Vibrations
1 Introduction to Vibration and the free response
4 Harmonic Motion
2
2
f Free Vibration
Viscous Damping
Response
to
t Hfi HE
t
Harmonic Excitation
2
Harmonic Excitation
Harmonic Excitation
3
Base
of
of
sI Wu
4
t
Undamoed Systems
Dampeld Systems
Forced Vibrations
Forced Vibrations with Damping
Excitation
4 Rotating
3
Energy Method
4
Multi Degree
Unbalance
Lagrange's Method
m
rn
mm
of Freedom System
1 Two Degree of Freedom Model Undamped
a stiffnessmatrix b eigenvalueproblems c modalanalysis
2
un
Two Degree
3 ThreeDegree
of
of
Freedom Model
Freedom Model
5
Design for
6
Distributed parameter Systems
Vibration Suppression
ViscousDamping
I
g
s
1
Wu
un
Y
Harmonic Motion C FreeVibration
K
win
THEE7 7
2
Viscous Damping
to
m
M
mu
C
3
Harmonic Excitation
k
WW
m
of
UndampedSystems
4
Harmonic Excitation
k
Focuswt
ofDampedSystems
m
run
Fo lot
c
5
Xcti
e
m
C
yet_Ysinwbt
k B
a
Mo
Xu
mm
K
Xrett
6 Rotating Imbalance
Base Excitation
c
4
Multi Degree
of Freedom System
Multi Degree of Freedom System
Xict
Kl
Ks
Mi
run
exact
run
Mi M 1kg
Ks
Ma
k _fa kz 4Nfm
run
Xiu
Mi
fax c
IF
ka
XD
I
taxi kamiXD mini
fq kz Milf
la
Kitts Milf
fa
ka
4
kayos Maw
ka
mimzw4 w
wueJwt
lol
IYai.ILYI.IE
suite
4 4
o
zeyewt
Q
ii
o
15
4 4
4 4
0
C fa.mk famz mikz MIKD fakztkzkstf.sk
0
a.fi e tazffge
cosc2t 0i
b.fi
Ans
FBu
µ
wit 16W 148 0
wir 8 16448 8 14 4 16
2
4 WE16 Ans
u
e
I
Y
4444422 0
0
UH Uh
o
2B
wt_Azffyedttaqffyedt.i.w
ba
I I
f
i
Hui
WE_12
mzwfacoo.lt
ki m w Ekittaks kamaws
McMaw4
wit
enI
4411 1
O
TattsMIWJ.lk ks Msw7 fE
MiwaKa Maw
0
aewtf4CoilYI gBa
e.x
ItfkIfI fEjaewt
kotkz
ksXz
i
HI I.es a
fmoiomjcw.ae
Mz
kztxzXD ksxz mzn.la
mark fax Clatters Xa
Gift
mo'Ll
takateaks
k2CX2Xi s
IF
mini 1 HattaXi faux o
To D
det
s
712th
Ict bi
f
f
U2
25
coscarst 04
Ana
cosktOD bzffy.co.kr 04
Ans
Xict
ki
K2
m
Wvu
M 1 KX
Ma_1kg
Ke 24mm Ka 3mm
Mi 9kg
m2
Wvu
Xeco
Tmm
Xoxo
Omm
K
Let I_µeJwt
CIE
way.eJwt
w mittatka
ka
det
y ye
e
I
E xx
wt
µe
23
qui
0
ufMztkz
27
926
WE 4
4
Uu Yzma
3
31
i
Uj
f
Uk
Uj
f 43
uz
4 6
9Uzi 3422 0
3Us U
0
1 1 Col
3412 0
3A tuk O
4.9124 3
la
24 137W 172
9hr4 54hr472 0
i
3
ii
fcf
e
3
2 912413
jw
W4mrmz wtmikz fawmz kikz
fczwmz.tk2
o
Minnow't Cmilatmolatmik4hftkikeo
9w4
Ico
is Wf 2
o
momma
ix
Xslt
U2
o
U22
s i
to besolved are linear
of any solution is also asolution
Since the equations
Thesum
0
w4 6W 18 0
Wf 2 WE4
WE12radIs Ans
w _IEradls
I µeJwt
y Ue
Ys
4g
Aisincwit101,1
y
Ww
Aasmcws.tto4
Aisincettoh
Azsmczttok
Xrco
Xoxo
Xeco
Tmm
Omm
O
1201 0
g Aesthattolz
YgAlSino bz.Azs.sn0z
AlSMO Az.snOz
E Acos
2
2
Aicost
X A sincwnt101
x.ae
mn tkx 0
Yz A
tO
smcktton
Oz
az.smcz.tt
Ai
sn
Let
M
01 02 742
I
7
A 312 A
312
Ai 112 0
o
FAzcosolz
Ai A 3
X YzHz Sm Ft192 g
o
2 As.gsoa o
SMFH
t
Ai cost 2As.coOz O
3 9 142
i 4A cost 0
AFA
4Aa 02 0 70 2 742 Ansi
0.51Cos
o
1 5 Cosft
32
snotty
3G sincatttyz
O 5cost
1 5 cost
Co5 cos 1cost
c 5 cost cost
Ans
Ang
Xict
close All
clear
clc
Ki
win
%% Information
m1=9;
m2=1;
k1=24;
k2=3;
M=[m1 0;
0 m2];
K=[k1+k2 -k2;
-k2 k2];
x0=[1;
0];
v0=[0;
0];
t=0:0.01:20;
Htt
Xslt
m
Xact
R2
Wu
coset cost
g gg cost cost
ma
Xeco
Tmm
Xoxo
0mm
o5
Ans
23
%% Calculation
w1=sqrt(2);
w2=sqrt(4);
A1=3/2;
A2=-3/2;
Phi1=pi/2;
Phi2=pi/2;
x=[ (1/3)*A1*sin(w1*t+Phi1) - (1/3)*A2*sin(w2*t+Phi2);
(1)*A1*sin(w1*t+Phi1) + (1)*A2*sin(w2*t+Phi2) ];
wiraradis mode cw radio
Myode
y
3
%% Plot
plot(t,x(1,:), 'linewidth', 2)
hold on
plot(t,x(2,:), 'linewidth', 1)
legend_entries={'x1(t)','x2(t)'};
legend(legend_entries,'location','northeast');
grid on
xlabel('Time (s)')
ylabel('Displacement (m)')
title('Two-Degree-of-Freedom Model (Undamped)')
If
M
1kg
Ke 24mm Ka 3mm
Me 9kg
Xeco 920
0
Xiu
KI
I yeah
Let
ix
R2
m
run
XI
ma
run
y ye
MI 1K
w
Ui Uz
F
14
z µk
ki
Ltg
QMM
kq.tkUkq Q
24mm
1
Q
i massnormalizedstiffness
k y wtu
kµ
µ
MEtkI Q
3X
qµeJwt
g Heat
3
I
wayeat
5
pacharacteric
Un Un
Mutua
I
XI
the eigenvectors are independent
1 2 GH
Kl
E I
Q
spectral
1
11
1.0
2
orthogonal
plc
A diag
I
0
UH Uss O
Un U o
UH Use
E
Uz Yrz
it
I
ft
it
s
4
AI Q
69 t Id fo
in tfwoiowe lTI fo
6919
ii
l
WEN O
wer
o
pet modalcoordinate
y µkq
go
Ltpr
Sr
HE
4
2424
s
Uu Uiz
norsmalize
H
L
40
Iii fol
o
o
F'KP
iii Xz 4
eigenvector
YTUs
a
a 2107470
i Xi 2
l l
t 1
at o
W eigenvalue
MINEO
it
E PE
PEtKPr o
ptpr.it
I Atf _Q
o
l
3X
1 17 6 8 0
p
KI
IEtk
Xt 2 12 4
wa.yiedwttk.ae o
CWAITE eat Q
y
a
Kailua
I tk Q
c wItkTa e
ft
Ct
mum
q
Unni
ptp I
F 39
qtMkKMkq Q
µ wµ
D
It
TspEfomatrixofkM
L
kMMk
K MkKM
D
Mel kg
kz 3Ntm
E EKE't's E33kg
MItkx
IqEtk
orthogonalmatrix
73 3
1 9
f M 439
1
ft Mk_31439
9
ring
1
Modeshape
Mi 9 kg
ftp
E
KIEL
Wi Wa Naturalfrequency
Cholesky Decomposition
f Mk
K
A mo om
tµe
w µeJwt
_Q
raise
i
GEDI
f EP
Mix Kx
ix twnX
X
x
in win
o
O
RE 1 WER O
A sincwnt141
p
uilwnisincdkhutttantlwmut
Y mm
Taco Theo
Omm
Xrco
f Mk
r
fr
E SE
i
oDXico1
iso7wz.smcwztttantlwF
0
300
f Lip
2 5 Pr
r
Lx Pe
Io lo XI fo
Io Ks
net friotrilwi.smcwitttantfwj.TO
B
ft
P
o
34kt
9 I
fact
PTLEPTPE I
s
pTLx
zygy
3
mm
ri fo
own
nets drifting smcwitttanifwln.fi
tfwj.IO
2rzcts
frI railway smcwztttan
net
3
Sm
3
smart 1
3
3
7
2
21
Ans
y
o
It
i
f Lip
t
Krs set
Ks asst
ft
f's E
Ifs tf costat
3G cost
Ans
L Mk 11 1442
K MKKwik
8 Yet SEH
2 51
L tpr
bog
III
Ypgcost
3
coat
0.5 coset1 2T
l b coset cost
Ans
W
Y
4 P Cid ka
5 S Ltp
f PTL
6 Ico ftxco Ico f Ico
7
4Wi
f2WI
7
smC2ttTy
3 detckXI
2Xa
Ft tanto
3 Cos Et
Det
gXi
close ALL
clear
clc
%% Information
m1=9;
m2=1;
k1=24;
k2=3;
M=[m1 0;
0 m2];
K=[k1+k2 -k2;
-k2 k2];
x0=[1;
0];
v0=[0;
0];
t=0:0.01:20;
%% Calculations
L=sqrt(M);
Kmn=inv(L)*K*inv(L);
[v,lamda]=eig(Kmn);
P=v; %check P'*P=I
S=inv(L)*P;
r0=inv(S)*x0;
rdot0=inv(S)*v0;
w=sqrt([ lamda(1,1) lamda(2,2) ]);
for i=1:2
r(i,:)=sqrt(r0(i)^2+rdot0(i)^2/
w(i)^2)*sin(w(i)*t+atan(w(i)*r0(i)/
rdot0(i)));
end
3
2
I
x=S*r;
%% Plot
figure(1)
plot(t,x(1,:),'linewidth',2)
hold on
plot(t,x(2,:),'linewidth',1)
legend_entries={'x1(t)','x2(t)'};
legend(legend_entries,'location','northeast');
grid on
xlabel('Time (s)')
ylabel('Displacement (m)')
title('Two-Degree-of-Freedom Model (Undamped) - using Modal Analysis')
close ALL
clear
clc
MATLAB from Laptop
%% Information
m1=9;
m2=1;
k1=24;
k2=3;
M=[m1 0;
0 m2];
K=[k1+k2 -k2;
-k2 k2];
x0=[1;
0];
v0=[0;
0];
t=0:0.01:20;
I
%% Calcaultions
L=sqrt(M);
Kmn=inv(L)*K*inv(L);
[v,lamda]=eig(Kmn);
P=v; % check P'*P=I
S=inv(L)*P;
r0=inv(S)*x0;
rdot0=inv(S)*v0;
w=sqrt([lamda(1,1) lamda(2,2)]);
for i=1:2
r(i,:)=sqrt(r0(i)^2+rdot0(i)^2/w(i)^2)*sin(w(i)*t+atan(w(i)*r0(i)/rdot0(i)));
end
x=S*r;
%% Plot
figure(1)
plot(t,x(1,:),'linewidth',2)
hold on
plot(t,x(2,:),'linewidth',1)
legend_entries={'x1(t)','x2(t)'};
legend(legend_entries,'location','northeast');
grid on
xlabel('Time (s)')
ylabel('Displacement (m)')
title('Two-Degree-of-Freedom Model (Undamped)')
Linear
Transformation
4.1
41
4
41 411
141 1 411
A
r
co
a
39
c
1o
s
a
so
dy
8912 1
H
TX O
too
Hoo
c Unto Upto
O Uu 10 Uk o
i
391191 191
no
coin
bill
cm
b
a
lol
a
E Mn
4 1112 0
o
c'oil
til
l
o
I'Ea
o
0
841 1 04
O Uu11.412 0
6114147
O
I IIE
39141 41
a
o
Ex
gftp.un
o
T
o