''
dya =(a +2b+2c+d)cosdd0.
-:
d3r5 =(o *2b+2c+2dlcosl69. f-;lf-\,'-.' -or
-, '--*Y.'"! slr5
-tst.vt.cizarcosttcv.
y, = |a sin 0, yz = (a * 6) sirr d,
y. = (o* 2b+2c*d)sing, Us
=
The rotatiou 6<lof the screw ca' be ecpressed in terms of 69
:+
.-'i'
;il ,i,i
lras
degtee of freedorn- Let d be the ge*eralised coordinate. Forces *,hicrr ,ro
nork
act in the y-direction at poinG I to 5
displacenrent sd,
f,d,
{Fig-Ea.4b}- For viriual displacearent
+
t)fo
A
6y1
M = p*10$s{o' ++6(o +0) +
4c(a
*
2b,q sy
The torque llf decreases as 0 increasesExam.ple 4.5 Derive the er2ression of the forie
P aod the moment M required to maintain the equilbrium of the pendulums shown in Fig-Bl-ba. Tha
springs are undeformed for 01
- 02:0-.The pendulurns are modellcd as uniform bars-) ,. ,.,
r
*
=0
4d(o*2b + 2c+ d)} +mg(o+
3_r
Mi
0t 60
V.)6e
2b
*2c+?i)l / 4ra
*[*
{2-,,r, s:.q-{r
trrI
i
k.tez-e,)
K<,,, '+",1*f,{le'-o')
Solution llre shall present three solurions usiug 6W - 0, 6W"c - 6V - 0, and 0Vl0q;: O: i
l- The extension of the spring is 0sinds. The rela[ive rotation betw'een the ends of the torsional spring
connected to the pendulums is (gz-0)- The forces acting on the trvo bars are shown in Fig.Bl-Sb- Note that
:
the'pair of moments due to torsiooat spring does some net work on the system during a virtual iisplacement
-:
J
)
l
-J
!
I
!*o_3
J
J
-
,
k.'\r
tj_
lj-1-^
t.
l'
u
tJ
0sin01'
y1
1* L2sirt6z, 6gy= L1cm0r60r * Lzccgz60z, =
lrr.in e, ce, - rlz;,.02602, 13 - iLrcc d1' 6e3 - -iL'sin
='r'r-Jr,* +rr.o. rr, ,r; --
,,.
n',- r.rdro,
fotces on the bars
The virtual work of ext'ernal and internal
0)150r+ [-e{02
(-I;Dsin01)6vr + M 6A11 F (0' 5W : P 6st* m2s 6x7* r..ts 6zs+
Ae'
q
+ t[ + r'V, -.
m2)s L1 sia0' - &0'
.!.;\
'- "''
":i ::"
= fP Lr cose L - (m 12 +
Y 6i0t and V60:
k'(i02-'A)1682:O
0z\rrr2g/;2si
*lP,1co;021
Hence, tbe coefficients o[
A
r
(n1 12
tr
i
d1 601'
I
O'}FA'
(l)
-
ie-'
*
m'lg Lrsin
01
."u""r"n"r.i:i
7:Hltx"lT:,f:[is:.fu
as.-,v 'using Y:
sprins and sravilv forces is orrtained
*
|f,2
cos02)
8r)' - \*'gL'cos01 - mz9(Ltcos01
= !t(6sin9r)' + i&r(0, 6tl:&6"sin01cos01 60t+kr(6t-rdtX6gz-rr,l*trnrg.LrsilrLrlar-rn2$(-f,1sin04-|L2sin036d3{4)
rz
f,ence 6W =
6Wo"
-6V - P 6w.+ M 60r-6Y70d{ibetl
6att
from eq(4t'the eq(t) given
vidds' after substiruaion
calculation'
"*ti' . ..earlier- The test of the procedure is same
of Yirtual
as constant f6r the PurPose
1o1k
zretre.ied
p
M
moment
and
force
3- The Siven
In thai
sease
as
tgitlr
P and M ate trea'ted as consert"ative
* f,tz'G'azl
l*'gL'"T": *2s(L1ccfi c661
- *rn:g& coo:
*gi;1e* - e'1'- (t'" + 'n:)eLr
v = -pyt- araflrto"indl)3 +!E (az-l-rlz -
IJ
L
Lr cu
momegt
1L
l_
L-
should separately be zero'
t;
i.t
II,
,t
I
(2)
&r(0r - 01} = 0
- &62 sin 0r cos h * i{ *
(3)
01) = 0
P Lac<x,l2 * !m29 I'zsin01 - k'(Az the
eqs(J) and (f)r are*respectivel-v'
**
is
lf
then
and
P
r*-rrorc'
(3)
ef the
vietds
Equatioa
:o('|
rnomenr equ.ibriu'riqulq
-obtained
^I:0l::lt"'
J th<j,
equiribrium equati,on.for lr. z
P
L
L
l__
L:
501' and' 502
-
il
:i
;::'f$:;";::':::':XiltT"iilX"i::T:S'ffi1':f;::i:';:""':i'i'1d
6yt.= 6cosd1 6d1'
tJ
L
IJ
L
L
l-.
L
L
L
IE
4' whereas
TJ
lJ
llJ
IE
,ti
L
tIJ
.t 1
-
= -P(.L1sin,r *
,r
,, *Tf **
+ *t(osla
"iJ
arc co.oscraatioc, theait is most corwcnieat
to
asc
!e
E
Fauations (6),(?) agree with eqs(2) and (3)'
titted up by tbe force trL from
Exaroptre 4-6 The platform h Fi8-E4'6 is
are light' Find the rralue of P
the pistoa of the hy&iulic cylinder' Tde hnks
for given
(5)
lhc c4uilibn.,- eq.,alio,.,x ovldqi =
0v
pLrcos6l*f62sin01 cos01 - kr(02-01)+(|rn1 *m)gLtsino1 =0
=-M
00t
40
aV '
=
a
-PL2c<x,01t t'(gz -g.)+ *mtgi2sin02
do2=
'
-
f,e sindz)
0
(6)
(i)
c
t^t
i
jr-OH=O
gi:c'oA=AG
for 0 - *14'
Mt, Mz, Mg' Tte springs are undeformed
as
resoh'ed
is
E
treai A,Ur,Ur,U, as constants' Force
rralues of R.
- Solutioa
[cos C,sin
We
R= -RcosCi+ Rsindj *ith
(a sin 0 - c)l/t(d C) - [(d - a cos 0),
a cos 0)2
*
(a sin
I
-
wich datum'at
The potential eoergy of a constant force F
'l]"
cl"lt
12
"t:Ot.:ft,"
rhe
*r.;;fi;:';acts ar, G- ff"
to'r{r iswlt2and
*.H;5:;1ff:A;.:fi"rffffi;.I";J'i.;-"..*aie
t64
..
I
.d f)l
.:s.-
',i,
:ir-:.
-....
,
:.
potential energf of a moint6trl/, aqtiog on body f in.positive
of body i. For the given one d.o-f- system. V is given by
;
direction is
-M;0;
where d;
b the rctation
ii
,, {iiliJ
v = -(-W)va - (-P)" E - ?Q * Wtl2];v{- (-M'X" - 0) - (-M'zXo) - (-MeXd)
i l-l
++t ,P'- ol4l2 -(-Ecos{)eg - (Esin{}v6
=-(-w\(zlsind+6)-(-P)(c+2acq0) -?a.yw1l2)(4osiud)-(-Mrl(". '*0)-(-M2Xo)
Rsin{(osin8}
- (*M';1{e)* }&r,.[ a] - ..l2l2 + ][,,[0- ol4l2 - (-BcosdXacos0) + 1e',[. ' : 2A
-
The equilibrium equation
ry:
a0
i
p
-|
*1212
-
qF-
&L
".."
*Y
,!
in Fig'81'7a' The unstretched
if the handle-srltfjo'*ll1 bv equal forces
'uhe nrechatristn st:orvn
* | r .. r
I Yt
"/r;ffi'\
o,tr
c.sc
tIet
Solution
obtained by
U,ty
a1c --
u2b '+
''t't2
=
u{lb
The componen0 of the velocity of poiut D in the direction of force P is
virtual rate of rvork Yields
6W
; F'ri+'t-(a -
Ln)u,6
- Pugafb = Q
uza
= uPofb. The PricciPle of
P:1ri+*lL-
.t
I
-t
-..J
ir
--i
!-J
I
j
I
*riting the r"elocity of l{ in trvo rvays:
.
_i
,-J
Che
the clamping
0 as fixed, a rircual angular velocitl'r'r1 is
be
most.onu"ni"nt..Iteeping
to
in this problem, since it happens
of the forces are shown
giten to link 1- T11e virtual velocitl- ccmponencs of points C and r1 it rhe direction
instantaneous ceni're I: of
in Fig-pl.?b. From sytnmetry, the virtual velocity of poini B is along OB' T'he
angular velocrry o2
virtual
link 2 is obtained at the point of intcrsection of the normals to 3L and 5' The
'rs
I
,l
|<- a .af--.-1q|i
principlb of virtuat rate ot *-**
force,be P- The application of
--l--o --'r-1
I
t
A-'(
a ---k- c ->t+- b
t.1
I
.Sl--
ta
I
I
.""1!, n.Jj,',1"","',-'
on cjre,obiecl
I-.J
I ,_l
i
T::l;:',;j'f;:";,1^r;j
is .Lo. Find the clampin*
tength of the
:plrrc
friction.
Neglect
F(o)
of link 2
i
ZWoas|-2Pasin 0 +4(Q-W1/2)acos0- lt1* t43
(l)
t2t,,[ 20 -x/21+t,,10-n141-l?ncospsing-Rasin{cosd=0
Example 4.2 Inacessible objects are clamped by
rf-
l:
TV/Oq;:0 yields P:
2wocosd + 4(8
l-,
L.
.L,
Ldlblac
U,
\l
-a
-t
.l
I
J
-,1
l
!/
-t
--l
\i--j
--J
tl
J
\l
J
,I
tl
J-t
*l
!
u
I
,]
curvature of the peripheqr at
of the top cylinder above the point of cootact is h. Let Ot,Oz be the cenires of
lo5
I
_j
-.j
..J
..J
J
Ll\l.
1'\-
.: . -;::-..
rt. ::-.
/
l=-
)
L
l-
L_
t-
L---
LlLIt
L--
l'l-
1-
L---
LL.
ILJ
tj
LJ
IJ
l.J
LJ
LJ
L-.
TJ
LJ
u
rj
t-u
t-:
u
*ui"T, P<il{4?'c.
rhc point of
"r"1""i tL"4qp-:?*4;i*-ti""."f
small rotatior (Fig'E4'8b) when
-ry..v,-?,.{l*3
:i;
cont"cii" D with z3oti=e-T.u"olup
"*aitiolimji[es
$7fualR2
BD=DA :+ -''R2Q=:Ro'tr'7n *
verticat' 3nd ihe tine Ab2Q is at *"81e
In the displaced position, O1,D,O2are collineat at 4gle 0.with fhe
with datum at Or is
0 + d wit; the vertical- The,potestial eaergy Ir of body 2 of mass'rn
(I)
v(0\=mg(orol)ced+(o26Jcc(0+cI = ms[(fl1+'Rz)cc0+(lr-82]cos(L+ hf F.:)al
v,(0D=#=_*onEr*&}siad1(Il:a?X1+Rr/8:)sin(L+h|.R2}0|
(3)
+ RylR)z cc(t+ Rtlb)el
(r.r
v,(0) * =-me(Er *.82)c0 + -.R?Xr .-.0 as the egui[brium configuratioa' which is
using eq(2), the equilibrium cmditioa v' = 0, yields 0
y"(0)
otherwise obvious- This equi[brium
pcition
is stable
) 0' i-e-' using eq(3),
if
ntlez|z]= '#(Er +Ez)2[RrEz
if v"(0)=-mc(Er aRzl+(r.-R Xt+
i.e.'ifn<Rt&ltRr*Ez)]=ll$lR|+l/Ed;....Ki
/(Rt+Ez]-al
decided by eraluating higher deriratives-liace
.For rhe casetr
=.E1Br/(Er+aa), tue statrility is
0- For this case, eq(3) +* /?:){cos d cos (t + ful R:.)9l
Vt'(0)
= -rng(Es
Vo'(g)- me(Er + fi2)[siuf
\'
L-.
L--
(ric'F:'8a)- Mark th"
on body 2. @nsider
LJ
L_
LJ
LJ
i]}=*: .*--4qonom!&tio ,0;0
rhe points
-
-
(1+ E1/82)sin(1 +
v"'(0):o
yrttt(g) : *c(11r + E2)[cos0 - (l +
V'* (Al = ,lr.s{R1+ 8z)[l - (l + hl
>
0
Y"(0) =
(5)
(6)
&/Riel
(7)
Ri&zl2 cos(l
*
(8)
P'i.lR2]01
-1s-)
Rzltl < 0
a local minimurn at 0 = 0' alld therefore
follows from eqs(7), (9) that'tb,e potential encr8y v does not have
the equilibrium configuration is onstable for this case'
equilibrium configuratioa seduces to
For the particular'cases given in Fig3l-&, the conditio"jn) :t:t*le
(]'is belorv the centre of curvature OaI . Br : oo: siable if h < Rz,Le-. if the ceuire of mass
2. Ez = oo, stable if L < Er3. Rr = -R: stable if h < Rzl(l - RzlR).
It
*
4. R2 = -k stable if h.< Erl(l - &lR).
'trr4
"
stableifll(0,i.e.,iftheceltreofmassisbelos'thepaecteoint'
5. 8r=0orr?z-0orEr=Ez:0:
lJ
11 !
?
(fiS'S+,S1],
;
at
A
suppo*
the
Exarnple 4.9 'The bar AB is piaaed to
c
A
tu
:.*,-,"';;%:;:Tfi;":is Ij:.jl,Ll.i at u:J:;"*;la;;]1('
-&a 5 is ^u^mY,
D
to bar BC at B. The bar BC
which
pioned l'o a roller
and
guided in a vertical slot- The bars are coastrained by five springs' Find
i
".+6f' *.
\"
I
"!L1
X"
;;;; ; ;'il';'";;;; ;*.iia :cquiriirium coonsurarion fl_*aeg, -r^/
i4}u- @u
"
of thesystembecomesunstablellki'B\p1'1lt<rf|
\1,-.
Solution Inthedisplacedpositiono€thesystemGig'Ea':lb)'anglesa '" f,I 1
{y
\i*
r
and { are related by
oW,
(.)
-r
,#' 6 A
,,
zl'i
-$=a|tb
8E=Dsind:csing- :+ uetg2ld.ordertcrmsip+ug
*"'*'
&3 Lquals {t + 0 * C) - r = O*The telative rotation betwecn euas of.the-tprsioaal
' -.' I :
i
:
s"H:
potlntial energy I/ of this conseratirc slqtem,,accurate.upto 2nd order terms. is given by
v(0) = ikLoz +2[]e:(asi!0)1+ ].rs(a + d)2 + *br|" + P[c cc0 + (0 + c) cosdJ :
+ Pfacxx? +(6+dcos(crela)
= tr*re' +-ezicsino)2 + |r"1e +allblz +^if7"otu12
. = i[*, + [.(r + o/o)' + e$2 fi1e2 + &242 sin2 d * P[a cos I + (6 + 4 "*Gflf)J -''
-
v,(01=[Br+ts(l +albl"+k*2lb?\rt-rtzc2sin2d-P[csill0+(0+c)(c/o)sin(adlbrl
v"(0)=[&r+ts(t *alb)z+k*2lbzl*tkza2cs20-'Pfoccxl+(0+c)(c/0)2cos(c0/!)l;.'":-t*:-i:...
-:
=.
19_6
..
,l
-t
..I
,tl'
'ing*ition'f,=o'
For the eguilibrium !o,
,I.,*'..r..,:: ,:.-
,"toii
foi* *"(,
0
The equilibrium configuration =
v'(0);0 (o'k) ald
(,
'1
t,r
:
\
,:
pla*(0+cXglq2l...,,.
* alb)z. * *oo21b2l *2kza2 -_
O is
stable
rf y"(0) > 0'
i-e''
t
ti
(
(r,
if
l:
J
J
P<tte,+&:(1+,,lbl''v*4o",1021+2k2a2lll4+(6+4(a/D)']'(1)
load Po:
the
minimum vatue of
p at which the equilibrium
is cated the critical
positiou becomes unstabre
eq(u:+=[{&r+!s(l+o|b\z+kqaz|b2l+2k2a2|lfo+(b+cXc/6)2J.(2)
If
I(a = 0,
=
lli;;.ff; I
&1
u,:,',J.-T:::::Jiff-:r'#"T-ril1t';t;l'ilT:"'*
orthe reg is stabre ir
" =:' '=
:L-,,,"";'i*l;,1;;;A);;'d;;;;";i*ongu'"iioo
I
,/i11
-- t,
ry
Zkla i'e'' if k > mgo/Z'
&2
= 0, te =
i:
&'-
'
ns <
}k,
rnsisting of 6 li(s
equrlru'ruru t::"f":l:l::::"i.L1':;ler
f,ne equilib.rium
!-rnd the
mas\
Exarnple
Exarrrple 4.10 Find
""-'r.'"i"it ity. Each smaller links has masl
(Fig'E4'10) and discuss-thethe
lengthof
free
ot""Ja*ther
All links are uniform and th<:
lit;;;;;4m'
bigger
each
rn, and
Siil:;:Tl ,ri'iltlit*ition
of the springs I;1
d of trre svsrem, the renscrr
*,
thisconserl,ati,esystemwiththedaturnforgravitationalpot.errtialenergy,atoisI
(1toe 8e - ccc0)2t -2m1g(acos0)
+
Y(d) = 2[]81(2ocos 0 -2l.ceLs]? ]tu
0)
-
v@
2ar19(3ocos 0l
=;11;;';'
-
r'"f
=
{a| =2(4t1 * *2}sio"
0
=O + (t).
Thus the positions
@
at
'o' Sf
J
+ coso[4(2m1
sino=0 (ii)of equitibr-ium are at
J
J
J
:
*
m2
(2)
t']"'("*o -cc0e)l
a- 4tn)ga- 2({&l +
&r}a2(cos 9
-
cos
(a
06}|
+*:)trGos0-cosoo]l:O
[4(2m1 *rr.z*4to\sa-'(n:
:
V,
JJ
J
d)
V'{0} --2sin0(4&r *}2)ai(cosO sinf,[4(2m5l-ta?+ 4ta]so-2(4k1i
f,
J
- 2',.9(0'541e -^2m9(3'5ocsg):2(anr)gto*" ,r,
* mz * 4m\sacso - *::
- 1(2*''+4(2mr
t mi* 4rn)gasino
cos09)
*
m2si(aacoso
-
'
1' d =
0'
2' 0 =
a'
J
JJ
J
is '\r
:'# ? ;
[#"::i:;x::r;x;l;I,'iln
a;*;1',:!:.';
J
(4)
aud
2rnr
3-0,='at= cos-rfi2*r*mz*{rn}s/(at1+t2)o+9os0ol l. 1t
cosOo) < 0
4(2llnt* mz*am)g - 2(4&r + [z)a(l ,-.., ,,
*
m: i-4rn)9
/
(aEi
+i&)c*cc
:
0o
<
1
(5)
positions'
these equilibrium
''
consider the stability of
cosg3
if
only
exists
position
this
o"ios"o(1}*" condude that
since
1;W"
'
"oa
poqition'
v,,(0) = 4(2,o1* mz
1- 0 _
e = o l"'to unstable equilibrium
ao
th."'ir'(;i
position'
oito,
e,
posiiion
"t
a- if equitibrium
o = 0 is a stabte equilibrium
,-o
;;;"(o)
n&
3d ' oI<0'
0=risan
b. if equilibdum posiriou & $od
"r"r,
*r*)r;:';d;;;itJr;
iL "-;;i
o:
2. 0 - *: v"(r)=:[n(2!ry11*'+t#fid-);;;'(;
equilibdum Position'
3.
0
=0zz
'
using eqs(3) and
@- qs1l*
v,,{0sl=44rr+tztrfin1
*
(4)'
t^\nnn-t(4t.*&r
art+(z*r +mz*4m)s o-2l4kt*kt)c2(ce
t
03-cos gol}
''
= 2t(4el+&2fsin'es>
0
and heoee 0'= 0s b a statle equilibriur-n Pqition'
lol
:.\{
,
\
I
-
a
L-.
9^1
<.
ladqlYy:ttre'-zgrro force memb!* in the trusses showa in
./
Fig.Bl.ll.
(c)
2
L
J
tt
B
6
I
!-
A
L
Solution (.) By successively drawiqg the FBD of joints B,
7, 5,- 16, 12, 14 are zero force members.
{b)
BysuccessivelydrawingtheFBDofjoints
K, D, E, J, F
we conclude
that members 3,
B,C,G,F,D,E,F, I,if,.C,we.concludethatmembers
3, 5, 10,8,7, 13, 12, 15, 27,26 are zero force rnembers.
(.) By successively drawing the EBD of joints B , C, E, G, H, I , s,e conctude that rnembers 14, I [, ?, lg,
L
18,3 are zero ficrce members.
:
p,lsample
4-12 Find the forces in aII members of the bridge truss shown in
Pig.tsl-CIa.
?
R3
\-\--
E
EA
<-vL
/
',
8c
.4--
---4
,L.
t-=Lrz
0
9
lr,
:{K'
F,3
Q:
':.7ffiil-
Solrrtiorn The FBD of the whole truss is shown in Fig.&t.l2b- Tlre 3 reactions from the support
deibr:miaed from the 3 equilibrium equations. for the
-r
:+
!1
o.l
(o)
'z
I50
l{s, = l6rtr -4(20) - S(50) - 12(80) = 0,
l?r=90kN, 8z=60kN. Es=0
*hJ t..r"" under coplanar
F" =
J?s
= 0.
Fc
= Rrt
are
loads:
Ez
-20-50-80 =0
The forces in the membexi are determined by coasidering the equilibrium equations ofjoinG in lhesequence
A,B,C,D,E,F,b,G so that at eveqr stageonly two unknown rhembfr forces have tobe determined.
cc0= 4l@'+42)tl2:0.8,
o-24?s
0. F- = -Fr- Frcos€ --0 :+ Fr = ISS kN, F1= -\10 kN.,
Joiot B
F,= Fz-fr=0. fc:-80-&-0
+ Fa--F2--l20kN, Fs=-80kN-,"
foint C lf; = F1.cos0 - JIs cc 0 I fu
Ft = Fs* .Fr siu d * fs sin 0 - fe sin c = 0
"*o<,,
:+
0.8Fi.+0.9?01.F5 t!9,
0.6Fi - 0.9?01& - -f0 =+ fs = 25 kN, f5 = 103-l kN
=
Joint D
lI' = (&-&)*u,i = 0, Fr = Fz*(Fe *fa)sinc = 0 =+ f6=F6=103.1 kN, fe = -50 kN
Joint.E: F, = -so F7-(& +rs)sina= 0, f, = Fr Fro * (F5- Fe)coed =Q
I
&=-25 kN; F'ls=-80 kN
=+
Joiat F: f'=F.o-frs=0, 4:-2A-Frr =0 + Frs=-80kN, Frr=-20kN
Joint O: & = &s -l Frs * Ftzc<x,l = 0, =) Frz = 100 kN, Fy : Rz- Fl2sind 0,
=
Joiat r{:
:
coso- 41,'2+42lrt2=0.g701 + 'sin0=Q.g, sinc=
& = Rr-
F1siu0 =
:
I
a
j
I
:: I
).-rr
tcB
1
I
It
-'i-.,a= ;; r"i;l :t' :"';)'
I
1:
'-:"'-
.'
''
I
i
Joint G
: F*= f'scosc*(6t- F$cos010 '+ ' 0'01?3 = 0
2ud check
eq!"t]:"
J
I
:
arithmetic rather
if rve had used exac[
The 2nd and 3rd check equations wootd have been exactly satisfied
.ftaificant dElts. Tlrcre r*'ill always be 3 equilibrium equalions *'hich
tlran roundi[g-off the numbers,to 4
since the forces in these members ate
pro*ide a check on the solution. Members 1,5,6,i, 12 are in tension
the forces in these members are negative'
posiiive- Members 2,3,4,7,g,t0, f f,'fa are in compression since
15'of tLe K-truss 2
4.13 Find-the forces in rrembers 5 and 9 of truss 1 and in members 12, 14,
Exaraple
o\i
^g*ffiffii'Eff; ,Tffi,
,"
"'
,,,t1ur?ffi#,'*"i
,o
lo
30 /--K--J|A}\*.r*btt
3s*X9.
;i
ffi;
="",
*:7(!\;l'7(_
-r"j
1 + A/i\l \l/:\ o--'-=7,\i*.i
/l\:[';fr
I.j*o,',
-Bi-(
-nl
-
iJ
YZo..P ..S'R
*ffiiii uKffi
hffi
:;Wr', ^:ffi" *i
*,
;ffi.'SKSFYy^,'r
-:,
i.
t;l;f"
%-,i?,+""
q -#\
t"--t -{f-
'i
rs
i"';i''=
:ifi,f1TTf^
-r rlnk-
!
"t]t"-rtj['l.
J
- $5 $:.
equations
Ttre reactions are computed firsr from the equilibrium
l'[s:12R.r -20
x4- t0x 10+10x3-30 x 3=0
=+
+
+
F,=R3t30-10=0
\ts
rd;
(FS-E{-r3c):
of the truss
=20 kN'
.R3=-20 kN, lrcs| =2l(22 11z1r|z: OS944'
8F5
-20'x 8-30 x3*Ez x {=
t
I
i
,
I
t
kN
i
I
oblained py'findi$6"the''point of
Thus member 5 is';having a'tensile:,{orceio126,25'kN. ,f5 can also be
intersettion O of A;,Fe *a'*i.,g Ma =O:
8Fs- 160-90+10 x4=0':+ Fi=26'25 kN
It[s =
i
I
Rz= 10 kN' sin? =0-'4472'
F, = R1 + Rz -20 - 10 =0
the
is better to take appropriate sections thtoughSince fotces in a ferv membes are to be determined, it
the
part
of
the equilibrium equaiioqs of tie
truss cutting Lhe.rnernbers of intercst- 'f5 is determined from
Lruss to the left of section PP (Fig-Bl'13d):
:Fo x'1+30x'1*8r x4-'R: x4:0 + 4f'6+30-20x4- f0x4-0 :+ f6-22-5
f! - f<cc0*Re*Fo*30-0'=+ 0-8944& -2A+22'5+30=O * It--36'34tli''
:+ f5:26'?5tN
F, = Flsin0 * Rz-20+Fs=0 :+ -36'34 x0'4472+10-20r'Fs=0
iI\
I
&1
A{p
\
t
I
I
0 +
I
4 members' Eence we have to firstfld
A section ihrough member 9. sucL as sections QQ and,Rn', cuts
and using Mc = 0' or find fto by
the force in one of these members- \ff,e can hnd Fz by taking section SS
s'e can find r'g by considedng the
taking a section ?ll" and using M6: :0- Once Fz or Fto is determined,
Consider the equilibrium o[ part of
equilibrium of joirrt D, rvhere now ooly two unknown forces are acting'
the point of intersection
ou the right of section Uf (Fig-81.13e). lVe fiud F6 by tlking ntomeut about
t
t
truss
C of the unknorvn forces Fu,Frz:
= R1x 4* 10 x 3 - 10 x 2+ Fro(cos0 x 3 *sind x 2) = 0
20x4+30-20+&o(0-894{x3+0'4472x2)=0 + F16=-25'16kN
+
Considet equilibriurn of joint D (Fig-81.13f):
Fe = 25'16 x 0h472--ll'25 kN
Fy =-Frosrod - f.s = 0 +
Ms
:
:
:
.:
l
ro3
-l
L
]J,
L
L
L
L
IJ
t'
L
1:
IJ
2.
*
For the
l-..
l_-
IJ
IJ
L
z.di1s
--''413
e.{ (rfg-Bn'rs8)
+ sia, = 0-8. cosd = 0.6.'A"".tioo'ke
'Tlre
The
-- --'*--f
--not.conce*tent'
a're -^1
i'ti'i;Jiand anv three orthem
. - ro 22,
oo 23.
o1 bei.s
L-i-d
amarrrant
coocutrent
14, r',z2.z3 nith members r8,
;H"[};";ffi ]'[ ";; ;;"*;
ffi'"i'""II"tlitffiil:H*-#:;il
. ? ,t .-..-^ .^ -iohi of
af section
wtion
XX
XX
MB, = 0 of parr of the- truss to risht
;ii.*ffi;;;il;
"oo".io.
":$"?riff
t':
_
f11=40kN-'
Me"-18Fu*7)x 18-15x 12-15x24-15x36=0 +
forces
find forces in inner membqrs' we first fiod
This is'the procedute for outer members of K-truss. To
of the joini wh"t"-.':T meet- A section Y\'
in adjacent outer members and thea consider the equilibrium
0
at G' C'onsider equilibrium equation M4:' =
cuts members 8, 12, 16, 1? of which 12. 16, l? are coucurrent
yyle:g-a('rg0):
of part of the truss to the right of section
i, - 18f's*20 x 18-
!M<
oonsider the equilibriup of
TJ
L
1--
*-i;;i;,
15
joint g1E:3-€h.r3[):
fitFrz'= 37.5 x 0-6 - 5 - 1?'5 klt
4.14 Draw the axial firrce,llril,' B-M' diagrarns:::* o:::t-*
= F6- Fr+ -.Frssin0 = 0 +
F,=&zt.Prscos0*5=0 =+
F,
g:6arnple
F6-l0kN
+
xL2-L5x24=0
"+frt.."
If-rat:-''
t!rJ
L
t-ffi
IJ
L
L
IJ
tok.U-rSs
+lL-J rok*l^+lm
_>
R,,,
10-40-0-8Frs =0
*
-31"5rkN
'r,
in Fig'81-1aa-
NB.,
r-bl
-'6[-..
f'1
r
(cl
.,,
l-fl-\-,.-4
i<-------. x
j., =r,, ,--6;i:,
I i-- 1;^, .6-1
-,'-.-.1o
..,., . ----....,--J
€=_ -.-...=:-:-u_.a-_,_.__;'',
Uott' sides- the FBD of the
supiorted-oi
Solution The reactions are detetmined firsi "fi.e the bEam is
:';',
beam is shown ia Fig.Bl-t4b- The equations of equilibrium'yigld
L
L
I
II
;
-i
t{s-108s-rt0=(10x4)x4+2Ox8=0 :+ ]?s=4kN'
Fr=R1+R3+20-(10x4)=0:+ {r:16kN'
F,=Rz*15=0 + ft2--lSkN,
r' For the first 3 ranges we draw FBD
We derive expressions of JV,S,i/ ficr five differcnt ranges of
trvo ranges
(Fig.B[.lab) of the part to tbe teft of the sectiori and for the last
to the right of the sectioir-
we dtaw
FBD of the part
i
O<z<1: .IV=-ff =-'Rz:15kN, S=-Fi - -Rz=-16.kN' l{=-Me:Etz-162'
M(0+) = 0, a-('-l = 16 kN'm:+
S:-Fi --Rz=-[6kN'
IV=-ff =-Ri:15kNi
L<r12:
-..-
ilo
1
l
-r-
-.t.t
f
)
(,,
A.
tt
'l
I
2<r <6:
+
-J
-J
=i
.-J
r^l
kN-m'
t+1 =56kN'm' M(2-): ?2
74 =.-lta-gE1:f 40= fPc+{0'
kN:'
-i" ::
Irr= -F1..--Rz:l.5kN!,, . S=:S=-B:+t0(z-J)=tot':36
l
,S(2+) = -16 tlr' 5(6-) = 24 kN
I
i
a\
i
Mhasertremalvaluewhereschangessigninthissegmentfors=toz-36-0'atz=3'6m40'- $(z -2t212'
M - -M,+= Rt:.+ 40 - 10(c - 2\2]/2- 16c+
N=Fi=15kN'
M = Mt - E3(r0 -
--J
+ 20(8 - r) = 200 -242'I)
M(6+) = 56 kN-m, I4(S-) = 8 kN'm'
?.J
M(8+) = 8 kN.m, Ir'l(10-) =
--J
N=fI =0,
8<e(10:
-l
M(6-):56 kN'rn' M(Xf| = 84'8 kN'm'
S=^s=fts*20=24kN'
M(2+) = 72kN:m.
*
6<z<8:
-J
S=Fi =E3=4kN, y' i{'1=E3(10-:r)=40-42'
0'
of tY' S' if for diFetent
i B'Ir{'
D rr diagrams
rr-zr'nlc are
ate drarvtt
d
in Fig'Bt'14c otttrg the expressions
and
S.F'
force,
a-tial
The
-^.,^- - rierrar.c force in u-direct-ron, or
l
J
:h:]:Hll;l;ill?;T;,;'i:,;;J;;*"::=:::::t:'",.:'",:,":r:H,'[i::ii"l#FI];ll
'i:r:::,::;i::"1;T:il#i*i#H;;il""F;:.-=iyr::.::':'t:*T,"ffi
rarving
:l*.T;
varr;s.euadrarilart*:with
,
rtrte(
diagranr
the segment tfitl't g =cons[atl[' ) varrcs
l#'":'ffa::Tf,:;'=x#:fl:::*;;;;;;;","*;:
..o,nu -r,e rrf
undegi:S-E.
of the area undes'i:S'E'
and A'l/ ctluals -te
-ve of the load on ihat portion
acts'
;;;;t
coupre
couple act's'
dis.ere
discrete
a
cidrr
6r
-S- As
o.vhere
s
"r,",,so'=isn, r
f;?t;i;i:#"":::J::ff";;;;.;;";"
iFia
F/
tS;
Cpse'
-- -r-^..,*
urhere
.4-h
-+1 ;-
Ji-i.
_-J
'.,o1<
equ;als
\ r*-)' '1"
..J
l
l< z -'4l'.5 4
t.4
If
{okx , lot<*
-d-51<-
-.1
--
--*;>*."*1.},r
-
i)
i:-
4+-
6<==..^.=b=56{
N
l" z'--->1-t'5 AL
tt. {<-- z,1 - rr
N
,-_
ai1
(c')
.
i_
'ii
I
. 1_I
+to
L+l
-i90
r?.s
k-z__L'_
q
-
ro-G7
., l,i
i <{l
A
4jL
|.-q
, .-/tit
,E
ec ==$fl=[ztx4)
|
f,*r,*J.3,9ffi+
,,<-x-c':rf
.
.-___-__F1
bv its-equivalent at point
-\ i!<
--, ii'!r
<<
BcDisreplaced (Fis'pt'lsb)
ror""lvst*..ri's;;o;;
!ft.- -40 kN-m'
t* " *o"'"nt f,' = 10x0'5-90x0'5 expressious
of
D: forces 90 kN and 10 kN in negative " ""d5;"it"o
lone side' We derive
beam is supported on
determined
.rn"
The reactions need not be
N,S,M for five different ranges of
r
"
"irr""-tt
trsing FBD's in Fig'84'15c'
lv! :-Mt=-42-2'212'
S=-fi =4*2t'
/f=-Fi=0,
0' M(2-\ = -12 kN'm'
5(0+) = 4 kN. S(2-) = 8 kN M(0+) =
:+
M = -*[t= -4x-2*/2+10(z-2)
N =-F"o - -?.5 kN; s = -Fi = 4*2c-lo'
2<z <6:
'
S(2+) = -2 kN, 5(6-) - 6 kN
+
10 = 0 at r = 3 rn- '
4*2cfor
S
segment
[his
=
in
sign
changes
e-ttremal value where S
O< z 12:
!
I
1i<<
}<<
.-..<
-r'-
l
ilr
M
has
ilI
-<
.-j
.u/-di:
li
ii
./ .l r{
';,'<
il
i:
-
- '-2(!kN'm, tt(l)= -11 kNrn'
4+2x6-10=6kN'
6<s<?: N ---F|=-zJix, -'s=-ry=
io1'- 2) = -6r * 16
(c
ird+
i--]
t-.\
[_,
*"
8':
<:c <
LJ
I
L-
ir,-.-+:{(6-)
-i^ : -42- (2 x 6) x -3) +
kN'm
,1/(6+) - -z}kNrn, M(7-\: -26
10- 10: -4 kN'
lf = *4l = -?-5*90 = 825 kN' S= -Fi =4*2x62) + 10 x (' - ?) + 40 = 4z - L4
M'= -Mtr = -42- (2 * 6) x (z - 3) + 10(z -
7a
?
-.
=
1(z-8)= -Fi - 4+2x 6- 10- 10+ 10s(8+) = 6 kN' s(11-): -6 kN
S
LJ
LJ
LJ
LJ
L_-
L-.
t_
t'
L
tu
U
1.
*.
.&f has extremal value rvhere
.t"a =
5 m, 10
i
v
- z (' - 8)''
14-s- (r-8)2 = 0
=: 10 rn is admissible' r^ - f--?I+40-l0f:
+(z-s)212
M - -t'{a: 4t-(2x's.l-(",1}]10(:.-?f10x(r-?)+40-r0(z-8}
s.ii{s
-:l'l::5
8)/31 = -62 * uu
,: +*8(z
1l:
-slJ(z - 8)t(' M(10) 1057 kN-m"''''
rn
ofwhich only
ilffiH"
r-3'5
kNlm
&'ig-Et-16a.
;'
'z
z^-i*,^4,%€
'-x
k}{'m'
=
s'.,lf for difierent
solution
We consider the equilibrium of
Me=q? + (2i- 2i- 3 k) - (2i en = 4!* r2j + 12k kN'm
?lhe normal to the section
-
thJeam
e
+
4$ + (2i-
&=4i-si+8LkN
2i- B x 6j * (2! - i) x (-+ll+'(!)
in
-i
negat'ive face)' Hencc
x (-4!) =
Q
the comPonentQ------t
i
coorpinate directions are positive:
dit' =
8 kN,
&=4i*8t'kN'
=?
S.I1=rest'of thgcomP- of
=+
Mr= ComP. of eR io -i
= -12
af
B.M.:rest of the comp- en= 4i+ 12L kN'm'
M, = -4 kN.m, M, = -12 kN, M = (42 +l22ltl2 =
s, = jkN,
*tli'+)l
tbr
(Pig.Bi-16b):
to the right of the section ai E
(a
at E io Fig.Bt.l6b is in -ve 3r-direction
poiniing in lhe
;rrc
ComP. of F,e
B
4-X}ryt
fW
"lV Ytrf1,*ty
^fi
^l*o* :t -
F=&*(2i-4\)+61-4L-'4i-0
IV
- -.-rt,
-6*rz1\
e1ft*/m
zk|\
(c)
;tC":e
"-
14
of N'
l.-.- '
t:
Li
l.
tj
L
s=
-
the oipressions
are drarvn in Fig.Bt-r4d using
Tbe a:rial force, s.F. and B-M. diagrams
rvith varying slope of'-g and
hnearly, s *"lo q,aJraticatty
segments- For the segment in rvhich c :ari;
E 26 kN'm altd occurs rvhere
The ma:simum magniLude of M
&f rraries cubically with varying slope of -5'
a d.iscretc couPle acts'
shown in
'Hod
and B'[{' zt-E for the bent beam
the axial force' s'F'' trvisting momeot
.
*
t_:,
changessign in thissegment for
lf1a*1=lt nnot*' M(rl-'1=
3
L
L,-,
tJ
tj
s
Llz(' -sI(" -8)
& = -8 kN, 's = (4'+82)r/2 = 8'944 kN
kN'rn'
dit'
12'65
kN'm
'
,
:
.'
rla
t
I
=:Po* *,u,
--
t,^=r,Yffi!,f;
4. tu=0-1;Ir=0-53m'
tJ
ih-op
*^\ 2.17t**o
]-.
u
Y?
^\.S(r-r *:v fcv-43,"#q1,H', "}(.S"tLr +"/
(o-)
u
N
l$Nczu* 6x 1$)--,.\{6nlx J}\<,**
tr
eAS(:: rkE*-'#K';'*M-s, '6+{;
+o,
H
,",3i?-\:;,".::r-jr"*
*
"*,
;:';;:T."iH;;:;;.;;;"or,,,i.i..,t
];1
f :
orequiribrium
.,r.*.iro.
are srrown
"q,"rio.,"
",o.r,lJi".to derermine the I unknorvns: 'rv'1.F1,N2'Fz'ivs'Fs'e'P'
in seaerar' i'l.
ifi;:::"J::t,1il:::f,:*t;*Xil';f:"H,:l'lJ::"J:"fr:::.'.:::'
'"
;i.*'xi-#**xl:tg:;x:"':tr:':i::ir';:l;I:"ff"loiorounacr
f
--l
I
ira
Ms=A2N7x0'4-F3x0-4:0 :+
It- N2-240-Fs-0
Fv:r\rs-320-
t
02N2=
we check rchether the assumptio-n
E
[
1f
F3:O'2N2
:+ N2-240-0'24r2=0 +
i;'rX:=lil;l];ff.
no sliq at
B is corlect'
+
N2=300N' F3:60N'
A'3=360
N'
.mption is correct. The
;;;#i4.,:tl'*;i*i*;:i;;'"]:r*I,T:Ii1fi,;.:TL;11i.iffii,';;
lJ
**Xi:'ir:,Xh:*:l*.**;:?:,'T:tilY,i#'::;"[ff':I'i'lapeadiit
N'
Ar =
- t6o*Go = o
*
F,=P-02Nr-120-300=0
MB=L60x02*120x0'6*3tl0x0'4-Px[-Nre=0 +
Fy
E
I .
I
=Nr
1oo
P-4{0N'
s-2'24-4'4h
tr
fff;j.":1,?$*;n*jj;,X;::nxHi;;,:::Ti#i:;;::":
1-I
and
tr
]
H,i
L,
l-.
r
"**
(21
m''fi'n
r*:i**n*rii-*;:**,m;::n:Hr"::l:::::Tril;:';ffi5T:t
tr j
1-=
(i)
&
is arbitrary- Equatioas of equilibrium
orrrnvnnJ-annY
u;==;:l::::ljJ'*300x04-Px053='
F=P-F'-120-300=0
u3
=
P=422'6N'
I
*],oo*,
=+
F1=2'6'N-
I
...
'
1 02' 1lhus motion impentls for P
The aisumption of rio's{ip- is o+. siuce l&l/X1 = 0'026p1
slie ai 8,_
*itr, tippiru of the blcck abour E and rolling of the cylindet wi!hou_!
: {if2'6
N
Cascsgand{llFsllx:=0.15?9}t,s=0.landhencethqTul?'igoofnoslipatEiswrong.Eencethe
tbc eylinder
Kinema[ics implics 94 = 9.D = g,t and hence
cylinder has impendingllp "t B and no slip at .4.
dorawards
is shorvn in Fig'Bt'l?f rvith Fe = lrIVs = 0'1/vs
has impending translation: upwar&. The FBD
and Fr arbitrarY'
.TJf
:lii
.,I'J
}J
iJ
- 0-1AIs x 0'4 = 0 + Fr - O.lNs
* r\! = 355.6 N, Fz = 3556 N,
:+ .lt/3 - 320 - 0.1Ns = 0
&=Na-320- F2=0
* 5, - 275.6 N'
F'=Nz -240-0.1'rY.=0
=+ N2-240 - 0.1 x 355'6 = 0
i'e'' the assumption of no slip at r4 is
\!e checL that l'F2l/N2 = 35'56/275'6 = 0'129 I pz = 0'2 is o'k''
will have
3 equilibrium equations imply that it
Mo = Fz x
0.4
!
i
rI
t
I
I
and only
correcL. The 4 unknorvns, M1 ,Fr,r,P for the btock
E, rvith the 4t'h equation provided by this coodition'
point
either impending slip or impending tipping about
rvith Fr = lrJVr - 0'2Nl
wc assttmc impenditrg slip uithout tipping. The FBD 15 5|16\\'n in Fig'Rt'l7g
t)
dow'orvard and
160
-J
J
I
'l
J
I
fv = Nr - 1.90 + 35.56'= 0
2?5'6'='0
, F, = P= 0.?/Yl - 120 ir's =
J
x G2*
120
x 0-6+2?5'6'x'0''4'- p:'v l1:-
lllr =0
*
P
i
420:5:I*li
+
z,
=
L.722
-J
i,$360h
J
tl
(3)
-J
i.e', $'hether
\ve check rvhether the assunrption of no tipping is correct,
motion
the assumptiou of no tipping is correct' Thus
casc 3: lr = 0.5, eq(3) + r = 0-032
at
slip
without
'{'
and traoslation of the cylinder
impends for P = 420.5 N with sliding of the block
> 0. H"n."
&=Nr-160+as.5o=0
.,
F,=P-fi-120-2?5-6=0
'1
= 0-069 <
i,
0
+
*
P=404-23N'
JV1 = 124-{ N'
+
f1=8'03 N'
for P
=0-2. Thus motion impends
:
W,I
I
i
_-l
-<
--
4M'?3 N
-<
rvithout slip at '4'
rvith tipping of the block about E and translation of the cylinder
roiatlcn irrespective of the va]ue of torque
Example 4.18 A disc 1, mounted on a shaf[, is prevented from
iu contact rvith it' Firid the condition
M applied in the direction shorrn in Fig.E4.18a by placing'a disc 2
Neslect weight-of disc 2'
so that disc 2 acts * u rri"tion lock for disc I'
;";:;,;;;r;&
,-*-)N--,1
W\ffi":
I
L
Cascl:}=0'53,eq(3)=+r=_0.0694(0.Hencetheassumptiooofnotippingiswrong.Tbustlre
with Nr' ]?t
stipping' The FBD is sho$'n in Fig'E4'l7h
blocL has impending tipping about point E rvithout
at .8 and F1 is arbitrary. Equations of equilibrium
Ms = 160 x 0.2*120 x 0'6+275'6 x 0'4- P x 0'53 =
I
-<
<
.ac=.1R
-<
to only trvo reartions at A and B' Eeoce the
Disc 2 is a 2-force member since it is subjlcied
tt. roration of disc 1 is prev-eutcd if disc 2
two forces must ac! along the line ,4B as showu in ris.E4.rai,
e(tan-rtsr andaitt"-tp2' FronrFig'E4'18c as20=(d-r)/('R+r)'
.{
Eence the reguired condition is
--<
.<
solution
><i-a
doesnotslipate"na61]i"u.,if
0
= !co6-r(d-r)/(E+r)J
(
min(tan-r
!'t,tan-t ltz) :+ (d-fi1(R+r)
> 2cos[min(tan-r p1'taa-lp2)J
I
l
:-
fitr
;i
-
a
:r{
!
Lj
:.
LJ
LJ i
LJ
LJ
!
!
:
t-
L.
Lr,
L
IJ
iN
N/(b)
(t')
I -t-
r =' '\t't'
SolutionThenormalreaction.lsw|LN/m.Assumeihattheimpend"ingsli':li:1?::.1'=::"':::
ihis assumlfioo ls not
i"-; *'"uD is as shown in Fis,E4.leb- Eowever,
;::X;";;;.*;;
bar ls rn rne s'u,c urtw
n. - F, r^L Tbe
-rB-- fi<fion
f-:+:^f^..force
ls not satisfied by the forces in Fig.Bl-l9b0g;not
Ma
equation
the
equilibrium
uus
sYu.^urru*'
=
since
slrlce
correct
correc.
:atrcr,Ieo :I ""t
1L --- :- impending
:-^--r;-rarrrinn
totation
on- Hence there is
partly
' - F:-D, ro-.-;1L
has to be distributed partly iu idirection aad
i"- idk*lt
.ninorvn.
- an
=-,,-i-nn.n
io Fig.Rl.lec wirh c
shoron
*
rso;
:r-;: |j.T:T ;ffi;J;*,il';";;";ih.
-Eguations of equilibrium:
(u
+P=(L-2alpwlL
s=P-(pwlL)(L-o)+(vrflLlo=o
2a2
-4La+
L?
=0
a: (1- Lld,'L
"+
p
=-1'fZ
root less than -' is listed- Eencq eq(U *
.'l.o*.;,,
with a i'ertical st'ep' is
9?r, with its front.wheels in contact
weight
of
trolley
wheel
A
four
4-20
Exampte
P rgquired
of tle trotler' Find tbe:i:rirnr 'iu -fioice
pulred by a force p (Fig-Bt-20a)- .C is tl" .",it " or mass
all surfaces t"t:T,::it-h the 8!9'1rn4
do rnount the step if tlre coefficiert of static frictionlat
"f
"5
;;r. t , *" ,**tble
lj
u
L
y#'#
;TY#+, &r x%
c
IJ
IJ
L
lj
IJ
L
L
IJ
r-.
L.
L.
-,,
-,,
A--
fU.
k-a* -4-r:r,o -*4
F?i.r
''-
-}'t<-15rc---+{
1_
t,
L
,i^=1u*1L\(iL-o)'12- (pwlL)o(L-a+o/2\=0':*
IJ
J
t4
to)
u
L
I
=-,,*.r:
o thibrizontal
valle of
,,t9a). Find the value
^.e tH;lar'r"stn,9,1a holzgntal,table (Fig-Bt'l9a)'
4-19
Exanple
- be uiformlv
Assu,,," the *ormal t"-T::"" from'the
force p for which,o,* * t"-i*;".ion.
T:
(cl
is p'
i*i.ti""U. The weight of the bat is W. ail' the coefficient of static frictionl<-o-ri/t-
z
1 0V;/
*lI,*
--- L-.*
(b)
i
q%.=
lr:,,.,
atEat E since tt trJn.yi-t3t*-"tt' tft" luip by losing contact
Solution
along
"
-TT::
it is a 2-force member with the tbrces actiag
is sub-iected to only two forces through ,4.and D. Eence
fu: "l ? *u @ acting along
AD asshorsn in Fig.E4-20b- Similatly, \rheel 2 is a2-force member rvith the
l{ence the FBD of the trolley is a-s showl
BG u:shosrn in Fig-Bl-20c rvith sin0 :32A.,40O: 0.8,ccsB': 0'6:
There is no reactioo
in Fig-pt.2m. Notice the in[eresLiug fact that the.friction forces
this coplanar system Yield I?1, .R2' P:
Fr=P-Ezcos0=0
& = & +.&zsin0 -W =O
+
:+
+
are zero--The'3 equations of
equilibrium of
R2=$PlT
R1'=W
-4P13
a
1500rv-3000(14/ -4Pl3l-400P=0
Me = 1500W - 3000Er - 400P :0
kN is hinged
Exaurple 4.21 A uniform rectangular'plare ABCE of weight 6
The
abour its ed6e AB and supporred by a light flexible cable cD.(Fig.E4-21a).
ai
hinge
the
but
'A
axis
hinge at I can only support a force normal to.the hinge
the
and
B
and
can atso support axial load- Find the reactions at the hinges /
'
*
P:5W/t2
tension in the cable.
of
The FBD of the plate' is shown in Fi5'81'21b' The coordinates
The
(-l'2'5)'
the midpoint G of the diagonal AC ate (0,0,6) + (-2,4,4)ll2 =
reactions and the tension are given by
Solution
t9 t_
B,
1i-o
where t = TI(CO) arid r?o ' AB -- 0 since &B L AB' The 7 unknorvns Position vecr?r, -. - , R5,t are obtbined from- Ma = g, {. = g.- &s ' AB = 0'
& = Rr i+ Rz j_+fis ki 4, = I?a !+ fts!+ fio k, T=rwJ
tors are obtaiaed by subtracting the coordinates
"i
icml
.i
- i'
l,t5.
u- iJz-l
G-O)]=
lrom those of its tip,
s;
r-F
tt
5tI
Lb)
lI{
t
!o
4
.1 -'.
"-
.t'
AG*-:-i+Zi-
E:
I
&,q,+ Es.+
L{a-
?+6k-q
6k+lC x T.+ SB x
L2-24t*4&+2.R5:e,
A.G
,-
krn,'e:-2i+di-ZE-,,'
&" - AB = (Eni+ /?"i+R.t) -(4i-
..is
.<
x
2L)
ra:nj-rt*
:
=a&._ 2Ra = a.
li i
a"=l_l i
l0 0
(i)
:.
(2)
EI
.,1* 3,_il-li 4i -21:O
85 l?l
k
_r
6
i:
j: 5-L2t-2&=0,
-':----l
k: -4&=0,
'=+ &-0,
l=0-5,
.' 2R6 -F .l?5 -.0
' Eq"(t)and(3) :+ .Es-.116-0 :+ &a=0
and
?: t(2!- 4i:48: i-2j- Z1 :+
Eq(z) :+
Ee :
-Ee --I- 6li = *i + 2j - ,{t kN-
:.
:.
(3)
T=
(L2
*22
12zyr12
:
-
3 kN
.Er!
:7-_
.f
R5
:-.
-r..'E
Ma-(fi2)q-p.&=0
The equitibrium eguaiion Me
.
a:
'.
tte
?-*
=0 for the whole rig yields 1g1:
- ftr}- wa- pR=o :+
The equillbrium equation M,
r
!
P = WqlZR
=+
,
R1= w:,1o+r112)lb
foc the subsystem of Fi6.84.22c yierds
;..0
Ms-iw'-BD{"-.)+ara -I'zc=o-
:+
Rz
&:
=Wtr + (r+rr)/2cl
)
-J
J
J
rJ
-J
l!.-.
!l€'
J
JJ
b\
:
L
:
:
\L
L
L
L
:
:
-
C)
r{
r{
Fl
4
t:
o
Z
frl
(n
N,
p
o
U
@
tl
@
=v1
=
G5
H
=
I-l
GS
ru
M
F.,rrrl
rln{
r-t
-
@
74
Ffi.rl
'ts'*4
rrg)
>H
J
trfl
/fl
V
M
\""/
6d
|)re->
f\*\
ii
L
k
IJ
tj
TJ
L
l..u
E
f;
E
i$
EE{; E
E?fiEE
g{
€t
N
,,N
.\ \
:
Or
$++f 3E d
?f$$tii t
g':s
:8f
iriids
"r
p
cu
a-(j x
g1
OdH
- o.=
Q-O
Oiz6
F,
arE
OE
-< f
a
8ts.c
II{
h
'i
rt
..
oit.,
d d-'! (E
H
l!
H
o €_dj:
.d
g 8.8
ia
d B*
u.E
B
s€ at
E
,,',.
*fr
I
\'
\t
x
rl
N
1
-6
c:
..$
'-l
g.xt
o
.C
F
6!
G)a
H.
q)
qEl
hO
'iR'
ii :F
.= cll
E €'{
'6
.ll
,E j519
flq
ts
g
Fa
c' +1 ct.:
tb
c)
bO(a
fE{ ts6
oi 3ryi
I
.E.e#ll
ttt ^g
f + [:
>g Pe
o€,9r
Earll
6.8€q
O (D.oZ,
o
o'u
H r.l1
t!v
9-N
t\ Hdft
ts bb<
I iin
.*z
H
O
5
85
ti
ts
-. d
\f;.
co
F
.; ;E
t frrr
8 -E
H
3t &i'
$:
zE 3Aer
0... -E.€
r.r
ii
q€ X
B
fn "qd.E,
6;'€fr8'l t
E E $t,s'''
\J
l5o;
V E.l E
'.
i a ;'$ir f,{: s
'.:..-d^- o 6.9".
*-*-{-s-c);EFqo
E
pi
; q E T T # i B" E.
?iEailIg?Ei
qf
tJ .$tE Et; t *fi si
':
IJ
1IJ aEiIiIi$fi v
u
LIJ
L
lj
L
IJ
l-
l.
IJ
u
lJ
tJ
L
L
ltj
lj
L
L
lj
L
L
o:
F{
bo
c,
t\
09.
bo
rl
o
r\
bo
\Fa
g
fq
r-
#
q
11
f4
bo
cI
-J:
14
ho
CI
K
it'a g;i El€i i
h
bb
H
o
c?
iiEgEEi?iiEi
fi
-trff
'us
I
d_
ta,
-/Q"s
^.6-
9P
tJ rc{
@ lg'
-{6
sltt+
u-r
I
'=i
x
c
t
.o
=d
ai()
L!
'J4
-€
2o
wH
+t0
!O)
€
€O<
ll e
,.d
63
'-r ;)
+N
Nf
(J+:
v(!
{-.8
$
.a,o,
I
CI
-<l
r,
.o
il
LJ
LJ
LJ
u
u
l.j
t_
lJ
u
u
IJ
tt__
tJ
l.j
ll_'
u
u
lJ
tJ
tJ
L
lJ
t-.
t-.
L
L
tL
L
JI
<-
a
n
(.s/ur 1o
E
\'t,
(s/q))
I
.tr).
o
sEE:;
Eg:1,
(,
6"
q!
n
cl
LO
frr
o,
d#;
E<
:-\.-;
O
A o;r
*E
'o
d v''C,h-d
E{5
-e
.$;
85
E*
vpr
E
clft. o
€
'(rC)
o
-o .o
r,.
H
U uo.i
;.E
E EE
n;s
o -.-::
!5
P
fEH
E
O)
c\
OJ
i-0)dH€
-o P
o)r{*
.ti
sdA
ijsd
(vOi
o,
n
kl
bb
cr
F{
.;
(o
f4
bb
o,,
.l.
t:
r-t'
'f4
!b
cr
,F-l
@
R cq*i
.-d
E€
(H
,9
io<hO
gH;
q
- g H.E
Cl€^H
.9
':?.-.
)
2
4,,
vo
5
(4
o,
t)
a
E:E
E Eo :€E
!if; i[
arut€ffii{g
stEE€
Eig
H';8s E"ilf 'E; rEf
EEEi ;9$E a+ i€;
,iifi
-tf't
IffiiEE+gIf
l't:s
E 3ff ;,:fiEEBi
set€ffgiEEIEI;$E
\
i
J.d
d
L
H
€g
d
C)
. (r,
EE
oE'
.7
^O
nc-l
.'
ril
(n tut
E -E
T;
B;'E
tsg
E
L) -ra,t*:.
v
fi F'9
ol +t
th
P .H
fr]
() .l,O
o)
,ql^
g
H
?,
Ft
q Hf
d
ir.
fqH
rl
bO
o)
gE
E".
ir
F{A
r-{
bo€
d d-r
'a
dc!
Eq8
^. atO
E
p -t\
3 H.s
9'H
;i6tt,
ai
.EE.g
"xEe
83.*
-a+rE€5
E
lo
I
l.'
1.1,,
I
,{
o
I
gr
o..r(..
Be{
I5
E-e
o .;i +,
.o
EA
Eg,E
(o@
oo
I
Eg
dt>
*h0
-r
eEt
dH
=ot
de*
6rdp
. ,cl
+:
E-l{ o ol
Et
1${t
EE €
E.
}E iE
E
E{,?EE;&i
+I il i It ffi E i 1i I lI i €iEI aE
,Hil,liilluiill:lil
t
.{
a)
Fl
d
{{
cdd4
f
B
. .=
g{B€
Ef,tsE
s5_EE
P
f'
giE
At
,i.HqF€
iret
i 8q 9'*
B
x.E o 9H
o
EE
E
?E E B3
9
st Se
.e 6,8
qgEE f
.d
iric0)
s
g:elE?g
.9
Fxt
E
.Ei e-*t
c\ Er,Ete;
J
5 H'E'E
r
q
SEEE
Fl
E
ei.EYaie
u
*f
,O
4.
rl
o
-o
,t
o
-t
.o.
E
E:
o_
fl
t,
.E
@
U
i : ff
e(s.ovrv.*vi EE
i
h^6
d-.aa
()ci.L!
i9!
'
COa-^(D
bo X#
Xl
0)
E
(.:)q
o
,\
4E
o -
i
g
3po<(! E
!!9k
a)
: e€€
^a
d 6.
X6d
=A
oH. .HG:)
UR8=
' j'+ '9
-
9PCl=
t*'r$
o)v0)F,
ocr
o{- a'-a
B
<I
o
d
;:r
I yt
xIE sE
E'6
()a6:Y0)
o
led
g ii,E
-3'E
19x!
e-oEIEE
J
E
bo E [Ig
a o.! tri= tr
il.
.i -d u+
g
HEEE
61 6
E o.9 A
.o H HE
<-s
i!a
7 -'=
><*
*if :
gal?
r
ilt
iilgiE 3l Er ii [?i
c{
fiEE IE
tgEs;
A
:I
..l
I
l
J
J
J
J
J
J
J
!r
J
I
I
r
lr
J
Ll
-t
:-l
:--
i._
!a-
:.
---
3.
-.:.
\r
L
LJ
Lj
L"
L--
u
LJ
u
LJ
L
L
IJ
U
t_:
L
l_^
L
l-.
1--
IJ
u
IJ
l'
L
L
L
IJ
L
t--
L
L
l:
tl-
E
I E'st"Er'
ts b il ts 8_3
iiFrE
;Efi s
a -u o.n-\a
$E
H E.E B E,d
$!$iiE
9;;s+i
*iEtB;s
i
da
ll o 5F
:$t{sfi;
-BEi-<&r
:iE3itt
a
c0
€a)
o
CI
tr
r1
qi
o
o
o
q{
,
q)
a)
U)
€
! c..
>''.l
-6
;5 >
+.!
4l<
7d
o:5
-c
€trbo
cid
O€
Xo
{o
: Ia
-
l*iIagiiiilt,
gi
co
i--{
iali
gltti*ige'{eiilI Bii
I gglIiigiliE
ifig i*
€itE
3HH'!,E
g
ll'iSxse:ia {
L
i
E*:HEEE:;
gdSE€6n€E
o
F-3.I
fEt
H;t
us'ss BS$
il ['l'l
iEE€
Ei€ t
i6'r E djr
iExTf;r?E }L€ fltlti $gE*
gei! EEEfi +i-?u b
iI$I
iEf,
4il
i;Ee*, *,
,4:i
LZ-'NF
.o\
8i,
i\
,o
J
f4
E
"lI z\ I'!u
h,i
Sg
riilffigaifuiiIIijIIml*
i,I[$
1-
H.
.ir
o
c)
n
rl
+
>.p
'l
'l
< Q:
cU
,{ C
.',
atii?Iff?iE
::ilaE
oE
FI d
'E'6
a+6
t!
.a .'\.
iid
lQv
6:
n!H
y.<
Et x
-11
€'H
El ra
d .:1
€+i
I
,..H,; gr*
.s 8.g
:€iB:l
E;E
:t
x H€ o*r
H
&3
i:g d6g'5
eiiEt
iI Ht j,
$iE Bq
8?E:r'
:s30f,
N
x
x
x
o
ol
Crl
d
-
67)
c:
bb
!{
r\
ca
c..1
:J
J
J
J
J
J
J
J
J
J
J
J
"l
J
JJ
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
L
L
L
L
1_
l'
L
L
l'
lJ
IJ
L
L
L
L
L
L
L
L
L
L
L
L
^NND€
6+.
..-o
.;G
€-
d^V
..3f
?':
Yl
q q ll j! i
o;:
'rdH-+
dH+rL.H
E
:1
&
-r
$=
x
it $; H s
v$
x.
3'fl i+H; E E
!!ffiss
;i-a'6€ -
.:j'st-or.*Eb
U
U
f fr_;f.fi$, f f
. "3;i:,tE er er
}}
{ 9"3**;,'g
qeY + +
of LI
d'
t!)
Ol
v.i
(O
Ot
fl
.o
(
tl
"a
q)
€
t
k
c,
Q)
,{
o
H
.
E
;
.=o
>oo
€tq
a*E
H.d
.Pub
g{t
;-a
€Ir
ryEUH-A
'€
3?€
esH
B
EiE
d.H
Hoo
.li€ o
} H€
E€s
;rr
-x
,.#a
z 338
@
<.
el
F
o
frl rri
EBri
6l
o
or,5 I
E'? 5
ll 60 .V
ra
z, <l
t00 g)
.il
1t
(
il (3
\
o
O)
i:.
.i ci ci*-SS
r$A€iEc sTilY &fl
+$*I g;,
,s
!)
,a
H
d
oG'
r6'
riE
-b-€,
F{
YI
€)+
ts6Eoq
ge
'1;r g
V
H.
tr6t
Oa
t)H
?
-1
I
r-l
,;F
ei
aHll
,Il g 6d*
o,
E
ua
tr- I
'85
3x
AH
sE E
I
I
trtd
8g (
aB
-E.E
(r
'i€
.f
5
3
0
IJ
-s
q
38 e
E;
iH"E
iE$€
E€ E [Iii
'(a
>.9
.EE
E -xt
+t.r: "i.EEitt
*-.
i flE i$
r{
; E :iD ci
F{,i.E
u L
gT
!.S
-L
g{t $i.,sirg;
-eE
il"&f€
irf
;EH-$
qi*ii
el;?r*[rj**
gte;l
a;
fl* $l _'
lt
-\
.L,'r
;ot
A*T
E3,co ll'*
!p!'oloi9llq9B9
Q)
E"i,
S
orTo,
borud
.-trE€.-ti
6g8B
od0)
(,t.=
H
EeEE:
€ET;f
#€; it
i
.iFr
3'
ro
ro
*
'r
X
ilEfEs
EHqaB <
.E* F E o
#:€;€x
a :'I'5 H:3 3
E$ \
}ET s
'E,i
;.EF I
sit
SEAE
!l o.g
EaEg€
A f o *k
x' SSAHfri
pr cr
Hiq;
#E53
s5'i;
iiqg'g;?
E'-rrrr
\
{in ho;i,to.h
Es,{ I
Hfi5'FE5*EJl
l, ,S
'&<D
ri-i3l=FE
L:
"
:s=Hl
i
r
1ts B€
H
t4
(n
trt
H
qi
oq)
_n
P.C
(d
(A
q
€oai
.tsq
a
4?
€
B
BE
a)
U)
U)
d
dd
q)
U)a
H
(U
d
A
t{^
€O
E
i
i
-1
J
J
J
.,J
J
I
J
J
h-E
;'E
J
s?iiH.E€
F' t'E.E J
x,* r'T * *
b ,, =-€#'si
H: g T ,r T tE.-E i J
ci
*?38
g
,*_,g$
[*
o'S'+.;
J
i.-,$$$t]EI
*
J
J
ri
T
5
H
i,g
$
ll
il ['n
= ii -s,qdl ,
ro
e
r
I
-gHEu
EF;;
E 3 '! eri';
x'f;di{"E}H;€E
Ease .Oa+a
fla
6E x=.E*
q8ooistSsEsE
dv
XdE
3";. itunq- 'do ;:x I q*Fr
.6
{ 'S.q H.3 <'a
&
"i
56.88
gf
FJ
a<
#i;?H
5E
d
s
--.!lH
€
c.i
'\
6:
iI
u
3* EE
^s.3
q,
. Hor =.E
m€ FH
^
3;d'6.[
.s
bb
q)
q)c?
d
l+
f ih E
bO
F$i$iEf,
;
i
38EI;Egild
EEf${iE
€
;i+q ;?isa 3>
{sis gE.Hr,1
eT; ,l f E For € +f i{Ef !
il€ni ffi;Eli EE
A
<ii
co
r,l o.
r4
Ei .o
d,€ "O k'*^=. .E
d
6
xXH
r{'o.
o
9.g
H.E
x
"$:";*€;8"1 o
;;t$E€.EE:E
'$?lFsrgTS
*:f:83€j$E
lj
L
IJ
L
L
u
lJ
L
l_-
tJ
L
tJ
]j
L
l-.
IJ
L
l-
L
]J
IJ
l'
L
L
L
lL
L
L
L
lj
L
L
H
E
qE{ P,
fiiE'E;
B :.g
i
e;g;+
st j
x?
rll\,
{
5-
H $EE
a a.; -?-,
;.*
i
i HE,X t
.trba.ts*
9.E, Ft
'H.E€5t
9r=H
;:; : J;
: t sE' ji \.'tr i' ]N
1q-{.
*: sH tI rA?r]:
a9€r
Ba *E i:
tl Ya-eS
co\
tS$Et"
g
-B-B U=i d-H".
.doDF/S
€D.isa\
E
iE; q*; e
{T;;g E
#{E;:i
s,
xf 6#,sEI
9s'il I'F:I i
i g-td'E€
_g*;::E€
;?* i't;$c
d€nIIsEr
3.I E: *'i,E;
o
flr
t;81*
q.o*3 e
=oro
fl3 is#EH_H
ii ei;:ir
vl
!-
o
O)
g)
E8 E:.E
q2
(a
i€'9H
p
u<
E
#ts
^.E
tRg-q
ts
o)
b
.d€O
tDF{.:i'F
ICIAF
d)
* qpcq.E
EEit E:
^Jo o.yH
HE€J€.H
EE.E.::X
--lq€
(!
€
PiE
A-
Ei
1
.3
I
L
!oo
a[
(J
\)^
t/ :l
..- ft<,
\
,/ *o
,r-
tv*
o
i;fu.
le.
t*4-
_
\s
Y--- --- ./
i\/= ,/l
A.v
l,/' /
-ro- i ,'w-
,/
flEE Efi€
T y€
:$g;Is"=S
t\
F{'<'.P€
all 14.rlI
\ Ifr; EEX
€
;
€8ts:;*
gfi8aH'*
I#sx E
qEi: i#
I Er€-3
e'eE'68.I
=
€
E
-
C\ L.= tr., o
.a+<
L
,tlo
I
rd (,)
d
c)lc):id
at<
r"'-
..F
{8E
€Eor
5d'd
g $5;
9 oi
n
55..F
# e;
rx{u
,rM
q
ES'cl
r
.H
+!E
$E?X
s:so
L
T?
E fl
.-.! o
'
E-Hi
<+
o'U
{'< ,*{E orEr:t
X il
? B'H 3€
a ! €':';
o@609
AX
E Eq'fi
F? e E
. ..=+,
lt(oo€
. c-tN
ad
t8
-tr6
Of.
a)
'5..9
6l
.d
COt,i
. (!)
5 E.E
HE
.$
h.oi
EE
*-o
;"O
fut
()d
€co
>€
o) =.4
€O
F{
c, g'I
b9
.ep
aO
'.EE"
63
.=
5€
.EE
EJ
or9
K €q
E
.
r/)
t.-
!Re
-i
taEitr
bo P.Y
'-AX*
irr
frl AO
';
d9
coH
lr 6 to
sTEg.E
: ;'E;:
I9si€
X
,,#,{,EE$
,t;:B*^'E
,i*fl;€f
E \s1EE.E,
gg;giA
ro r,...SnHag
:3iE
.s il
e r'I-9
ErE
jt
.E
al..r{ d ca
tt J
{o) o h
gi
,
I
-r-. -E ?
E .:'E E
x$xi[i
i
,
.1 A
'oa
id
b0
-d
r/-)Q
,ad
(g0)
g€
d6
>r.=
ar!
o
pc{;
^H
(d -_H
COY
E.€
F.
I
o
I
-J
#
fi;E€
eEtrB
<o {q-9
H r:
ca
"o.y (g 6
6SgT€E
;.
*gtiEi-
5
6,.{1
-l
E;*EE
Erf i;
.99'-9'ug9
t4
I 9;-g'HG
3€#F{'o:
i e:fi ryE
rEE: E I
BElr i'H$:
6iBEET8
6 o o'S
gS H8E;
a d '5 gi'i{
oeH
O
l+{
t
,+. (9
--t
3S$i E:
.-l
n 8a
e E.s
E 6.e
<iP€
g.E H
.Et*
&EE
o'--o
ts:O
r 3.=
x,B
o"'o)
'E
o)*
9:'
E";€
':
d
.;:i o
.Eq o
@ -6
cD,=
>
'
tgiiIEilillEti3ElEIi
g
tiffiiglgtttmlgIil{gt
oll
l*r --{
(^
]J N
o 6
C) -o
c!
o
c{
a
-:
;3.9
c9.- - P
=
r-i q
x.-
05 1o
.(,.=
O
O)
5EE
E {'i;d $
boq-q
tt.E j ts .E':
B AO H 8 gE
6: : 6'H E B
.ep 9.qE s €f
r".g'Eqa
€'t
ruo
69HE
.ot4
I s:_; E =df i
*;:s
rat
x.t Ut 6n'E
L
X6 6.i eqE
;€Er,.ErpB
. o)H
ri-
s'i
F-ct
iEor E- E;
Eo.id:.8
o) H>-
-5 ,-
!
T
Fa)(A^i{€
L u, H'- H
€.E
acB
()-i
@
Ir)
' d -x'(3l*
();0.
EoPl
d >rr
EEi
r€.-E
-
H.4
EEI
uE.
-o.)B
E. EEA
E 96
? ;.Ef
fl
Efl g EEg E.
;.: 2Ea B-<tP
..i E 4.9 9 H.xn(
Cl;!6.rd3s
,EI.EEEE{€
Eh \)H*I I E
E; O d:ET
ii € E.He "e 3
< c.:E
EE H 8ts.4
iI
d
-{t
-Ul€
AP
otr) -9-
h6
!lr
.(D'.@
d
f
EET
3ii
(D
-a i6t
5-d€
gEE
fr aaE
6 t<5
E6e
a' <5?
bb;?
B6.8
-
r'
.J
__l
:'!
--.,J
.,1
.-J
I
-J
J
.-j
-.J
J
J
J
T
J
J
JJ
IJ
-J
J
J-J
J
J
J
J
J
J
J
J
-t
U
L
L
lJ
lj
L
lj
IJ
L
L
IJ
]J
IJ
]j
L
l_u
tj
lu
u
t'lJ
]j
tJ
tt--
lj
l--
L
L
tj
L
t-
g
-88
'g
e:
E
liI
Ia al
$"t
oq
h'
E 'E.*
€cl
.o
tE
h.=
q. Ee
:E
5
6
,.s (di€a
Il{
a
':H..i> 6.3
€
€EBE
.O!+iB.
BiE
:o9-E'E H
EEE4d
Ui*6sr
B5;i ;
E;S s H
.3t
.-8
r{t.N
hU
r(t
o!t.E d
)
g-3
,q
s€
.gE
YD
5":
.qE
(Dad
r.9
o.o
9A
.-H
io)
b,H
. 0.)
,{d
B.E
{J
gE
oor
-a
or
Yo*
Al6)H
Yho
*'Ed
i':'
.'br-,
a
,
o
or <\
! r+
-:+ rr r0
' :(+
O lia
-
.
L.
i
6E;3
H.e
>6,
€.E
'o+
'ocoH
d<i
ii
19
d
d,#
g,t
.$t&
"i
E :'.9
E
d u-o
o
3f3
o
€
d
t{
0)
a)
()
()
€5EE
*.it
o
f ;EH
.:IB€
r
c
-t'['.
EEx.E "i
8s€e
{.d :5
.:tf.$
6
g
.E0
o -'E
ilsF
.3 dE
nr
Fl
.g
k
ci,gt
tb
fq
(r)
Fl
rt)
f4
H?
SEE$
Hcs 6
:EB#E
bbE
I.
I
o $E€ E I
a8H3, (-,
60
fr -'a
.0
bo
c
d
3oY
.:<
t.1
6a
>Bt
H?
ad
!Ph
HC)
(E
F!
a)O
dA
E+
-o)
H
HO
rr{
ci9
+.H
od
qr&
dtr'e
ri
9Fl
qE
(!€
*.i x
+ql
c6
-J:
Fl
c,
bb
f+.
\o
.s
?i
1.
!b
cl
t\.
Y
'F.{
Fl
bo
CIr
t\
iEtaE{
sB
Ee€
'
$a+ ?sEEEAI
iiEEi?ilgEgEt
EgE
tltffieffi
9,,,
$
E
i Ekn;:
i;et EiqEaf3Iai
;i:ig;*F"E:Es#
EsE as HEr
qEEo
${iEq
o* uo
i
qE 5.':.$
.E
E H-a
?arHHvrt:
e S sE
EI > cd -..!Y.:,.
@ r -- sd'
d
fra8 $:
E'El .t :
Ir
e.rg
iS
*-l q5o
eea
E#E:'
.$fEEE
g
8 3.:e'j
4
Q)qOtrl:P
P O rr O+r
EEfE9
€< s H &
F?c H{
;;E*E
rO
-{'
cl
t'b
fr
?
.!r.J \J-.,
>lr
.
l*
'3 ..rcl
ogl
<__-t.
.-51
\,
!'
\t)
rl
D
6
-q
r-{
bo
fz.
Fl
E
gg
E
*'I:
t\t\
>
f
l
J
s I iESt
fs
!: s::s J
Eg Etroot J
J
i a tietEg
6E E 5#;if I
*t
J
J
JJ
J
.-t
J
J
J
o
J
gE
aitf?f
B'E q eleb:A J
g&
rit8i;
J
J
t-
a
obo
fr{
@
C)
{I{H€BfEE{
g€; itES-EF
3.FEf;e8
?:.f
opoi
H o 6rr
-cU^L EhlC,r-t€
es H Et s
H;t&,€E
HonE;.i I
S
i
, l,-r.p E
t,ItS
$ ri;itE
EE
€:f;EEE
r3Ifi1:
^ H I E.3,3
,<fi
3^3 Hee
(SP
k
,-:(oqda-.
JJ
J
JJ
JJ
J
J
El i,Et-e
J
ii; J
i'tiiI Eq
'33.iita
h*5 5 *EH;PE
a
.
e:3 e:iE
gEEEg
$5c€ E-
J
J
J
J
u
u
lJ
TJ
t--
L
tj
u
L
t_.
IJ
IJ
L
L
L
I_
IJ
L
L
l'
IJ
t.
IJ
IJ
l_-
L
f-l-f*-
L.
L
L
L
A
rn
€
E
c)
.bo
A
u*
.tso
O)
So€
r<
,9ts
dH
h;
trf
d-o
oo)
€+t
A+
.6
u-o t2.
Rd
ijd
x;
.xE
bO
cd
*tj
o5
-q€
it{
'i ,r+{
A.O
trt
HOo.
o)O
,=f
({()
ll 5
.Eb
(d>
-i
o
o
a)
€
A
d
a)
11
U)
+,
IA
trt
.d
g
oH
E'I
-5 d'
xO
d
o:
p3
Q)+i
€66
l-o
do
_o
t6*
TH
oc,
.S uo
5,E
.qH
rro
.il{
qH
o;
o)
,= -o
p
oo
u
E
(J
t-
dl
.oo
g
dir
t
o
.o
.
o.
a\
a
,<l
4-+
o
a
{P.
\
x\
1.r ia/
3:
(A
eo
s1Pot7
<L.yeo,
o
T
E
e
HE*:,iiI
:I
a;.tET€: ?
X
j t"'E:
r EstH#iI
ll
r\ Ebjc-{ 66 t_,
-!Pgclosk>'E
rr{ ; E
E#f$€iY
'E.s'E I ;8 ;
iEIh,EEt n
tEi.F;er*
8.8 sSQ Ho<
*;SE;:EES
t Et reiisi
*
gEE;+{ii
';€
:BI
a
J
.31
slpqZ
ar
l,l,/l
.!€
)ee
ql't
oo
Lrs
'la
T
r.
I
E
j
I
lEt
(I)
--q,
tl
BE
(DrJ
E fl.g
tr.gf
slo
E9
aE
0)
r',.6'
^ ogB
.x
-8.g
+Q{
8H
f,Q
*,
'a<
{c
8(6
l\.9p'E2-4
o.-
i Hs
q
Hs
E'
r{
o.i
iibo H+
qc)u)
dts
sE
ri
+arE
c\ bo
rO
iI igiIiII i I$giEgIifi
r
; E r: * 3 i
g,i
r x€: =
II
'
i
,
ll ll.!g€
j
r- <dri 5
'sd€ti'g
dt Ht.i"
E*s HEH
<
:I I gS;
-eE;8E
r-e€E
3c!
^o:i.gE H
a3E:si
i',a.H
t-l
q
o
a?
;t";
or-8$rts
I
_1
:
o)'
o
{cDl
>
)r.^
:
3:,
I*
\;
h
ii
tt
lt
rz
+
I
E
c
'rf
LTJ
d
cy?
-
c\:
(o
bo
FI
q
14
fiiiiiE,EIEiE**{E
fr*
bo
.ol
tt -; ts.fl
-'c{{aQid
IEET-EB
nA?
i?s *f,E
S'Si :,8 s
l; EJ
si ,in ri *
di-^tt tsES
or $q? 3.:l 'E
.X
-Jo€{lo>
"
5 E.i
-"
.SeoXE
dv^€
F d
;i
g
.c
(J
'a
<
J
J
IJ
L
L
L
L
L
lj
lJ
L
t_
IJ
IJ
L
L
'iEE +H dE
f#3"1 r UE
;:
3;ds$ +f
EiI $} EE
! IS *"r
ff ;?i : ee
$sE $gE$;f
rdfreifi8Er
f EE€3 E;f {
.t
\'/
;o
.3t
,:. bO
lld,
v.i
N
,$I
3l.E
oE
C+)
ai-i tr
-()
*e
c.6 .;<
,.j p
F..f
.Sr.n
Sii
f:. i 3l
tr qlo
.-trd
s
o-
EI:i
ocdf;
HBe
x 4+
h0 AE
q)6(U
-i5a
>c)
ir &.d
:EE
*BU
-s'
E.l:
s
5 -E,E
-E crd
)4
^k19$
vH'
6);r
'*.tr5
el-eid
co(,
1
l-- EifffffIE.
t_-
IJ
L
t'r-]j
TJ
IJ.
l
L
L
l-.
L
L
L
L
L
L
E
.=J',
^
5q
qS
ra;
oco
bO
3E
r+r
ca -. f\
uoz
x- H.-
5
H
-q
Ee;
HxI
d
t,.f e
IEs
c0+
uEe
bo.Btr
E*E
-.!Hi
8.2
E.E
SsHs
JE;
F. For.3
bbq96
E
h'€
iEi'
.i
fr.{ {: d
,E
N9k
oE.9
$g$Figigi;EE
_1
t
E
o
x
..
j:r;.ll
8.2
<e
_(o.d
Ets
h-
c.t o
stsl
; g"E
d.=
.s s.E
h;.I
Bo,
trd
8E
HE
c)'..o
HE
I
+?.
. O
k.. dt.
Q.
&.EA
E-t
'l5.lo t -H
E
jE <-l
q og
.j':
::.i:
.9P -i
t\
bq
..oho
EE
.'d
kc
a,o
'\-
ho
(O
ca
.d
ok
gY)
ro
c.j
F.l
bb
e.
{,
v
e
Fl
p{
t\
Egii
-&sE
9l orts g !?;f,
i'E r'e *e
:t8 Ht ii
r: as;
E
gE 82fr",
t.?c.-s
H?
T.Hg*€EE
.g >'6A;6.9
E.E:SE g 9
t'AI LsaE
i 3E*r*r
s€
PiE lEge
i:;*E
e'{
B"E
ii i;f,eE
s^E
stgt,E ee
38" I* 9t
E
*sHitES
Egr"boo>
n: s€ €
E
r\
sV9AU-).a
h'r,l:Y
&-
i
i.o
O
i ;EI€ fal
o)
^.
rv ()
EE.E,
X :8.E3
E,.Eifit ?€f
fi.iei"fi ,.Ef t
cd.rro
:ri{;6 tat
E{i{ql
3tOPor9.o 3::
E -,EEf.E Ig q
II?;sq EEt
El*IE! E=g
Bf iEf;#:"
Ii€$i*.oEcs
itt
Efr
iHt
fr
E;:E€f :::?g
air€!
aEE:lt€
*t8 ?[,its
!,$,rl€!EEE
.-r, rt.t:-
..r.,
*r-
-t,-
E
E
o$E
.9a.r9ts-3
H
$rg;€E+
*,.: H3*31
q,Es.E€f,T#*
HI; i EiBE E
si c*EBE
;; i; H:r:"€
Is-3.si B.Bt
€
q)
H*lI
3:la
u Hcr $4 d.i5
.sERE:eEf
,..8 r'e dd? 9 o
!8"?-*g':ti
A o (2 d'6 tt ':*.2
qiilEs;xt
i .n . { d.d .e E o.
E.E$EEf -s E:
.9o oo
E'] e E n
.-l
W"
*--w'3-o
bo
F{
u)
n
q
fg
j
<,.
bo
q
14
(o
u€i*ilgga€EEgEtiii
aEB}S,IH?AE1fli*,
s?EEE: SE
* H€8eiE€
o
" gE+X3E;A
o,
o
c(u
o
-{
f
I
J
J
J
J
J
J
J
J
J
J-l
JJ
J
J
J
J
J
J
J
J-J
J
IIEBiEaiEiiEEIti[ J
J
,ilitllqaltllm iJ
J
J
J
.J
s
9
h.o
uE
eE#
.E
€.(1).Fi
.g E 6 .d'
I o -ct
I€H€;
.9P-:.8.o
H
H,l
E.sq#
tEE
ESdE;
H
;Eie$
gEBf
EE t n-E
s+tS *p
'o d o.i, '"...o
E I:f EE
olooF>'e
G).
o
0).{i
'is Ea b.=. hg
E,g'EE#E
I'J,x u e E
.i E H=E d
# gE; B,€
"eE#Sf
l
4r
{-
.,].
o
f
a,
-t
F{
g)
n
q
to
r\
tao
m
L
iq
d,1
++
^
g
t;
5f
et: ara;;
HSEE 3E
€"€HE s'i
E
E
XEBE:iTqH:.S
aE:i -gfs{
ta{I $ttl;i
IBiI
aBiiE i+}E 3
- €*e*tE
o
-o
-
dtr
Fr.c)
E.t
-€
o;.
oo
-J:
.€<
()
=i
5
.Ag
CAO
t+a ti
oF{
.
!.-
.9
t\
..>
'i6
O. ?
+) \i
.
t-
Bt
E €E
#
.io
-t
.=6
' Ei
9?
-.k< .tE
b€
rd EI
tr
t)d
,1
8E
aO
a>>
.a€
p'6
(o
rr)
[irE:'i,*t;{i
.iEir{'iEtff
-
/.o
6n5 -r{
d : e:
.s*EBiH€{hE
t
>--3.g
o) o.
.E€
frrr
(o
Fct
#
";8
!.d
:
i
.
E
t 3!
;
2AZt*-E
fr€ l E
tt, X1<tr,:*
BEexaEo",
frEEr;,p'
air€qEg
9;d€
Y',,d o
ia ll o*ta
(D
x>q
i3 6o'
itiEtsE
Ete? su
Qll "6
iEc.isEE
gIt
E
*Er $lEt
€f;1;3E
fg{;F;s
:
bn
TgfEE {
EE I a*
E-d#t*6
d
2
or
!i
F
c)
o
€
6l
o
-;3.
ci
3
d
Fl
(El
6rr{(g
gE
E+
Oo
€.g
.s A)
vaB
S.-
[,, ,,'
.3e
$
t,;'
.i
(o
(o
J
t_
,L
..J
I
I
J
J
J
J
J
-J
l
^J
J
J
J
J
't
'rJ
J
-l
!
-J
I
--.1
,l
J
J
-J
a.<''l
.{
:<
,'
EE
,.o
shorO
36r.
l(a
H€q
r,'d
co
Htt .
.$ E
--<
:-
tsG<
tt()
orE<
E H?
;-q
--:-
*E'E
4,2
!3r
q
r{
q
EJiE
14
E
]..
#q
te
86H'
-.
).< bbg
<{f
gEE
'E
i
bo
E.9or
EI{ oq)l
A "l'
.ca
Err
ON
96pt
o..
.Bf
€H
o.r8.
a
3,?
Fl
A
bb
t{| r!
U) !i
-t
\
L
L
L.
r*-
t_
IJ
L
L
L
l_
l_
L
l-I_'
FB
bo bo
,ri
*i
dd
Oa
3
.'5
\
uo
'*i!.;
.j
r-i
r.E
f..g
oc6
()c
=ta
ES
P,-.\
c'v
HS
S-i rI
'E;
5r;X
s rt
.i-d
V
FO(D
>
aoQ
d-(g
L
.=Hd
TigE
Ed,0
lr
aa
tr
--€-H
f{k
"i€€
\55
-a_)
o
1$
d
'
t,
.E
.8 ,E
F-
-
.E
6's
X
as
S,
H
A
E.
--
nlo
vs
l(
i-i3
:<.
zE
ts* P aP
I3
r_A
T.z ti
Atdq
E
\
ro
(o
E
o,
1{
{o
tn q,
Tu:
io
rn
6
T
-c
ffi
'tqt,;:
=C/o*
.:C\l.-
xC/o h
c
u2.ts
?
a0
c.t
o
f*
(o
6.i
F;
E
L.
bb
G
E
s
n.P
o?iH
>: '<l
tu
c.t
:* '>.?i
d
9 '"o-ao
:f,:9€
.c 5
7
!-.iH.v=..)A
I
-EcE*
t
EE
H
5 "6E;,Etqt>
A EisSBE;:E
\J
EE:"8cil;
x **;oF,H* f
oi
;x.3u E€+ t
Hf
E r; E; Y? "+;
r!' Eorj;JqH
o-ub",'4,1f,€H E
i c,f r.=I
atri} ;ii*Ei E r:€€$E?
-.^A'd
cr .sieE€\;
O
f,l sE;i;'€;ir El;.>'E'El6EEtr
.LJ
f!.f
8.g.;f
N* 3i;# ,
- iExH
5
E; e 3s
6 6i
E€
R"si
q_o
r.-
E
iri o)
.{=CE*
H(a}l
>a
<d
8o a''
v
axH
v
T B'E E
€t.3
E s'
- xb -.8
E
+jd
- 5ii
E'- E 'd
o
o,a
tuuEd; = 0)'<
*Ei: EB
d..-O
g;Ei
* E: EQ #o s
Eco Eilo=.
6 iu 'i
^
3 E E;E El;
E
ie:f;;str
EE f
-E
; g I8-'EE
f{tf*;ifE$$.IiEgigF
r.i
ffffggis:,lF
*++
L .E a€
L
ii
i
*
ffgB$ $r$+.I $ i*;i
L
L
giffif,sl
i;til r++iiftl:* iiEr
L
L
L
L
L
L
L
tJ
L
L
I
t
gl*ll*latgiHffi
mtffiEffilHHn
t;eBi?e igi{aa; tEte i }
rr
ii 89 i2
:€;tit;a:E:; eI;E:rEs
8
H
EC
e"E
6-o>
*E
"a.q5{s
dql o6g
cl€
.9cu.?;-a
:?E
-A: IE
E; I',
gl'fn
Sig-1
CEa
[,ig jE. ;
0]
-At.EEp,,,,,
=:
6Es s.B{BEq
I6*siisEir
-_I.rels€t7
E.:+8e8ts:6
f;E+'*?i;Ea
?Ei,i;AEi*
9E
p
kT u g - u
E.SiiE:;"
qBdt6Bet?
; Sxidgs:a
}:
?E
*'*-iroga*EE
gFu -2,F
* ffq'.a *,E ";9 3
&
l*
I
o
-1
c\
c,
EO
r\
-o
6
t
E
E
E
:;: ef si +
.E;s€ >E€ S
A
E HE 3+.8 E rq
E:cE*.ee
!E.o'E =*;'E i
T's?'rE"E"
*jEEE ?E
Ag€ig€E
^ 3=E€g€8
tr
*
i
ag;eii*;;
*"=Z'c6q 7
o
EESE!"fE.t
E z il tE >E=
EE18
iE;iT
v?.:='e.9>Erc
H3
r{
rq
t,
@
cr
6i
EES
E{;$$EEf#
*'i gE
..1
:?I.S
i<
in,
I
c.l
I
..I
J
JJ
J-J
JJ
J_-r
_-I
J
J
J
J
.-I
_-I
J_.-.I
...-I
_{
J
.{
J
_J
B{aT;;::EE
lEE:; t+ i J
J
J
-cs-*
.
J_.-I
{
.-I
J
t-.
f
I
8#EE.E'EE
8i iHo -a rs
iffiiEi
\R,
iE ;€i=
a
<)E
>--
.
;BE
};{E*x.E:
cb ['rX;=
ffi
=oH
lJ
.!p
f&
,,-{ffiI
'll-:j s
r--4,
a\
.9:J
H
5B
€E*H
c Y:e
;; E c 3.I
'io:.P
BE-.:
Q h'= [j "^;'ts
:
O d
A. .r't'd
H,gE€
e€;#
x t-, E s,€
H-38'iaxeEe
EET
.d€r'.y.d;aCi:
1e-r'xP9I=6(J
-+A'-o'a
s ?=E'i",'io'A
8;i2
I2;xFsc\A*
d
3
G=rt
€ A5(
-'ro
iD iD o).v63.^
o hn* > qe EE.E
r.,
BEtE
ts
"q€;;
iZE;::
p.=-Qg.e -'d
gr-E
4"=-d
;*6.*=9Ie3
.1'c
a - 7\.qqbC>7-.:?(U.E
i8Eai-E$tg
.il
E-2;Z+ &g
6) dj'o
!uda)
i.o:: iu
,: d;
,$,i{€*
i
T
A
q
.r-'
o)
E
4d--4
i,#E
g
ts
E
d
d fi.e
d,Eeb uE
,se*'E#s
u{
7.na
c, a Es *
)H!
r,r * I.i.=
6i I'6t I
+
{:.E..9
o U<
=
J('o 93 I.]ClHE
rA;.i !E
rqqrSY3
tE:;s€
e'ELGF
d
.lD_g'E?.tr
c) tr.O
Il{ ->
?
D
d..a
n
€ fir< fr"*,E
El=
>)F0)(D-.
tav
!HHdl'
d.;.l 5 i 6
gg
EE€
'rlPc3d,
E
E.Z'i
o.2
> t{ 0)sl
a'rrs
trOo)aJ
tCa4A.
avaPU
".
3f€il*RE=E
t-
.^
-J
-t3.s.
'\
q € foT E-a2s
. i'; tr
b-s bo
v-
E
1,i; a-E^E'ie'r'E
.(3-.9*
6 €:.d
iJS
=
.A*A 1EF7;* i;y; E;
iF-r
ai
d
t
E
x
E
(o
tr
€tt)
.J,
o'
il
n;E '6
A
'+
I
C)
oP
()ho
,{
,Pz
.Citr
+bO
l*o
OO
;BA
D,-^
--€
di
o.9
g€
E6
Eg
a,
o
c
<J
=:?*t o7
a ,f tri
,E
= tEi
E;:
!l 6-ll
:;;
=lH!l
a- r'
AOcr;G qa
Le-.$EE:;<,8
;
islliiis +=h-
ii-tEla ffiEii*laEEE
o-
StrfTiE:
,N
,$ t,*?fli;tc
;a;ili;H
rgi gili H
I
H
{;a;;;E
E;Ser;i
;;?:?Be
?
t
r-sesj ;Ai :tr.
*\.
I
L
I
a, ;{-
=j&
fl- EI
K.]]I,N
tx-ztl; :::
I : ;
im
tr
tn.xc
;
87"
=:;
H
{ffi
H
L
r
E liq'
F
:;
l; !B
l-El
t-
i
i
t
'
,
o -tri ;qq
E.H
)so
g
ai
dG'ts
d6-l 6
j*s
a
l;-8'6
i'E *
I+E
j-r
.il
.d
^
o bo.Y
ot.E
^
tl o09
HJ
o
?ii,
^o.=
i, I&,
AzZ
Q-:( 'Eo
!;":4
cl
(0
6r
brb
lt;fl
a
g
' litB??i:?€iili3i€;
f-
i; ;r; i
'-lEiiifui*wifil1
.J
:I
l-l
J
J
J
J
J
J
J
J
J
J
.J
!
:"1
--a,l
']-J
-t
--l
r
EiE€i? it:B
EI:,E€ *qlf"
'€,:ii:
J-J
E;
t:Yq i;;;
I
*1a1ili
EHE*' EIl liiH
, *s;I i?eir -J
i'i3i J
Pr
;I$5,I ig+;EEEE; ; E; r E1EE,
xPt*'s itEs#€ frEE * :l ; i''"e i zeuu*
r4i
ri[n
ltr
:
iJ
E iliaiiEi
l
J
1il*Ei1itl1iiEE H EiE'ii'; J
J
}rEr:EiiiE rErB
EtilEilllEl,Ei E BiElEEi?
aaia?ai;
et* giltti
ie; iEi:$
c, ?ii *n*i;i:t r
?ii; il?gE?aaii E*E Ev,Ee
,\
$
c!
6I
G
Ei
bb
lt6'?.'A
oc+
Et&
Bl
o
?.nta
t.Hl
E
€+E
H.+
6rH
x .1,8
lilEiiqEiiEi
Ei}EiE?+3B3i
!ITEiEii;iiE
;IiEEIiEf,g:*
fiiAlil1iEt; r
EE*E1EIE;EiII
*a*EEaHfi'EBE*E
\
1-'
/,
/l\
ol
dt
llt
-*
tL)
(,
b
E
,1
(,
a,
co
J sgl-
.
7}
g.5
'r{
(2,
tso
- c.i
Jrr
q
6
f\r
oi
co
&
tu
tr1
l*"-1
_/*
or(,
*(*
.\_ .\
\ [-u-
v?
cr
o.
tb
fi.
F{
6r
(f)
<i
$q
t<\
.-o O.
.E,
:.$
[.h
o
o.
o
.'D
o.
tt.
uL
Q)
X
$
E()
.r:
c!
!F,
o-o
- as
cfa
ca -o
-(}
G;
Eg
ql
=
.q.
.": '
=
H
&9o>
1
E.E
x'?,
3bb
A.-
OEr
uaQ
o c.i
L
H,^
E€
E:ftl
f&' €
;
'
€6
(t)@
6 f!{
..(rv
,Ad
'o.c:
'Eaal
.
-.0
iig
ca rl)
!
ecE
E l€-=
g
ExFo
EgT
bi€ ; i;g
3t* Ets+
+
".IE
i:€
. afi€iBit
iIsgiE?E
H
IiEeEtIEe q*€3-o^'-HE
E
t-
;aE3:*T:;
*:
_;EE
h
3E=65
ts qg q
E:
3
o
+IE E
T
-otod.:-b
8
H
B
E
bo
EE.E
ET
Eb .6
*ii
E
eIEE*
TE- EO
€aEEE
E::E:'
a'E
rE $+3 :g
b'E
E
-s
",e
Iil s nE
sflgtI
S! t €EEE
=dd
';'EoE";
E*
slts
v
E
I
(D
tl
ts''
- 3
H'E
661
^io
r{o:i
o6l
€
*c)
to
>
- L:"€
&5 -d",I
7: c,
'6q
El
dii
v
qs
'c'
Ittn
H.F
o'c!
'=
d
-=9{
O tlr .;
a' o,E'i t
Y
\ ? tbo
tfl €v
;i
€a
r<
&
tb
fr*
!a
o
_aa
.!a
o
o
.+a
a
,.o ,
_€
..{, u)
'C.l
qi.
€
.:.'ci+1' o)
.€
.$
ff
Ft
cO
F.
d
o
9'
H
io.
fi
*{
(D
o
'sP
fr* ts
d
.;
'vt
,9,
.a,
AR
s(D
el
flt;:t
E€T EC
s
rrlo.
'E-9
gI}
I 6a
sE
o e
I ()\-
=tG+
a ESE+
";
ii gii s"
.3[
ori'6 oi s
,o
.d
6.lj
bo
,F.
t\
c\
o)
o.
ub
Fq
.
.
..
d
Ai
o'
3E
€E
=€l
G)H
.9p
O. c\i
'aq
d
^ 9r.
6;
oq
9d
€q
-E*
q)
-E
E6
5o-o.!
?tt
q,€
T,
.$€;
h
x'E
v>
sk
d
(6o=.
.f qr
,g
q,d'
td
ga
.g€
AA
sa H
--(] k
A.-
.Eqor
60L
{g€
H<)
cO
O
oi
d
H
tb
A
f:{
z-A
:t
J
J
J
J
J
J
J
J
J
J
JJ
J
J
J
i-
!
Lr-LL
L
Lb-.
b--.
L
L
L
L
L
L.
L-L
:
L
L
:
Lri
li
t
-/
,,
-,
aO
id
6.e
nr€
tk
Ov)
g.tc
'ii-lo o
H0)
C)!
a@
.a
a2
*,1
Y ,t-a
ts!-o
h=o
HH
0)
* Q.9
".5e
-.
c)
.x 9?q
.x:to
od
v'-i
€
Y.hd
xo=
qtrtr
e.= bo
oocd
d#
g
.>qE
:
fr€
=.q.d
IE-o(trlH
v@
m ..'U
ryod()
.0J
-5
irr,E
a4
oo6
-q€ o
rroco
+l tl t.-.t
l!
. -rl
I'
b
iiiiIa{il
iIiifEi*EE3
It;iEttifg;U
aE;gIi$EBEEfl
tr\
ql
l|
u
@
fo
Ys
(5
*-t
6lX
aii
rl
\-A
Ho
ESS
rr *E
p.i'€
a55
qq#
o-o)
EgE
qq €O)o
8EE
g,EA
F
rl a*.9
tO + YE
!d.-H$
Iq"E
f\trd.
bl)o
s.cH
Fl O(X
f,3rO) q
o.--. a
t{6q)
A6A
8.!"
}lAa
?cu)
(/lH0)
o ".!
oAv3qE
5qd
tro)
.'
():N
>5,.8 -r . AIU
EA
L
c
oo
o
t'r
(,
.D
xo
-tu,D
E A€ E8c
-HOt
;: E
3 EdE EE3
oi+r9A
bSE; EE:
o
E:x
Eni.E
39€Hx
ea'?
s 6ri&
AVH
- Q':
=co'i
U
ai? lts;e
* I HE ?") g
6i
H
s t rg
HE
b-E€':
f H+ eHt
FF'.8
tr
E*s : EilX
d.'--;:sz:, €OrE a
r=;Et
i:9ib g'fo,: *
H,E'
oi.or
i'[;:Eg i
T'EE; ItE€T
fl.
f,=FE:Efi3s€
t,fi a 6 H'E
3gEE3 sE6
E
,fiil*iiallg?
e;?;*il; fii'gtra
-l
(a:d,=
,
i,o
""_$,
r^
-6
?6
o
0{
Bri *
iDH
-ood
I3=
o6d'
fl-ti
-oE
H
X;
A.E !t
.$.q E
a
ti
.E-5 E
.-^ oPq<
:i ca
H..l O
42Xor
hi E ;
*H
Et:
*EE efi
€'x
e
-&96
gi€€
Itd o p6:=E
a\ E!a^
or
6 ;39
Fr 9E
a
F:EB
663t
-i<
R HgE
4
J
8
p
5
3
e
C
€
c
o
CLe
{ .E?
Ef.
I :}
Fl
\
J4'in
.e{.
P€
a<
g€
f:
/r
(o(\l
o
-ot
-rug
E
x
3o
3
o
o
r/)
l,
xl^
a1 d
clai tr
-rat
* -lO'nl
gC\
-t.5\\
;,!LI
$
->&
I
dt:
''Fr O
€6
ll 'rr
r._l g
N gE
et
l'1
ol rrho -c
€,=tl
tr{
>5€
.4.*
o2e
s'5
;s
u4
o(c
.A
;s
.c|,d
rH
;E
cd
FqH
#
c-l
^o
19 uo
H
c{l d
"iE
# 5i:E;€xE ei:ii
E
E ig &r H fi bo{iei:E
? fs*EiE!:i:PPs
*i$ai i
\ iEEi:q
r tE3sF?i:iistE-
i
i
iHr$Hi[a€eEEE{
o € GF
E9lllo X FY-:E hE o
i;SI*fi:f sE; EEi
il
o
?
. *?;-BE'H-H I tt x .
i{
#H
E ss.PE iEEs s:3E g€
u
: ]*aiUg E= RI s H i;t
+: *i; * E [ i;s
s ;tE r
EtlliflgIillfii1
:T
*I
-t
-_J
-I
J
J
J
J
J
J
:l
J
-J
.-J
..J
=.J
..J
---.1
j'-r
-{
.{
-d
-{
!r
J.
:.{
r<
*--'
--.
\
L-
f
LLJ
L
L
u
u
LL
L
L
L
L
L
L
L
L
l_
L
L
TJ
IJ
L
lJ
L
f-..
L
L.
l-.-
2
o
O
d
o
O
fr
o
&
tu
Z
o
q
::n,
-';.d'.
11it
fiH tfil ;lBffi*figl ;;II E:fi
ttEaifilaua *-
!!5; fiiilrl;*
En
lgffi
iEii
>d
FqO
U9
dE
z>
HH
'lv<E{
HE
Ets
(o
N
c:
?31E
LL, IiEi,iglitfifitIg+1.
L.
f-'
a
,(D
:.
i
6o:
v6l
I
e'l
o
€
*-tr
il
*^
+S
c{
'iE5+
E;Ei
ll6ll
clt r
.-
i
^'a ii
u..;r
1
E ss$
p
I
_^i
Ea
(]
'z
I
o)
'F.'F.
++
ll'i,'ll
^
3 ,x .S.ia
:
X
:
F
u?
c\r
zg
Hk
.9€a
;!o
(l)o
qii
G-
,,
7'laa
I
('
0,
B-flt j
t'l
I
rrlf,--r{J_f
f
e
o
G S€ddll'it
.gE€u*q+
S EEii[+
;EEEErsl
E 3 &EEEI$=]
g
,
I
E T'€
t,+"IgiEE,$+t
crt
E;isfE€E
5
u;iE
,gHHfil**!l}i
i
?i
-I
J-J
J
J
.lt
J
J
J
J
'l-J
\l
J
tl
I
-_J
i
;.J
,-.1
g
d>:
,. bo
ri
=t{
riH
n.-
.:-
l-
:-
:<
.-<
\<
-.*
;<
.d
-o_
-{ 1;
FrU
t9
_o
,Bg
AP
H
';
.dH
g_lr
;rl
'aa dH
or
\'
AP
Al
i{*
d>
3-
Q)a
EE
tg
o:j
H'd
.do
a*
o.bo
q.;
E'S
7a
fig
'E '*'B
,\ 6ll
\
6c!
,L
,
.-H
B'
gx
-
.!x
=
BItE;g:Ei .- E$$ c*a
H!€slBEs EI+o I;A
I
1Jt
,\j
,{
torle
?'
llE1ltti.ff**iggpf
o
ju
E?agEf i E B
8c-
--J
*l
j,
\ < ,\
T
t\
b
Ea2
fuA
f.
+t
-.
:
L
L
L
L
r"
L
I.-.
rJ
L
L
L
L
L
L.
L.
L.
L.
L.
alEi:
i*;q*
.;e
E 1A
Et;*
%
H69 q
I.B ojlao
€Ep3fi
'i6<:sE'
sTif Et
rE,e€ H!
H 0)
i;;
,-liE
>=P-9o63
o
o. .-ii€'1:
AdlHU
. tr E[.:qii
o
s€
E
uo
;
s€"E,l8a
(6 d
i Eggf
E IQF a.HElE
t
HT E EEE
EE:E
II
E-E o E E':
<'2-; s.E I
el#B! Ee
E
G
H E
o
,t)
r:
-o
.t
rZ
-61\'.'1
1\
o!
oA
E
'*
o
H i :H
:ql3Eg;EIE
;E
8qi egIaElEtgilffiE Ig? i$B
e#
-,I €.i,,'F
(;EYi,*fl
=
L E^I f
r;"'aEtEEiq
L
3at*.EfBi{t{irEiiaE;EaIE
L
L
L
:En *{;EE;itgi*gEst**rti{;
!t
ig+
L
L
L. i= t+il EI lsrB:?Eg 1$irn Fip*i$r:;:E
gE
L
EE
lflII
+I$t
BiEI t}E ?I Hi${HiliqH
L
... i S l
L. .r- rL. 3t; *E f { g
[3li?{iif$EiI
gt;E
1Eii3Efiat
I
ag
::
;s
L.
f-
'
'q
E$€:s
[*o#e
u: >,e.: n.e dE €€-S E'
r^E"E
Hi5
tEiE€;:H
eE:B:fE€ EI o8
,+
:*=
:T;B€:? Ei.E Es*_ErnIu;
IE:g$
E.i;E:-EE; Es? $i
E1ig lE re ;ef
Enur::i:-EEtx*
a€ei1ite1t;rI
f stE;a;a E;:eer
r{
F.
bo
cl
6l
o
cl?
t\
bo
cr
oi
c\o
0)c-.
@o
u7EQ)
P
c) 6-;'- > A.
^
O)
X bbEt
HE
*-ae
!HA'
nt;s
r:':
',
(
(
\
H'}l; B E
sgg9sZ
g* 6 ETo
;EEi:.8
E'i
E*BEaE
EEEgEt
z';':': 3 I
r E B*l g
H>8OEE
EBoAaS
: EG ?:
i€
a
I"8.(s.Q-8 t
uoFt
3'o
EE.14P'P
i5
f,Sii
*
Sta E
H.:
- a6l
E.eCl6.Z
5E;i H s
q e i s'!
*EsEi
iEiEliIEIi:E?E
.o
Y
*.i
!iu)
H
v
I
.i
,H
,q
d
o.
s
'5
o,
6,
o-'
o.
O1
fi3
:..!
.r1
oo)
d>
4.s
r'H
Por
E&
G=
o
li 'd
6)+ L
otd
-dA
*,,4
''t'
:o
;E
>€,
o
)
E
o-
iE
*tq
q?iEE:E
I*BEEB
gtl?€glaailgit
E,t stEE€;;€ag€f?i
-{
:l
l
-I
i-I
J
J
J
J
J
J
J
ffi ;;frFiBsEsBaaE€
fl'*
HE
f;
Eeta ffiiEHi€
E flEEE
J
J
J
J
J
J
;a;Be?ff*BEiaI
#EHI;,EBiiElAE
J
Bmt,iailtiiaiillql
,;
o
o
c
o
o
6
0
I
6
i,Steat:A:lr;EnXEE
E;* EEE:EiIs:s€€E=
J
J
J
J
fl
-H
J-J
J
J
J
J
J
*lliiil1l1ElE1
sBeEH;
J
J
J
--l
f
L:
tJ
f
LJ
f
LJ
f
LJ
LJ
LJ
LJ
LJ
LJ
tj
lteE?iigi;
aflEEiiiIgf,gtI;EEig?EB
}HgiEBg,+3Ii
:;rEIB:€EgaE
E3I:EE;E;IH€
L:
L_-
LJ
8-c
da
p.=
dh
>>d
A.T.A
{\
{\
//\sol
;\
rr-'l
t_
iv:q;
aV
-E
^
A
c'EE
ta
JE,
o
+t
€
o
0)
-hoY
+,
.H
E.{t,= A
=#{E
.: -9 .,i
)
3€e_E
f tr E'o. H
.g
Br..9Y
H 9 -x.!9
.1E
6EbO9$
6.d
4",Ea dA
4(!.lo) o)0
>$
s E'Ef;
(!
H
.Eite
.[TEE
-d.boN-
E. -Es
,7
9f
Eq;Er
EEEE
.:d
L. e db
L. E 8E
tl.'
tj
l.j
LJ
tj
tj
tj
1-.
t-1_-
L
)-
->l
E€
Es;:
a
l
=
::E
iI
5 EEf
it
;; AE
?
ri t
{i{;;ris;
g'Eq
;B3;i ?tBatgi;
*a a;;l
E:€Ef
.'
-
'
b'r3
Aor
do
HE
b
5S
6?o
oa
9k
'60(6-
+> "
Oa':l
'H.
A
Eo
'3+
.4)
u
I
d.
ts.9
-s
Ld
L/
v)
q)
(6dl
EE
a
l^
fr+ X
.9 "m
Q br
9qaea
c.i
-
.;E:
, .c€
'rPEE,
B,
^El
^l *'I
cr
<.E=
d.aH
tHs€i Ees=u,
"Eei€foEEiJoii
s+ FI
r flf e
-AIiiBiEiEI
tritEqa;fEg*p
?gE;EE;E€ I E
itrfiEIti
n[
,:g!agte:iIi
*;ffil1Eiil#*
c'i
-:!
-<lco
<
bo
CI
$6
,
?
j
:
';
,.j
':i
':
;.,
-t
s
:f;EETEgH
E:E$iEEE
;i{+;gfg
i€; IE ?E E
Tgfi:HEET
tEr E iESs
.*: a;;
€
E:
qxT{€;?fl
IEI:tEi8
?
'?|"
JdkaraCe
H
I
EsJ
7gi
*si:
s
a
oifi
sra **if;?s
i 6=
iss{EA
3E{iEigt{E
-
;e:
EEr*.igt*ii
;E lIiEE igE
ESg;i;EuxE
$tIIIiIgIiI
:E53E#Etat
EE;Tis EE YH
** sHniSE#E
3.e"Etrs
'": o u x'-
'E
of E ce*
d
4.a
F
s
ttx:.
aEv'o{t
I8f 1.E
xr-3,.9e€
ttr{E€ +ta
O^i o-
a
t.$
HgEss
o
d
It;r
Tl 5ti15F{
sE
i*;* t
'E
<zz
-.o
o Ho
h tS
tttie€
I.9= rdiEtqX
o) ->'.=
l\
.i, €.E I !'s a
ra ef.e€ B p
B€; .E a€
x dE€tlE
,H*
I
E3 "
Y$F
--r,
---+}-'
r,
s!()
d \-6
t;{ ?E
E
€;
i3E
E 3;r:uo=
u?.:r 5.EE
8i r. :; El.E
e
,vI !p*^E,i"&
EE c! H'
gH
EE
cI€€e:
E
ox-{l
€r{\1
a9d
3*a
^6
o.o =
5d.x
EAb
E
(g .o)
,:b
A
6
otl)
PH
d
.{ cD€
B
3l'o
lH
aiqp
l.oo:i
F;^.-
E
r E'H
',31e
u
f,19
A.E
+ il-q
oeH:
6',9
h'- cr>
= e!
Id_
BH
oo)s)
e.+
F
+?
"r oJJ
gp.8
B#€
,I I
E'I
=
EEO
8fr *
E.* s.
EI
,41
Fl
td
bo.;
l.-
{r
r\"
E
rf,
00)
\,r
,q
C\l
bo
cr
14
eo
rr)
g
g
bo
ci
o
frr
,iQ
>
i.!6
'>a)oE 6 d
"€ ,
T** eE
ti:
H'
F
E:
ETE
c'
H €I
.9,-e€
*'J
rrr a*'I -E H
XF'f soE
cltr*:;-g
HE
:62-3r
'#.i fiE
Es^. F I .
i --a 9 H e
s€€EEc
B.
f :-qI "rE
oiiaXtr6
a,rOvA
s gts xE a
? i i*E EZ
E
e;aE EEU
tijE"€
*E€dtE
lJ
L
L
L
lJ
L
IJ
L
L
l-.
u
IJ
L
L
L
l-_
L
L
L.
Ij
L-
IJ
IJ
lJ
L
L
l-L
I-.
L
L
r:
L
ggisgit
BIgIiIf;mI},Ei
Q
f
.f
o-
a/
*6
U)
o
"t
iftElfE[Eijf{;l
Eg B#$Eg
E a a;" si
cFr .!i,hHEH
(ue
f:.# o-:on
p Z u.O
&5\r
g
i c\ 9.Ev.:
g;
gE#f E
HE t.E.EI6
tt ':, 'F E;{
:X.'al'=9
^\
-'
q d.H:
v2
'5'.=
g€ ;EE'H.j
aE $: Iqi
Hsfli
E E BE
s'i H
H
- 00) -epESxO
jE'- -.
'Q pd
i 8P {...'
yEi'EE.E t
dtr
8z
Ei E:
gxH;E:E;#
SqEtsE
3ri
s#85.8';EEiX
B
{8 B of E;,t
<ri.r<-e€ q€-{
p, *
g {"o
s
.
€';T E,! E\;I*.q*:
ssR€ E.F i;e:E,EE
fi
E* Ee€s
}E:i :{EES:EI$fE
i€; {i:I"
-gig stEfifjffff
u€s*i€gEEiiEigg
o)
II{
o.
X
i.-
fr{
hb
p.
6l
c-.t
(o
t\
bb
0.
oi
ot
I
$gIggif*iffigggi
H-r 8€ j s;*i
fr [t C;#
:iE;i"$e g#+;-*
-;
H
p.d
H'<
t,
v1 6
.+ '.E
o-9
.*eH
poE
- -(l)cs.-
.+(
.tm aEff,
s
:
rJ HFIE
'o()-
,,F,-E
bb
35
r
s' E;'8gH'
f 3f
ii,
E E#:
H g{€
dr,rp
E sel
7 -6€:=
sE;
F{
i;;
ec
?
tr .aii
EeE
q ,3E
6\t
t
.a
,.'ia
o
I
q)
bo
c)
z
tt,
+
o
.v
o
0)
€
0)
c,
a
o
7
o
o
s0
l^
g
.C
o
0)
I
Z
+;
'8,
d
I
t
or_
ao
,1
9L
r,
E
aE
II',.
.d botrlH
dHv
8
EHEE
a,".4
=(1)G)e
,,q*€ 9
B <r Et
T .E i.He'EE
bbts;
'il
fr s *a
sEYE
la.a9"
a
E
ie &e
geP?
i€ u
alz
-v.rr-H
tr::.tra.o) H..r.
d
e 'oots "
,:
F E Eii &
ts-
F
q tT.*o
xd o?-
.9 A ,
r"
.E
td
Y-O
BI s,
<€;E
^-
i) ..
o:
Z"-4
s'H€
I
co-c! d o
g
EI EfiiEiIg
I I;iE #:EEE [ ;
-HtgifiIfiEfi
I:E
EE
Ie;g
dHlffi
f A; ex€ i cI F BsE
lk'"
i[;x
()ri
{:
(D
\6
(r!
0,
----d
J
D
,r
o
L
oi
:c)
o,
Po.rql fuo]
CL
'
o
o
r>
fi
l+{
a
*c
cio
asl.6
'n
-=e
E.E
oo)
()H
gb
E$
+" 'a
H}
*
0.)
1-?
€d
f,
o
d
#
€
g,s
o; frr
?q
-9 cl
o.bo
€c
{J=
-!
b6
Ai)
;E
.
eB
+o
tO
*r9
(),B
Q)ir
<J.Y
.x
a-d
ze
d9
3E
(du)
t
0)
aa
oo)
() 11
HH
-I
:I
-I
J
J
J
JJ
JJ
J
J
J-I
JJ
J
J
"-I
JJ
J
-J
J
J
J
JJ
--"I
J
JJ
-,I
.-l
-I
IJ
tj
lj
tj
u
L
IJ
u
L
L
]J
IJ
lj
tj
lj
lj
IJ
u
l-
o
0()
*
8.2
>)e,-
6d o).;
€'o.4
t E36 {
2D(6
A.a.\dV
o.Yn
E
d
x &sa_l
7aU2egi
H.\H-
o, -6"8E?
€{:*,:83
H.o'= t'r H I
o *l ! 9 d,"i,.o
y,"9
&9-3c
g;;rt
s
.gEXt.H?
s:btEE
q"3?;r€
ts€8€E
lEr E* s
ts os
=,1-E
k:.i
i..
d) g
E E t 8€ I
boll
yF"s€
=E
EEEA€.8
0) +
E.F.
s od
E;*t:s
iEE
gI
oo
>99
:?,
Ea
9d
!u
:r*ias'ttt
I
i il
gi}:
o
3.H
d I
*€3i!
€E
le.iE
:: e *0)
E
s; s Hg €
"Rfris Et
;Ee*aotE
H,e;tE!t
:tt;
H
()
$IgIiIEE
..$dHE=eEE=,
P.o.dnt
F
\ E;!ruPil
ts S;E:EEpE
iErH.il3sE
€ a* f
tg{:;EEE
'= -^d-* - o^q
,d
= ;*,.E:EEt"r,
b,
a.sEi EcE il
"
I E i EI ffiEiI
*I
,l€ieIiE
tj
]j ;3:{EiulE-il*iif€{
IJ
IJ
u
lj
lj
u
l--
IJ
l-.
L
ilffilllgiiiligiiiil
It-
l
I
igIE
I$ E{if
E€Ee,:'EE=';? EiEe
; ii:€i
f #q3 HiEf
iti?i ;tai l
ArES
#;f EoIt"qEE E,E= "
E{EEHi$i Ei f$iE
,Oe1iB;,-tli
r', o;e8 a:
Iar
it ii;; EIgf [;,
j;8*erge+f
E: [r I
I3::E'
igii;E}tai E;;ia
2
*
':1 !Jv5
"Ei'Ea-g_3
iii ltct
3nfiX,a
8.5 3A*tT-E
E$g;T+E
:ir
xolc.i:6Yt
!:E *;
EEEiEa+
#'E
o)
E
g;tiEgI
?Ea)-d'lis
3;Sglg
'-
€;
IETIEi::
g;iE
B#
3Eg":ESEE
'r\l
E
)allY
(-)
(6id
i
o
tr H.;g
(o g)/\SosE
^r(eX
o.4l-U
.* ESgB
tu
EEre
g: o
=.!
totE.l
F$V
; (0 >,q
=6d
I'ff€-
=
.ol
5 ,r! c-.+
. b.- s2 ca
rf)ar!YOL
<t -X .y o
c
o
c.i tr
Qtg nP
q
k?
bO:po>
EE 3€
l;.
,^oai
ii-o-lJ
d=
d
.fr-ou
o.r-q O
EH H?
N
q)
*.?
tEH-3.*
d
-l-P
E i--r*
^.9 v o
ol
sJ
Ef#E6
ts #Eq
ub
f\ =!i;9;
EdSot:
#.sb€F
.99dd
V art-'-
,'Q,'': b
<,6 5 tr. ?
f
,t
.
T
I
J
J
J
J
J
J
J
J
J
l
'_J
tl
J
'l
J
J
-.t
:..'l
3.l
U
d
$ts
--l
do
as
-lA?.,d-
ri
--_
-.--
-<
-{
o{
.d
.-
--i
.{
#o)
-tO
\(J
ucd
Oar
oX
oo
strP =i3
c.i *9
t+ +i5
bo oH
s.
AV
t*a-(o
I9oo
-:.
,,H v
.(a c\i
do{
.E.*,
!e
.Y+
T6
L<(3
.DO
tDa
ad
AO
Bx
!s q'E
F-.-d
.996
t\
nE
thE
<d
v,
co .a
-.
\
L
r__
:*H;f,E
lIet;
iEitiEiEEEgi
EpxiI:aEE;Eg
*;;;tE,ai'i s'
ii*ift
iE i{ i
5
: f;..=
P
V
r<
E'
EI Hs
Ee;I
(t:.Q
(E
., t:
.s
o
'd
-r#
l-/
(D
l+--'rt
l.-T -!,
ifd. o-
l.
I
1F
-dl
ig$3{Eg:IHIi
i
e
da."*
g
ts
@
tr
E
itoi
Ef
lt-l*
SI
"b
z't o
E$
E
J E. F*3.
l iEp
3E
a
"b
,<-.<\
tdlc{< ll
il 6t lt
6 -^ *
L.
L
E;{t;riit aiE
l-L.
L *eli flFs
TJ oi,i # :?o
L r51t $r;
L
L
L
,3) -
s
co
xxI I
iH€;
I
o rlcrro rii
=
E
(E
t,l
E E E€
l3i3.0
*^ 3dE.r-
;"ETE^N;E;.8
I * 9'@ q H X
-lN.)d€'--dC
T;;;
j sI op<
El
trE
s{
qi
:tE;
.rlq'b
:E:';
f..- 3rE:;
L.
l,---
L
lr-f.-..
L
L-
yl
€
nl
o
T
E
tl
!\
..1
E
g
*
o)
q
c\
q
ho
k
t.rr)
c.i
q
b4
t,
ff gatii ttlgg
l llu3lfgi igtrl
tI?;
;g{
EEiii?liiiiii{iii
tf;+n
=*=
;!gt 3;
fE
IiI*;€
';Ft7.z=, eixF
irr SE -t€
H.eE-H.P€
lgitffiltlgtlll
-
E.EE '-co
TE
db
e
(,]
-i.n
oru?SdAB
th fi.=
:1:sE
B.A
Egi*fi;
bO (/JOi -'n.;'ri
'€,
tP. e.E 6 3
5PE#
c;i;e=i*
ita;rrs,t€g Bi
*rYii
FE
;;tE8,E
e
$agiE
x.d€tEi
o,60
EE<$
r I*E: F,;
b
rri
t
E
I il g; Eisi.
EoT IIgiEIEiB3IE}BE
#{ g:g;
:E€
:.E.E,E
j5H.t*..o
E
ffEaE€EE;g;3g; i
:i':trE-!il$?;i
S .oPE"E: +gxt*
E
; Ht:Xe
il g:i;;
*'
to-odbOtre
o.;8 E.g;E
,,
'i
i
E
t;i
a;E;iE
;; iEEE
;B 8t#e-g
Engtr*EI
E
E!tEtqE*
i t o ' q6 uoQ
E.EE n€;E',t
g"eE,ef B*3
?IgEicE?g
eEt* 3
H'€
;
E
EA
sE€ q HE
.s'En
iH{STES
HZI'E E g
;9.9)"a,,niF
E
;ii HEA
e4€+;Eaa
sex
e.P H 3ra
*F3TEi?t
< *t o'5'6 d
<*4''
-Pi
*sdoEf,€;o
eE
\
r*r i$___.-;
.J
ts
3
E
t
,j-ri+f=t
1.,;i'
,:,,-;{
EE"#
g
:
'-*
c.l
>i
q?
C'I
.1"
F{
oi
(O
fr{
_61p{
*l
1J
.{
I
9f4
{o
6
5-f
I
sfl
4L
sL'
(o
<?
14
bI,
6l
q
I
ci
rO
lr3
F{
.tobo
a
:1g1g1:ggt1
,HEi? iaE;i
i;iE: EBEil
iffirl ?t:i,
IttEl ,itEai
iEfit a1gB
"*eTa# *EE:
.P
E>
:ai
Ptu
o
(o
.d in
1q
lbo
.{.36t' .r
?8, f€:\
6
5
:
d
4'E L
?eL
ILJ
-6>
Lr\-
E^
t a
F=r{
-eE s \
O
E:E
*\
----
O)
f,t
oE
Ei
o -'ri
'E'E
.l+e
ts;
efl
.E'* +r
-E .e
-
E}
G)
?E. .:
#Ht
lrl
'E
()
v
.i
.rO
9r
Cill.t
>,;'i <i*t
€ll ll3a
.t6d< $.-r
aE0)c<iNc6d
r.9(
.q.i,
-9 ..c\
{.8
O:i
b./
.*
q
rr
g{r)l>i'
'\>)*oJFE
^u
-o).dH
o{
()
Ezz'.
6ll€lj€ fr{
IJi
r-o-.d).:4I 'o)\
P-
,,
o
a
t;oc'lil,'iE
o: Bcctri
,; Bc
'E-t
IEEI
i8 <
,nl D\
.-or)\HboE
I
i
c.ii' OF :.{{c) r(d6
o
,Z
rbo
E, C)l
,.& tr H.
tl
H.!
rr).
EO;
c
o
(2 lJ
tu'
o,
(Dr
tt fr{ i d{-q;
l8t 6
im.
:ll
iE"j
6
!C
0
o
5
ql
I --rd.
ra -O'.
o.o
i bo.:9
-d
Ho'xt
al
a\j e'l aG) D^
t:
3
ItHr
tr'
iEit i5:
u
!H-d 9rr
ll ts)bo
oj_d j e.?g:
-'44 qZ
=q'(,
do- -9 !oY
:o
Iij i;iE(ql
"dro
ii '-r
lq?)d o "a
OIJ
.=H
C)
q)
'
)J
JL
uo
ca)
cr.td 6'- d.} a
(r' "s
€te
a.t
dd
HOJ
P"
P9 .0
.J- nw
cf)
'olr*,
'olr*,
db O a
q
1d9 4 Ttstrtrl
H
c! tu
g; ::
rA
q
€
s
Hia
d, Y4Ir
bor{
w *O
o prfr
f* H9
4P
!r
oii
,.o
d' i6 o
)'(,aVr*
o sl
1N
() q'a
Oo)
fiA
(0(,5rObca
(o
lv
-1
-q 3k
dQ,
_9
*_{t+€
-U;i#P F
trd
9o
U
L
L
IJ
L
lJ
u
lj
L
L
u
]j
L
lj
L
L
IJ
L
l_
lj
tj
rIJ
]j
L
L
IJ
L
IJ
L
L.
L
f-
;EFgEEt,if
,o6.e11"e.i5=
rE xi= d g I
iE':rilHEf
EE*E:Ei g
i
Id;:€#
ei6E;?I9;
aE
HEE
i E'"
.I 8€ sG g ;E&rB'E€.8
€
E
ed E H'< E H.E
€ Fi#safi;;
H€.S-dE3.g
'
HLett-
i:EH.E 9F'i t
aE;
i
;+!EE*:E€
;Eb:E::!E
d^--9: n o E 6 *
-
[
,
i:3EBX
aE
f;
€EEiEE,E
H'6 or.=.=.o'= ' n
E
s€ I
r.kA)
e
8!r:,
*Eq.H
H{
'd -r X'i
E
E€ EE
oooi
()E>o
.H(t)d
k
Eu
..oxsv.(D o
J
is€S.
5€Efl
oq(U
Pf.a
d
E:
* f,BT
EE
tt a
;:;*
3l
# p-ge
c
g
s€ sRs
o HE
-c!,q
*E{€+
t .*-'rilE9
;8.E
.tEfrr sniti
I
€
cr^
5
(rl
A'
'.1
{k
!sts
d"r.*
E.i
0)ag
:IE
6l ()
'cXbo
.i soa
HSB
S .HE1
d
:s:{ci
{L d.E
'',(.4Q
.Yd
5>r\
ir$< -O o
€-ad
$
-A
'9H
O'
..
.
s.; Ht
< oE
E b€-
o
tub Is
i[9 s
iX;E€f
il=P.
'I'o
SIE H
S { drn
.ibokr'-g
-.atr
H€
*'asB
P
T
,!
n
!
sI
rr
8.E
€€r-l
fidoEr'.
8Eg
FI
hn'
€
r.t&
H.dd
Ho
t
>o:.ox Q
o(o
'trc\ (J -
-d.94
6.9P
@
q ,4 r!
t €5!;
.$ >? eE
k-(B+i=(E
c
'€
ro2q
'-.!
6E
E-Es;
I
=€-
L^T
fss!
H(D
,-^6
!'! -.,I
-;'-s
B
EEBE
d? F
H
-Et 8t
!: I o'6 s
F qEor9
SEt
t+
(O d €i'p
Ee
.!p < E {..t
(L
ir-S
orE
-c
"E
a,
6
,
do
rl
$sEE
dBE.gE
EE#.E-l
E
9S+.
H
t:eE,q
v.a
c0.O
qSTEE?
E'EreIE
ada
n.=i gI
O.=
d P
p*rfi
H
:= !2
-ad
-
dLo).vFa
P6't I
ra
.
E=i0. -O-{i'
e-t
,tr
f ; ,P.E f',
,:,iq*xE:',
Fi
$t
E* 6€
. o
D-- 3
>d (E
sg
il'*
E
'il EaEE.E
H -p >
< =r,)(l)6d
3:EE
.g a s.E
P<r#'H
g
!P"eT EE:F
ECE
P€
HE
€.{,}:E. 'EE
e"
=;?il ii;?
fr
H,
re
;;;t
I- lag"E s€i
t ri*
E*€t i:
d ; -3.-8 &:,fi
qPgI
E;if imi
€sl5 {dE?
* *e?i
R *:s
F EEEE
f; HEHo=s*.€
R€;gEiS6*
:
siq{€
#c* iE";3t
EEEE
E
ati*
EEIEI
iig'tg;EE
Eiig
ilit,fftggs
:EEiEfIi{EE{
ry
E
6 .E
Sri?
EEt 3#
tEi
tE
H
HU
s
9
j} ;;3,!
.€E o
-,x
: €+{
rEg j,E"E,
,E
,,,,'
F,.$.E
+E
:
F E-P forBBx
*'EE
pH
e Bd P
_: ..€ E
",.H
Ee p
8Iy
o:; i
ry;
d
P
EE }=e'E.fi ci
t)
ti
c)
c)
+,
ES:E iE;E o
fEE$!E;
0)
?EE 3{ EE .go
lr
BgTESE E €
,FH/
F
d
g
.g
k4
x_E
v\
A ta
^F
o H€
0)ti
*fs
v5.
$
E
.E
6i d
O. -Hh
or 'ii
HE.tr 3
ri'
'
tE
A
-4A-.HU
Eii =t
s tS.E
fr3sE
ct
o ..'{E bg
Et 3*
ots€
o;
p HH5!
r e qa3
trEfifr[
F_:. g
P
-. lr, #
#i
- 6.oro
s8E *
,
F:E
x,c o
#t-E
:'sG.
E,*l ,9
e.s
E
)
6 sh.*o.or
:
o:
o
=
d .9 c.i
o
-fu---
?6
E*,P
A)'tf, EJ
ca d{
Ei -A
-LH
iuo
9.d
E H.E'
EHE
il Ef
,ff *e I
;E I
-
F .EE*,
c.i .: RiI
O. qjg v
iith;lott
HEE
vA
d
< Ft
.34
FE#
J
J
J
J
J
J
J
J
J
J
J
J
JJ
J
J
J
J
J
J
J
J
JJ
J
JJ
J
J
J
J
J
.-l
:
F
fi d
.;
€ st ia
EEE*
C\
r-t
;FEi;
llll
O^
()
!4m
>i^
6U
8.E
cd
v)
llI
clc
cto
o
(o
aJ
1
I
I
Y>
Ei-itEi
EEBiEEg
s
*** 1IE
€ $$$iBfl
x";H+'E,;'8,;t*E;34 =a= 6H;rFrca;.E'
$ [i
+
q- {s tfu
fil* gEFfu-*EB
$:
rrlll*+fi+fir
S 6-} , EEo ttt
;+'e
.'$tu;ii+*+=s E* ff f f,it
$ t IB
gfla
t
3€E
Eu f: EaE
1$ iE
.$; ;* IrI
"}
i
3E
?}}*{,Ht:St
g
lI
l_eii
tl=fl=El
e{
6
-tI}+$$$,it,
;d
!91
do
.6()
rtll
q:ts
a,)
ol-
+t-
6l
o
,.,NN
VNN
I
F
{t 'ft 'io\ {
'AS N6l+q
dO
8'6 \
roC
1*
s_d
\
EE
ltI
Ev
gil oA
ead \
il
lll
-a.g
,
"
ISB?A?E&
:(A.E J8 ;'IA E
*si'*^fjr
gB
$ffitqi
E
E
' ;;t?g**:
I ix
; 3€-$E .E.s
E.E I.E
€et;flEt;
.3=3-S-..Eo
A*ai;fre
gt{!roHgH
.iitvH.HH-
?.
E"-5*
{
A:3 E I;3
l;+sE-*ete
*E qs
[a'rTE
sie [: i e
BEEI.eItx itsS
$:f SssE ss
BiX ar
E*nEEE:r EEgE
;
x
fE
s { #+,EP,€E
$
=aeg
a+ettE?itEtgtg
H3:EEEIiflIiiig
IligEiiiiiffi,
n
co
o
tb
Fzt
rJ1
rL
-{.ila
t
V,N o\*
'r&*
>{
o3
(-,
qc
Ull
o.o
,,?
6E
or
c
o
o.
N
g
-+ y.b E
E 3 HI
cits-o 8.
d.F
bo6r Y'J
f4 o.
vOxA
N9vH
-g fr.
E "oP 9
\k.i.D
A
vEE
bod@
t
>,
-
roaE
c
LT€EE
l-q^-
,J
e6i
Ets
*cs6
o'=
'3=
o
AO
Hrr
raN
A\5
€H
-Ellb<B
d
Cr 31
bo ,o <,
Adil
1
AO
E<s
'6
qlt
. >c.t
@
F", E
a.i
t^p
"a":-H
-vd tr-1 boho
E
<!r4 E Ll*:l
-r.Eo_>
lr.Ir
_oYc\ k
<6
,Sr- O ^o^-o.
d -o
xH.1
d
A F^^-
bb .9 -q^*a
'€,
s.tsEq
1g .:ola
H r?g
\r
'd
Xc.j
16 E -a, *CP
J
s Ef r#
tn
E
o
<r t, Nl4
ox
clt
clo
:*
f4
oqN
J
:J
I
-J
-J
tt
J
-t
I
--J
-J
--l
*
:.r
-<l
.l
.<
--<
.<
::-{
<I
\-_
-{
-{
L
<
-{
:.
:.
ra\
d@.
H,t
+Ls
-Hll
-{
:.
.g
.rt
!
.r< E
(1)\
tso)
F{L
?or
crt Fi
\
l*
l*
L
L.
l--
L
L
L
L.
L
L
L
L
L
L.
L.
L
L
I-
o
o,
O
.JE
O.,
;s
ii?ffii
EgEE
{;,b
UI
fEaEig
-€cr?
I \e
o t)
U
(9
.4
;
#fifi
$Effi{
HI*iiiliI Hgltt;
I?iiiiillElliffiiIil
I --,
<--
frl
b
oon
Fl
oa
0{
bb
94
T
$E iE.T
_Z-
{' le; E'
b
-B:= -o.E
TIBEBt
E€IE,:E
il g B Ede
H
B?HErH
E H{EE
;':€ExL
>,5 c'Xe!
3rsfrB:
k
a.or 7 6
d
::fr$q'E o€
TiI3E#
E
od
o)0r-
5*;
g'-rqd€
'E 6.Lt a'E
E'trle 3 -3'r E Ed "H
# El?
blt
!o5+
ov
.6-
vE{
.. 6l
cr!?
E
E TQ-
Ete
H}E
i
,E'-g.Q
t'.:
ti.
'U
fiEfq-+
EBE
+TE
(go
E!
.E
uH:
r #e
F-*
<.fr
. =-.8
..eP
t#
rBEI
.fl
,fi
oEE:9*f
,.9
6
'2<
J9:
o
T
6H
FE;
lI{
"rtsE
;
it;O.
:
'rH
'0)
0)
o
+,
0)
{,
-v
.0
a€
o
A
,{
€
.do
I
d
o)
t
A
o'
I
g
H
,H
o
€
H
H
o
d),
(h
A
H
h
d
Ad
boH
JH
HA
ddi
6l
61
CD
E-
?J
po
E
+
3
ot
\
lt
N
a
^
\tr
\l
*It
'ts
,N
N
el
-I
t
$
N
a
bo
\d
s
v
8 E fi€A 8' G 8
-v
--!'tEr
o)cocaEQcocaYca
v
Ei
v
L
dc)(g
:
q
H
A
+,;.*i"
A
-=
;Ei?E
.EH"T
iA ..3*
f Srr S E
+
=
q
i
ii ,Hll 9i#G
0X 5 f
:
=
rffHdE
Bs6:Hf,+;
I;"
I
* **
N
N
q:
.EEi
[.!
o^N
.oc/J
ri
E €'
E
*,+;"" S SEur Lil3
G
X.
B+; *
'.;E'x H EBc -o 5
:
i,'€I#B;t
E : -^ H',dA j$re €
: ? ? sgg rti i$
€ GG 93; E
E"1
E
i:
EE:.=
0 3I
EEL
I
U
H Y
f
,iEin E;
#f
E.i
S{
$iiiiiiiffl+
E
oiticdPHc
E.EH*fi
d.g);;
U -'u.# o
M T T:,8
E;. e-go
EE H.?9t
s-3:
g
rJL9U
.= 'v -. r.
ryE
s
I=E;i'P
cd
s aE'+
-q
'E
E.EEAT*
tr.-ftE€fi
c? q4* 1E $'
'€, H E x sF*;
si
EE'E
o.:)FdcaY
,
*
7 9P-q-Oe
16 0 0 E H
.I E.r s'- o.
tr
!F<o*
sE
-' !.5 E.E
6
<r'>,(Da_d0.,
o 3H Se'i
-^u
; I
; *a:6;
3rr:E 1
E-ubH33
Fi
@
A x eo €.{.
iEii Tigiiiii
3i3BgEEtifl: ssiE.1?idi?E
es
m o E-
cri
ilH iii
,ErItt.*i*3E
E;i€ltEtEE{E
I IEgfltI $EiEE3
E
IEE$ii{fliiil f
E ;aiE;E*niIig
7+d
ci
or
X'
':,
-$ -e f
A*::
,
z
lP,
{
- fSrE*I:EI;iE
u-s
B
c';*If * ts..l E€I;
E€EiiiiIIE+E I
, E€;E s;; $€i€ !
.AE
*A EfE;#?3I*[BEa?:
i;s i: ;
ii
gggi
$a
.
.'.,'t
co
CI
6/)
:$E sEif EE{ EiE:€
;iE tu igtiiHg
€
{E
EESEI:H!E'Aii
€??EE
=1
H
<l
-o
rr
s ii t ++l Fs
i
<;:
3
g
rIE E{TI tt tEi
Biff t, E
+ S't. i: $$;
+s,
$+*if'f,$eg$
q) e e '!
.i.a
\"
J:
rE :? €'Sfit
*s:E;
*3,aff,gg
otr
o)
€.6
t{i
H
€
H
o
ts
o
lH
o,
o
+
H
o
'H
i
tffilgrB,ti$i Itiit,itigi €i E-Eii$ iI
I
o
=.$--;i-(
,
F;
R * ='?{
E 8t:
I
E
'
,
;;F.ul€
g{iir
;*Ie{-:slE,sI $ $EEfii$Iil
i
d s,o IIs$iEii
{Oei
_ i-
gi
E
II
qE
iigigat
t;I
IB$I3i iEiIEEgiI
il 3
giEi!ffi
ag*I
tt
i.
+4
IE!*i
>:
r{
E
l.a
)
€
.d
P-.-Bt
,-i x
iEESE
iggIg
a)
H.:tr
q)-dl:?-d
L.rH
FSit
l!
o
=1
E'E .E E.€
H
Ef; E EE g
^
F,?'g d
d f-{ 't
!!;s??
H
H
.
v
'
.i
6
.t
fge:;E
..Lo,g?.9.-
s-
uE EE#
Ft.6 d 5 >'O
H
:I;sii
x.^Hru
tu I t 3E Hi
d
?E3a#sr
;
#g;e:T€
uilSar:
.lr ,I
s p'fi €'E
s>-lB'
E
ts P
€BBE
fiiiuiEaIiilffiligEtl.ii
ig3iI
d\'
€*E$ts
-X Iat
N
rr
tfir;H"
6.E a'i 6
<fitx --9
El
Tl i, h,?
1E'E- g'L$
.H'-r:5
Et HTfi'!
.o=
Eaa
i]
"r-9".j 5:{
Et$:
ET:;EX
dt;lq#\
fl9.3f
:;I a li
H
o9-H-Bprirr
*6"
eI3'-
E[?ti!+
r H*€E*.$
I
-oln
\,
Td
d
G
a
ts E.q -i
a Etli
O
# ; rHq
o:lE s!
€ti{
IHT
' E
o'E 't ql
;ft;
F
IB
8.8":
=-E
E
EE,
F;ItE
€
E
Fg$I
t€ii
E
A
@ -rQ.
<i
ct
c,o
;e$ia
(g;!
g.-H F+.9i
-H >€ o o ?
pP-d
9D -A
f'*,9 H * *-,E'^
d - F{ *..
EE,-:H HH
ai.*
# ei? EEa
q-. () l1 Y a Q 5
O lr(-I:
<6 i
g1
siriE
iH
(v
vv.li
o
r<E,rt
6
A
oi€ o-B-3: i
Fi e€€ e I p
co E il&-8i€E
-8,*@€Q,-c:
&
s 3'tJ-Ed
abE.=v8eE
E -iEE;P'Ec'
X H'o.: d i: c.
*E $# l$a
eteeii*
9r:.,
d*;d
E dE
Et 5
e;*fr
E E } H''>;
a q H€ p q 3
-EEEiEE
=tE.E
-J
J
I
I
-.J
--j
J
J
_J
:j
J
J
-J
J-J
J
J
J
J
J
J
J
J
J
-
i-.t-.t-
L
lt
L--
lt--
taE
irnrE,;
5co"'E
i
!
o
E6i
q
ro .E?Ei :
i
.' EsFi
o.-do:
* P.1.9q;
5'aqEB*
g.g
.o
#.*q
r 3i3Ei
EE€
i >d
L-L--
:X
*.9
if
9Pd-o E
'-e#
X B 9.6
d
E{@'i lx
!i h Htsf
;HHLH+
o:
o.>
b-oE
lAcE'F-H
5l
v*r*H
.{@DH
'
J:;
E€I
>.{?H
u?x9
! i q
i=
.g
i€H
€Eg*EE
*r= d'* E st
5"
LJ
LJ
LJ
L:
L-.'
L--
l.J
LJ
L--
u.
tj
LJ
LJ
LJ
LJ
L-.
u
t=
tj
t--
t-t.
L
--{.
}.--
a"))f
,sjZ
(D
co
o
c:
ho
c,
fr{
o).
F{'
co'
CT
ho
fq
o
c
(,!
co
L'
o
o
c
o
o
c
o
(,
o
o
a
.o
a
o
r,
o
ff'
r\a
,\0
-.rf
l*
E"H
o
'lt
4f, --.,d.
( trcr i
t-{
c!
bo
c'
fr
,,f'..
tr
o-d.: o A.Eo I e.€
bocko,.do
,aa,no?
." € E r€fi
co
g
+Pao
^?
a if r F!
;E BET
Cl-Q 'i o *
.$E d
; ES i}
Hfr
I3
.Y.g g
,,i3-XH:
: # H^.;€
EAs€
CI. .e
?-a;
EiIee
EEglq
n gt9i
E
uo >,.= c E
trE
o 9.Y
s
: yt E,fi
a' Ex.i:;
=
s
ii <;g'-3i
6d C)+rJ
ci- oP
cE
@ O (!
r{Fl
:
:
I
I
I
I
I
Ei+{iEiIi=
Bd*EE ;€i.gEEE*ai.o*
BBi?i
fiffiffiillfifigi
IiiEgE E*sigiI:tfia
i;EEEEEfriEiEIIAE I
;€iE*€qtfa€;;+I
;il'?E:;uo; EEui-i=
or
.J
-l
-l
.-J
€'E
dll-tI
^a
t sEE
trl sod
E
-
-
-
:-
:-
r-<
*..
*.-
i-
-
-
;-l
r-tl
-'l
\lJ
J
\l
\t
-J
-J
L
-j
-l
\l-J
J
-J
g
EEEf
EtBg
iX
rJ
cB )E
,-or $.Ftl w.H
{QE'o
u#.;i
E$* E
E F$E
N tex..8
?EEtrH
.*Ee'P
i;nE
BET:'
3 q+?,?
0d
*3s
S
N'ui oo
E E:;€
Ibb<Edzlq I
E
6
('il
E: E Ei;3.'f;ry
olF
st
c:
crf
d< 5 HqE
ts
ff t *: ;E;
=:
;EErflg:i;;
"q.EisFt;FEE
*;E
tiE ;ei:st
sEH#rE
i:s=Bi;r€
;t i s;E?EI'
a= .
si8 €ti;!,
tEsiig
ct iT?
t.[q B*Ei?}
bb
14
co
cl
C,
bo
Fq
C\
c!
co
bo
cr
fi<
q.) ,
ultsaiit*
*€fEUi
;Fx-E.!t
N-3 ca'g N iD
-t
\
:
E H.E E.E g h 8.6'
f;Er.'E=$Z'6-
; E'i aB:, B'f :
i
r:f ,;i;;e
I{$HE$EB
E
f F'l 3€ P!E
EE;Ees;[IE;.l*i;€
aBt
?: o: a,a;E;
EEE EEggEI{
;*'sEAF:e Hs
+
seSnE* 1S.r;
+)
iq
ffi
iurEtEf.stE
i; 1r,
$E
E
ll;r taifg?
e#€E:a+r
lst;
; si gI EiE u;;
;$EH€ €€EHI$,gEfi
i ilii+.t*:BEEi{:
E
j
I
_t
eiflilEaijiI3Elfg
*iilI€i
ElEffiig3$
Igltgi,itgit$iiii
;f ;r s g: ;:E
B
I ::-,:
E
Ef,f
a r;E;a 3#!E
*'ErFc5
e €,-E +g'
{ g E,#$ H;
isgii.fffi3
,e EflJ [i e:!Er
e'I3:;5'EI;6{
sgitrigift
sidsEeE*:EB
i''
,.,i
-ASE.HE
EEfiB
H.is; t
iEE sE
' lEE;'3
.p'9ts6E,
E€."87
EB€Hgi
i: tes;,
#'- s.:
E
;,H - i'Q
r^
H
H
E'?:
EefrgE:
'. .Y ,x
). .E'B
E! d .i:
3
u.r
EE€:I
.Et at cts
?Es€tq
* 9S H P*i
E E H:
e2ZE
gE
EE
lJ.
€
-:. I.*
q d ()6
€s"lf
ts.H
-
or
.SbEE^
E*
^E
S:E ,H i
fi
a
.E
e?.:i
?E
?,**
e:
-e'3 # e
d
Gl ti
$sEo !E
'A.o
;HEb
d tr p{o
ab PEo
E E€.3t€
-E3Ei
E8E8*
tr :5 o'E
U
s+88
"+ EE..
ca H ad\) d
frer
if-
+\ts<
a
-J-
C
Gl
@"
I
d'd !2 0
s6=
.AA<>
4'aa
; E S3
F : - 6 flor
f q BEg
.E*.EEg
f 3 H H-E
s,E
r E*tr,,t.E
EE;8
cr
r gA
ff
E
#n:3e
I i;€Be
*H:
e" t
3t
EeEsa
>
E
aq:Br
*o lzi"
E .i.s, E !r'e
-i -tq ti d a
q
H:; E
#;EH:H
.= tr:
Fa
E
I'
.D
1
;-o
g &E EE
o
T
r+
lo
<.
rpts
r-93--t--a
f--
{;
<r
cr?
bb
co
o
f4.
60
CY?
Cr)
rb
c,
f+t
a
$
.esAdHESS#
.rE"gl.- gU*E
u"qilir;;fI
E?:;5ii
e:
iE
eiEsii
{a! +i}
E
;$E
E.;E;-E
E;;f;acS€ EE
t E E'. €'-^B e I
c?Eg9<-5e
iHEtr'r.8Ei
-'84;d?Er.A
sE:€e.l A8€""
E 3!iTE E€ag
.EBC Htr;.A 8
fr EEB;E:.BHS
q E{ : aI e s,E 3
tr [HEE€:EEi
"ri
€IEEII€ E iE€;83=E
gE,
iE ?Ed;SE
6e
EffiiaEtl*gtA,,itE
E1,ii1EE,EE.iiEEiEEB
#E€;i€ *,#sEt€EEeu
lEIlgffigiEi;aiag
*i
I
:l
l-J
I
-J
-1
-d
:<
-d
.<
I'
.<
i<
='{
:--!
<
-<
-{
:.
-{
<
-{
-{
-{
}.
u
tL
L
L
L
ETE;€{iEE
EE€.;
E
iiEgl$iitugiE$6
]j
lJ
L
fE€€g*{giE*EfEE
HJ*EfUg,fI
lcl
ts-
tr'
+
c+
aj
o,
.lJ
t.
.o
,I{
€_
:42'
.
:
>:.
'cl
H
-L:'
H
u
an
t,
q)
t)
u)
.d
-q
+
h
E
ilE:ilE
€r:"b
4
E
AH-.A
A
9
,Y
dI
r
f:<
t1r/]'i
d'H
.d
hobo
=E],H
G)
_-C
grE'E
E€ET"
t
ryEts
f; oc€
E.AE:
d
-- ol:6
.g V,F ,t >l
x.^' tr .ad
^:.d o'l+0)€ O x
{,r
@
H
I7-a
H
,0.)
(n
.q{
E<€ H
a bb gtr -d
B
(o
r+€ EEE
d 5 d"q
v)'
H
+5
B? c ad 'E
H ;=.x
-t2.d.I
6 pEE, k.d
* - dE €do
F *rdH*
€'* .rtr Ed €()o
H 's(rl
o
;EEf E.s€ x
(J
q)
o=
o)'
C)
o
60
,r
d
>> bo
(J.
o
o g 6 9.p()
S, xo
<.t!.t
As)
F 'o
rrBeEss? ? ts-s:-odb .=,q.
s:i=*E;? H sft €c
H
se::;sa; *:i*
{'€#Ef;E€
xfi3-e;f;
- i#g1r€:
'32'.HEc*
E,;$E g€i,
#rt:ss=$
GcrN:#.9or
;;fEE.,l:
9t sd.9E
* ic !-tE
EFxo
E
doldH-a)
Iigiiiftiff!EgE
c.j
tr{
L{ .:?
'foY
;,
,=v
k-
6H
d+
tr6
>tr
.9o
xtr
Fc)
=*(!
oq)
5=
o
O.
O'>
-v
<Y
ro
C) h^
-r
fi
ig,$Ii €.
I_-
IJ
tJ
tJ
L
L
L.
l_
u
l1_
lj
l_
IJ
lj
IJ
L
L
L
lJ
L
l_
u
L
I*
$i;=tEsEg;.E }
rf!66gis:aE{3
\
f*
o)
ca
c.I
bb
o.
co
,i$Eiii[Eig
x
I
(I)
a?
ca
bo
A
fa.
Ei;iatiiiiE
giEiiia;ru-
tggtfiltiii
p#€=:EEfi'f;tEIg
iiiEitB3ti3+
gil'EErsEsxs;
E?g3i{iiiliiii
giEiEgliii1g,gi{iii
N
E.
,4
fo
dti
s1
-cd'
-r2
(6r
oE
dtr
cB
o)
s'
u2a
o)
€bo
dd
dbo
tF
f:l X-4
sd E€
\'6 c
.9'-i+
a
- tE
'
ul?i;iiEEii***'i:*
r{
F{
ho
cO
q
tu
>,
-€
+c)
B-a
qEE
3l
rr
col
a.P:
'-'
hb.9p
'rE
<>r
€
.oo
cl6
Fl>
f
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
*-l
L>
L
l-
HE EE
HA
EE
GG
Xh
-ol
E
B€E
*i*
€,9 'oP
H.8
EE.
€r
!t;
€€
Hrq tiiE
;:;
EB €:q
gg
+
B
Efle81s
- Cr,- )+r9-ut.
H
AV!
'
:E;I;
B;?HEE
5+Eofi*
QHsP
e H 8e B h'
c\il k
$r
n.
$
g-igtiIuEIi
E
o
p
rn
8?
oo
xll
co
(Jo
'
{EEE
E E$€
EE H:
?f?n
EBsc
E'i''
I
.EtEH
f siI+
n IirE
'i'E;E#
RE
.1 *b -8.v:
..i.... . o,
.-:
d.
o.= oE
eo
o-.
.sE
9
,
qo E
e E.E
iI,fE
'H.E
E*+sor fr
.4 Xor
H.6_e€p
3E 3E A
t:
o
Ct
il--f
.=
---i
\
r-l
(o
bo
c.i
q
f*
E
f\t cq
=
\o
-lf
-L
--1...t
ry
x
o
\l
L-
*t
d
co
gJ
bb
p.
tq
a.fi.9
=IXd
^@
El
a1
4A
\=
E
H
#$;9
- 4 b
9:r
H
dof,
HgH,D
r-..i
--oH
F'E H 9
6 E 9!
6tdig{
8
()
HI
d
d
c-il
rd
E{{r.Et.
-'.
x'1
Hq
€aO
o
o)'o
bod
Lla
(!d
v?
6g
ax
o@
gE
do
EE+
E
E.rrfi
a x,E
* H! -
0)
Efi p:
3€- E'E ii
r
E
E.S
k o-'tHEX
-Hat
->
'd
qts€E€
gEar
(!€EA
SY.-H
oH€ E
a 1.9
-b5tsn
h s6
=&.EaI
s+ HI
o
o
I
I
\rI
:
l\
;t'-*d
; o'b
q'H,8,
.9 Hs
o (u.
).3,
,'.";i ,,
)",J,
*
;5
=s
qe
.E.r
.!P
frr l-{
96
e.
-or,
9r
6€
€o,
tD
ilF
F+O
b. -g
.
:
st
EEilEi[iig
ifittt*€
E€}iaiE;E;E
tBtlAlE;Iilr
IatlgHi;ffE
,lt
-o{
a
I
--*giBH
B:;g
$ El! 9 6.Er
6!
j qf
g'E
soOE
.o
6 E'6
E-o
gI
EE,E€
3EeuBgE
3r s'-b
.sHfi:sH
>.Ee-.rj.gi
E
s:
;EE;B:'
bsEEEg
fu
:HE E B
R
t E 3sI E:
# BEclaE
o o ts+?rl
(!) o'"i@|vAo-.x
-3€o? s.H
?.ot HE 8.
o\ft"
LO
'o-
{|\-/
tn
c>o
i.)or
c.>
E:EE
Eg
= .I0) €'a^
. ,t
E f:l
.$ au:HA-d
6ftrOErEls
FIEiEB
E*s{E E*E
E
riliaaEil
l-Igtilit*
r: tff,EE€;
+ eiBEsF;
lo
JJo
1F= (,
tEi1iiEiiffil
ll
tttl
9;
c\d
n$
6lO
e(3
}. tJl-
gtr
al!
i:
rO
Co.,
bH
rtr"-(5
{3
ull
o
313
f,s
3rr
ll .s
Rllrr
-i:
GtrH
eq
-e! Q
bo6
,Xo
['_1
Fa lJ:
EgE
.ocil
Rdc,
T
J
I
hl
I
t
<o
C:
60
hb
P{
tu
c:
rt)
CI
c.l
-$
bo
co
q
Eq
Yo
acn
c0)
{dE
NiE^,o
612
^-:
cd
bb.s^
tr{ 'q
6\)
F, dU^
6Y
v)a
au)
ql(d
5o
c.i
*
c.r q?
61 ts
ll< o)
bbo
a
\-
|i 4
*'*7
cl
-e-€
P. Xa)
k(d
:- 'O
F.
'.*
aE
o)
0)
ii
_cn
Cil
N,
.l
J
J
J
J
J
J
J
J
J
J
J
J-l
J
J
J
J
J
J
J
J
rJ
J
J
J
-J
=J
=J
-J
,
*J
a
Lj
lJ
IJ
ii*;gi?EEieiIEE
IE€;f IEf;iEE FII€
iEEiE-ItiEiiEIII
L
L
u
lJ
L
L
IJ
ig;; t r ei;x
+lt
aE}E
lJ
lJ
lj
L
lE
tJ aIlffii,igt?lalffi
L
l_
IJ l t:;; a ;i ;;;
t
t-
IJ
L
L
iiii$iiiii$
L
L
t-L.
L
l_.
L
L.
s;
-€,
6.;
.'6)'.
-!
..d
,o+>k
E.
o)
H
{'
o
F
.<b
:qi
-9
1
€ $ ri't:irf
*; g!HEIEq
uIi
ffgttgaA
-EiItEllgii
,r i:E gil€;f;;Eg
g;ttli;?E:E
co
$
N
r: .*3
;Ea*€tEeTEI
c EiE
€, eO
\
$,sEErn
-.(B:'.'r..-l-.
Erf
rEslg I;e
d
#
6
^"'s
.p
l:EEEt Bq$q5
E::1eE
EaE;&+5iEI[
€ IE
il;;EE ; Ii:tH
-Pit
iEe
;P?xEr-x o*
,
:
:
IAP
o gI gI A.=!
PE.E I
A
H
),
*€gEIEt t'_v
: E H#t E :
fiIEBEgfi
IEEEEIE
6.e'[s
E?tcteet
'-.Eil
E
gH;E:a:e
fEE?EiEE
:gitl€tE
t
.
el
ts
63
.{
F.
I
tr
o
E€.egtsbg_g
IH as;.arr;
ce<i3Gtr P{
8E'it
IE
E
* e E;;ci*g
F a,n:"# ??E
EH";; a!E g
[€EE;itE !
d
,}{
ao
.-a
F{
4a
ct
O)
IEET
*
HEaf;
g:g"
ii" BE
g;
cdcaOcD
V)
S -:
r-t .- c7)
* ,!'s ge
E '* asg
, g'E.i.E
#
fi
<, :i 8d,
F A@_o
, fEit
L:Et
d.q Et
a
# EuEi
E ti Et
483q
r{ > O-ri
4 uc I
q ss E'=
: gE!I[IH
H= H'q 5= r*
f,'3tr'E*tr:E
=
,,, ?H,fl,g
F'
-:_
*a
< -EA.
a .:5 bo
SS E6
A
:t
.-l
J-l
J
J
-|
J
J
J
J
J
J
J
J
JJ
J
J
J
J
J
JJ
J
Fgf ? E:?;E
J
J
t ;;E{HT:E
Er;ti;[E J
:d:iiat: j
;t; I J
J
fO
tc
cO
T
.o
E
-o-
tA
s -,1
.Y
oA
rw
J\
u)
a)d0)
d 6,8a
ri.d
g
BE
6 kfq
().o
OH^
.D<-
.;5q)
Et{^.
{J ou(,
c"? ;9-e
j_Ho
giriE
A< Ea
Aaa)
{-9o
X+k
U
.EA8
I
jj
rqfiq
5
d bOq
oii
o-
Av+r
3E.E€
B *E.s
tE63--9.:
Fl'I
d?
F* d'U
:?,E*38
6
9 d.=
EE:E:
fq
3
o.
SH
arrx
O.I
< fr€
.i o.d
E XE>
g)
J
-.1
{
u
L
L
L
EEsE$E
silE€f
; [IE;*
L
L
lj
t; g fo;
l-.
sIq€
*E
fl$iE I*
- Eo==;t
g- E b !3
H E.*
E g.e .a.E'ts E
;-g 6E€:3EE
:si.,tEis
H.5fP-35 H fi
B
E I t"ro 3 *
EoH^aE.E i
ifrj,{#31
:tssapEa
:; i*aE ,
q2 b4 o *'.=
€
'il,t4tr+-'
Ad^tdU
EgB6";I
L
iE$E!I
lJ
IJ
lJ
L
L
l_
l_-
L
L
L
H
l-.,
l- t
IJ
re; gE3
i $; II
;E:; gB
Eg$:
?E; f
E
EE"'E,.,--ro 55
- d C p^r.E E
I:,E Efli'E
L f
IJ sir i; I;: ;l,,
L : ts5;3E€r
L
i.t
L
t;i;f;?iEi
L.
l'-.
L
L
f-
trL
<J
k-s
i'
--{1
,ir,
->
,a
d
(J
-*o{.'o
*"'t
Iti3EiE*EiEIf,EEgiEi
lliiliiiigitiiliiiga
uStiiiEi[fili#ii:3
I
8<'Hd
6.tt
.iB?
X :6
E
beR s
{1.
o
o-
I
d
ffi3iiE
i:il;ttEet EI;EE atE;gaE
BiIHiiIi3{-iFEa
o-
*'E
(
r+{
RE
'\
oii-o
'3e3Ya
F.E6
.hqRE
il :tgi
r ieE?
qr
f;n Et
E obH
,
K.H
(J
H68f
EO* o
.: TEtE
lez
# o .Gd
Yo)e()
gEsI
f :8* d
r?
o
0)
{ *a:l €
in ttr€'E
H
0)
EE+:E s
_Hi!-3 €d
H*€{
bo
rO'"d
$€sa
'3
1
\
i(
.xi$
Fr.t
bo
'r')
^<l
o
9q;
q
dt$o
'?i
(i
rc
e
d'rc
a
i
E
E
H
a'
6.4
E
$EE:
:fi
s:
{>ul .O .
-rY
'atEn
, botrE
-3€;a
E3qE
61eH
E€ s
,.: d E€
d€'vk
d
Bc,l :
BE3;
HYdd
^x bot
llrid
N=E
f,6E=
B.s
# botl,
.4.;{(d
+.e
o.A-!
!'Ss
H 6.i
aBEto
serS
tr.bo
'",= ()
$
LH.'H
I-D€
sIx
;g;;t iE;;a il ; i €{a; : € a;Eg.
o
3-1
\
oxs-r-Pr
"6'€&f
i;itt! .^68
#iE:E
HEA
iHqtIB
i$iif
H
s?r
d
i
Eu
('
EiitgfE
E:EH.AE.f;
*;93€.Es
i:t:EE€
toX-^Hl!
d')<d
tr , E'= 3'.9 0
dd-E
k-Xo)
'i4,249:or6
E
xE
iE;EiH*Hr
::€rE*$
E a I Hts€*,
EEESEEa
gEEEEdE
l c
H
bo
<)
o)
bo
a)
,,4
\
;
+,
fi
EnEE:E
F-E;fEfi
.sB
'
>EB'e€
gt gat
E.IE FEt
Ea;E€.fr
I
E
?4.5
.tEE;:g
}f EEE
€E="lli
E:>eg: €
q,
*rd--t' a vvP
E
P
>. ,.
EgEB*q
ed
,o
0)
+,
k
EiE+++ a
Ise>>> o
.a
<
i
:.
<
:t.
<
<
<
<
{
-
:
<
i
t
\
IJ
L
L
L
t_l__
L
l-_
1J
lJ
l_
l'
L
L
L
L
L
L
L
L
l-.
L
1-,
1
Et
t+{
tnH , ts8
..
I
ls
EEoE
,.E
l..q iq) .g
o,
t) 5(0 rO
60
r.a
,
L
L4
dhd
g'E s
.gEE
t
;.,
.d
ar) -t0
ci
E
x
x? x6 o -. iE
(!t gd
o,
rl36.
ca
o
ArE-d .9'
. o'6
E.=
tl) ffiE"'Eq ?au
E{d
F
h'
6l()
,tr
t-o
.dU 'oPq,
n
.,0 -q
tg d,,--
t.'
aH
€)r-aE(6
po
O.r\/P
€5
+,
'Ergi.
E
dH
U)
CU
qO
a)
tr
HO)
t4 +'
() 9 ,{
c.:
bo..{
bOr
Q4i.o x
t
o.
9p6 H
.H 9 ^.;r
5A
d.. aB
Et'E L
H
ad
o)
'c,
9L{
'q)
d>
(G
3
oq,
o
'o'€
(,l t
-Jl
ge I I
qo
d'
..qi .v9 o
oo Esi-q
-r't'
-o
Lq)
t{
;EE
.rttii41A
'39
..
=.i
..Y
9ou) Of^rbO
A^7-E
^X-q
!€x:g19fi'E
iD
ts'i* f
4.tr
I'
cd
.q
L
€ rr ,,g.AE
o)
tr'-fr
u)..'-.o
ho
q) rii gpE:
.(D
c..
E
lQ(8.
f.e
'o
'6
ul r4
E,,F
6d'
.a:'g tsr'
aE;l
O-.
-r<.. (E
rEl
,1
,d
.do
C)
xo€od- E &s!
E'L=-.
d
o()() E* eE
sE
.1
,r '.8'
d.
(J
>.
.o
c')E d
Ee9
x^q
.i .o5_
I
d.E
q)
>-Y
8
n'*
L.
()
a,!,
E
-$)'
o ts;.9.g
I
+ 6fi'6
Gbo
€ H6tq)
E
4r! cqi
0)
ro oo
o
o.f
b
3
'58
.- o) "
^
\/ka)
*To
-:-ts
^i
.2 &-, #
Ed tr,tOdo)
l<
o)€
('ET:
=P-"d
,2F,58
o54E
vl
H
Et s
E:J.f;
gEE^E
o:!Hl
9t
lido'-
i
*c*€
€'il c x
E.sE
-xrv \=
oo
.?s or
; HE E
d ^,9
>()<b
-.
sg
ur,
:c,iEE
:=€Oir
F.:7
6 ti
i
5.8
=
=!
i ^ot=r l
'-+
q'r--l
tfisgEs JstE;
, O H E E ..:
O^
$f€iair
-t,'-f*
f ;BE*: F; *;ET UsEEE
.i;'.i
Xt
of€
c
{)^
,ooi
! o
boo
bo
c d
dF<Co.r l-.
Lr.
zL O., (D
g'c.i
r(H
6d
3
da ".j
.9at)o) a)oo
A'6
oo
.:o
@.c
;.:5
Q
€'
€
O+.d
.o8
a=
! uru
oa)
o.€
tr,! Htro"
tr €€
o'
o+
5ho! ii
a €
.ii
o
5
3 .o 'o ,a
,61
.8&o) $or
d
50 o
G)
w e.= €
$w
qd
P@
a)
H
da
d
+o o
,{
.:q
.q)
€ Aq)
o
tr dA
o
o 0)
0)ooa()
o
a).9
ss F* t-tr.:
€
a
\a-
d
.6
..r. rd
Hr
5€
3ii
o d'
Aa
aO o a lo-i
tr
fr{
c\r
t-
Fb
o.x
.rJ o.
.odd
Hd
EE
d
=..X
uoral
9o Sc
9d.Ha) ho
.o)o
o .!E
o
,!o
.o'
o.v)
o(d
d O€g'g
v)
o
c0€
4@
c)
tr6)6)€0)
V6
trido
drdco v€ioobo
tr
boc S s, 14
cd dd 6o6!€
(A
'E c.l
ct' o b'il .a)
i Q, A,
a-4
-u)
O iil n
E;g l8;;{f SiITEcE. a
ot
E6
;f 6 IEE liiEts;€iI trp,!o' €}H€P
.d
q)
d.:
() #(o
a o
t4
AHCcncd
Q: o d()
(gO
OE
qH
€ ho
d, ..q
€O
vd:
;;g E#E€r!; !iti? ( €E
ps xo8f
ite iasf€lE"itffi _(,-.9€'6.!8EE8
E'Ea
:f,#s;t: j'lsEr=l"iE
e,BcrTt*# 3#I'E'
EEE HE,.=EE;
q E*r"
=:;
co
p<
bb
f:rr
Il?
6a
bo
P.
frr
cl
oo
ks
ilE:sns;fEHeEESaf€
!
i=ti#i;xsr;€"
e iEgEgt{igl
85f
{iE
"42_sE;l;:Eg
60
P"
fr{
'+
bo
P.
h
oO
60
-q
bo
A
fr{
H
I
EEE iEi?gf
F-J-){
iE
gH E€€+ aEeEgSE#s
EEEE?qEI
iE;*
EiEBgi
se;Ef{
;ifrEci
*td+EE
U,
E E< H g! s
; ;sfE€.g
f^ It5'e&T
E bb< 9.:
gr,fifrls; f
g H 8 g a;.agt
as
sii*gr
;r.E-E,;X1E
€f
gAt!
=E
3:EEHi€N:
i{E:EfIE
p,B'I -i
r-4
g;s I et gE
H$
Eail
c'iH=adHO-,,
eI
B.
at; $il;s !
iiaif*etii
l
o
5
O)
fi<
h0
,q
cO
p.
oo
14
bo
ca'
q
U?
tu
tb
cf)
<?
E;
iiq*t
EE$;iE
a; s H Hj:irE"A6ss
v/
k-.,
(5)
-{
:I
l-I
J
J
JJ
J
J
J
J
.-I
J
JJ
J
J--I
J
J
J
J
J
JJ
J
J
JJ
J
J
-..1
...l
;\
1-1
.l
{\a
.-i
E
o
"r,
^.4---
-{
i(Cr'
fiiii
.
J7
l"
'tt
16'
:rJj'
F6A
gp.
EEjEEE *
g
I
gEr€EEs
'*5 ;=;€E;
#
EEE
€ t E; E i.;
€EEf; E;;
8E ?i.tgr
X'i'*;'f;-H
E
n€; i
: gsqEes
EJ:€EE;
P
ts'
CN
'gr
:i
i
fr,if,p;g,8fi '
f€
#-fg# aepg
'-E €agE
3EE-E#;€
<
---
F,q'sxtE
'a-Hst€:
oi a
sf{Pi;*
t\
bb
A
co
c\il
(,?
tu
,bo
q,
(o
fr4
bb
0.
-$r
F{
t^
c
j
N
5r
c.j
-o
o
0)
.4
t
ro
x
d
tr
frl
bo
l+
€)
q,
€'
€o.
o
o.
q)
*i
o
&'
a)
d
€d
.o
o
o
s
.d
o
0)
<)
€
;
H
.3 .E.E
I8dEJ
T
I E€s
+E..q6
3 iiet
.{
oi'o 9.2
.E
*^
e H?E
I '"r,9
d:
B
a
r ;9I{>"Jt
4'T€
E.E
"36 €'d;
oG.9*ts
,€E€.9 j_
gtr
E
E> E.E E E
.a oEtI:.E"
!if,fr.o,c
E ie€AH
$e; S3E
0.)r{H€-i'
E es i,;s
; }:'E,H
'E
! 21ry
*7
f.<';ur'E
E
€)^
2
e.t
\r..
)r
rWs
T
E !B.si;Es*l!
E ugFf Hira;
E ;8it;teE,}
E Ba*,*;g; * s.
aEEBgEEfiEq*
-
iffana;iilg
ffiiilE--
,
--
@
q?
bo
co
q
ft
CD
F.
q?
A
&
14
lfi
E
i si
liit s
.! s E F e EEr's'{.E'E E E ;
R
IiiIEEIffIgiaiEi1
II
;5r:'H'tE iEIS*g;.aeg
iliErii*EaiHHi
lH?EiEliE
t:
,/
7
----. ;'
*
iEifiaEEffiEEefiEEEg
u
L
L
L
;; i
L
f;s
f;
oI
tr
r{
q
to
'(,
.v,
oIt
+
ffi
:o
,G.
., {l
o
@
bo
3i
q
fr{
o,
bo
r5
o
lr
(0
H
o
U)
ao
A
d
o
()
u)
b
J
A
o
(+{
c)
o)
.o
d
F
IA
rt)
d
o
L
d
o
o;
t-
CI
tr
o
()
li
o
a
6d
€
I
c
o
+
o
o
a
Ao
t, 0)
o.Q
trtr{
H'
rd c.;
o.
ra
Hb-
E#gli{$$i
EEiffiIE
is:Etx;$+t
E
.oF-!
l*l
H.d
iq-,
38,
ro)se,
frc
-aE
aH
q
qo
b-;
<.r
C)
<-g
(o^
Ets
'hb
6
L
--k
{
<o
.:.9
F.- 7€
ti9?=c
rc*
'A o
.99
:
fr fig
.S Enq
hts.,
{E
F SSqbO
ti-
EE€IJiBsH(t
ile$it t*f,
tigffitffiig
t€,r
c"i
q
{EE
h
@
i..-
c.i
bb
o.
fr{
f;*
bts:+ EH *-l
EsI:: Ei
rroEE;E gB'
pIIEe
2-;.s.B"o
E
E:-E
8!f: ;f
;;E *
E e fr siff#*
- #EE€{;;i
t-:.E*BoE
sdE ES*, E s
Hd;5
5s#:f
s?
;+qtf:f €
t-€;EHf€f
''c{
.f .,x, ,E=HsssE;
4
EEs
EE
}6i: fli;r
s? ().?
bo!
Es?cOdvo)C) d.u
:E gE
€3
€C€(
_3'E S'f;
EH
C.YP'i
g
E
g;.EE
E fr,8
p'r
i
lIJ
lJ
]J
l_ ili*itiaEIasiiti
IJ
IJ Btl3IeEfefEi€ffI
L
[n;€i;B?#,i:.g,$
lj EEEs
s: Ei; E:e qEEf r
L- s gf;Eg*;*s;six5
IJ ;
IJ €jIsIEE€fiEt'$;Hi
L
E I
L
IJ 3..q! E{ c*
Ij --t€?'l
r'
g
Hfi E 9::
Y'r?. Aa
n - (o Ll
E^I
o)
E; lEpEss€
.F
"€ =r'i.EE
5ET;,ET€E.?
jf E,s5;=-
^A X
3S'e€
"El
1H# ihf ?
o a O
€€=
E€
H EHar
F=.= E
nEu
E
E:
E;8
E-, *b'8f,5
E^E
1;Ett
IJ EE€ #E;f
:3i
6 6 o, ! o'd
f
EEET
tj
lJ
L
IJ
L
L
L
L
1:
'
i
j
:
^:'
',:
,
'
.d{
,
:r!
.
d 4;.
EEI
3gu
e)
F3
(a{, E
.db
rElr
EI Q.I
(,,lI\
'a
-E
HHS
i
F
*3t
HiJ
dca*
H.d
g.q E
H;
-E
ig\3
o.# o
:8$
.EE
=
te-8'
r.'$
a
9
.Eg E
ah )*a
9t+
yla
o
sf
{:- ;
s& *
s
*Bt
Ee
E* 9
hE;
r3E;;Ei1iE{iiE;
-iEBtEBi11EiEIEli
BIBEEIqI;E;IIEi
,lillllitEi'iIIlii
-7-',
-o
I
-o
I
(o
og
P.
bo
tr{
rr)
cq
rO
CI
c\r
ho
q
fr{
Ets:ts-E€a E Ets
d ...."9--d
:J
-J
-I
J-J
J
J
J-J
J
rJ
-J
-J
I
-J
J
--J
-l
.-/
-J
ir
J
J
'r
I k*Hj.--
r€3E;Ei E; f,
HE:g:3f :E;
_-J
eEalE*E ffiA
Ei;E,E;fi"igE
:"J
tj
#E*i; EE?€f,!
_.-
_-
-rl
!r-J
-J
-iffiiiiaEiil
-{
-t
1
\
L
L-
t_
L-
L
L-.
L
tLJ
L
L
]J
tj
i{gEI$€f;EEEi$f
; rgisiu E I ;ggt u; iu
tEieE{iir;gitiE*$
;t:;?E E{ ig lir g;j B
9ix ryEsE*il;* },cf es
€t=s#; r€ }t}s€€ f;€s
t- #riiff€ffiffii
IJ
L
t_t-.
L
l_
t_L
IJ
lJ
lJ
L
L
IJ
L
1-.L
l--.
L,
lL
,
c
al
ts
E#€E E.iH {8.
.H'sF .E;*
,€
5;s*; sil ;*
F;fE{;-; FeI:
i;E: f=a
Biiifuff!=IE
bb
(0.@a
:i- r 9,fd
=+'
:=r\HE
,dv
.u)
q)
f{g
AVA
o.:
EsEX
o. -x o
SH 5.*
a€
*f; EE?8
o\ (
.<J o
or.9 <ix
E
Hi sfi
E-tra
bO- r6v
4) (H .4j
I H-e€
snE:i
r/)
c\l
EEE:O
; asi
Ht a:
E,:E
)
Eg;
E
oq?e€
co
s
*.g{gfgggeiIi
# d;,
[E;E ; [$s
og
.:...:{z.i
rb
P{
oo
oo
s, *a€sEf ssfi
c_
t
o
I
t-
og
60
p.
r\
.o.-V
e
#i;
irrE-
s *gg
lr:Eq'bE
o.ql'Xo2t
a
gi€
aE$lfrEEtEIEs?
si
ea=
d*€ iE
y !
-
ur.9.,-
_
[E?E
fI?II
g'E
;
?o E e.e
'* :t
H ." ti-;€ 4; *
EE A* E:.tEE
'E
X'6.'o EBsx.E
X-od=igdX
&o.sf;"q x tfi
fEEs[*:#;
EElqi;r?*
5E#tlor€€g
si*=B e 3
EB.=;P'*9333
aE-.
Eg
if Brf :=
.g= e Et-TE;S;
E
iE
;E;;si*E:i"
sH *siEg$
E;II
D
L(]|
o
.j
(),__=
qh
-
IBEiiBEIi
eit'{;g;:tg
EAif,IiiEB
{ H;E l aE;t
IlEiil!EiI
<f
rl)
cl
}i3
* EeE as eqE?
-3EE€?iBIEffiBE}A
I-t
II
J
JJ
JJ
J
J
J-J
J
J
J
J
J
J
J
J
-l
J
J-I
J
J
J
J
J
J
j
J
{
lJ
f
L
tJ
k
L
LJ
L
tj
IJ
L
IJ
L
u
lJ
t'tJ
lj
lJ
l.
IJ
L
L
L
L
u
L
rr
+? o)
O{j
El
8.8
H
rr't
E
aH
q{D
.go)
a.9
4a
-5
F€
otsq)
'd
5
tH
5E
q)ts
f;
1{H
-o)
_rA 11.
'
.
Eo
s Hd
.(D .- -
fiE5
fi9
tLo€
(aso
H
tox
9 o'i
.399
ets
o
'€o
(l)
d
.t{
o
o
:
6l
o
r{
r.j
b'b
A
Er{
- -:'i
r{'
o.'
F{:
tso
ao'
P.
fE
?:iE€$
H.p I g E.P
; n {ega
[8E
HE€Ef*
BE E
I * oEE f
EffiiEEg
:;.,E P! 6t
EA!€fHE
ESEEgAf
Ad
=e;EeEE
4
>Hr€H
d
l{
*.EE Ex
8g^6.8i
3: E-=
tt '-'6'-
<.)
-.9
99
d
t
d
"Y
E
b..lj E r
P-E
d
E
n
bo
tr
B€
g'H ?-u
S
Xs E:i€€
tr gq.Ein E
sIInasS
,ry * E E E HXT
t{
i'ts
"frPEd
- F:8! s';:E
E 5 E_e#H€.8
E gfr;oEEE
Ei€; A€i
S
;{
n? Esf
'2.28-.Ei
I
fq H.9
F cruE
r H E{ ;s
e
irH{;
t
q t:3
.d
E-,;
€+)
'dO
' .O:F
,4"d
HQ
.E .g
F{€
€
3'd
vga
d
€o
a'!':
sE
i9
ts
boi6
d@Q
^E
-_\ o
No)
t€
-c{
)
f
c{
a)
ET.g# 3'f
rr)
r.r)
cl
Ee
tsF
i
{E 6'*' a.-i
!
-O o *F
'E
*kl * Eo,
(a aE .: Q .9.q
f.'9
FEt
=
E::*
Ef
5 Je
()
0,
t f E*3 et
# xEtE fqii/8'
Ufl'E'E
:-c)
o El'.i
,i-'(,.
E6.iI"i€
R
Eca x-E u
" ,, d'i
'I'-:.8
*q idt',Q 3 g
ttt)-v-Fl)aa--'
a
-
H
F.$83-€ E g
Ert H u'F .tr E
.
-'E3saE;E
oi EB--,HJ4!
frbb AEEEE
EE
6
E * 9 H- I
7i.
tu'e H 6.X.=
sHf
-dl-tr€Fd
83< P
(d
_.iE'* t ,5
tEH TE3 B
Fr (a ag (l) ()r-l
:
il
Oo
o!
oO
ca
OF
-.=
&
c8
-J
,.
orl
0/l
)7
,-
Ef
^;
ul
o
F
r
-5I
sc
a
E
,o
o
o
c
o
E
q
c
o
e
o
o
Y
o
I
c
.G
o
E
I
i,
c
o
2
c,
o
o
.
OJ
'nl
6l
9l
0. I
:l
o.
oI
sl
b;
.J
I
o.
6l
rl
5r
3i
I
I
I
I
El
or
;i-a jil
-Iol
ol
u!:
\
I
I
/
<T'
l,
3
-gEIiIIiEIi
I I
{f,3EiEIE E€iSiiII
g.
8fr
d5..
bo
t
c.it
c:
c"?
tu
<c
IJf
c\t
E'- g
uelq)1? € -..gEE
I
L
{
I
=
!{
aa
<
L
i
-
\_
<
\<
\
--<
-r
:.
ia
<
o
d uoa
()Es
ra5 <r
r(
<
-
<
<
<
<
.<
<
.
tE:
>oxsd
-* 0.6
g Hn
E;ts
a€ r
Z--'E
'E'.3.'8
vd.
6Sl
>P.
H 8.3
-o'q
Etrts
bo.=
eH
E
o,
.fxe
,o)'
.--
-o;31
I
5l
,.lJ.-P
*E
$54
E
6 x'X
'r:..9
OE
<
\
u
lJ
L
L
L
t--
L
lJ
L.
l--
IJ
IJ
L
tj
L
l_.
IJ
L
l_
lTJ
-
l,
u
IJ
L
L
L
L
L
L
L
L.
L
L'
t
Eo:=
d.=
8.8
Ft9
e
gt
H
s..
ea
rd .<
Ha(U
.6ut
.7,..9
.:E
i E.?€'
3.IEi'
o d.r
.:
gaE
>.o
-E
!) gtv:i
.E
Ooo)Dr
H.>O€
P
s-<qE x"
E'Ea
8
2.. boH
N.E
.E
<EE.
+q o &H
.; :r{-dFi o
o a.a 0
^HH
H5g'
tuj-
ri I
€
.i. etr
ho o8
f
"60
!{ 1,H
I
\c{
pta
6l
pt
ll
al
€f."I
dTtt
.
.vad
$
.a6
ol'
E
;Bf
!
t
d.lo
1ElgIi}i;ii
se
E
;-.: 0)
€
6\a
\c
ol 'd d -c{
e
FO
:.o. p€ cq
+'13 o E
+
-ic' .bo doL< .'.e
p
p
.e
oHtr
.11
6l
jj= d
-e :*ll
(dH
6d
h
0.}=
p[ Pl -J- ()'
+
I
F{.
t{ .::
q)
€
j
o:$
(l)I
tr p
"oa
(ts
ct) o
s,-lt
o dc
<da .iI E
It
+E -.stsl
'o
o I
.$n
6l
tr e
.ql
'r,
:O
0. pl
'qf ll
1'8.
.8. E€
tF0)
E llu.E
3 ; ;x
t J EEI +lt
isa.9 E't a'
E
''to:
.tr
t'
()
a
rr)
co
o
rb
tr{
o
+o)
oi
ai , E;*E
(o
-{
co
si
3
=iE
I8.g=
o'o
E€ Er
H
VA
-.
EEiq
'6lioi6
E
€
d
a
'q5'
O
,l
d
in
.1
g2.
fi
e
tr
a
.ra(D
.fl
o
()
O.
CA
{5
{6
(I)
(l)
a)
*a
g)
:1
..i; E+
E
,€llEa$
b0
bo
,4
o
t{
d
e
ek
sN
sN€
a
an
?,8
t-O
E
rH
+
ll
l+{
<)
<€
*t
d
ah
-i1
-o €
H
!
-o
al
3 e.t
o
.&
ll
ii
o
+5
o
rr
F{
J63
ro!r
q)
I
lI
iiliffii
-o{.o
(..!
*dE
ut -A ts(aE
.
CN
t-
ca
>r
3
..;
E:EF
<(-q
ol&
Joooi
.: !.6
"AET
a=* 9
-3 F.E
0).d
4APD
q,-9
(,oo ets
.Hv$€
.$r't
E c,.H
a
(E fr.
E9:. o3 -54
dco
<)
*
*3r
If,Eg
a#:d
::
}
8E
-c,.2 vd
.lgldI
H.E.E
3..1r'6
trqr-o
-d(a9^
O's
f4 ;^9-Eo
d4u
bo
5:0
€
o
?:s;g;
t iE Elaf Eg
.iE
r':€
fl$ qE€far
;t iii, ;i
{J
a)
O)
()
d
$t
q)
o
i+{
+
a)
o
.V,
()
{i,HE:$iE?E I &.
$!a;$i gEE*8=
ffiBiiiEilliIEi
ij
tr
E
T
(6
tt
H
..
ff-$i€3gliiEtt;
). ,
w
sg,
"
xI
S ,,'€
_. r-
i{I
f, Ss?;
t
.rN:
E
.'..-ii\ 5 '-l. -.9.
Y
6:s oiF
I
E E6 --<
dd
gllilt;gli
HE;€se3ffi!
.E ^E'3 E
-f "r?
|
3
dY:
, si E
co ! €
S ii."s
+!$
Et
T -6ila'
ts. E{lEiffiBiaHii+
t-
'
:a o €g'5
p! .g '€
- Ei a', Hf -ri-d'.9
?-sE € tsto o p -E*€
ilb
p
tr
€
.i p x:o
p o
fi
-{i
-d'-*,GEd
e
eu
k
ll o -.s
I .9 - -|i -:? E +t -O
3 l,&
llllod "15
=t.$:
E
d
ts f$F6iiE
a
oC
x $ *'l-rH
c.i d
j
EI-!E I E E
6
p
t{
E-'* s$$g -<ee@
-a) #;€
€
.
';.i
,.F.kllO
':'€q)i/j,.'
'-.t'-'
: :l'''
;:'
!
<-db?Et
?q* igE
Y,5 a
-fr:-a -O-df!.
gff.;iE
..\-.
sE
(g
e c.6.= E E.6
€.Ign;"if
s E.I5oEI r,'E .dE'E
E
-aA A 9-p
E'e€! aaE
ro o
dvr'99
t-Y0)oq+
o.", d 5
P.Ht!.r 9a
F#E
!;Es
E,! .= 8'r *
r.r*iaI,6F
g.egfi H'8
E
il,i
{#
a'"eE€'f
+, or O. H
E.e".a€EE
Eitt
cl .2
€>'=li
s8
.o4"9'ar-E't
r6E
a
iE{
€-E-E!
8',,,'H
+ HE + E
=
i) €
o
-EEgE€EEgiiEBEI
E
i!fiflgataatita!,fl{i{!
qli
i Ig i1qlilaltfraIal1ll gtA
#'';
E
t ;? : $ { E;EEu
aittglig iitit+H,g
:ff siiaE:BEiE €
,tfiiillxffiglaiffiim*
p"j -
\
H s*€E o=*3E
eiii?tii:?
.:r6E-g.Eo.9Poqr
:6uFq>,g9.A,EE
g:e;€,'e €: 'g
E
l
-
I
-<
<
f
t
-r{
-{
-q{
-I
-{
<
-q
-q
_{
:_{
rrf
'<
!<
.<
<
r;
effi?t;Eai
-<
i{
i<
-{
<
<
.
;AS€IE EEEE
ru*gggH*iilffilti
:
:
:
:
-,
:
:
',
tlr
I
t-_
t,
L
-
LJ
t,
L-L-.
L-.
LJ
L-^
L-..
L
L
L
L
E;*.EEE
EEESH ?
bo fn
tr.E
: E; gEE
o
E;.,'BI
.= o
5
'I O.E 5
€
5:fTE:
g
:E.e:;E'
ri H:f;
H"H;8f
6.d o'5fr d
B EBE;fi
g'Q E#-
f
E
E
bo
;E": 3,f s
H s6! .E
-:.?P orcD.=
t€ olit'agf d.Il
g
Ahgz'i'?,,g
E
fr 3-E
gtcq*.o
u
E
E egE
a8 *
FE€E
-c o" Et
':.*€E
-"1.E
q':E au
s;
.E
fEalP
;+Er:Eg
O-{CO.f)
:t
.28
Et
.9 ^
E&
.g
Ei
Ez
se
EE ts 3ifit
8e' e iEir.
gE*
'ovo+i E I;:E
ix 8. d Ps."HE
E3: fr s$Bt
igE legf e
EE: E fdfiE
3 E.t s
N c). gH'*
X.H,
;
EE"E
X'- o i
EE3
x
-F
(A
o
tao
FE
s
€,t;E g
#cEoI
I .-E
5e
_r
H
E HEri{
:$:
"3;E;
E€EE€
#fi'c,i
S E HEE
o.-:=
E;{ii t
fi
H5 g
;.E
OD..-dH
. (i
k -xTl
E.E.EEE
,- E E x.S-B
I
\
\
=-r
t..
"IE
QJI
l
ti'-
v)
{J
'ro)
'=-*
F+
()
u2
83
E.a
at
o
o c.9
o=
c,iEo
qE9
#tl
e?o
AA
Ed
o.2
oo)
ET
l.rFl
E.6,
€E
H
/-bo:
60 0O
I cQ
r. ca
F- EE
bb
rr)
E.9
'c,;
O. Pfq
9iifiE
-dE
-A
P
d€
Oo)
Ff -O
F
c't
EEt.E* eg
E3giIxiHiEgfitei
rfli
*e s€E t*
iE[EE;iru
" Ees;€#Ba;
*
.e.s,, E s2, dE#rB€::5i
iig;llllti*IIeil
E E€EE
d
t
oFI
rE:tt=
G.
SEtI
a':
I..-=- --l
,i€=E?;
f,'r€; EE
a
E
9
o. o
rl,
c
o
a
s
BliiitEiEliiiiEE
H€itttEe c€!Ttsa
i
-_
o
€
o
C)
EI
Et
0)
q)
ta)
{'rI-/
o
(,
lro
.o
I
€
o
a)
d
a
0)
tr
H
t1
I
€
d
l.
L
p
3
'-
(+a
.9 ,'
8t
ot
,.
t\
o
0) (a
d\
B
q,
o ,o
€
$
d o
s. *o
o .H
H
.q8:gEg Eo:E3"j
;*IIE?s EiEeti'
3iiIEiiliEEi1B
o.!e\-
K
&
>lr'\->\,..-.
o
\ ,o
\>N
ol
o)
q
@/
.d
N
o.)n
r+l
(O
c.t
bo
O{
fq
r{
H
r/)
bo
c.j
q
f&
Fl
$
C?
q
ho
14
ligiiigIEiiti
=EEiEEaEEEiti
,Iil;IiiEIiii
e3f
'[ie?EaBf
-:EEtEEEi:f,EE
H
tt
J
..J
!t
I
I
I
l
,.J
J
!t
J
-r
-J
J
Erl
-d
-j
j
- __)
-i
-{
:-
l:i-
-<
3-<
E
-l
-::.
r,-
:.
:<
-{
\
f
L
L
L
l-.
l_
IJ
l-.
L.
l-.
l_
IJ
L
L
l_-
L.
a
t
,.i.
il
q)
3
o
(
U)
tJ)
<J
q)
ts
il
Ll
3
p
,
0)
,
EiiEgaEEIi3tfE
,,-I
'I{
. ..,
rr>
6rts
E+
->da)
r'I
-IuF
:" €ag
(d v=
I+!
X#
*"8{}
L
i H'.s
a)
€o'- EB
.13Po
p,I
9?
'.aA
lD5
E3 >
u,
t
EtrT
as#
t
H *.E
b.,E€?
l\kkq)
E€e
c.rj
€'l
7 -'
E PbY
c.j '< '6
't
SL
I
dc0_
{ '-clorto
+ro
:'d ts
E#;
f-
rs
[F*
W
/
$
sr
Ft
cri
b'b
?d9
q >ii
.H{q.E
.6
a-h
f cro
E
EO
Or -:'nl
<'=
ot
eO(U^
*'^'
.9
E
(D
B{.e
.t+
Et
.e
E ll.?.
$E** '
EII E
Ab
tr
tsoE'I
o, \ 9..
.gdtr
A
". fii
c!-()
*ofi Q
P{ -E!q
n
co .8.*
6l
14.
b0
p{
c.j
H
6r
c\r
rO.
*&
ti
i; <'3v
.l<--rS.t=l
tu
N H=
c{Y=
H HP
,
t.t6r
,EEEE 2
i;EE il
x#
EEr
.=
d
!j rr
c"i L.-99.6
E
oq
HUl fzr Fq
Fl.
aB-
H','#:3E
b €
',;.1 d ...8- 'c
.'':""','p'[oT
,.:;
i:,
;
:
,..''HE*8,,8
l3 r i
.E
EEEE Ei
-f #'I
E.,i& €ilEe: EBEE
tl:F
:s[32?*],.8e;
i trEEi
<gnq6
{ix;iEqiE
€ }q'y gE
E',1
5 a*:
iu.Eeg.g
g.
E
F .'b
iB;+.8€t; stai E Hgg l-_g---JgHR
irmX?
Y>
E;EE
€€EB!
BEEBi
iEEE E:.E3:;B;*.a:,
<Xgft
<h
*G
ielI Et.EEt?tEl?E
EaiE EEE, ui u*:*
o:oE E;{EsE cts*t
EE€BHT ;*BE
AE"'ES
EQ;'i
EE'=EI.E
E€a.I ;EE=S-l
.Ex9
o)
()N
lffiE{iiiBi - o
1iEliiglEEg
tiiaaglE,iim;
E:E3IEiEigEEg
.o
l_
;;E;
r-.
r,-.-
L
IJ
L
L
IJ
IJ
t
L.
L.
L.
L.
f-
€ fl;5fc;
E:
?E
;
ilEli*E tE{ffi
E
6*3ii I8 ii
e. H*efiE:sEt€;:A
* i#s : ,
?E{EIiE iEEIff+:
?83:x' Ia=EE;:,i?E
BiBE,{r!,r H;Eer:BE
EBE;i??t
E?;: s;E;
g
I-E<=
*i E €g:;IS
?E;B?E,?i#;;BEEi$
EBf E
ElEEEiflliBiSa;
Ii3-E E$flI+iiE$r?
ttI;i;r
rt*iH; lisi
;E€+i il?*€E*
[g;S::;
ttttisEt;Ef ;'#ta
f,;iEEE{SIelq;s6l;
E+ilE
E?iEE E
?i33gE E
g;X;XSgEEEEE: gA
AE
jt
4o
f
2'Op-
W
ffi i*iiii*il
!; ia;:;*xnr
-iEiIii;iii;
H
EE-*
.IEriliEIaru
'e
cO
|-
CiI
2EAflri"a E pFE
H.o3.E3E:'Ei.9tsd*a
fpeE E Exse6?f
EE. 3. s?}E:aI
:l
J
I
J
J
J
J
J
J
J
J
J
J
jJ
J
J
:{
_r
-
--
-i{
I
-J
-J
.J
J
J
J
J
fi A: I t':*
i tgi E,lf* J
B:i
t*
t
E
E
F EiEEIEAiBSqE
o I H.iH.i :.E H .e e;'
: REEESEi'Extql
F igEgE{trssr;
E
r6 ,uE,EeiIa:,;I
InEnEad; s8€
\
L
L
L
L
L
L
IJ
L
l--
l_
IJ
IJ
L
L
l_l__.
L.
l-.
I-E3
8qg:;$l
r'
L gIEISTI;IE
$$$
[€Si;*,E-s
$mrBi:€t
IJ
"fi€
HgEEe;'cs
$$$
IJ f?i
IJ
lEB
# 1=
E
$r}
i rf, af;{g
eia
E;s+eis,if *'
EC€
E-'<; *
i iaatxagj
EiES>u-o
IJ
r_ igt l5l
L
l__
L
L
L.
L.
L-
tt
ro
c.l
:;
A
!-i*
ti s EE.
(d -fr
.
H
d a A O O\
:d0)€><
VYoL
EiatEitgg EgqffiEEt :ffiii*Eti;i
f
3 4;'
e:c:i
e
0., c 5 c:9
.c
iii L,.o.dd
;-
d.e
rI
E Ei,E EE
h ll,^l
A u
.:Y,
o
o
' E
E9
H
:f
ai.q"fi
:,{t€gE
*f:f€i
;EgEgi
i
gE
'.
rJ i " E.Ei
- p
B 8:E Et
-eSEadgE
^ot 6nEtEseEHts
q T8 Ee ltr
EEEgEE€
HH€€t Et
J-J
J
J
J
J
J
J
J
J
J
J
.-J
iJ
-J
J
J
-J
J
..J
J
J
J
J
J
IE1H1Hififiilfil, llHltlrlHlfittsE J
J
EtiE:giE a:aEEf,EI1It x ;EEg?c+?i;Eif;:;IEE€. J
J
J
.9aH
;,-T#
>tta
Bsts
o{sa
c',t :Ot
€T g
H
x g ^t".E
H €iE
E si e "
g {E€
F ;T;
E ?EE
r<ruEE
tu .l= c
E s B€ \
Z *AE
ql
o
a# g
c! +'ii,:
h'-
E
CO !O drJ
J
J
r*l
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
l_
L
L
L
ItJ
tJ
L
L
L
L
L
L
L
L
l--
ffiE ?ESEE;I
EgIgEigiii1
ElIEHIIEHiIg
i€ 3i€ g*:,er,:nE
gBiiEEi;3EIE
cE;-;*€=gqTEI
$a€e;;E a*rsE
tio
<r2
q)
a
qi
4
o
a)
0)
F
fJ
o
aa{
o
i
C)
-d
C)
F
A
€.
T
a<
'
r'
*agaitttil:?igli
H€{t'EE Einz'a'a€'EE-
''
tEiie;, ftiiiffFgl
i
EE a'fl.f*€?l
* ls xi€;; E T
eEEiTl *{s$g:fin
* fl*EI$
gE
3;
"EigeE$
!BE.ffiEtiE
E
E
! e 6o!+
b >,.9 H?
.H >#
dl
E ..86
P E'Z'2 a5
9 {Eqtrl
cE%*
I-;Eo-
(r€€
hnk
H,= O
sr
EEEE
D
V €..\'
t
A H d.1'A
E't .Ha E
k
i EB;EE[;EIEgEEi33
F ,i;
€8:;aE*
; . i6ttsc"E
3'e r.s.*)
'Ei,,Agt"{
.q
F- et EX.H ex
c?
*. *
P ubt<
i .,=,
o.0
o_o"
- t'1.
Fs;f;#E;;
lr{ g gl o p 9.9'o
E,Eigx
.El 11 -o c'-Y6d02H
!E'Ht lEs
r
HtIo.Po
or.E E.ll
.9 L
T 6'# ts;; ^-q
-X
H
c;5 fot g
St f EF
a:a(/lP=
av9
9.* t.= fl
o.€ooE
<Ftp&
^.\ -a o
E
bB*s
I ae;
X
6.H U G(H-.o
B
E.
E.r xtEf;6
s?€:f
9
> tr) 0).o
E PiEE
H€'
E EEfi#;
EX*r.9
c' H*
Efi-gE.*:6
Eo Etr
';5a-'-EE'\
.9
E'E H T !:Eu
E;TgEgE]E
E ?E ; E"f E!
.,p H 3i:E,EEH
tr
i;tsg.^tu2+
E';a'[,e* Es
<.F E.: a.=,EF
r E; i'; E n*3oEil HE€e
\
l
I
I
I
I
I
I
I
I
I
i
i
I
i
i
i
i
o
o
a
o
o
or
ll
$c.
i|{
<D
rh
jn
B..i:
.t,
d
'ia
Es
ell
.9o
C)(D
c)o)
t) ,o l.{
E.E
* I
,S .a
:E.E
68
Pts
138
(Ds
9d
oB
*i,
ea
-d
clp
-qd
>E
.0)H
G- To.9
.ti.
t'lJ
B"
E.g
H
ca .d 'd
q.E
6
b9tq o.
5d
qda
€€H
c\r
fl3
iflx
3 t,
*#
@
O)
C\r
1
I
J-J
J
J
J
-J
-J
I
-J
J
_-J
J
J
I
J
J
J
J
J
J
J
J
J
J
T-E fl
J
E E*au,
'*Et
e
t-.6 E t
Hg
@:= !4'rl
€E
f:;.e.
EeT{
i6
€.d
>
B€
3B
,1o.9?
? 9-e:
o+€
Oli
ilS
as
{= Ets.n
B
v|L;
< ESEF
3#3;
1,.
J-J
J
J
J
J
J
J
J
J
*l
{-? C)
E_d
xtA
hO'€
d
'6*
5E t=
q5d
:$
!
Ftq :
.6Er.S
oJH
6l ,
H
=
-i-ts-tsh
(-t.:
q.B
E.e
e
in E.E,
X:H
I
#;
.=
s
ErS y'H
r?gH€E
* Etv*
S.X
HEiisi+
E E.E x
E ;.E fl g8
+'st;
Eg
'EH*3'3e
f,
ffi'
$
oa
e!€;sE ft
tr
J
lc
:l[* ,
E:Ei'E: "[h
6?,+
c{60€-on{
(9
6!
I
hb
Fq
o
I
l
I
o
-o
tr
(D
€
(l)
-q 8'
v€
0ro
.$E
A
U)
d
(1)
#>
=o
bo
#
I
E\ .*
't.l
aD
€
o
tr?ts
t'6
II
:lp
T
o
o.
_t'
f{
(o
.d,
tb
G
h
'.
I '.. :'
I
ln
o. -,
a
.ro
f.r
.gir
<{d
og
5bo
i
I
bo
94
r{
1ii
o
I
T
c,
O:m
l=
O.a
l.r
,<f
bo
Iq
E
3
14 9.<r
. h&cf
v) )d
gEr
PV)
.o*
d.i
h+
,bo
ro:5
$++
EE;E ?a ? t ii
gifgs*KI
'EE,gE;*+ t'nnn *,-rT+Eilgt
t
#tsE €_EE
*6#e ;Eer
.HEE.E _gd HiiTi
eBtf s*iss
si**r.e.E xflX
frHst#; s y?.9
HdqsIfEE;g
I[i et sr FgE
gEA€!$,tfEf
i8f,A{taai s
IIliisiIgi
*e{E$s+s€fi
E{ g;E rys
lE!s ffi:;7
EE
B*I f+E€E
Cr€'E a A
9 6.E 8
E
l;Es *'6:*+
p*Ei ;:ire
EEETX
6sUIi
fO-.H€
E#E; E i;E3
st;t
Yode
b0ol:ilH
I 8 rgi
6Es{ 9r;4{
Eil€
gE*
EnE
E<*'Eeg
<- E E[*E;
o$888*E;Ig
=l
dl
r1
kl
Itrv
ttt.
j
_a
a
(
-rt \
@,/
__lJ
-#Eet €8 $ E. gE tEe
rt;ll
6r
t+
:I
-I
J-l
jJ
].'l
_.J
I
-J
-J
J
-J
J
J
:J
J
J
J
J
J
J
JJ
J
J
;: Eg f; El{
-ilEi=l;*truai-t JJ
J
J
:{ii3g!fssa;;{tE J
J
*$a?aarigIsiiiis?iI
Nn
r{
r{
bo
14
<,
o
r{
frl
bo
o
F{
f4
bb
<,
o
C\
*:igiE:g*iffgliilag
IR.Adt I
; E-q dtr,
P33.sQ7
atii
E:
ot
*-?j6
HEo(Dts-
d,E{t:f E
"-r
3
H.
$E., O E'A't A
.q
sE; sg
,O5t'?E
BA
3SE
'_E O
-o x.o,ii
d-.ro.r-"
E I *.i lI.E
.$
I\ E.s i &,8 e
.
'
-Qr.
s e# fi s s-e
.i
)*
)
$ fcsE E:
X
r.{
e H€
i"EEflE
s# Afi i
+ E3rI *eB
e 9Rg s*" *E
';<"; E EE E
\r
\=
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
fk
f
L
ff
L
L
fa*
E8 3E
EFAE.
*
P g.E€E
Bc d
o){r
O.
EE
E;
E:
HF {188
sS,EE
-ht'' d'ifu ets
:cE Ida'r
.otuHtx'rx
e.9s rif-E
lsf;EesE
o o.o
p :h-.9 .:- .g
or-+
€n.hEE 5 J
;EiEggE €o
gi ,o
.li.
S;;S
d E.o o.E arE €oq)
E
bo
0)
H
G
u)
{J
o
tr
trI
o
,o
c0
E .o
EiEEai
gEs:;:!8
tE.3; s€ ;
ni,E*HEB'
'^..
f5
\,
6_€
P3
.i45
-
o
o:*YolI
Fl
':
a
die
$gIiiIitIii
; dElE:
Ei{iEifg'i;
- ! E.Issf I e::E
g
:iEi?::B?EfE
:: Ets.eeE.E*
;AHass;:i;
6.W
z
!a
,Lt
oi
-[':1
3
o
4-l
oo \y'
l.--5
E
ar
tl
.9
x
=f
t E.E
(.,
(o .af
ll rl
t3
oA
q<
4
.1
P3
6l
$
6!
G
'r'b
.h
....
Fl
6l
.{,
c,
bb
1fr.
o
fr{
bb
gc
-1.,
\--
t-
EEE
5.ue
EH*
.o
a,
a4
)o
-r/)
Hd
€r{
14
qr
c,a
"Fi:d
tbo
s,
q
h
.d,
(o
bo
p"
Frr
M?
f oHc]
-$
?E
ns
oA
6
an'ri a
.
co
bb
pr
14
aavv
,r.;
d 6d - ^9
0) €6H
trlJ.,
+ oet6
AU^d
c ddoqjE
bEsE
€E-;.
o)
t
dr oi -r
:
rtr'{t
'iH ol ti
o
'=ca
* od
^d
^a
36-j
f, a;*r iEZ 8;.
.9p
t{
pZ4
@o
f;
iHE .2a
E :Hg
-t
E EaE'
*e'E
=E,€
-e€,4
HEic
d -^ts
.f 5v
E ix ?t $'fl;,.; , .5o!
€;EE;3 g ; si ttrEEF'E
-:'i
tge;gt
FiiIEf,HEEE
:#;g;€ ;:iE H"iS<
EE *o
- 6..1
rr{
o
3.e
')i f :EsE# * .ts; ..9-b.E
tai [i >r-
6t
<,
5 EsEp
: 3f er
6
etir
:18;EHI,E
* ErE;EiIiIf
nI*SS:1€,.H
7i
't l
4 <{
fo
-Z
6l
o
t
-*t'1l--'f-+ j+ -'l
:
(o
I +t,3H
--
5' s
E r.9
.9, 'F
S €EE
Fq
88e
@av)
@lP q
.ilJ(.)
€ 9{l
'i.(D.--A
C"tal.o
d ;5'-o
O u). d.
F
H
+,4
Oq-A H,P
-1 U .rd
,x.E
\r t 6.9
6! rr.
#EQc
H
ts
=gt
6t&.o v':i
c'x
a
() o.=
t{€€
-oag(D
-d
,t4
=
TPE
?
A.I.d
-.i
=EgH
q
,1
{ I'o
H'E
IJ{ A*-o
61
oa
eE:
ji.6 €
d
i
'
3,8 EO
EE FE
F* r#
fg
eE
E-g Es
?; E;
E!B Ig
rl=iir
jiYA €,i
E8
ct
;f q*,
t.)'^
v P
Q.-
(,J
q
rE fa'
-.-. F.
d
o.-9
&x a a H€
I
t
T
E#3 nBE
t
;;g:*r
:*i'*lE
rO
i€#: d x
;?E; # #.E
oH':g
.1
5\E
sp e
g-
jTEs
E.t'E
<sii;,:€E E
E
i:c
i
fr
*,8
H8
! ri ir'e t
EH*ijt.{
HEbtET
8tr.s#f
E
{€sEE;5E
a€xsE EiB
t$E a5f s:;
:E'iEE:;: f
+s$
o
2v
o
,>\
I
qo
€
t{
q.l
an
u;
H
t.O
6'O
I
#;
!
:? i#: rs*
J EsEEri
F:;sx€e
F
j€EfgE
;€ aE 8r
HE€ $;E
te a;E?
InI i
a{ TstE
HJ.v
H€ Po;
-d,d(uL
C0
>acr
3i
H fi tfi
+t!
S
il
3
A -oE.:ZE.'^ o f, H,x.t'TE
E
j
=
? -.fra"u I3s
a:t:{px
^,
H
sdgs *g
hod -d
Eida
EdE
i8s
HHE
pIE
k
s
:
sr$
(t) Gl-!
at
[etj
a ;E5#;
.
8T
[..E;E
x
E
€ ;E
E;;
*iiDad
.H
c xFlr.H
t6E
-E
E ubd*
Ir{ 1t q
s
+-€t o
EFtsc
EEBi
x'6 oo
!-9
oi- a
t
c
tt
ltt
?--.
hi*
T
,
q
r|,
'!
*I'6(i
/-q
I
)-t
(i
.tl
o
r-t
s,
tu
'.bb
lzl
Ol
+
bb
P"
f4
bo
dH
^fq
.E -'E
3 €r
aeE
(0
o9
5 ; E9
<, .9 T=i
?E qgib
l\
5' aE
Q).-
o
H
H*ttg
\ E4
ta.t)
g
hO
>:..
HH
^o .dHtr
. e
o- o
af66()
O
o-
O-
'A
H
*
.r E+.
a ir
E.EEOu2
E
!l
:E
"
.,.-,.
€EE
-.
#E
t
E^E':
bb ir=!E
I
(\- X<r'(r*€
.o bb'i'€:
G fu'6i,,tr.q
'{U
'Eo" 3 Z
r< .o.5
tu E.'8
H
r-lctr H
ettl
€ foE dE o
't'F
6
-E 'E
sfr
.E .E
: 'fo:
coi
E'"rL
s
O
;8.H\-#
E
Eq*PEE
d< .I
,-,_I.,E
..iF;,c.9Po
H'Q
qiil'r..
ie
?f :
C)
A
,S.-E } E; ;
,'8's B
.,
t
a
r{
-<
-r
-{!
<{
i!
:-{
-!
.:-r
--r
-
!q
qI
-
<
\
.<
<
if Eiig
- -.
=0.)
Hg.E C 6
E
d
-{
A
H: rn.s
J
J
<<
r<
<
E bE
t€
'= .'.^gQ *4
;
rrSiHE
: H'qs3m
if#ter
E€
ET:€i
?tYf;
E
tl
I
f
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
IL
L
L
L
L
L
L
ff
L.
L.
L
r
fl,-,
E-Hts>
U. .=
el..t E
AA
g
ar
H
Ert yii .'
ffis!
.9s F
u
It-o i'E
.9 H P.*.
'.dxo
-
HEBUS
5*.<@'
-Q
o il
o.L
.
ET EE$
cO H{
id., dc\ Ei
q{ tr c: -^.X
'*vA
iEiEE
ErHra9
3: E il'ro
d
}Igf
E
o-'o o
8'd*Z\
Xflc:E
'.
TF'ffr"r
-.'
F{ >'O: x
or-o
tr tr(x
j
x
(gt
Io
to
d
o
.,.<
t-
-:
<.
q
bo
tu
-Y
..-&
..9
.p.:
{JH
3B
.o
Bi
.d
.9e
b€
Sg
tD .a.
gF
YAa
'F
?:
gEaE
Ee
8.
- q){ I
.E€t
H
E E:5
d -t€E.s
f,,:ESE
:EE:
fooo
X-d s:o
$". E 8,
sEs;
,:-'$ E:g
X
c)
t"qHd
-l,Fq'i ;
.(r, .= -- cg
E
il .= 8"tE
Cil
ia
5
,r ho=A
.o .li
c{ >,()
9A
fFo 6q?:[rr.tso
€ ''i3rEae
o:o9H
tbb ':f t
r+<
o
()
o
k
a!6
€
€
k;-oet-----t
l,}
t\t
T
II
I
!a)
3
A
g
n
o)
!G.9 V>H
9
P
;;TTAH;
X
B.EV€IE;H
T
0. - F 'iu'6
>,
o.>
rr{ Es:'E XE.E€.
v U) -=q)
.*'<ru5-9 a I -H
o-b5ai*-q
Ld-4!€
^(D o-=
!+H
8.2"a>:ie
5E:ori>!
S
q)
'Pr e : E ? I
sS;eHn,EAg"
trr ;i
-.X ur c
.$ } E: HE-E
r\
*9bE.;-g
tErsEfiE
trjjd+:'-.trt
9A_-
S
ila
E! Iii
EBaffNE=
o-'E'E
,,'.8,'-a''=
E
, qA,f1.-. ;' o.'.II ",..,
E
ag
aq^'.:i>
-
'-4
I
*' * E',Q a's I E
i .8Eq-#;
F'E USaIS
q
.s <'-.gPs Fs
tu
-\ Y
|>.=
-sf"gR:
o)-1
: } :YS-H d V
co
r+
rr)
,€s,
"i'ilEE
or, d >: o to H C) C C boc
EE fi€T'g€€E
,-rE RE; E*e-s
lE FI Ereir
df !g::f t i
ti ii*6ttfl
::;EEE;;iia
*tagrii*tEc,
H II{IiBEiiEi
\l
I
1
+
o.
+
t
bo
{":
E{ .P -.{,
pi @.
.tP
,
o.+.
=c)
lo)
OE
-.
-l
B.E
ir
b*
6,"9
AH
Poi
d
'a
,F Abb
E'o/I
(O
Fl
'{t
r{ r{
ciB
E:
q oii
z
j
.5s
H6
^o,
,oo
#B
€.- sz-
clo
.-ql*
fr<I
'€},-O
io
i.r
i
.
Ol
,t $
t s tti ;.q
.rr : iO
€
d
-.1
E^EE'dE &r
bb
(,
I'rt
;I e ES{ f4
H d- 5r E{ 0^u d q€dJ
E E.E,g F.l' b{r
H'!i.d
Ef?TfE
I sB"A-iF EL
qi
trtrl
8,9 HE sE srL
A-z
gE
uoB
H
P o)
a)
{ 5e
r.= 3* A:.
" qia91*-5
.d
tstn
olYtr ^ -d o
a'x8ddf; 5l'
E it
c)
'eE
a' €
ry.5€
q)
,,1
s.e
6 a
xa ;9
-g
rh
+rar{
Aa
A6
iEt:EE
Ehp {*
fo-p
rJI
o-c
AP
g;HdH* ;g
:
.Z
)(
o
.n
L if
.H
$
o,
lti
l
I
vtl
+lq
.<
d
1al
-'f,(
.0.}
€
:a
BE
<.
b0
c"?
t'a
:F
-{
,.d
d
bo
C)
t{
d
(h
IA
.o
€o
()
€o
,9
,<
o)
()
L>.
i>
f4
bb
r&d
.d,
co
T., c\
JL
€st
B
H.
A*
V)
E€ q
abG
dco.5
g rr I
bo o
S
cH
3'l
9€
e
'cdo(a
,o€ c)
.o
d d.
raE
cd
v: H=
af E
pF
ar.Td
d
(o_i9
.+r -{ di
oo? or
6Si
Ro.?3
'8.$.t
of& *
()vV
*-oA
9H9A
AP
Os
il6E
€
=ZB
'(g
o) 4o
A
o;
bb € , .hsH
*c()
14
Cr'
ho
U2
o)
(a
d
t4
86 bE
o <E g +
(J ;{ €
EE# ia
c)
o
)t.
'c) 14'F:
,g
cO
H
-B
()CN;^-
*
*;:A{
.E
f;
E
aEzO
Et
Hf!
to
gt#EEE
'5' c g'3 3
lE s: gE
E;TEE;
H*E ElE
o'EtEes
x'3qE dB
€*Egi;
fsI;;:
I'i:
i 93' so
.gdEBEE
E.go.e€ f
EfgEE
.e
E
>:n
q)Eo
,EE
orE
(E
EI E't" c0r
o-u
o)
ts',:
d
ql!
E H";.r6"; P,=
c)o
9-1su4)
EP
^o{:NdAN
o
-d
€
HV
ij-_, aE
13
rf)
C.l
<l
c:
.+
tr.t
bo
q
q
bo
r\
.(,
E:ET;
:Kis€E
tP-e'di=
}iEi; i;
!€ ;sE
[;€
x "' q'rcd"c4 E
i3 ,: * l,"f '*
?, E..I tr rd:: i kE
sv.
A lzt ,n .Ei:'H o:
q
E
pH
a
! .F: B;;E
j'€
?x8+ Ei
E EE E T F€
T EEiT Hg
B #.i=i ir
Iut s-rEE
; I9* B *;
) .=..'a,If t -g'Hi:a'g
ll*=I:$i
'ii bu# E.$E
€€"=E<aq
.E.E ,.gX.tr6
EEXEdKxE
Z
*
-9,
l!"t
I
{.
,2-l
I
I
O
€€
--.P
'1r' o
.
n +{r
'€.
En
6l
AA
E?
EiH
:il .O
Er(l)
EE
hg
;E
rs
3 E€
...o'.
ca E
0.r
-E
€
d
.P e Ag
I\
E B=
T? HE
tl{
lJ* S.$
o
r
, E:EE
firr; ^
, 7a ''o,'o
' .. {jair.9
5 B HEE
d E aqf
,
I<s
cd r:
nE
N
HH
,-<
6
d(E
ts::
9d
()t<
ca .H cD -A
LO
:..
d
uo@,
ii,tfi
q2
-d
o)
ar2
. .d do)
,:' HO (E
:,,",:
'. '.,. ao !
;+'
O{t
o.(o
O+r
aa
(a'6
t'-'6:6
b6
ra
-54'
<r
qt
'a-
{*
./
NE
c)ta
€Q
.q
,go t3
F{.Ed
l-L
RrEq.H
,<.ad
E
N Xgit d'
,p0,)rc c
<
rEf€+
\
-l
-J
J
iJ
J
J
J
J
J
J
J
J
JT
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
-
'
'
o
F.
E{
--J
o
t{
t+{
(4
v
iiiE E$iE
irggEgffs:gif,gggg
.Z F3€E$$E;
tz
Aftl
9.
F{
Flz g€a}fliEiilf;ifEsEf
.Y-
l-{
Eh
t'5
F
V
n
(o
&
!
\.[<
g€E
I ts r,.E
m€
o.l
fiXt<o:
E€3;
o
to4
Eg
E
E
Ht'E*
>-*o
.(J-o
E: 0,
.HS
*
t
F.f E{t'
:F E
g;E;
*-*fi'g
$ gfEE
6a
3 aeEs
O: U BEB
# Bq.E,,f; I
I g HEj
eif;E
EEEE H
s
,,,
i rl
Ei
"g#t fsiEE'*srE,fi i =r,
j,i* HE ; : i, eE ;iig
Etage:r;;eEiif*l*,:
rgs*EE{€ ji
f+l'
}l
E
o
d aa;EE;
E
r4t
-',.',
39!E,**
't{Z s:; ar;aI: agE;tE
"ee
$i;Eg
F,$r*r
;gf EEflg
14d
ho
sii
or .6
bo
otr
€o
o
.* .Il
E8
Y_
Ebo
o.o
dz(
>.8
+N
ooo.,
F\
HH6
g:E
.!9i
€€ Euo
c.: r*'i
!o5
6I
tJ
U
rE€..
CEH
H
ia-
tFr p
-0)
EO
g<:*
HVr
L'
i
<B.Y
c
:
*co
,=;0)
q61)*g
YO
z
T
ol
ol
l,il
l
lo
t>
l+
ol
ot
dll
Fl
Ir4
-r- Eo
j..fr'.
f-.l
E€g+EEU
aoo soE soo :
r il,
iiiii
g
?8"
zN
at
J,
{v]
.
ll
,\g
C-l
+
=+
F.
lco
+
-o
C\f
+.
\,
=
q{
\3
+
,o.
c'l
.L]
F{:
i.
6l
lco
A6
T1
Fq
ho
rq
s,
@
ca
@
/
d
-61
trl t\
Eil
,SAo
o\
FI
dht{
liliHE
tr + *+ ti
+E
EffiiE*t.t* aBi
Ot
Egit€3{' ,i,$ s +g? }$ $ ,} 1
E.l*a*ie$$$r+lirE
={?+?I I 'l 3+i
?EEE:EE
E €*:
B::ea;? ;dtrsE B E;
;ieEIAE?$+'i*-t
i*EtaEEa TT ! #EEfi
gEta[Ea!
? + Efg?
g*Et
?*3 €E;i
E?I€EEEa
EE**Eqer
o+il
5sgE
{iEE
:E;;itEE
iiiliia{*t[titgiis
a
i-
.<
.<
!{
:--{
-r
:-
<
-.:.q
rl
--I
.<
-{
<
<
--<
i{
-<
-<
-{
-l
-l
J
i
I
I
I
I
bf
.ir E
oo
o'f
()N
_do
'{J
-q
q)€
>o
oo)
Hts
*z
o
;:
dE
d*
o\
AH
Bt
8.
SB C\
.t
gu)
L.
F{
.a{
6J^ 1,
Eq: d
€O)
d-d
rq€
€bo
oo
O.
0)
{)
cy.E dA
()liY
.H
€oF
td
I{
(i>,
B-o -o-A=
,co €(.)
@(
€*
FR!
€;I
6'
'A d
-o
6a
es
+
?;; Es. t
*-9J*
6l Or
<.i g
EE
tja
\rok
.s g€
*iE
H*
.9 -g
.E*
,jI;
r4i O
nd
o-bo
E!
o'd
di",
hi-a
xobn
' U
@=
E's fr',E
g=8
<t 16c.1
9.E
6 'IJu
b0'c
.d.)aa
Fq tr {:.=
ill
H^
t (!t.E! .C\
r-- .+
- c,tCl
.
t- 'o.$
cft.
61 6d l-i
a
3
Ve
qx
?'t
A-
1,
I
E
H
e:L
lgig{l+ixagt
EEg+$I}gE;rEE
€s; ;*ys p i EE; fl Er
s i*p;l;
EIg-3tu++q{E[aI;
gieg*+,,*
Ff g;;g€? !Ef
E:
-EE iEEig.E :3ffiE
3,i€
atrEEig. EE?338
-giIEiEt$igIBIE
;g -"as r.:f i;83.E gE
$iE$i;t
f E ,,t gE [#€
SE s:eiE xt EE:,I f,E.i.s
ggEigliEtIiffiaiEg
:.9
eE
a
=E
E*z
g?
iBie3?{tg
#.8*,,{.j,6--oE
{1,i, Is.E
€ tnEie-tgt
*+ $€ Ei
gs
;:E;
+i
E*
i:.-o -g
ix: $F:8 H." 9.:-e\': E'< i
- a*tlfiE;€A*;€c; r
lo
05
'{
g
"
i;flE;*}
E€;rE Ee;H
*.g 3EE: E,F:t
:E
x A8; gtie'
ESEE#;rs
q,:nEEEeifH
#E
x
i
I3 d t.H"l E,E
A.E
EE# f,j*;^: :f
ilis={E 3 ;;
a;if,f:+I ig
€ : i; e;T: ;:
{!?E:i€: ;d
{IEgE;Hi-sd=
"
l--,
*l
A
EfI3I
E
E$ftii$iEiFl
*:;:'EE":
6He
I Ff EE;
€fl* ffEE
rd>,
Y 4.i
.939.
HcS#
-3\E lp
F.l<.=.=
-.o^9
o. ('u x
6tu.2 *
g'x FI
! 5 or.9
X
Ho.
E*
Fftsat-
-,.dEtH
.E.uo)
H
H€cr..
Ctr.o)
g $E
o(adai
6 fr:.E
ESTg
E'
ts
E3;€
qJ:E
*H't{
Br
5'"tJr
E:f I
or d cBa
#EHE
x
o.
I
.,o
^6)
()
8s
b"o
bo lo
,{ H}
d?
oB
X
l\ Bt
d&
>rli
ttr
-di€
o0)
La
h,.5
fuo
'o)
)A
.d
o.-
o
72
A .E.H
9'Es
c.r
I
bo "'=
f\;k eO
:
l2
s'
(tt
Q}
*
A
C)
Gs
co
.=
tr<'
x.$ €d
cah
Ei
$EE€ESEEiit*gE
EEasE=it qEBIs E;
ET€Eg I;3ET
Z;?*i:i $ €$EI: EI
3 !#g
P;
iErE;EI; Egggi; IE
ggi gi;alt,
ffiggi+: a
gigEgEEIa HEEi:;EEa
i
$
-
@
co
t€:atitE ?;;efiif ;e
E
E-ob-
o a bo"i
>q)
AF
=.:.E E. 5H
r,. SgEf
o?
ts HG.'
s fir..bt
*
rYa
? 9E: Ed
EB
E€E oo I X
f\ 'oor6E
>, r Z.i*
q-..Ii
H
<r:
- d)
6'1 ?
=..E::.ci
]i.S
X AsDd€ i+e
g
*a
a
-.p !
p
EE;
H#
o.>
il E'
E ts 9E'e
g:
I o"r >
ae
q
J
l{ EsI
E.E
? 6€
q 3 6 Ijir
-E;€ s
I i,Hz
-L'n_.\.d
F ii
hn , A').H,=
.{6-gE E;:
!c?i'c.,.9q
s Q&a E€ H g
"
*-
E€F..T,
-;
+ g.g: E€3?
# <;E
=*?
s 358,9x
J
:I
J
J
J
J
J
J
J
J
-J
JJ
J
J
J
J
J
J
J
J
J-J
J
J
J-J
J
J
J
J
J
N 0) 0) 9^
L!&/#
t.. € +r Ov
H::EI
E dix ".H
;g.EB
6FHrE
-
t E.- ;.3
,,=
R'E!
g.Lr
E
*
LrE ort.E*
>.Y
'(* -.x'i.c
cE*t+<o (E;'d
I
f E HgI€
3.rr fi xe
ij TEH
*g gs#s;l
o.!*
?F qS HO
Ie;:*
E
?<i H a 9 x
-pT-v.E E 5
edE E+E
A
g
b 8d
Ee
a<d
()Bq
.gHFsi,
E EE€
p-E:9
A
-8EE*
odt E E
phof*.
c-:l o
A',= "
E
q)
€iH.E
o)*r d
x-€
{t**
H H.Es
EE:E
x'E- >
o
?58 r
;f;Eci
'E PS'E i
P.g
!€fi
'aU
Hf;
Ie s
?Es
E.€
+I€Y€
A =/.1
HoErrS
hov !1
f,gegE
*EiE
o=a-;=; 3
xs+il.E
'U€-
iBsAg
q Hf ; *g
d 8g
E:3
glorEE
iibb r_tfl<:
rtA
o
-q
lp
er H
E
8,*
t €f trg
:
H'4ts I
.$
r\
E++ 96
EEE,
P
<d
s€,
f .E ef-r
ao
ffi
:I
:E g;t,i€ E; IgE
.'H
-gE;ff}ffi€lgi
n etIg:gtEt#EE
.*€.q I .*.{oe.b
@
co
h
bo
<.
q
tu
bo
0"
c7f
$'EBrE$Esil;$E
E;'EEE?
I?iIEIffEiiii
iiiigIiiiiIg?
€€Ef
[;cEE:EE'ilEqtt
sgE;;
@
co
v
bb
A
f4
lJ)
co
-l
P.
r\
bb
co
0..-
.{,
t\
H:;5
€Avd =^
E
P{
iigiggiggilig
6)
O
?dE
var
.r5€€
I xEE
fr EE'1
E€o.J'
.,
1..
Ei;EiiE?E?EgB
gffi?iiEliiIE
P'.g E
C,
ra () O'U
8Eh
HDDJ
I
H
16
rr
{.*r
.I ?o
E -*s
tririv
?A.d
i8
8€ o.{
.i+,
Ei",9
-r
-
tha
H.d
d
Er$$
Y7 ad
rrig
q
E=;:
c':x
B. E
o.9
> AU
5
':r.E5
E3E
.!llTT
31"
3l-
U€
€E
sH
o t*
M?
6-
t
# €3
.sE
xP
.i.l1
go)(tiEE€ ts.^
$ ;EEH
gE
b
tu E HE H
I+
(D
xE Eg
f
,Y'=.8
A6A
H.d.A
(d
e-:8
d-
l-^ a
til? A
^
o>
.H.9
"ssdd
E
UN
:e
dd
s
i, n'$
'+..H\k
.$(D>irll
.iHa)
O" B
'
=
.9'"
3E EE
9-g#;
dq(n
-sfg
=-e
x€Eo
*etA
92
:E
a'= E€
d
o o<.=
9()ft.
Et ie
F€.E-
6#
agt=H
E
5-afiE
-.t7Oe
* '60; I
&€€;
l;..rt
*;t;3g
5',8€:EE
,
bo aot"6
'rE
ol
u .sE 'E
^ft c9
s-d>.95q
i I lfi
clr -'ob-q
A.
eq.rs
q,i? i
-<'F e
€<,5E
loil:L&d
n,
Eatx?
I"gilr
*3.3 *-
+)
-!i\
9.* 3 ii-c
s
^!Z.d
F
r G{'3
F:*;f
= H € s. x
5,,
-E'6+f
Dd
I 8€;
-€El@
s&'E
d.Yd.n ()
Et€;
€ i,I[
to.'EEE
E5'
s
,*s EE
E-ii
e
<nn
.=.8-H
BgE
bb >'
fiss
qIr) 0.
ii;*
4
aL.
A
.=€e
frfr
Fq (a i'to
ff
iiIgE
iitg:g
;ci;Ei;E:I
eE*!::'r::E'I
ff€B:
;;Eg;;EIEli
iEE$-EIiiEE
aB€a
EE i_E
t-1._
Y
$
q
.ho
tu
$
<l
.+
d:)
\
H
b0(.)
S EE
q
bo
r\
{Ei [Ei$E €Ei
sBEEg#?;iEi
n:
c.EE€
IE g}
; agE
jsr
ca*
; EgI;r
tt
s€
Ii.trE€E!,Oi s
!9.-9*
t
J
J
J
J
JJ
J
J
J
J
J
I
JJ
J
J
J
JJ
.J
J
J
J
J
J
J
J
J
JJ
J
J
-T
L-
L
t=
o.<
a)
E=
€6
wE
o, I
cc,
HK
€H
o=
d 6dli
-{o- Q o-.>t
-dE
Ha&
i€..r
0)
35*
E^o)
g
.A
gt9.8
n?o
{:Eo)
a6
9'-'S
dC)
;€ g
:Y -.€
-.oq
u0)o)
-C>(,)
€5o
o rg99
.:H
.O
*'o S
.Y02o
*
'cL
'd
E
d
q)
o.Ji E
d6E
6)- a'
_oE3
a=
i€j5
E d*
---(l e
>E
<.1
€.V
F.$
I
#
'5=
HgE:dF
sE,8{€
6 E H.1 oEz
sEi;g;
Ooq)bO*.*
g
ia€ Ef
bO
f ;.E=
o 7 _O 5.tr
H
sf H;;E:
u2
EE
'3&
f E{e Fs e
is *€ i;#
gI::
s'EEf*3t
q()dJi
:or
SJ;E;EE
u2
r,v58 it8
;."EET E;
,Ft.l o - @ "
ila.fifrEg E
,\
lO
It
-,}<
I
'
liv
a213
aqd
HE
s.E
'd{ =
(o 4s,
bO=
ro
X 9e
.$ s€o
f* -9
4f
H
1t
-q#
.s
?
f,{E
p.E
#a
g -q,€
'bb
.hUar
1>;
9.1
r.rj(Dav
o5At-i
c
.CUtri
=
.^ qri
y i.q s
E:EE
f;
rJ-
b-99
U!
ro
EH
X+E
.E'H
f* t*
€(H
F---1,1
I
i
(o
a
E
0)':
i
1-.
'9
,C)
€iH
itr
o.>
fi
c)
(
a
,6
oG'
B
dR.E'b;mj
; # #;::E,H
H
€si
E#
Eit
;;::s:sE
fif 3;,
iip f;atE
;tt$E
\
eE; g IES*5
*EEESEE;
sfSStsde:
HBsadaEEs
s#*t;;EE;
E-9.oPri*-6,=g
tr-O *.*,C).XH5i
f'- (a , <:L !i.q o
J
\,?
c a C.e.,E F.i it.
;i;rsgs
d r I'E.E's> frAe
ii* gf
H
FEE'l"c€":&
$ g HE E€
srr
H
sHE;aIcuE
F:g
ssE s= u
ss
HdHf; E x
iffggg$gffjf;g
;l*'e}ll-BE;En
gifEFEffEi5Ef
fgffs ffg€fg
f:figE
gg{i gE*
ee€E f,g
ffi
:(
!!
r-4
q
a'N
od
VOtivk 0)DOoQ)c.i
.v
().()
O+,pbo
o;i -Q :.i20(/)>>
>F
dEEo
c)
.io)E.+b2
-o
0)
0_)
-o o
r li.
\-(
l- Nsa) {"dD.o
=<\€
3:t
'o:o JVI
'l qav
I ili
.q
'itlrO
L€ -rg
ve;cd- o,
,d
lbvrlol.*:o
P{ :bo
t<(Aaa
boJ9d-odlo a€
) 'r1
E.t.A.,,n, o)o )E
fr{ #o) .rirt.
'o.
.!
0)I F++-Q
tH
a)?a rbO
<J
c),
lri o.
iE:!d
Or
b0
al,
l€
o
. o'rd
rd ),o {D,
,o
',O,
or.
€
0)lo;,r
' rOr
L tE ie
iq)
r.(r l€r
or'1
tg
€ Er Fqrd,
t{
q)
'o(iEr .,1i, ()o
'a1+
', o't i€,
b( d1
(J
o'i .Y\
Qr (;
a
.O
,3j f'r'!
'i'. bOJ
<{t
o'i.5r
6/, bo
Oruiq
lr?
'> .q.)
-;
+
€(
o
'{i €
.(,{
oI
E at boL
Fl 6)L lfu+
A. €:
r(6
?bi
()'dt
bo
.a(
O>>* iiva
f-ra r.El
Eo
c)
a).c Q.(ll<
(D.' s*
rnlJf+ € dc
o.
tA{ {+
{((
d
v<..c;
<.
Oa QF. -Et
rr)
a'i ilo
oco oo!o rC
O)+ <.
c
S'.:
€# €
-<r
:$ t*P(J'rl(.)(j :'a;b; bo
rocli<
.H oi gii
o(ii.ft.!
fr{
BEt: ig?gt;
iiae!E.fiei
Efi:iai8:{I
O.Hl
is -dE I;E;*tE
O (o'
rd €
Eti+seEX:i
ilslEi;;$i
E,
a
SE
i;is I*'EEE,
$E
uo
s6f aEss{
en
rA
E.o E
H
'H.P ^.6 rE-r-'6
E;:f
8E;: E**s€C
Y
d.a.
ut
o
p
d
o
't1
r?iee+;FqI s
ta8;{E
i
e siE
d#--'6
oH
s
€ AEBE s
u2 +'iY'
=-E:fi'
iEEEEE
Et4 xIE
s:
?t Ie
xH
d d,:
o
-!
E'Ei
A3
E
!{,-et
I
.EE;tsE
rEt
fig.H g#E
^^((rd):a€
*s;i
ol 'U ftr
**Es;E;
EE*fi g 3d
€8;t $.EE
::iEs€E
<t b-
gEEEES€
E
I
I
A}E3E€If ;6E EEBAI I
titEEltai+?
e; isci;e :f E E€t;:s
*iilgffiI
n
:EE:*tcei; E: aBct;;
I*e E..gEys;E;As;3€Ts
EE
,iliit?lgllalilE}iE
gTq
3#t;-g1g;gggga*l;
s€i+AEEx636€fE
:iiliiIiIaiggtIIlEfi
S
<
:
<
<
-
:
:
<
i
{
L
IJ
L
L
L
L
L
L
L
L
L.
l_
L
t'L.
L
l_
IJ
lJ
L
L
lL
lL
L
t_
L
3 i.E
e,..e I
-:.c.9
dn€
L.=
O
j'8PE
ff
3
C,)
.A.r
a
::a
-'d!.It
€ q
#rsE
u+.1'i
qdMr
q
-E#g+'
-& @ *E
d o,=e
or
C's'E
? *3.9
3E
EE
o do H
€tr 6d
ttoQ.
3'E; l
6,- Hlo
bo.= .'1
=O.tr ^ci
e-38 +
d'-8d
3e fl
CPE 9C.q
* i Ho
x
Igs
"3E€A
ok
^.cl
o.Hd
H
EEi
rr
iY Eq .o-.
'UHV
to)
-e
$
+
e
rfi
(n
A
*.U
.*
o
a
(+{
o
--- tr
o
Sj
'tJ'
d
o
()
I
'o.
B'H
+s2
{d !t
ttL
-t'.9
E+
ET(l)
aa
o,5
o(6
r*
€*
s.ff
cS
'.s
(o
itgi:i;
,liiiiEisIIEifi
U.a d., ll .E!=
A E"oEf *=i $
E E;;iE#€.E;
g
f 3.EEi
fi
x l :T=tIF€;S
F E siiit8i Ei
; :AgIE; r;E
e
3 -$.8 EEfi iHf; 3
i :t;8.*lE *
e B*{Ei
ilexs
rXi
*isg{agiiiigg
;E;il€I;f
;.9
o
(.)
o
t{
,9
co, €fi
iE ,o
tH
d e; €
^.0)
.i*r -1
bb 9.9 o
+ rE
tri .ob
tro
<t[()
at
S.s:
U-
3d
c)
0)
t.6
.!a o
.= 'a)
-OF
6F1
ofE
dr
a)
-1
d
f
o)+
8.
$
3*€
BXeo
c(x
E-(.x
d^ €
o\.
o 9€
6tsx
frB-u
ag{a
E.E
:Ei
F. E6'
<,
A UEg
*
bo
ta,
ct
N
<,
Ebo
p. F.dd
=
G.E
i? fr{
bb o) rd r-
fr{ )9
-,J
H
hO(J
Ho)
<5e
:f,E
frv0)
T
s6
Qor
E€
,q
Hi{Ua
.i'e
- H-<
br{
E
Xo a)
a .e -a'
E,ei
E#s
.-i
d-r." -ei\ '6 ,* ol
'4 oE
co .P.: €i;r
f,+ s.'-=s€
erE €
tsa
o
o)
5
?J)
'2
!!o/
gE .i
0.)€
bop
-c
t8
(d
d.E
Hd ,r
L4 g)
O*
dt (A
(a^
,o
o.x t=s .5
il;€ ii
a (I)(Dv
*.:1 -c! o
8E;#
I ;.8
b 5E .bo>7o8! EE$
E sg'0)
z 5 crr .lj
q
+{
8; EH
o
t sB aEr
!;.1, t
.4, :'; *^
x
f;E.xs
d
dt
t\
<{ H'a{
E€ 9E
fuaf
l4
FAt
s6 €u9 ..s€ s
o)>5,€f
(o
€
i
,:
i.
ra;g;lfsfil;
#€f
i*eIa$i;rl
E
E 8-3
SSef
*
.Yq)E9
f E#E
: $te
E*5t
sEEt
- k.*o (
:€
I
Vd
'
E.g
-c .EE'6ts
H.H
€EE
e
i B# ;
E'E;
E:3
r-:tlq)(g
E
,qEF
-.(: aE L +
-"orE
!G'*E
rI s;u
ndEs;
I
I
I 9r
r0:
w
o
u
o
(D
€
vt
'6
o
€0
o
IA
=x
!o
DO
f,'E
oA
6E
c€
JU)
lo
t.-
<;
bb
P{
k1
<l
t-
+
bb
P.
t\
L
o+
orA
(1,
-ul
o c
c OrO
o
c$
::c -
tl:E
.,4, := - .-f
*Nf-
-ffi
tf
.
c,
ur
rTg
\>
,,,.
4E
th
i_
J
frF
2.
t-' {
Ui
.,\n:\A
L,[j *
sa
v
=f
ca
-J
_J
J
J
J
J
J
J
J
J
J
j
J
J
J
J
J
J
J
J
J
J
J
J
J
J
.-l
J
-{
J-.lI
-|
{
-
r_-
r_
lf--"
L
I*f---
L.
L.
Ll_,
f-
ts
d+
tsor
ffEs
.
co
S
+
q
'E
s
6e e E
s 9!
$g a'+[ 'E
gE HEF ?E
igE I;€i H 3
;; ;:
itEEE;E E E
ic:€EEF E B
IEiJEE: F':.
6:EiE*s B E
f*ta5EE 3 E
*EIgE
to
oi
o?
d
g
bo
14
d
a)
.o
t)
o
a
&1
o
I
@.€
CO +{
bI)^
s,o
dE
.c)
o)
TE
H
d
c
(,
g,
!)
tr
o
€
€
.O
tr!
fr{
o;
c€
OO
-15
F
T
E
1
-lF
tr
fA
S
.s
cr
.+
q
tr
l_
l*
ilo
+:
o
oE
T
z
l(
@
E Hll,fia
; 3 -EE3
* ,; ilEE$
; E E[;*
fr.-,EE1g,
gg #,,:
e ta.o
8'E-i.,EESg*
cq E,E
.tEaB
E
HsH
E*E$;€r;A
X
*$EE7A{88
3 B;EEEEgE
€ t gE:tt;
q ; , ;',gtE E?$]
& fr s€EBEE€
#
(D
z
l_
L
.t
g
,L
rs
iE
'rrt
U
o
s F{ff$
i
'
,
id
:6
;
3$
{JlJ
B*
Bbb
-orJ'{.
.4la
a.e
rafr
EB
o
9'a-ci
EB
99
Er !t
tld.gi4
ho bI)
d.e
'4'4
r{ lr{
''<a
8o>
EEl
ee8
H!
tdi
-Ee
'tr
(t}d
€6.
o
Bc)
c)
dq) (A
f{
Ak
t-t
o)
9E
<r
c'j
<.!
cr
bo
tu
H
(h
6J
()
.o
c)
q
k
x)
€
td
d
bo
.o
<a
A
si
d
d
4
a
q)
+?
6
E
Q
crj
tS
at,
C.r
.+o
-i
o
bb
fE{
zj<
-,
-:a
z
a'q
Ed
d{
9
:-€
lEr C
-
AHH
P
<r
.- q
iJ.
bO.=
_..d_D.-D
a. rr. .E
*i
d
El
hdt'D
:s;E
itH
€t
sO
(O
-iEs;
l.-
.9.,
tu4
-;d d..
AE
Po
EH
e r€
iit ?f
n'
H;a
o 33
$ sx
r<1
A
tri
c
a
E
d
H
0)
.d
I
rd
d
fr,r
nr
J<
o
6
Zt
!
!*
rrl
8_
z
iro
ln
rn
z.
{
*
f:
.to
--T
l'
]
I
-r
Nl
2l
:t
,l
o
{:
2c<l
l@ll.
'R
E
E
e
.€
..
b.-
l\
(D
r--
Ft
FE
+t&l{-J--,|
#
x
l--
t
v
t^
!4
E
E
:
E
d
OJ
b.-
oO
t,
ho
<.
F-
.bo
vra
1^ q*
<.
zlc
g-g
i=
I
_J
l
I
I
--l
-J
-.{
J
J
J
_d
:!f
_j
I
-J
--J
I
-.1
.../
i
,-ql
,--
<d
-l
--.i
-r
_q<
O
3:-
rJ)
.{,
-
h0
c,
o
Cf
3
o
o_
fi
boE
(11
Oc
sQ
H.D
o€
E.B
'4Or{
trl\.
tu1
3ta
F.d
rrl
q
NH
,ri
$
o
{
Fr. ts{
<€
.{[ -or
aE
bo
s.
q
.{"
b.-
t
bo
.+
q
lJ)
-t
cd
H
I
1
€
x
oz
<,1
B
'-, :'a.c
v)
-t:
*.u
7
s* -D
I
trJ
EH g Ni
f.H r,I t F
EE
E Hx
8 -B Q .*.efigc tr gE' €y
<oZdQ fA.g*
<l+ita
O oE
&*rE E !# i{;.
*'e
e;€
<€8
tt€
j
a
$
fr
A
bb
cr
o
-.i
.
bOd
:i \u!i
[I{
<ia
g€
$.t
d
E
>
o
.4
v,
.rA
E
2
.u
f4
i
J
+
Jh,
Jl-t d
,-r
T
-l*
-x
(\t
E
2
J
o.
c\r
)
€o
J
-J
1
-9S
o
;\
ts .E -tE
B
E E.q
.2
ao'
.EA
BE
g
ts' q Hi
EE
.=
e2
qR
.:P .H fr'A
trr
?
g
t!)Hdo
or,H
Ets
oE
"j
a)u(g
o
5 iy
i
C
6)
o
F
O
9d
q d
a
d-\
ts
s*€6 'rZ
;s
cd F 6
d
t4
c'r^=4{.H
Q.;
N
€
i
2,2 :.;
H r€
..4
E;€ sl
loOOr{
d
^
;qa B E€^
H.i
.i
u,
.a4y:il;Ca6
uo
Hiit-eld
* - H
E
.li:o
n&
f i5,
<o .!Po;
^A
nf,o
.9
*>:
!isSFqE
H
*
('z
iiI
co Scohoo
vi
d
c.l
It
d tJ *u-o
4. dcr
'&
\
)A
(!
r-E
-J
r{
z
J
aa
zra
oo
Flr
(O
6
<i
bb
A.
f*
-(Oo
dF
o
f,"H
O
@
.+
bb
0<
t\
\
vo
*zo
-rrt
o-
E
ra
.,
f- fne
f
-1
I
-.
@
-f
e
I
T
f,
co
@
bo
<.
q
tu
.F-.-
ffi
!/
o 3.-
-i9
(o k€
E
s'i
-<)d
s.E
{l)oq)
.-OH
€A
*{ tff.=
ai
'+-gEd
qTEE
-B'.E
*'
' E
.9 o.Hfr
,dE
9t\ -
PA
i;;
E-, Etr
.-.n
!+H
a 8'*
.,N E
.iHH
' o 9: qr.
. o -.dq
-i 'ia +'-
':-*68
:',$itt*
A
U
€'
I
{J
.61,-
!+{
'o
'd"
d.
r_l
:.
.c2
(l)'.
-o
E
or
C\t
oo.!!
oe
.{, q) ..'
q
9a
.!p €
$+r
'@o
.\c
hq;
.eb
H.r
14 arS
-q
+' P{
bb
I
-e€)
co
co
frq
.9
,v
.eg
<,!
q
88€
C
d
-D
r{
!$r
tr{ d€
Q.E':#
log
.6
d
o
.!9ooo
hE*g
-drlt
.t{
.'
:
bo ti
.ftt
.H ^A
qtso
€*
<d.
bo
.$
.J T\
<
.d{
fq
tr
{
E
ctL
la,
(Ju,
rt
9
3a
g>..lx
o
-o
@c
io
c
'n 'l
o
rr
3d/
,lo
c
o
i-''
N
o
t!(JO
-\
rfi,"
iB,ET ;E igEt
EI€EiB
{iffig#i
Ei3Et
' lffieEE-iti
P3: I: E!I q*Oa* $E
*$fr:;t;egcI#EB
sB:sflt=s **eiEit
B!gE?a*tttig?;€
.aTB-H'E;€;iats,rEq
P
r;Esei;as;E
Eggf
;h:iqaes:ry;*T;E
a;6r*-EE€rS;+Il*
XESgEE
i qti*
;EEiE+iE
r€lfter
EgEEi;atIitt
*
ll1lHi iI?aI 'l
iE$llii lffi}
't
o
o
tst'
-0
8q
LL!
"
5
F=*{
I\
-o
c\r
'<l
&
14
bb
o)
l3u
T
I
.3
le
l_
6J
aa,
'Qo
YH6
.€
Et{4H* :*:g
H€?tBr- sEf;E
1
J-J
J
J
J
J
E{E11+giEi+E
.J
-j
Bgffii1gtfilt -.j
J
J
aeeIagtEI!!I J
J
*gaEies*giiii
B
8€38
or aoS
;aaE
*E
*o
t-'. H'*
f-S
boo o
a s-A"Ha
;;e*aAT* HE;c
rr)
O)
ct
:€e€B
sdAtts
:E e€'h
tr!-
. o.e'O
4 d E or-9
dd d.9Eo
sHE;ES
4i *)2z
(O
^l
;.EEEC
Bfi U.9R
E:E
B;
o o= d -
+ 3o
q
PE
bo
r,
<{,
Oj
bb
P.
f4
ge; B *
tE e {E
at **S
t)F-l
,. O-C
- c! {r ,"--
(g
IEEBo
rri F'i
".o
o)HI&
J
-J
J
-J
J
-J
J
-J
J
J
J
-J
.-j
J
J
E
H
a
g
2 X'A
Q)P
-
(JFia
9E'::
? a.E9H
>o 5
EA g
a
E<
o.=
-o.o
j3
C
EgH
@ec 9
dtDP
4{\J
.Yd
3
C)
o.r
'+ A
(H
\4.=
o o) J.l
boF-'
=x
>ET
qi-o
E4a
9o o
i.o e
c!fi
O
€H9
Or?
{.t oaol
ooo
.S+
$ut*
J
J
J
*E A*
€-3-35
F{
rJ)
o
.{,
6b
O{
fr{
<t
o-:
-{-'
<.
q
bo
Fr.t
E I
u-AE
i:'rA*i
IEIq
*?,Eg .r j*:
EI;f,,;EEEEgE
6l
q)
bb
A<
<r
r{
o
IEiIEiEiIiEi
Hsir{tfiiiEI t\
-o
I
o)
gg: Eil g,iu;re €d
.
+t
f.r
i
T
-i.
;EEEEg
gEE€
b.aDX/
S
e
E;
iE EE€ ;
>-t
Ei Ex aEt
_9
,E''i'i# E E E
q
.d ;l-1
,S
eGt E H
(:...x'-dc()€
.E
Igf rf
E:Er8I
;E IAXH
E
€ sg E s;
gE:Hffi
€ {gdF €
o g E€ag s
Fl
d
bb
O)
o
Fl
{
.tu
bo
l*
@
-.{
$
hb
F"
fr{
i;;
X
E*j*
<tE s-eE EE
E*
;EIE;tn
;:j
5 3.E-3trt
$E?f,iEEIfe
ii grE*;E?$ arE
{€riflpTIae
r.o
fl
A.
E
z
b.-
.d,
!b
A
{\
fo
o
F{.
<,
p{
bb
l*
.i<
t-l
O
o
r{
bb
P.
f4
c.l
o
;{
bb
<,
p{
r\
oo
co
,d
id
.t-
'o
i
.
o
bI)
J,
C'
a
tu
l,
o
(l)
o
L.
gd
o
N
1,.
H
+,
0)
Fa
a
A
5.
6
r4t
r{
.+
&
bo
.d
fr{ A
dF'
.a r.{
r r-{
sd
bI)aAH
g .gt
f14
).d
Ea
FS+t-l
er s4
an rl
-1,
ZZ,L<.
e
e
Ei€EAI,
a.y.or-.EE
-e.-E
;e H E i5;
s et
EE
.O 0) X I
:
Ei HE g,:E
EqstslE8
Ess;€iEE
j
E'i{ s,E
E
8'A
ag:*€Es
.E p-o
>1
29l-1E
u, a
a'..o 5vE.id* E!
Ta€
ooHdrioin
E3:E{ea
B
@ao
$fBEE:t
456 o-=
6d a
3e:E*R I
"iEE EE;E
H
HP
EA
c;o
9+
d'j
e C-l
d-l
*€
3
-t
T +
L o. Eocq
{t
.9PHn
fr{ E$
o,
.o 04i;
g
fr{
-bO
(I).d
€.E
kD
EE
-,.o
€E +!*,
H.a
€:
3.9
a8
Fl€
E.
c,
gJN
€EEB$
uoqg EE
tEEfq
F-.0)c!{+
H
$*3*{
o oi
x
€*flt
' d >'Ylr>)
,
E€EE?
*cl
(!t
H
E
lJ1 - h-o,.l G
E iI
E:,E
J E'^E'iE€
B
rr{ fr?EOE
-s-8aBE
to"o;-8
?ES:H
5 E.E;E
# g EEet
E -E'tov
-^ or ci.B
'6
B XE-^_3
cr f I ueB
=
<,
P.
T EE::
bo 'HBr4€ gg
€'
grff 1
c.a
a.2
.1 B
+r5
ho
r{O
6r Fi
t{s
l& EE
5+
H=
'1 c>d
t€H"
i"
lil'
H
*H
-J
11
E
ito
Eat
r;;uist
3Eff EIiiEEi
F
{iI EgEE]CS
1111
lHffiEiI
e f.i^t *t'TYt:'ei
E
3
Dk
*
BB.
E*
q-
t'1 Err
a)
aP
$o
lr{
0)
-+ E H
O O-r
-cl{J q (
@
XTEE
^d
-'Fl
uidk^Ago
i E E:
si €€
' =
gE
EE
S oE
E#
6 r,1;,
tr E
F
EE is
t 6 €t
I
EI} -E aE IB;E E
-iE;€;E€eIEiE
3
.9
ao
o
8'
.E
c
A
r
ai H.3
I
i
I
-J
J
l
J
J
J
'-r
-d
ar-
=
,,..-!{
-<I
-{
=
:.
:r<
!{
<
:
:'
:
_
l
,.
s'6t;_E
:
.s6"tB
i
l
=
4 38E.3
ho
bO t-l d :l& 6
.=
'-d
tzl
3 c
* +i:-"
o, X
XFt\
-{'A t{ <
Ff
-
.
d
.. il
IA
5cr
e<
€t
(!
Ests
ol
.8-
'i -lc
-o
i*
->
,
Srt
os
gsi
€-L
o)r
>t^
!,
ot
5.
'or*l$
E -ie
q)
iJi
H,q
sgE;
d
I ai
.*f 'F
eesi
.E
.ts
bOY od
.t-LY
(Ddij
tr
Ftsoc
g6t
e
OecH
.H(AHd
f 9H=
G,
I
Frr
EIEe
€(\r€€
.i
oE.
.H
{'t E
i
':s.S;
EP qE
ETE.9H
tI aEt
ft$tE
< sH >,8
$EE€g
T
JI
i
(\a
I *EpiiE E *
r!fl# I [, EF
1 €;flg+?i
.
?
Io BB{ggtq€l"i
5dE
Jo s{EEE;eE+
L Ei*: EiiE,: IE?
leEgtagi+gi
E
E
is r
6?ttru;=iu
q>:
$ E;glE:i+t+r
j I:*E
t ;g [E
-+
o-()
lr)
oo
l-.
--
l*_
L
r-..
sEE
E €gE
o.C
SU
oc
LO,
C
:agE
H€
*{ u
€; figq rE{€
$: E el
iflEEil"?EiE BEif,
IgffiigegF
c.i
1r- tt
.se
lt
,9
iSe
'.1I,
m
oo
.tJ
,t
d6l
(D
(D
oo
- o
'se
-
E:3#f,
Se
B Ert i;tgt; f i
i iEEg* sE€FEs ; ;
E?iEEiEE€I{igi
d
.U
€tr
o
p.
rd
o
(H
r-l
!O
o)
+
E
.;o)
dO
s
.-$
9S
g.<
€
'-6 ll
o
-p's-',,
'd s'
5riE,'i
.h'5 @'E
h5
h5
-o ,!
et.€t
s
"io!o
€:r; :r
h
!D
.E e.E e
(H(DCi0)
Ei€-.-€
ga -\
ri
>hs
.E
rirl
c'E o
E'ffi-.
,E<F<
' (€ "l-\r
.rrE
qrEBf I
E
>.o
=-oE.E
q>Axi E
6-Q >is; .
'rE€E
grEt
-r- x.o
=a
a di -o 'd -o
o.o
'
1 dg?.E
O
ocd
'()0.)lir
!!
cii>
9d ll E
€trl rB€
Fri..h
ho
B'
o'e -c
---)
il
0.)
.i,
Ul Oq{
a(Joo
0)
9
O
EEEi EEt*t
rygH
-rz @ '$
trbo 'E
dri.r'S
fi
co
Qp
qo
,2
O.q d.S-,
rC)C)
>rilrO
b-.=
gtIffsgIscgtEaE
srfiE;rE
Afr{
f
.9.9
.:EF<o (o.I
IIf
'u
q)'v
.=
fl€EE
tiiBiE$EigsitBEEi Stlt$s'g,6gd
--.B'9
is
'xd
gft
it:r€e**;Ea;s$l gEF s.8 sa56 ag==
S E;gIE ;EE:€sE E*i
*ig$giEEBL
iiigtig'aiiirE.ii
€*Ee
L e F *; E €s ;i!
:!
€
E
sI
fg
L
A;EE
;,rgSila
L
IJ
l---
L
L
IJ
L
L
-9
t' tl
tr$
U^
6r #:
ll -e 'E
!
&€ e;
-e-E
ts
adl
-C (l)"
#
g .: Es
:4E
ei
.3v) il .kE
eNST
.ol6i._
s x€
I te
"i
E -1f
a L r:,
c{
-lE
'i <.!d
9rt -5'
E{
o
:Ei:EfI:;lEi
l--f*- ij*;telt=i?, 5tii i
L
l-.
L.
L
IJ
L
L
L
l-.
L
f.--O
=
6
I
E<
.aN
L. E T L;I
L. f J
L. E -et?
L. '} E€i
L,
L
.
i
'
'
AE€
v/A
H
E
.r^'
EEEg
't.E ? o
o.EEA
i alt(\
g,=ge
i.
.:-' :(
BeEE'
ilsae
EEg'g
gsg*g
E X.q
HgH*e.9€
a
'e
O,
A aA i\'
:r6-E
#:1
&EE
...rO)
.'i
Ed.Eg
n"t
gEE>,
E
>
t: E;
kt'q
?eE!
.Fi€'E a
*";
r5 -3 E
iH:.t
t*tlElt
q*
;r
k:a
JIJ
e-B
H'=
E€?qteB
60E '=
s a leeB
.o fHE
E^c t
r.S'i,il'%6E:
..: .= ti ' € 16;a -O
:
;
*?; e
tu B.{'
e g #98;
qa E g< a e
E
HegT!ot
o9'6 rtstsF
dro'E-5d.Ba
9' A -1^* *e
B ob.E E".E.E o
iii
€6';
3 aei
'6".1€**t
g'4
E
I -P
*E;EE1,EB
E .q f i';
B 8E ;,,dB6;
(,
t
S.g?
o-
E
-c', i:gEt*
tE
o
H
gBt
EEt a;'
ats'i'3sIEf
t E" Et;.Jl,.E^Et;'
Jl.{
I
tii*
Fi $ir{
€BffiEtt{
q?
tilleffiit$a
S?tE{IE?FIIE
or
-AA-
EE 8q
E.EEt
d Qor
o61
>
c>
te€t
IH c
.H
EieE
9 g.E
'=
? E*-E;
't erEt
E0)irvH
+o6)t!
'F3€ts
E
i*=
.EE;,
E.g 6P
B.*E E
E+:
---{) u0-ij E
5
.g,En'n6
E H'fr'e
8eq#:
"o
,ri 9! .6
Cl i H &'
B
B
3
th
trEE'
dg
E€S
*
.I
l
J
l
l
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J-J
J
J
J
J
I
J
J
J
J
1
:
:
:
L
it-.-
t_
t*
ttl-
L
t-
LLL-:
LLJ
LJ
LJ
LJ
f
LJ
f,Es
=.oi
r\ qL
.i
Yord
{l
E o
H,r
Su
AAA
ikH
-- >3o)
i?
*Aar
a or'
()ar
d
EtU
Qrs
ds
_t ts.e
AV
--d
-.9Pig>
1=()o
2.o
E*E:
t-o B il
rr
EiES
gIts
$e: *
tt
;#E;
;Ye
EEHq
.s
E5
-aF
il
d 5-
T€ 88
.
B'; H E; E r!
+A:E:;'EE'Tilr
a
Eo,'EcH-'E**{
qdrE? HE*;€
€EE3fsEeEe
HS:EE:;SPE
g:;fi8;;iE:E
iEIgtgHaE;
.9.o.*H.9-*8c'i
€_ei:.Esirs€
E
oi€EPoEEf,€
t-. EBfgi{g:;{
;:rEg:liEp_
tj
t-.
t-j igEEg;eggi:
L
-I
a
d
o
o)
E
I
H:.
d
I
o
q)
,i
-ts
$
r+
La
6':.
.i
iil,g'E...
gE
P
{J
->
Fiho
,9.
.0)d
P4
gil
.'r g.E
.*E
r() _c! E
cl -- .i
f*
ta iE
t*'3
.d
'q)
(oF.t
I
()
a
d
{)
d
NEH
H
bo
,rj !d
o' \/
ll
HS
o'o
*>.
Eo.
oor
.9
g.
ga
.dH
r4
bO
cr
rl?
l\ cl
/r
F.:.
-tu-o..d
,o(6
{
H, Ae
E{
Y6J
,9P -g
€-
EE
€g)k
tr{ .6
C)
olI
Ft
i
-'
r
'*S,E,
ro
narO
<cr
=ao
lE{
'a
bbE
C'-
'd
.sts #
jolr
'i.:'-
:'i
-r
-.:
:J
t
E E".E
d
o"c,
.q
+() o)
io"
in5
or
r-
c'
:.$
rri frr
s{-rt#
*JEfis
?{i8t
EE.Fe.I
&
.gEoq
E: ll O
€,8? E E
.tr ... rf 6J y.!
90€ECr
E ito*
t
.ts'E;EE
8.€-6.e
-*38;E
4s5-gs;
8:9t 8
EE.""8e
"
E
=QEE
FoHt
IJ
(J
-.i >
-.=E
x() 0)
.q'=
=# o
.a>d
ro
*d
.v^e
{)Y-{ (U
.-r-O
.E
e9€
a.goH
3 Bg;
E.E
.o E orI
-dd
Eca
oEi
ad
a3.g
EEE
.o F.E
trcad
()€
AHq
.P -8; €
{\
j E'4,
c, A
r& €,9oF* -E
:; l:r . aE
.4v
S
Fo*{
effilx
.s Y{,
I^ l,
To=
+>r.
o
EE8.
E
-s3
gtr{
' oa)
I
.E€
=E
'r'
.',
-.
..a
':.
,
i
laH
'(trfr
o)
.P
E
d
6
d
90
o
{t
TD
'E
9<i
,50
d9
Ets
.g i3
E.o
r.E
s€
AA
YJ .)
14
€
.:i
d6
,o
S=6
-ta t
&J
Ho
fpb
rJ(D
aBk
()J
€rH
,1
c,? id
qb i,
.eE
o
a)
t-'
tl
bo
,d
g
r\
q)
C)
d
d
pd
o)
,tr
{,
o
ri{
o
:a.th
rbq
H#
dqi
oo
€5
o'6
tr0)
o6
vl :Q
la
Quo
o.r
9E
rE
H
'dH
fr. t
A€
\f
.
cr
[. H.HT
fur,Xii
.$-3 b" 6
8ilH
14' 9'O
.s
g3*3
3€i
E
€E8E
bo - bo-,=
'€.
E
-H.s3
SEt $
E
=r..a.S
Eei 33
EE Hfi
a
o.d -tAA :l
D.;.
: Et,' $
e!;;i
*E! sX
I=6 Bil#; t
.=xpo
tI;;
E p;Et
s
A $tt.g
aZE.sE
x
aK
Ef
(J
if I
*(
J
-r
e
J
a+
d
E
;t# et
E pcg €:
BE EiE
ss"ir
*glg?iE,giEiE
€EiEtIiEflil;I
-ggiiaiigiilli
3
.s
Bggii,laliiiaiiilsilg
$iEifIE,
=rI8ElEi
--4
.J
J
-j
'l
(J
.<
,-:<
<
,-_
::::---.<
-:r
-<
i
.*.1
^{
I
tj
iti
a €gt:€*f €EEEaai
? 4€IIE*?EEEE
f i EEFf,f,€E€fig:iE
L
I-
-\
a{
E
d
)
.
.?
€-E q E
cJ
€
geE;
.= €E a;EE
B€g Hf€g i!xE
:liagEi*?ai*
ug$gl:
=
e* q'=e*€S.uEEg
s*xr
u+ag;ffii
EE
=il
:E.E EE S&
eI{ Ei?I
B
*
ErE;tix
;E's.=aE3
EE€8BE*
r:
'' -o9 ff' (.:.
i,E 'P'';;
;';:-. (,,<
I;
aE,F-:E
e€a"E.i
oOI
':o)s)C. -g'u
-o
:E sEE'I-E
H,:;g]!EfE
*tiiEe
iT;E;:E
#E;:*EE3
-n 3 e 6 *3
;;EE€Ei
E
sIi;ii
8ii
EEEEE{C
.= E U 8. E s
* €t o ts9 EE
=rP
Ee*r.;
.gE;=t
3 9'o ,i's,
tE Bea
Ea g-t
3;-eEg
Ii
I:
s
-oqu-_
@ > o o-=
<'1
E.E -:
-Y-!pk
# €:;€ I
a A=a';
Ei€;*
E
. +v*-X
fl "s e-*f
g xr
l'J<+-Xod
ri du
d
.e EgHE
P U'I s-o
dct*'-Oi5(
HOi
E',3!*
s-:! H
tfeieEaE?Ei?iEsE€
I+tEEEt€EE*E;e=E;
cd!
4c
-o
o)O
ao
aca
q)
.YO
cd
;=
aa)
.=>
t<
o)
G)
-Yl
yo
:<
-.
-C,
^(. "i+
t-OL
grts
-o.^<o E
p
'-o
tr'e
H
\_; () €
0)
^o
- U2-v,
r1
q-a
-<
;1 "o
; bo=
q
;
HOq
sLo
!
,
i.:
j
li
'.
'
',.
ij
,.+
.
k>
-oc
_.6
9-6
br
gE
;';
=
_e
. e2
a= I
-s"oE
=ofc
o Q-c,
b
-.-
6
-_
-='
E
E.23
-,a
tr'o
' t":
H.
.(l)yk
'i1
'q.o-c
''Cl)-*
'-a
5
E
'BBq
-uE!
c)t'$
-. j:!li
.€.
-._
,
O..!a
.:..
cb ,;:-';=. :
l': d X
. . 'o.ts-,' '.couo*E
.
-.i:..
'.
:,.1.8
.-
.
'
'5'E
_3"
e
o'O a
g.EE'
.aalz"
E
aE
e
6
i-t{=E
s'a*e6a6.-l*
-bo7.J
:.--AP
flg Edra
Y) .I p
").t\ =Eo.fo
j]
L:--
i.il'
Se
Y9'.- o
e"rr E
.I
' '
r.
i'*,
t,. H,E, €,.$
H{,ffi,.*E#, .,
,"
I '' rg,f .1;
'.
,E
S
.-'-o rd r:. a ]q)'-[i
I
't'-
8
$'i'dlt t::
. ".: -=.c:
bI)0 k
...:
,'
.E g.E'g;
h€E H
e .cEtg*
* s;:gE
€= v
i€ f
3E
g8 HEE
i; Ee# Ha
.Il
8
g
.4
-z
3
tn
rq
€crC>hd:
i=
>-c gd
o 4Q
-9.
** P.E* P
*T ET} F
X-c
*.E.€:
@ ->./)v:iO):.o
; *E
q
2'8 €3do.
tr :€ !i n'eEE
S ="€
;tisl
i7--* B arr?3
q
E B"g>
Hq.8
fls€ ia"c 3Etr\
EE
fiErEE-E
i'8{}E a€ssI
r'E-g=
d:- -J E
[e'qx
c .:4
,' fl88?E;i*'
i
q)
bb.E
A!
ji*
2
<-r
a
I
()
11
E
AE
ie'*hB; E
E-B_s
EE
G)
€o --:
d,q
,-oe
I
Hf
-9
.a'-l.
H'B- *3
'o't
O
: {,9 E,;
-9o
; E-:
c'I^
ct
,o p'a e'
; Grllt €':.
s id B'T
8q.b;
gaE>
U..()f
c
()o)
0)>c
dk
a.e .2
3ss3B3--zE
.€9.Cs(Ul
-'d-=
q E*"o'.6:3':
-9 E;q-: a;+
r'. i$,r fE E st
. rr)
'
':
.
.
'f,o*tE b
'i 5 l'!o^q)
E*E I
ttt
"E
.E-'aF
7? +H
.! Hi
E tBo-.3
F ,- s? *
o
HE5*r"
E
h-3.iEE
F -G
kd
;i.E-E
r-rE h to
"5k
€a
l-ba}-oo
-
---hDD
-=.r o o
-Da
jFhs$
. -.=9
-i=-=\.c
o'='<
#$
F E-
^
E*
=F.6
R'1
;bE=
i;
5gP-E
F-1
h-F
'Y o;
rus
C:' '- ',,^aa!-
=
'=€;88
=ESEE
'' ,o i.
rj: .... s
=
^a"EEi
R E e8{'
;ae5'H
tr;iige
E:
.
.Y
H
:
=
ta
=a.a SoRS
o) --E.Ed
-q O- T,
':H
E
bb :.+-gU
6 Po€ I
€. t={E,y*
'O
oi
---9 U H od
.-sHu
o 6
+t -::
=-9G
F.:8<
q.X.X
B-Eti E
d
6r bo.i! 6l -q
o: c.t=
()od-;eo
't€=
c-o-9
k.- o(J
'
.f
=
bO
:
u)
: .Ed +.9.i5
€e E, x 8:
ut c.c H-
E"ts
=
o.P.^a-Q-"',
E
3 s-E .SE.i
E E:
a. E.<
f;
.* q: E ;g'::'i E
E
;d€
fii
ts
F*:E€{
.E
6.=.E'E *
Fh. qtE'e
8f,
ziz.Ef
€ 6 --6 E-B-o
e:-E= >* g
"q{EB
E H:
a-g I'E x ET
E EH.a.E'53s
E 98.5E?Er'
tr E X.H
c)H
H
E
H
# € E t" HE
9.f E g*.9
:E::tr;
: . EEt. ?"T
9t-es=il€
.
!I
J
-l.|
J
.JJ
.JJ
J
J
J
J
J
J
J
J
J
J
J
J
J
/
I
t
t
a
:11
,.t
'4
.t
I
L
L
L
L
L
L
tj
lJ
L
L
L
tj
lIJ
[-
L
t_
f
1_
t.-
tJ
L
tJ
tJ
LJ
L
t-.
L
1-.
lL
t-.
L
L
E
)>dbr
k?)o)
o,-d
<rU)€
€roii
*fo;
o-Y d
o.r
A.9
*tro.t?o
-'t
a
^=
- 'i
A.IEO
lt
Y;
S
O p u)n
fr
5'ii
ar \
'E€et
aa{
EsiE
EtE
g
3
A
iB.t-
EE{
E HE x I
E ,u.EIl
#E:
Er
s
iq! .9pE
(D!-'tsr *
o i 6B S
:"; EE
uo'io'9
U
j.or+{
rr f;.
'aA
I
a
I bb<.: -
t'5
:i'd
-i:
T
NES#fi
IE3:
$e€
Hi1
i
E
E
"d
O)
E;
d
.. gF
€
€€
H s3
xi d
EET
n' ft{ nf
sEiSe*d
EEEE HIiE
t H f,i.:ia
n ieE: s:
i;8
f: E8 g,ts
t;qE
Etr': d
'I;-9-r
3;:;
15'6 R;5'
n_ 0).= sl
N
E TT.EE
# E€{E
s ic3* :
6)
-EZH3
EZyE r
oo4 or o
9D4aH
gx)t
.$a,c\ts€
" iHJ. E;
!r) o
tri t-i
d-€
p{
CQ o=
{ 3i;E:,
*
EiES
.!.E tr
'.
!.8! c'.ia
a ,'i
'3
='d
E
'
3
3-3
o.>
t{*
*"fl agE
e x8,E?
'.
EoE
rr.t
o'.<h*
€E
,d-Aa
*-9iil,n>
?HA
H(0dO
> A.9
'u F AO
H^€"Et
#:'o!
tz'p=( U
4.9
:EA.E
.=
><D ()
(,)d
gii
bo
bo!
otr
-c(J
o8-E
5ho
.d
ijo)
dz\
*_ E
i''9 a
0)xk
'o F&
99e?
tri
r'4
.9P.E
{j+
E;; €oo
P
dIldi
vdD
E Et
.!- €F c
-o
*9
EOE
HA H
6{
I
19
€$
L'i
<d
.YQ E
Em
o
o-o
7"/
o
\J
-t
.
EIE
^4t
e'X
ia-
E
bo
a
E8'? E
rr) 6I.J X 3
A c0. -- u
s Edlsfi
- c6$
A€gTV
H <r
...'-lltE
,Eti..:6
Est?E}
-EE;f;
sf d:E*
F
i
(D,:.ii
.v
H#
A+
I-i
" EEJ I E.
E +i - .q
5;
*E
E
G-iU F.E
\ flEE;Ef
'rE
n
*;6,0E;€EE
d fl 6
;5 *E; I
3
E d.e r #.8'
Zo. "E
hlFa--rE
o
.:-!4
BX"
E'<8;ll'*
€e t.E
6qo E
€rH.:-H
o
aOtj
l>€E
or'll'or o
6
o)
E
B
X"s€fl{flfi
F{ll{.='
r.5-(
I;g'KfE#H
EEggf,E?
S
a.
.H'E SEE; HE
E"6.E E lEE 9
o>
+
:-O 6..-ll:z'bi,.6!1 E'i
I
itbpH.E*.E
,\'i
at
P r*t d Es&
€ a ff.:* N l,i, .E
Sctr' H6-ig.E
E;9Hao
c,F 3
k
;
4
Pr
x.ri "tsi+61
.9 --= f, c o.*
S '-,?,8
H,eE{#;
#
R
E
tr
E SE:.E EE
E;EE
B<Er
&
X€KSfrR A:E
I
i
{iiliIiu'
Biu,
*;E i;tgrlegr*fi
;Ba,:iEEEEE3E#I
tE* Ys8re€ E?'i ; H*
o
co
(?
Ic)
.oo)d
rio
f,
=-qr6
a5 * ?'E
>
C
BiIEe
ft- .*:db8
F'V I
eE
H
EE
E e 3*a
sZAot
g
Bt
q€ P#G
g B 86{
eEEEi
r
B.E gE E,,','
?.e 3E e
td *'- ),E
E 8q3d
r.-cl€s ?
=
.e >85 t
,': H iiE #
E,9Sq
E+r"ci
JH€/
O"
bb \(Jara
f&
:
C\
o?
ro
bb
F{
14
!-{
rr)
EO
04
h
E
€ I()
@
Fi#
. -\.*
Y€
E
Bt'Et-8
O
vcil
ro
{
It6,€Eo
Frc "il
':'ga.':"
r)'Ere6'E
X E:iHES
]d Es6';Tx
a ;I*sE
Hr'j )fi.8
t
.-dOV
Z IIEq.E
Y aa(EsB
6vfi.E=
B H "^cE 3
si
3 Ec;EsE
r 8YE€"E
H S;E +;E
fi!ifiiE
a s sI{.xE
t,
rf Er,3r'
H
E;li;f I
q)
O ,-=d 9i!E.Ep
E,='Fo
'dll{:5-
q -9m
(o r+.E.Ua'EE
io'd=-t A
€;!{.
I
ffi'
'Gl
:s-s-'r=
;6€EF
B.E-
Igi ,iaa
EESSE$E
e:I
"EE
:Hi 6,;
=
l'
c.)
u
ffiB,sl
gFt$f;i+
{a,g,i$*
*i['* $i
i+**riEB$arfl,aE
?
}{ fF*
j ii{E.€an
i
BsE*
EsE E Se ;r, : H EEB
rEE; 4s{ gti I$s :l it,Ejt' i,
ief gi: tlltai*=;iftti .g g.
iEt*
s
9s#
ffiii*
EEBiE
Et
csff
,I
fi;E*e
'
iIif€
'iiI$iiiIigi;,igigiil
I
I
I
E:€
Ne
r.(u=d
9gfo
AH
5
8
A
H
--o(Et
HE
EE
'38
g'
,
tgd
*H E.SB,.-
H.XeE
d E
EE 9
qq
X=
rn EBE
fI','E c :,
.E',H
, do 6.EE
.
H Eg;
I .:gl
rvl + bbp
!r:
il @E*
E?E
.
F.
d
(0. H.d q(FiH
L
E
1o
to
ro--{
rC
stJ
3
cs)-.
J- ->.>
€eI;E
--_-...-|
-
6t
ffi
HIFI ,E:
$€ f$
tSIS
E€
'-H.ra'E
!;.,88 IiPi€,
i* i Eati c#iIiE
ErEH
;E
E;:E; BE;EtBHis;;;
Eg,iEfiiEEEI;{€EI
IEiiEi
e*r; fifE*iEsie+!i
fiiiE3iiIii
EEgs €?Eu{t 5:q
iirs;i;;
iIH iil#isfiiiiiiiii
al
!.
o
>r
€
A 0.)
(D'
'd) +J
7it IL
0)
ln
+
q)
H
|l{
C)
ISdHHH
sE
i€it
tr
I
E{tE
-
'A
X{
H €-
P6.E o
d) .' O) U)
g
6.:'-*
,*?
€
EE
UYA! E i
€
r::I
AJe
Ir
AE'T;;
il.l
?
a)._H
H
:J
-J
-J
-J
.-J
J
-J
J
-J
J
J
J
J
J
-J
J
J
J
J
-j
J
-J
-J
-J
Lt
ri!$* 3f*iltifffl Iii ai; J
-J
giffiaIg*g;ilg Jr
;gEg
E
'ir
t
5
o
.H
:f FXE iIii;{EEiIf
={
'+E
bo
<i
o
r\
an
91
d,0 fE
I
.E.gf€€gEF
q)
6.E
; s* i g*#:;
$fi; ; i
--o L) Q F Ud O.i <J
or{
c)
OO
O
63
€€
€E.g
o
e.
.9
C.:i qcd : 6X
=5's5 o'a x.H H
x0)
'3
d.dE',E
or
6s
'.-Dd
o
'=#+i'{jo
oc
8rr
o
€q € ,{
I €
€>
-o€ f
-o{
t{= ^a
oi
ox
o
sY
di ,t{0) od. i.E
,(J
,{
t4 t{
ll
D: 0) +
\HEHz
E
:o
6r,e
.:f €' 0)
0).d '^h= 061
()c
O
IT
.6f.8
0).: c)
A< d 0)r
ZZ
€
HjA! €El
I (tr..
lHa ,o -- a) #.:;E
AA
o
H E oE{.E .st
o^ 'o
'd c.l
Ets I b E -.: 5#
o L ,fi
':\
k.
:.E:.f: b5
h
ot
Q\\
a
gFo
'Ec
€ q)
-(
k o
Ik
t-{
.ir t
€
€
+)
i6
.n
ng
6,
oc
€ €
.'d
.E
Q)rtH
k
L{
'oo
{J,
c) o
k + €C)
od
o
q)6)
OC(
a) co q)
s
ho
(n
D
=bo *iEEf,*=T
o.E
o
..3,o0>
Ed IR;3
fr E a SaE r€
6d
o^ <HHefr5ri=
o
Fl
dts 6J d.= r{ d
"iE,3T€I','fr
='.-eEE
'jH v2\
*{
J
J
JJ
J
-L-
u
L
L
L
L
IJ
rj
IJ
E:
I# 1J
I*- iiiffi$iiigI
L-.
t-
IJ
L
L
l_-
L
L
lj
L.
L
IJ
L
L
IJ
f--
giiii
iii, iiHsi8iilEi ii, ii* ift EI
r<
+
.+
iiii#iiii igiiiiigi; tig; iil I ii i$#t
r-L iiiii.Effi3,iiigii{iiil
l_
L
L
L
L
L
L
B?EEEigEEI
ggit,c !aE{E
i
Hgt*fEsilitEe
.*EEA;He€E:EE
E
ilfEEs,€EiEEil
d
fi 3?tt€fiit,;s
E E [€ HrilrrE
ri .$; >.s E B H'g ts b
E:.'i€.:8 5d f, 8.s
o
tn
UI
€
';3
.O
t5
ruUI
!
v)
3
@
q
.\
H
{t E
E
'" i
6-t
E;
ss
Il g E
o
.*,'H
n+rdd2
--
LJrr
u
9dH,=e
I2 €E€:
E
'8
pd iEes
H hE"
I
,.E
E{E*E
;z{-trHqr_i^, a=;
H B.E
H
ai- de
g
iEH.s
'H
FnSrG;.
?
#g
E
Eg E-ri,
t&EEu
9
6 BIi*]
tr E I.x
gIE.8'E
sfi
3u
gB Ef;EIg
s
s
H
(oijo()q)
q
a_
a3
{
+
It
ts
\3
ll
..l|.
*
s
e
!d
Fo:r
a
,J
r_l
|
r
r
L
J
J
!
AEJ\
:S
EilJ
tY,
;eJ
sa J
fr'* L
-^E
_d
'
<A
tsl*
LD!
l\tr
w-
-<
4
}
-AH
RI
H
t--
ll d
+1.
<.9
EE
s9
pt 5
i
H
o-=
-
-
r
iE
:l
LI
d)
.<r
5E
ll
ic
}O
rE
s<l
-
:
I
J
-l
J
J
!
G
E
+*
I
3'
{ *s
#*, .E g=i.- '
?r ,:,
I i €= 9e
f
:{fb
q
#.a ,t
s}trq+ti
r .-
***{{uo.r,,rr
*t }
iE
"Ti{ }[T
fl'l
nq
;.
"1e
b-,i
,
cl
ll
''i
E
H
x
$$,* q r
E,{
.9
Z
o
il'
e
ht:ri, [:S
';
lB
I
6"
s6l
eoP
tt)
raL
o-\
o
oq
o
.$d
c|
i
*{i',*$
${BE H
^'A
B.
S.u
r8
I*oli rr 8
Cb
6-'t
o-
lD
{-.S
trl
N
N.
.
./) .a;
ar|
c\r
PB rl
NH.
q\t
q\r
;8 il ? Et-
+
}t
,}u[x+$+}Fi$ +ts
tq' IItiffiE$H
ff n** FEBt+1{T} l-
#S+BHai$}1{L
-qr
I* ss 5?: t
gg jHg
it
g-a
[:
€
??H
+
l,+ ]
5
5
't3
$si*i;isESt.€i*5i;*$rigii*g
"i5
$*rA;*{
;
*
i*f
f $? f : [$
*i
t i
i,j,;I u.E:-- Eu*T#s{
djoi;=*
:itt$+;:
l:
8 zo9
n:*
*s orts
ap
3;'*Li*:
e{ .i.'d d
j
"E+
ilBggi,{{
g}EI;19
;+
H,B;His*iff E ttff= $ gl
''E
H{ttE.i*{$l
lt
C.r
(b :{
u)
a(b(D.l
tJa
6d
sS!p
ira,
8tt i":,s;E i,
g &*UYS H
5'6
p. "":E;T E $
:f .HT'? e Y i
3cs
(,
+EatE fl g'
I
- s.,o
ao_
d
ID
aa.
t
or
ol
5l
rD
ol
H
tt
a
6rSt
I .6!1.
p
d,?
U'idp
e.l
O F{'.
c!
ci c/t
a\ol
rl
ll rH$fS] **
:c'*s
It lrH i$Ei lrtt ffiiBtu.*tr 'gi
E
' i g$t#'
t : iEEBti$ fi i:iPt+ *
{iiiB}}:
*SSSg
d d dr
' jc,i c,5 {Yt ci}:ddgSISE
E
E
L
L
L
L.
E
L
>
L.
L
L
fL
IL
If
ff
I.t-
rL
f_
a;
'-
-p
"5\
\\.
ao
>d
tcA P
+f
tg
-\o.9
3q
'\P
sA
9p
-J H{
I+
-l
cb rt
E
Ov
.o
d
t-?
31
xd
Nv
U' A\
\"o
.'tfrF
d.tt
cq ll.x
qE--
F.:\-
o.
cok
,
l-{
E EY 5
ot
.5i
H+
r
isk-i-
:."i"ti
'i"r*.i
*{
N
x
HR
6l
ca
<i rO(otc? so cr: ct
ll
@
co
CA
,
ao
c.l
g
6i-
p
I
6r
G6ld
L^:
.1 Le
\o
O
rr)
t^J.o
c.l \ !r
\\Q-'\-9
c.
N ol I
Qn .-r
-l- D^
L-.
co
o)
.
1
=c 3.p.
O, Nc{
ll
-vr
I
---% ptr
*nc
*t+
?
=GN
'.J-.P
vE
l-.3 \
.N.-
.iua
N'\€l
i_€{€ Sei
C\ ^.9{
s-11
5+&
+u E,E?,*,is
i
l(3t-:
r?L ''.-lt;'?
$-=i,;-i =ai
-'C
:
f ;: s
-t
Ft
'co
,EiHHiii$riiiffifi *
(9t- €
N{\I
.-r
E .q=
? I *fl 5 ;i# *
iE+ $ $',H' Bi{ 3
j$
Yt+
E $ f,r$
!,
+THse=
2 t 3*igr *
1
s$
ii il $T;,6i h
} tf+ji
E F"+F$€ 3$t +,;
i,i;
s er
i
*.I$t$t
$
:i+ Ets a'
.?",r { aig
cu:*t;,??
rils,iT
r
i
STs;J
Tg{'::tuEE,l 3' t
;. _.Q,.*ii ,?,E E? ' H :
aA* lt"l
hi^e f?gil€as"1.s. +
>?&
J1isE,;X?eltfT
I
Hro
tr E=
60!ig:
66A
€'?dE
-.
s$:sEEi:SiissI i
-t-=€' ;:;;:Ss;;Ea;i
$ $$s $ 5 s R
ct.i
<r !t
,
B
rrf
+
6 a<
T
; il ft r sr;.lrs+ 5
:5i?q1Y
is a gt'++I$$
ii3r
e.|
!a
dri.;rf *'
'{'
6l
ci
oi (r)
(\l
f ,t;; ; *$T,t$i t' ii.
itffii E,ry:i*T*:
E .iX
l*E
ItEtt
SaS;;
; c{
d.
cc ci(\
i
i
t.
H
ot
t- i
-f -,q
:
${6 E
oi
=
-38
ts
I
?*iTa"qafrTI
c{
;
iE*
I
;1
-s
iu
at
g
3
gg}f k
$utE: Egt 6Eltli=:p" ;i;1st;s
iti [ f,E **i* *:ti€st= csu*f'i*sgflikt
gtg
Ep i: malil#iiag+tB1tll+tItxg
qu
'ldl
t
E
{t r$$5$r *1gttt
oi.,d I "',
*git+lltflti
B11
,icic'i''"t5<dt.d
HT
tB
;4 'l.l J4
.5S.S
SFu]E ; I ;,i.ri.;
B
1+
uri
,g,,
6
{T
qx ? E..a**$'Eet
r€s;liffi*}
-r:,e I Fd,i u ic
i55rs
Lsg .a -q &;s*+
E
'Es*?
iEffii:$t
;
{
i
E
l$ii$
+
EaB*}i.,. ,, =i,
Eii: $ g ElEgtb+l
in rQ rRb.;
EB * y a i *r i $$;*rtE
Eilii$3i
T ! s?f ; s;:3'S's5; ;t 3tiT35i ft ? BH
+€teEri ,i
g=*g BEEssi$$$$ 5€$
J-J
J
J
J
J
J
J
J
J
J
J
-I
..J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
'.1
{t
r
L
u
L
L.
L
L
L
L.
L
L
L
L
L
L
L
L
L
L.
L
L
L
L
L
!I
.
$t*i ii
ns
g*
igl 3*t *t
a)
6siE
?Z,E
i
)nt
E
8
EE*
Eu.'x*,
=
ssE!uH $$$ -"" T:,+',*i ;f,
,yi
tFF
tl
ffi Eiil
Fld
$i ;
(EF{FI
trE-
6I-
,
^
5 S59
.-l
:.:
53
3i
.
3
Eggi}iffiffii;* $II
E*,f; gA
.:.t vMi dFd;S*S
B
f.-
iffig$ tHgr{$'{ i$$$.TEtEii
fum**m* $f,ritgil+ttisg
a1l t,19il
ci
r. 11,f+*g
r{
+ st B i;€ xI F+ s
.yr
{g ffiEBi1
'j
#,
t{
ffiiif$l
l8
r(D
L,dA
6 t':
i€
lagr-1
a
*e
tx au t+
ttas + B= E
*E Effi
.lissE
k#E * a;$}p
,,$
-t"i,?8^iEE
S
*
g?;etgiq,ri
i:*
'!
X€^}
l^
d
n
H
:'
E.?
g
g
q
*u
s---+ 5--
€xA
{
NsR BH sBsB
i; ;
q
B.=
:eE? E :" a3 *
wIE iH*}: c1*t$*'.'E1at1
q.{
1-i
no E
E *i*ti1+tittttF?t1{!ii$;: Haga{-Ei
H:" E ni
u
:n
6'
i
ii
JE
s sB
a,q-) tg.5.$"3E 9f,
s
TX
=EEEft
Bil
ss
BHB Bsss$$$s'H';
t
S3{
as,, ; r,rr
'**
Ai
$
E du
rt .ee
$i,
=;q
ulE
,*
q
fr- ff6e^E
BetH
iq}Aen;
-o6l
c.i.,i.$..i5;;;.::;.-;:*;"'
ot
J
=J
J
I
I
J
-.d
L
:<
:<
!<
t_
<
Y-
r{
a<
:.
:q
a{
i{
<
:'
:'
-i
:'
:'
:.
:'
fr
{
{
'a
.E
d
na
6 tj
Ei*
";;;
niroro
EH
e'
,
eee
to
6;ti =
e{€-"to)
>^6'
d
to
I
ll {
?gG
q1' *<o
-rt-\o
{{
k
h
r,F*.
to !a
to x
<o
":€
Es
ix s
=
:L
Sl
N
,+
L
N
E
(\d
O)
6t
I
t^
lN
IS
s
co
H
B
N
e:
E
s'
6{
+
:F
tr
a
N
d
6t
\:
=tE+$i$$uiisiattritgi$
3s
k
L
L
L
L
i
-x kF.$
L *s
'?!g
u
u F t-':; =i: t$t E .lgt f6-B i
L
t
L
ff*t'*u*
tJ
tj
tJ
]J
L
l_IJ
L
l--^
t:
b-l4
(o
q.COE
o.l
isis lis
*t
i,if;, I, =,f
^E
i
iii
sS Y' t EE€:*'i.*
JB+
+ ei: :r
+td;€
-9 .a i'E ,
,-.1 -
r '-:
Ssf .Ei.
ilEitiEffil
E?s rA"
c-
gp
gll+:913:.i
saEs Ts$ispe
$$ **
*{gS- ,,51,'
.B T
3'6E-\
!'63I
a:'-.'JS
ss"st
E+iges$ xs:i i""S€H
e'BE-i+\3
rfi>ry-j*:li F'^i
-i€€ $o? : {gY
09
co
6<
oic,
-iora ,,1{4-9
$i,'$
V.$n'
-3 <i
{i
t{t!t
IE
I
<r
o;
\c\r
l-c
ls
a
il
C-I
N
tr
+
ss:l+rs+u sss+t+ii{ii6t Irls
l> s
* j':i*s
IHiEIi*iffii s $f,i=$& iiq*ttfti'
uj .E*
!.-oic{o,i co
Fca
a\FsiA
l
J
.-t
B 3-E
;=3+=i +
qC6,
E
EIis*i+;
i_l
*i? E+:* st:ipAg
:i
E
i
l
I
=*
Es n sH6 {iil*
{i:t+rBiiEE
i? e ii d 3 $e;: 3 ?-=3t ii*;iF $t =
*E e i; ii "i i,i*t. . E gtis t;"*'a::Hi:
i.L= i":"8 :: 6:tq$? B E fi-*iEBLE"J;31 *I i
S€:t p;t: I A ?t**:Eo ; E ,ge{Egf :g*;tc,T?
EE;+
ris H"5i
i
;r_F "6tb8;T;S
TE+i:{j*-=Y+'-*ig}=
r
r
s?ii*s**"
=
r?Et E
E
*Hsiti.p::;
iltl$+i$=*;t $itEtili{igtii*{
r qa.
5*i+
'd
:::;:J::*T-i";.,I
.,c{ cri .i
r^
:
-<cq<
=
d
..
.+ irr *la _
Esi E i.ie$ii
.H3g
l.E
K
grg
iHlItiiial
itt*$I{i{
E J
J
-t
J
J
J
J
J
J
eEs
J
s*15
J
*i* J
J
=?
;x&'{
Pt;r 8- *s! t rlTf*.gtgg: ;tQtr"??rs?**sr*r -"i
yF
[Es
=i
is$;*j-*isFEi
s*xfrs?t6xtit$
{
$:**",
c.i s,.ri <i J
:t+*
rp
E
{
* Ei 5.q E^i?B$rci
S
3n.i : :a ii$=,
3:$itB*ira
i r ii,
g
t :g
oq
EF
H
?:
** Bt=lI
ri
il ss
crd d
dl
(or
-..l
I
-
t:
t-.
l-.
l-.
9:ii
'eS
={t
5
t-i
d
?
i
?E
.o
N
cll
H
q
=ttr
l+
(b
LN' c
'|
Ln
+t
E
gHtB
+
.I
*ir
i
();
aE +- - -c-J'.E r-Et.E*lxE+t'6;8ts$
iL
6B alllat $
Iltr
r*1'-,TFE$
;S:Tl ,
it=:t:ErE***;lE+ Ei
I t: q3 ]:ri
*+s*tI*fi€=;x o$ }rq'lssalf ?E$f** t++
(b
\-\
r,N\
oN
* q:N
+
tr
-:.fl.,
Nry
q
R
6I Ef'r
t*.
f t-To.
5=
B*i
:
+_.
E
<+
:r*:
{R.1
S? C
r5{;
7
^z
1- {?
Fi
6'
1
{nc
H*3.;, + r
s:i
.E +**ie
tj Lf *f t+f
qE r;
tr+
9Tt"t
$F r*q:
tj =r
+TSS
8e. E ,S.+ s ore
lj iE *t-?bi 3g f5*? \ -.li= s i ?$*i $={ i
t-.
1__
tj
ttj
d
.i;
!,
i
vi
t$].r
]l;s
S;{L^
rl
dFodoi
ti r{ r-{ rl
.4.
6
c\
ci
ci
E
? i 'i' =
+
ex
.a'
FTf
i sr
I
\ t..
i:s
*
R N R Ex
H
[:e*.Lo',oEi 1,, iE
c..'
o)
6
*i,_,
<cr
€[33t]$ HF*$t itE+
$?
t-:,
.Er,
$ gl.;*tE;
-x tt F
r. a a E i i: Y^+*L
HRgigs*gE$*$ffi
.l
.
:rrtc{. :
-o-r!ocDd .d d .o;dJi -. 3 E\l-r^l
.d
t__
t_-
Lj
1---
L-.Ed
u *€i
{s
p r$
*_ -_
3.irsst
ao
HT'.*
g
x_l
,Ii
6q*
af
#tT i^
=a?*"*}$fft
l_.;5g*3*$. F;
lj
r
s
c{.rj
ni'r
,,
,
Yi , ,,''-.
F
tdd
-.
-i
:6,A
;
Fr
s{s
'F*=E:iTBBEE
;,.-^-?
B
"-.i€$H$,-t,. 'H
tj , "Lx rEf,+'sHFo
='i;,
g*
$.i',*'E?$Et+ f,+.iTE€AB,?,F,H.I._
tj *_i
l-.
t*
t-.
l_-
t-.
L
u
L
'
;;
,,
LESE
9H
o
A
Nd
ad
Ni
o(
a{i
.lr.liH I. *6
'=
-l
I
f-s
C
ci-u
I
aE
6t
3
Ja
Elt rrc
H'6q
rsE I d
t
,lsHr
n(
- l(€r -ld,*
'
It '
q
+
cl
'c'?
:a':':
l.F
+
^,:c\
.nt
;rg cEca_
rr- i- ts+l 5la
ql
Er
E'5+
F\
lsd
ol
nE
o
+>!
+; .r.E*
+'
rIA
.loH Il-|
\Fq.
-lr"r{
aQ-
o
\l
AN
r€
'4q,.
(E,,\itr 5
lJl
H o.
'9.i,
iFFE
0 *!.(a
of
.51t
at
dio5 'r
Er
r{
3? e ga =o>or*ii
?rfi€XX
r- .i tE *
tE
E
E
${
a+t:
i,i
iiffi
€
$ $E
si Ei tT i=ci $ u i
*tiH$i
ATTH' sT sg:€EE$
}i*1i
$
i}i
*i
,
t *tgrj*Ft tlt --qtFs:gg3ti+
Ei u *^ i * -i*t gli
it
-59 E iB =.q*€E$_H$
_rin
eis;
isri-
rH-iit*I
gtJ
l.sa
i ti i=gi tts{s ^=i.F $$
,'T Ia
r$r
,
,,
Si+t
*
t*€#p : +=E
*I:si; x iiii+s
rs*!I*
*_"
au
:*
_&'BrEEEi ?FE$::t 3;E s d i
::i?s" q+it+$Efl
gci5tif+ii
rx
g.qJ- -o
F d
d
i u+i
'X+-plt*
EI
I
C-ld
Fl
l\ > -i.
'6
th 8.
Z.rcD
l2-c "s.
d:
N
c dr
t\
o or
ts
aq
O.l Hs'
)N
d.l :lLE!
:(b
B-&
p
(')I
I rr
il' /q)uir
>t!)
.d!
-S€ 6l
r.,l dJ<i
o'lr.,l
v\i
al"
$'
)H
".tsat
lvF
rL q.
qi(b,..-E.o
'oo,gH':q>-i
;lE
a5
or
F\
tt IJ )^
>\
:0
oP
\r
:'\
str!a
.6;.!
9cr
'6ll
\p
tE
'<v e
:s :lS)
L6r
El
l. to
(0
6\1
oid -i
b-qi
c\ c{ c\ co c.J
E
t.
-t
:-
:
:-
:r
_l
{
J
J
J
{
{
{
{
a
.-,
.r,31l
€
-S;
B t?$
its EFT$
()
rD
(\1
^-'--t.
rA
ol
=
,- 3
+
'3
<
:8J1l
'nr
d
.aT
Fi
c\
* +5
5..
..i
,
c!
==
.i -.o-:
nr
6l
.
!i'-l
I
co
(\l
(a
N.1. C \
ro
N
?
dt
tr()
'!:lE'
.60s
ioS
q\1\
^dfi+
\cr
d
o
Qc
lor
- P^,
H
u)
U)
Oo
OoN raE
.I1 (
I\
tth
O
v,
v^'
lrj
Ua
I
'
-tF\
O
AV^l
Gr
N-
. '\ [r"
. a-ci
.ic-q
.F t_ r.)
€€f
tr
D.- r rf)
j J+
-{-
--
a
tss
-\
C.l
t
H So= '!
g++
\\
8?ii
.a
i
sE'
?F=
d'
o:\ o'f
d
sYS!
o
s .a
s;9tsu
ll -+,
t5-
(t)
+ C^tO
c^'-
.dt
"i sl
rr)
l*
.
I \r
\).
Edcl
iO
o :L"L
5 i-s^'
N
S";
F
9.'oNE?
N
OA
o .()
o
6-y.
o
()
ds E'
i Ep
t- t 3, S f,'H
COr
i'd.E\
; il {Qc\Iv^,a
A^ d,
:,,I e$
+ El
+.:
l
>Jt
NN
\d
ca
co
rzl
;-,
@
'vt
(o@
6l$l
-tl
F.
I I .S6i
X.d
<,d
co ci) vco
trll a
.d foE?>go;ll
. .€q>'S
s
rrf,
d
co
5)^
t. B I vll
;:;{., IH N; $+S^-r;-.
X
@-vr
%igj;
*r t t *l*z tsr
.! re.lE
F('
\
P
tr'
v.N
I
\--i
vt
rYbr8:,BE'
tDj ..= '-l I .--: O
ill.r.l--rq.o -' <.
iUJ(O'aa
\_ r:l F
hs€',
*;Tor * .3 R vL
:
llll o_%"3 Re'E
'nt r .o Gb #l;til
\
d S"l
di
,-h cla. ll ?ii F LH
*,
O) t- "l*^O)D
p(
A6?*:
tr(d €5xi
Ed
5' :u5,[ftt
{ I I *.-(o
llT-e
ll;
ll
ll
rQ 'E sHStt
--nQ
rq.E
<t
€
6r
iio,?il_
-r
+s'-x
: -l- h-I.c|
{l^;
sy tr
Sra^.1.-9
-:<X
\ialo{co tt .3N
Et
t.;-5N-3 I
orrlJ
.'3
g g'ii{ G=
"jii is .d
lbs€'^sE
*x
$,_ir9E i+FH
f
H*? Eiss =
{s :+*i
9s$r
*
E ,.$-tjg
5=r *E*'-i r
xF^?i;iEs
siii+i,.$$sq
3stffi6$ *,+tp{i$i$rFT
tti
"5 .3?I lT+*
Fffi$EI$fffi+$i+i
oi
Nca€n
q1
F
€.F -:
+i$E
N
$l
jd
, .-,.5.r s'e x #s'
g.E 6'i_Efe
I
iFr*iiEd:=
+
's
I
t .{I *f *f +3
t{EtF ffi €- 3w+ +$+*++*$ €$ i A;Y i.
G'
E^
;
it
'
+';'{-o;-.,-"
--:-9,rC
?'r-,.r14E?$?$?$SSsg
- -&"
"1*Silni?
d --rt 1o.q "i
.d
-d
Q;
\I
q"o
<l
cj
.3
a+
ss5
6!.-
t->
oT.4'
d>
E'
+.+,o.aa
;f 3Q3aq
B
R.l,S'i , tg
t*'g$j,fs:
;s 5,S
E
iFarus*r
sji-tigrBx
5++sFiS'Ei
ptti+?isc
'jN
63ri
3fi#AaHntE
tgq?"'iYq..
Fl Flrl6l
b- 000)o
HP".iHSa
co
il
I
.i*r
ss
ib*
E,
4>'
ti,i
*T arcr
$"
Ir F? l.r
ls
..3
s-:
$3
<t:
g E
F^ E
E €;
i
iE E
r*
SE' Ag'd' o
3
-q
3
=""
iil E"x s q rF ii r E+{itgsT
.s*ll
r
se,J
H
**
iti ,Hg^t} ,E $P;fu i*t$ ::s$tt g-Is
; ll
--*s
$€
$ffi$Ei$s[iry$ffipiffiffi[ruffi
d
*>t 4
;' *ss BilsS-;
b^s'.i?.-,,€'*s?s.o
ilt*iai
iE*tEi+f $
*Io
l$ii 11liH*.tit*it,t,giti ti,
**i+iil'1i[I i{tifl I i+*;ti {l
\
$*r* iH+ $$ffi ttiH$Iii$u:
ii''
*H
Sg s! PI E[
'b.
*
co coea
{
3
.<
..r
J
-J
A<
:<
.<
!!{
<
_-<
<
.-t-
.r-
,.:'
:r
<
I
I
-,*
:.
.<
:
:
:{
!
{
lj
a
L
L
C)
C'
r j
t E s*l'H
Y
69-Q pllo,
F.i
rO
.E
-
_t I
S'
:
rti
Ss
.l-6-
c\
-n
?-
$-
rr
ll
ll*-
I, Cj-.E
di
?t
d
s istt
ll
d
e €
F.S
?
L. GtqFi if
3E $F iEg?Es$ €B=l- t
Tirii{llf
g
n€
!ii*a
1.- BE q t G* f, SiE
i
F$
ti3
gs
g*?Ti+
rj
*;e..s
f*6$*irstsrt:"
iLF l
isl
ti-}
t
$s SIi+iR+i*I tssi;s
t-.
'a
+ E 8Q; j.*i
iE; f, i f
q E
.-a i- a
iri '\ n
I's iie'
$ I ;tr
YR i$
rr
-io-ll .: --8r
' ^ors
,C, l-):G
6l d' r r
6I
b-
Eco'N lsll
--61ouiu,
Ft
b,-
'ri
rid.dds
ole
.a€
'
dri
o
da;q,
! .
,O N c.j
uF,.. + s
o(
3E
ti-'N
,, lE?
',, xg.a
r')
ro
ru
_U":
Ge +
J *'8
dB
€
gil$$+ffit$$f*$$
CQIO(9}b. b.. b- b,-
r I}*sii,g
c.i
c"j
@@
uid
@@
t@
a
Eri
sJi j?,s
?-16
€r='f*
$sf,*
IO.-g.O
", .4.
E( -r-:e
' --n
ddc;<o
c0
6i .d 6i ri
.;
6-
,S
si
+$cg$:€r
E
lEEt** ?r',f
\
^--
SB-t--oi gTE
e
>
:?s
'tES
i*i $i;? Eii+:r$ $i,i$FH;$E :€*tt$*
s*t
Ilg $.: *at
cD
H-r
o si €tft f,
Y's'iIftii$iliuu
c5s
s
s*tffit$smi$i
,&F}i iis t;fi ::"-r16r**
iirr"r?"r :
3it
i€.s8H5t*ipl-$ i!*i€s;
L.
l_ {++*rrs*t;s[t
tj
lu
l-.
1_-
t--
ll-L.
lj
l_l_-
lj
lj
L
rl.-
Ij
lj
IL
l-
u
i
'
;
i
CO
er
s
B 35 3
€?[ jS
p=;
s
g
},
S
:l
-I
-J
s i
Eq i lS x
S?
TBrHE-o
s
+.
s
;
J-I
iE
tt
*
JgpEH
ffi
+
E{
oi
*s
'{
J
s.s
ga
tr:
Ftx
s
r*S*$s*
$3
B*
..*
t
$
J
B* $ *=*+*si gT
J
i=lHFi t*..gi J
ilsl $tlti+*"pi t[;[l+g
€sattEl ii*us *ssdr J
J
J
gg1ffiggmrffi$$l*$H* JJ
J-J
O) FrOr >
,e'$F€*s s I
B,€
-,S
E -i{Sf
:i
.*
g{ EE
@ A I tr .n ^s €.a
=E
a[s=
:{.ss
1'P
J
J
J
J
-J
J
J
J
{
iffiffi1*igiffi$$$.ig+ J
S+asf R
'Ft : f {*i *!3
E Fi
E rra 3s- .tt F'$T,b $*
a+ ei fstYi
-., -?,,* *
Y'
s i3
Ht*ilg
:t E HF:
4
Jt
;ei
B$
r{
'n
5l
d
(d'
6
r.
b
3
t$rE+E$gi+iFIiil }ffiEgTIlTt
cj
3
I
{
{
I
T
${ itl :*Ei iu diggg{{E$i:i* q rBgii iti +i
F
rs i s
iE'.sTi
STts':
F*u3
oJ
;$d
*ior .
b-
I
fE
L
L.
L
L
L
E{Rf,
sicr
E S
€sLszt
I € .-i E T,
irJ rJ
appBtANa-,c.lr{
T:s d ->
.{
EFS^aE\.-jBr')
rr
O
rO
r{
rO
*i
6ld)
rOrO
L.
L. $eti,e
:,*.$iii €*'
L
L
.L?"*TE
O
rS
I
r{
E S
t{
F{
Sl.
9t
E
E
'i
J
#g* S
?
-(a
E
p
6ici
Or{
;t{
a=ls
oi
Cf
!-{
Efu+iii*
i
Tgt
ldF
OCf
r{il
HT**slFe
OO
r{F{
c'j | {,rj'r=?
O
t{
5
E'Et Tn
r-rrF
5i}
s'e I {Mi d
xi
t+
!$n{
$r$
e
{-
rl
.o
I
c
I
€d
l&l'6
l+
l*
-^N
I
<D
El
'6
S
6!>Js
t+ f**
}t.E
gi:s
E.nr
Se
Ee
+tr
-' 'A
dq
8s'
i+E
{,
F{
trr)
Fl
G-
:13
,.
;
'R XEIE-"t' ?
+ .E
+.",s
e ='olq
x'.E: $-'i{-- : ':""
.SLqt
;f ,E'{+,?-tl,E'"I g
E- f t''ir{EAi
"}
8
s-:\ ? E{eeS^I,
*+++i*
-S
S^
-HSEErs
tr
G
6q
()
.o*
IN
r>st
6v
A-{t
*^{
g.A
-V.
orOU)
^c)
N
c3
N
ail3lrr-Ott--3
i I ;'k :'i
c{ cd
!-o
t{r{H
t< t-{
T.
6i
--\ 'qs'Fjx $srl
t': ats'( * FS^'a
; ! lt?-"*S 3
aa
6;
-<3
vN
t-
.g
-i
rl
rl
q'
F
t I
Ta
.$ ar :;
Strf
c'i."i 'E*n
i H$,. iTi-i,si+ ff, ii ii$ i
'r:
L.
L.
i$$ig,:tH*$grul*,mi$il
L
.!
FS,.t"ld'.idd;,;"t&
rqe
L.
L
po
L.
3
o
E_
:€s
e
CD
Etl$
+.1 {
ert
L* _{:s !i H=
L.
L
L
L
L
L.
r_..
L.
L.
L
L
L.
L
f-
€'s^?
gc,)B??l
-\\
\*o,
-l t
-(a
::
A
_=-
=.I -+-
gt
H
,g
a
Be
;." #, q riE
:?$58
Fs
+
6,E
5f'
r{r{F+
r{
FIH
B
a
-E$3€,H€li'BBE-+'r*IE
#gti,B+
a.REs#T#-qiB>p:{
(i" r 6-o"- oI,;I--ul:S-ilF{';
-n
Fl
-
*ja
o
o
lt
(a
s
o
*!
.c.
Yll
'9 ^s
r-
or 6a
NtF{
-fN
g
I
c)
r
I
g
r
*'9
c!E F9't\
lt
-p
' -ri*
IilN
\'5.r
r'S
Vvv
c)
o
ocO
- xfE
*<{.
d.
-(b^
*: 'il
li,r
x=
y"$
'-1*16.r S
d
rO
6L6t
\€rP
c!-o
;
3
e$;
ei
ll
q
.3.e .t,
t2
A,
ol!:
":a
€
+?6t
6rd
J
B
.& E:T
.st
V
ccp
-vl
?? E: LS"i
ll
t*,'.tSE;S
TSh
-' &&'d= .. ; p.E
n
Ji
ro
rO
tro
Ed
O)
ro ro
cO
E-o'q
il
Et
HfTTE
HS+Ef
E
B;3\: rr'E
.-T
.o.s
$\to
dl.r
rO
<iridb-A
c\oiNN6!
;i;i-tErF{
Ir)
oo
s[S3:+lg$r,
'e5-Q
G+ JS ;g EEi
HS
t$.-*{tti;+
rr
On
a u
Pt
d*v)
o
I
8?
<- : -,+'*r .r
E#
ir**E; t'gtt,€.A
:
6i o'r
(
8,* HH?? HE'
,t j,? T E"'E'?'i
,Sr
E5;,BH ep:
1.r
s? ilqlgil":iE gfBa *
E
$s
-oxr
;$
Eslf$riqii$iff$9ii
\\
E,L
j4E
aE
?E
t-{
c\
fi'r*?ra;3=,BIe.lt st i
r-l
6;
6!
rl
rO
c€
o
6iar€€(a
t{
cA .+
ca
co
FIF{H
di
a*
,is
iA
F
;i
sds
_{
Y
EE
E"E Ei'$ tsf,c
F€luri'f fs',9 g X+ E,?
iE?=xsis ie 3 €J$"si
F*lt ?;Et : u l** $$Ei
E it,-*+.ss
3"8;
5e
j"E={5$.5Ig[
1t $*EEg.a SEe,t
.i
po
=fgtllttlr;it+ilttll
i5'
p-g
:EEs
ES E
A
I
"o
\l
€
v
\\
I
d
d
eg
d,tr
A.d
a E.
B*rr
iEFI=
'r d E-*:,
*e 5. gET
oi + X,;g
i4
r()
E'
f,*
gs
E.E
& Fd
gt
g
d
k
or
oq
rd
o
m
rt
A
E
$
T .A
:'
Il'i
P
r
'Fl
d
6l
.ril
(o
F.
.'1:
tr
cod
Y-i>
loco
€-l-
r.6
-ru
I
?J
()
a)
+
\3
+
CF
Io
N
.-l
Ic|
U
oo
<{
ro
a?
I
\o
L
6t
L
J4t
k
.t
_['nl
-i'
o
13 'G' t.
rs
€or
€
I
\3k
^L
'\3
'6lld,
NIO
,
l^
o)1
t)
vl
TN
I 3-b d*
E
F{
.F
ea
d8
od
3
.q
F{
,,=
o -Yl
r.6
{<'
j"r".t9
\=i\ .oal6l
N -r ilar'
o
-rd o|,
co i> ,ds +'
.() g$
c
tca
<|$D.-
I
'--$
q
+v
@
ct
4:, d.{,
sf
H$lcaitro(ot-
6l .{ q:
s
t,*
mo
_r.l
d r-. p-q
+
fu
.
ool .q)ql() .I.rJ?
ci3
6
ala
'-l- os8o
$xr n16l
*.
.- Hl(O o
3"j
I
*-o.- .+
E, .-'!?$,..o
l-.
()l
,.$
.o
6r
6.r ,t^ ts ,.^
ts#
t-{
.*)
6g
sr.
6.1.
at'rii ^ -l rlolilct
Icf (o ,m
rlcl.
i.{. !1
<(
\'\J
d'
+ (q
++
r<
ffi' .-+
\.o
,\J
s
!<'1:b{'
, Bj*r'lj
* *i. trlfo
+,
A FI'C
t;dr
a
erS
t
'q
--q
I !5*
xtzil
..
i
H
ll
SE€}?
<-E A
J{iE
+
;f;E
3€
E SI4r
3
'|
E
+-StB
g€€Ft
'|,
\+tsuQ8
s
isst$${t *[: :" *5i tr.-j n
.b
5.,
+?
r)
.of
.t
s,
*-- fi
-, .g
i-F-- oisi
rr G ,{.
g;.tr .\r!.!a--..
a'.E 5r ,$" ,
;'.-L
'$.o. . 'r
H
+ .'E' H-16l i. -s
*e:P €*;.H
E, ,*l*?:BlE
$,iffiigF;
$s
oj
till d
'i toin:i3.irlrO
rO ro
F{r{r{
f
G i'
€Els
Efl+S
;
T F{3
++JE
.'
1i.i!t
E
*E$ii
,i
T-,i-iBo
df$;? .:
f* , {"85H$
I+EBrE ,_q s +:., ll$3 E
,ffii
.gglfi
lg*3E*slt $f:gl,
r5h
S B EE? E
-*r cei
q
N
ol
o
3o
I
\p
ol
no
a) .c;
m>
rcp
A-9
r-'t .J
ru\
rl
nQn
0(
-\
3u)
\o
6r-$.
<tNcD!i{u)(cl
=
'=a
'6
1
\
+1N
I
H6l \\
LJ
t.
N
ool
6t
<%
irE
A q
ot
ao
o
\p
f
P:a
E
.ti
'.i t{.
<t
r{ t{
uj <, F.cri
\tr !t <. nr
F{F{t{Fl
sl E. .E^e
i€ ii*+"H*+tHe$E$gIt Ftb ++sh+,.
tA susi
E
$ssif i * fgp*txI t ;lfg,dlcl*3gru
r*
b;;5e:j:::STF
*
i-t*r?
u* u
htgF
oc(
q)i s HGG'g
ils
F'
o3
.) tr'
13 -icr H
s [\g*
tt)
-L
I
:lN
ll
i<
d--t-cddc,
[j co cn g') t{
t-{ }' r{ r'{ r{ 'il
co
,
.
'!
-\\
:v
an
€'
o
F
o
I
6
a
\
+
\o ^
Sct
o'd
.9L
.o
-tlg
E
p
S,{ h'
.v I ,-,'Jl
o^oq
Err"r,r.to
33
iq <i
g5 >3
E
s I'
TT H3i
$,*+sgj
-s-riE+:Li
:;55*i
E-
'a
$:a ;$, +ti
a(
-;
ge;
t
Eah
i | -?
',eI
-L olco r
t? aitE* s i3i
::
cio
SI '*^Sn*{5 XH, ilsH:Tq
qr
.v
r{r -t
-{
-l
F{
sY)
{
A
I
-o
o
\o
G
I
6
ll
J}
SI
--{.cc
()^i
(]ts\
jrr
^:ai:
r
I
t+t 0( '.8
7N
|
rt
n(
rr
o
cl+-eb-?.
3
g
gE'
.
,
"J,s
E-?
-Y
>
D.Y
,rj
dl 3't
1r:
of
.1, .t,d
.S
d)
F. o
irc.t
r..:>. '; P
'
.-
E
$
a
z
l={-:
. ',
'-"
ftl'
-:t 1
o
' :,. [.1.
p'
'r.
-r n
i+ +i**i
=e F* FgI'
Xe;';E
rE$;A1E3.F+z'5
g
P,
gs
fligatgEetg eiAI*Ss*l il
:33 RXSnxxsxn R 3H$S$
-<
rr)
O
N
J
:I
-l
.-l
J
J
J
J
j
J
J
J
J
J-J
JJ
J
J
J
J
J
:tr!
J
{
gss
d
cc;
c-I
cri
{, vi
N ei cr
ir El ..1i
$+_ xqiRlf{F$BN o .*5i
;
rr ?TT g
d.d
E Bl ?.o
T ''5+'6.8
J
J
F,,
EF
,,, *ar
J
1F.1, 1$
3T *g+$I
i,*
+E
;ei 3s J
b
E g"
Ei59+
r?--B'" ;;
;Si,
s
-sg,
3^*9
Ii
+?a;Iq*E+ tt T+38 :r. :+ {S
3$ +lS *a',
-*E,*
EE=t *4935.=9t= ss il p"*Fioo5lBl
.
y.aVo-^or--g9qH:
',id
EEi66q-g-i-",&qr,6i ;-'i"ll*Fi5;;siEsavai
s*tEHsB $*u* $$ $ s$t$
--:.?
€
++
+
q)q)
a,)
g.t"
EiEi
oo
o
d
A Hrz
-z
ricddd.;oir"j1i"iet
;\
fr ci o co co co co ca co
+,
t-
c\r
c.l
<r
o
ra
v, o. J4
q&.
Arl .xl
+
rl
't3,"IJ
(o
13,'
i
Hi
*dl
cDL
.v
*,+.J 'O)
*,r
-{-.. l. s. .o.'
.\^^
..o.^
!
r{
6l
-,SSS'
(.o i> r> li,
z
| ::; ' +iA6
i4 c\r
(o.,ri4
H33N \'.r3.i *' !nl.
+vr
+
*d:t't3'I
eD .: H cil
S ttq .ril
(D.Yl
.$, l>
O.Y
=Y6l
.l
5iqr, co<Oro
J1r)
F 'Ioo
, ^l
I
ir
U)
o?
'-l-L
'-l
ci
+ It
+,
d
ritH ^++ ,d
nl.nl
f333
aa\i, o)(Qor
o
ro.s.s
o
4ts,
suf,
o'
^l
I-, o) b._ Sr 8Ei
'\3
q
J-S-{aO o
*<,, 2.tr Q.o
+<,
b- ro co (o q
\
-r- 6r .-r
@
-r(O'
co .Yl lcs
it3 '-l ,1,ol
E e.EE, ir3
3ll
-!o
_tt+d
+
dgu"u
+zo( ->>
^5(a J1l ...ri
-vi r-.lvO
-r
.-,
Xl
ll
.o
o .-r il?
Xs sx = riO-i
--.? cfJ
('q
('
l,l
o DI
YOc/)F..o,o
-{t
-{t
-ji (o co 6l
oo
,;xq 0cr
-r
t'-9
d
r.- -l-'-l
Hg'1 $ 6.-rJ
u- co
o<iYo
r.-rN
ci' ,-l (o ll'-l
u
o
P.;
?qi ^ s--E i'tr
rl
id)
*Y
tL.l,tt 'v+oY
ct
YcD
ol
c)
nc nq
.rl
P".ir
{o
,0)
(3 c.i
5 Jca
sS Jl iu)
'.ca
t-, 'l CO 6q .atrE I
!
ll>
FrNca.dllo(o
r{Ftdr{r{d
,-li.-t
.-n.OO
3S3S,lSEt-L -l- ,ro OC
r4
o) ro
O)
Fl
ci
?o6l-{
*doa ;-.t.
o;
-!
,,
r
u
L
u
tcoZa
i++T
?
tJ
,}lt.a v? o
rf: 7rI)
,1
{"?
-- -ci
E
O io ol
.q'$.}lilgn
I
r* fs+i= *l*g,fu
[g
H }t lTf;fi; tl S$s1; ,l+iT$+; n**
[]tix;F
E*ssiluq
j7
-11
::
:i,u+.i
u;=r
Tii'E,p
?g
B{i?: I&'
t
a
z 3'L. >85 A
--{ aa
lTiiT=i?tr.fu1'
i
: t ,.{s r I 3*a
;=
i
;
rL tr:g
{a! ;
L
L. gII[i g
il;R+t!
L.
L
u
L
dro
o
\N
F{
;F-G-n(
e!
ca
t$ z
f Ja 'Y -fi ,
3tr-G-€
?5
,i>
dr
O-
(o
:$i
F.h
vu5
b-
H
a
l
s9F"
@
B
h
cil
rrf
,-aR a
tO
I2
iTIe
s d 6
AP_\
OJ C) Fl e\t CO til
FoOAcOoO@@
E"?
L.
'lifx: rY
'
L
f$ri[
E;+3Ei
il[ili*ffiiiiiilliffiugi
L.
L
u
L.
L.
u
L
t
$ttt
$8fi
16-
H*i+E B*E :
=s
S'ogsHBHBEEssEsB
s EsE ,€EfleS P
iffifiE 3ii+$gig*l}i1
a
* h
,,$.e
i-"
L +,r ,= *i,E r,I ftE't*
i{T ' E?
i.
,..'
, .it.-$pI.€',*[$* K
H E'
L
* .g'$;iBE
fl&
- 5'$$s
u
s
a=
L
1[] F* stt+ti p{*i$ *s tt$*i I
L.
=r
L.
L
L. ilili*,il3g[iiil
rL.
L
I-
,,,
ies
i r ix 3fl r i
|
u)A
d
H8
Ei
aq^
v
a
3N
o o
>.y
6r:
j
3
;
.::
s
d
H
z
ooloc'l-
t'- -...
cn
H
xZ ^T
x*"8
c.i
fi.*
^alor.
E , tfirs.sllI-i;.o
3 +o i{ | H,oci
,{
! ,,' 2i-fi5
p
,' {JS'
BB r{_
,"#E ir
c\orrrfl\l
€B +ts
ESE$
Eo*
oQ *x
E iis
iE f
i t; TT=$EE'
dB
;
5
E 3;j
=
€
EH
:a
bo A
.
do
3 ir'H
-9
'ts
3
*^^
LE"'
qE E
.E>
i-E r5
SEF :S
4
sEi
j+q+E*i[.
i*i
=;]*ti+* ]s$;l*i*i
l6r
c\r
la
I
I
I
I
I'lCn
l-r
l^
IN
I
HH ==== =
=
E t+}s&AE
lE;"; i "1.{"i'it *;
=
zx
r'}
\3
6l
S
+l
6rl
+.i:l' .. .$
.:.1', \.
I
lt)r
lv
lHl{
I
.l
ii'.::
O
-._::_l: C)
l+
IN
oT Its
d |
i
glEil
I;*t
*riffi$$1 EEEEiEi $$l$t$tE31
i*
-<o
t*j
(r)
t
6l
fr -r:i.'' ^ld
E, L.f--::: {t
Ess
l<ts
N
dB
.<r
.'^dq
-l#{
acD
u)FzOI
I
rt
|
I
N
(O
6l
cn
+
c.l
>
Ela-
.^CO
El\\
u?
H N
€tj -1,
.i ct oo)
o)o
ll
'('rO)r-d
a 9 -..-r'l
=
d
-6r
ifi;sE
di; 1;J Tid
d
t-l
r88i B
=--i
'{
srslisSSe
d,iriOE c S,i,Xe,E*#at;; dJ
g
xT: euY;{,
ti€ ss Ii*iiFiiH$33 6sEIiiSt{
;s
9,:' @<,Y*l 6q r-'$l5ilistlls sr
39r
sR:=liri?
ot-i
E,
d d-Bq Blt -. {.eY-r-ll 38d'E I flH :9
E .. *ti
YC
-.r
E
Y
.tc,--ls
S,3-eof
i
r
l
o
I=EE SH:F
^ d\
I
f"f i-i:ts3.
H}. ;n" F t j
*{-H .X.=.
? i;-I$st
T,3- $ [:s
-alt
I
;9s E iPs
j3$=rd *\rc
j
=
=;
@CD
-
rt r^H ixi: -&
B
fr
E 3c-- Vl +
"f
oi 6 so.
"o
1ts[R lt to Xt4
:sqlq
H& ts HTEEX' 3 ?ly?i
<r FrHi "_ r.
sBRiH
cd
5? u;'S? Rxf
lrl,Bs
dc,
rti 'boo€
--
I-
-{
i
-{
aq
--r
+{
-
ir
-r
=
-r
-.
!{
=
:.
-r
-{
:.
-
I
3
ia
i
I
-.d
.J
*r
.{
tr
L.
L
L.
l--
L.
u
lj
l-l-l-.]j
t-..
-.
c's\
-aB !'.+
+ :
t ;.$ilt
v
,I
BT
ri c,
i
t{r{
b-
qi
of
ii
d
-XnuF+T
Si
srffi*lgffi+xffiit
flHts,gs:,il$ EsE'slEFt$
=gFdE
& -t{ €i B_t Ef$} I FE
d.,
S: fr x:i+l**;{o,q{$ii'FY{
E
s'ixlfi;igfR5.t
niT
s
:
i
* s-9{
-ii
dFlFtFl
Gri
i+*: 5qn i
;FlFlc{
Ft€\
Fl
6lc)
$
$
rO
; IBF,$$ !s.$ s E tf- $ft f
I
<O
,Nd
:o:
,H
d
.H
-i
I
.ttNd
o++
Hi
di.,.
_\'6t
6-
^
.il
o'e";,, ,3
-ro .()
11 o'.6;;
r\!
.ri:!:.-i:,
v?
61r
=
Si
st
€
;.
I
.tq t"? t'
I*sie.
ri
g.f* r S tt
1tt5$
g'se.-<
I .-LT
liv
c\
.{i ro (o
-+
E
66
@o)
\--!
.H
ats
E
F
,E
A
a
<
E+
s. ,i
-\'tr.
kll
c{^
u)
d
.a
ll
. d 'i
o.9
o
r{
J I + rl-.=. oril ca'
+'si!
F'3'R ;I : YiE
E F-l$tr
i:s
oo€
+ r fl6^e{- a=(l)(o
O H -j 5 cB.A. E 1+
lfeq
;--so{.<
?BFsT
+t IT.r*3 x- =r
I
.o
{5
-Ai
Edr
s-F{'.:++.i{,
:
"o
e I't'; N
'' .
'
. ': '*I ..i .'t',,':"
',ot
.. rlt-^-='i{r'ilJi"{r
,fr1 .$',
€f
.'-.'
:r.
_
_o
g
1: $ff+ i+g
L
L. ttEii$u pl*ur* ; sglfl: i}.Bss$$B$E$HT
l-l_
l-^ x EE *a .E xE ir 9
u.
t_l_t_-
Ij
u
l_l_t:
t-t-.
t-.
u
L.
L
,rtgN
"€+ i
*EF
=*{;
<,
i
e
B
3
ar1
";
a
-ic{c'i{'u'dsa";S
*t
d
'$ tt T o '+-e
F nng * $B
: Bf,t
*tr",
3€E
-iE-;
XK R
tl
otfh
i
T E
;
ur
{*
6l
AG,
\8.
..:
uE-
r<,
.
J
l-J
-J
J
J
-j
q
o_
:'
-d
-{
<r
:-
:-
<
<
<
-:-
--J
.
__
4'
rg
a
rr)
C.l
i<
<rt
Eq
Hna
f']
EY
of
os
de
q{
frf E
Ho8
I
L
cto
ASc-ro
{' fie^
<
--'3
*<", o.oi
\\
d
t
E
k*3€
e{
s
S-'
, 's'
-L
>
+
o
.i ci c'i
r{ r{ Ft
-\
if;9
sGa+ {a
figEEli ?
gE*g
$tttt -tlatE+{ is+"
* qt.,B-T; E e iT iATq*B
B*u ffi{i ligdg
tt+t
s i E:
F{
ir{Ht{ei cri
*t{:Et*{qttgffi+++sEEE}l3iE3i
ttr iit * " qi??::{?EFFI{88
S 5gie3R*RR
L=*f
t{$gf}E
d
E;'
:
:
E
t:
t-.
t-.
u.
t--
L
tj
l_-
L:,'
1-.
t_-
u
t:
]J
L
t-L-,
1--t
t*
l_-
*€'
9H
,'d
o)-
Eal
{t}
'8Td.s
6IO
'P
fl
-P
I
ch
o'
HA
4
,s
Dd
AV
H
.H
.g
dl
(1)
!)
H.d
EE'
€^j
oq.
.dvt
E
iD
.k
'..9
:\.
,d
rD.l
'd
o
dp
.dLg)p
e-o tr+
.E
Edd,E td
itrh
o
3
srE B
'sl-c a-i1
9;S-f-X..8
.E
c.i 6.j 'd, rrj d t- cti
orererelolNC\
po
lcYSoSE
'u
H
o,d'E
EEI
fr
o, \\
E.
a-
G
tr-x
8
a :E>
5
i-3'^ >{
E B^q\
ft?-3
E
ExF=sHg
E
%
\A
B
e-E
5<;
YT
j
.(O
-c
'tr
{
g
6l
(frf
-r:
,rH
d.s \
't- 's'\
€.€
\€
Hg J4
l!\
.^.":.
BH L tlF I
d -i ;.S
c'l$o
Ft \.C
ii ^o\^ts .f
<.8
i56Eil
'ir
S:
q: L
?
.Yi + ui c,
NCrl6\6\
i-6
(\lC\l
r
i
6r
'\'f
^
-O
d)C)r{
NO?GA
l'dui?
-i*rqro*rqEi
'1
u ;; $ =.;5 lr=3
.i Of
c{
C.f
"i-
6l
b
ot
tt
+
o
+
L
;
g:
,*-ot
hs
hs
'\*
q-l'o
c')-l'o
+d'
%-t
blb
lh-
tir
-o'
I
C\f
Gi
H:
%B
lo
(',
*rs
u)*.^
6.1
Fl(O
-.-cO
I
:*'
O
O
u2
d .S
F,
a-
^ll
>
rHE^*H
oe.
9tgt"
iE
*-
+3 qsa
.:, {s
g+
9.r.t};
I -ts!.od
=
'... '3-3
,tj
c.i .{l
d) ga cr)
galoq
o),
..o
o -f:t
.o
E
cY'
d
*i13i*,i**
iEEi i [
ri d d ?g=* siPsi, iH gs33RF
co
F
E
lll\r ^. H"-R'i-i H 1
lF B=d e +lD
s ;siT
-o) g*sl gT s,
i
9i' t'qF s z6> .-.c' .t F 1 -\i.-\*o)vc!ff*f*-o),
Ekb Ei i E*. G-t
ur
Shee
F;TIFIU$3
g:^-L
dlcog .o
"Bs,S
f;;?FE.'3$-=E$?g as i ^q-T3t
_bG:
] A,rrq-- _b
!=$Tar E aT
+ ?r
Pru
SS5 3ETF
ilE E E-} flH -";E,3 T*T
ETF tt--\
ol\
q
^I
htru{slFi
-E
-.. *,2 l- q.5 s>
qSvs
tt € 'l,"re
8ES
il's
I ;.r"El^t9s!,
ri el 'E>l
d t;
,i
o
rB
4.U
<,
,F{
.H.
'o.'
'o 'o)
E
E.E
o=
u) o'
.<{
A:Q
co ro
A.
'5(o
oj
.o
X
6r ,o
(O F{
oo-:O) .r()
I t,.3 t
*E
F E <P;5
"o-€*n'-f
Ef TPx
l rJ)
-d.;og6i - T,i
{ut' dF
i*$$iii
l-: E**eiE il r iu at 3=
l-- tt
lj 16
i-t- *
i{sc$e:3{g3-E!
s*1*
b
3: t?
Il---, !€Es
L *it;;< ciF
l-- ;.*'i S$ 6 Eo },?B}BEE$ tI tI}E; E$I$3* E
l_- *frF: ? gE
l-- tss**s F.g
L
l-.-
I.*
L.
u
tq *b
[s
3t'e s
n=
tt
J-
sa ei
ror a
ttr s
;F LJEeq
'E
F[t*t EB[*+*
H*l a.I**tffig
Bl
eI
oi
s
i.*
d-i
co(y)
€*
€
3.E
'\
tt)
.lS
i*L-qs-
El
to
A
n
-
aD
tri
J{
cl
'6
a-
id
c
t
=
A
I
6l
rB
a-d
C.E
t{. s2
o€B'
E
.-a..{ o
6p
H
E
E
H{S
_*5G
$ss$ssqu$ssilssxn
E$ I
Bi HB HEB
*EE
N
c}
<,1
ci
cD
6r E
s
:EL
BS
E';'
cr
kLl
..-
i-
s
tr
:
H
gTE
+q$$$stqss
fi
P-r$$sig?Ee l'$ :
H5T3T{A{E€[},.
$ it{ita*t.i*
-
3c!\"i{ t $I
+
=,
Et?rsb
g
;s
r*lti-F- +s-.i
Rfrtt$
L *^s
Ec.q.5
.l.sxip
r- \T
-
.A
* 3*
!.9
IItgEl:EiHE$E$*fiq q$la$t fu,$$.
o;
$p$
ci
tago
rf
-J
-J
..J
J
!r
J
-r
_l
-<
'?
<
-
.-'
-.-'
-=.o-
-k
=-{
.,:--a
-
<
J
J
I
-.,1
L.
:c
I'o
S-\
h€
-5' ^
o-Q
-. c\
-L,
'-f
.,:
'c(
A
cn
2
q-ko
+^
_2
-o
b
o
-qt'
:-r!,
8-
b
o)
o
e.
ro d
39-5
C
,E E
t;i
i E;h *f
'-E
=c5
6a
sr> 6;
Sfrr '.. c\a!9
{+R-
-x?
I -'-T
ixlt
,olt7
t"
,Q,'Xi
?Qi
,s
3'>
d(r
e**/c\r ; r;3t'i\+ilt
.E;
-E
oc;^
ts=
'l r?p:?$?',ggb+*
l=-l
-r- * -ic
6l
^-\*
<,
-HC\lo
I vo6t
a.
-..=
oo
d
raj S:{
- .,C -|- a
has
tBp€-;<
:{ _--*8..i---* E
+5oq
'+
5
a- 53=.==:,S3ft-'Hl
g
\
r
",
"i
v)
E
H
"i
o
sq
-
cil
EuE=
E'iS
$s f,i.$$ €t€i
S+Zdl
?'E
\dL
k.ac-:
-o
+-lt;
Eo-e
.d.g1r
.cilH\i(J
+
- - .--i
l-- o
Cq c'f .<r. rr)
?Nil . ir l- -H 5r5:-'-ko
-rq
ls*
'-- l-SiSqHtt
Rli€
*:i I
+l* 1 tl .r-l-^ q
<ts...l b"t 53 n E'-i E-<i3
g
".qSS,-cD . i
JJE =I*.=i ?t
J'
o.
@
c.t
tfi-sts-iiHEEsEE:r
8* ?:'113s3+t
S
-LN
o
E
c
a
g,
d
o
';-
s,-'-q -l<--tavvvv
c( ;+<B=E-Eg
-c
"g
:sI
H
icd.rj
co r..) (o
:.
--:>'
-ob a:
'i
'-g
.6
5
1l
(
=
' .,'..' .) Pr\ 9
€E
Fr-nE
o
- .lI
'!q,
gr--<
ol
r,I
EJ
fr.
&
(o @ (c @-
o
t-
.jci'
F F
ci.qvi
b-.t- t-
d
b,
'
s.
(,?
.3
+
g
..\co
P€o-
.
-6'+
5'o
.9-€9
'
....5
.,-.oa-*1
| 6\-:>-.-a
E
<)
c:
q)
-+
H
d
s
,
:T;€ffi*,tffi
T_i
a
T.i€f
'1
)
.i
d
3
t*e
n
s$;si I+ t
s{
E-'si8
t- tsb-
d o'O q,-t- ? .S
F did
c;.j
q)@
i*Hx$*sss---;'i$ "?iQ*=;
'>
<oF66d
F-*i$ip g,t$=aI'iS+
i"-E#,
Sd.lJ.6tf'E
?t g?{S €r: E :fE€S Ni {{;;
*tE'
$ _{I
- r-S' -g
*=s_i.g EitBe*j:S?
iTis3;s n g
Eii gB 'g
- --i
trlorY ,,.
E+t
Ei
s;-9 - 3F
s
f, at3sE $f
e, c o. I oeS E'q
HsXti.i{ $
E
db-dtdct.i"i*;*
@,o<r'6
ro ro ro ro (c
f-cil
f)