The response of an airplane to a dynamic load
by Rodney Lee Gilge
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Aerospace and Mechanical Engineering
Montana State University
© Copyright by Rodney Lee Gilge (1970)
Abstract:
The response of an airplane to a blast load is studied. The effects of rigid body translation and rigid
body rotation are taken into account. Linear and non-linear solutions are compared. The non-linearities
result from use of a non-linear stress-strain relation and from geometry changes due to large deflection.
It is concluded that the response of an airplane to a dynamic load is definitely influenced by the effects
of rigid body translation and rotation. For large loads the non-linear solution predicts a larger wing
deflection and a longer period of wing oscillation than does the linear solution. S ta te m e n t o f P e r m is s io n to Copy
I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f th e r e q u i r e ­
m ents f o r an ad v an ced d e g re e a t M ontana S t a t e U n i v e r s i t y , I a g re e t h a t
th e L ib r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r i n s p e c t i o n .
I fu rth e r
a g re e t h a t p e r m is s io n f o r e x t e n s i v e co p y in g o f t h i s t h e s i s f o r s c h o l a r l y
p u rp o s e s may b e g r a n te d by my m a jo r p r o f e s s o r , o r , i n h i s a b s e n c e , by
th e D i r e c t o r o f L i b r a r i e s .
I t i s u n d e rs to o d t h a t any co p y in g o r p u b l i ­
c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a in s h a l l n o t b e a llo w e d w ith o u t
my w r i t t e n p e r m is s io n .
D ate
Q td ir ^
( Ql, f f f j Q
THE RESPONSE OF AN AIRPLANE TO A DYNAMIC LOAD
by
■ Rodney Lee G ilg e
A t h e s i s s u b m itte d t o th e G ra d u a te F a c u lty i n p a r t i a l
f u l f i l l m e n t o f th e r e q u ir e m e n ts f o r th e d e g re e
of
MASTER OF SCIENCE
in
A ero sp ace and M e c h a n ic a l E n g in e e rin g
A pprov ed :
H ead, M ajor D ep artm en t
C hairm an, E xam ining Com m ittee
G ra d u a te Dean
MONTANA STATE UNIVERSITY,
Bozeman, M ontana
D ecem ber, 1970
ill
ACKNOWLEDGMENT
The a u th o r i s in d e b te d t o th e N a tio n a l S c ie n c e F o u n d a tio n
w h ich p r o v id e d f i n a n c i a l a id f o r t h i s p r o j e c t .
G r a te f u l acknow ledgm ent i s a l s o made f o r th e c o n tin u in g
g u id a n c e and a s s i s t a n c e o f D r. D. 0 . B l a c k k e t t e r .
iv
TABLE OF CONTENTS
CHAPTER
'
- p ag e
I.
INTRODUCTION..............................................................................................................I
II.
FORMULATION OF THE P R O B L E M ........................................................................3
SYSTEM D ESC R IPTIO N ..........................................................................................3
EQUATIONS OF RIGID BODY M O T IO N ........................................................... ' 3
LINEAR EQUATIONS OF WING MOTION .
.
,
. . .
.
.
6
NON-LINEAR EQUATIONS OF WING MOTION.....................................................7
INITIAL AND BOUNDARY CONDITIONS .
10
DESCRIPTION OF THE FORCING FUNCTION .................................................. 11
III.
METHOD OF S O L U T I O N ....................................................................................... 15
MODIFIED GALERKIN M E T H O D .................................................................... 15
APPLICATION OF THE MODIFIED GALERKIN METHOD .
IV.
RESULTS AND CONCLUSIONS
.
.
.1 6
...............................................................................19
UNIFORM L O A D ......................................................................................... . 19
NON-UNIFORM LOAD
FURTHER STUDIES .
.
.
.
.
.
.
.
' .
.
.
.
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22
■.
.2 5
.
.
.
..............................................
APPENDIX A.
ACCELERATION EXPRESSIONS
28
APPENDIX B.
LINEAR EQUATIONS OF WING M O T IO N ...................................................29
APPENDIX C.
NON-LINEAR EQUATIONS OF WING MOTION
...................................... 30
V
LIST OF FiaTRES
F ig u re No.
T itle
Page
1
The A ir p la n e Model .
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4
2
F o rc e s on th e F u s e la g e .
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5
3
Beam E lem ent
.
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8
4
B l a s t P r e s s u r e D ecay
.
5
A r r iv a l Time o f B l a s t
.
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14
6
S t r e s s - S t r a i n C urve
.
.
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18
7
T y p ic a l L in e a r and N o n -L in ea r Wing T ip R esponse f o r a
S m all U niform Load
.
.
.
.
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20
C om parison o f L in e a r and N o n -L in ea r S o lu ti o n s f o r Large
U niform Loads .
.
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.
.
.
.
21
T y p ic a l L in e a r and N o n -L in ea r S ystem D is p la c e m e n ts f o r
a S m all N on-U niform Load
.
.
.
.
.
.
23.
C om parison o f L in e a r and N o n -L in ea r S o lu ti o n s f o r L arge
N on-U niform Loads .
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24
8
9
10
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12
vi
ABSTRACT
The re s p o n s e o f an a i r p l a n e t o a b l a s t lo a d i s s t u d i e d . The
e f f e c t s o f r i g i d body t r a n s l a t i o n and r i g i d body r o t a t i o n a r e ta k e n
i n t o a c c o u n t. L in e a r and n o n - l i n e a r s o l u t i o n s a r e co m p ared . The
n o n - l i n e a r i t i e s r e s u l t from u s e o f a n o n - l i n e a r s t r e s s - s t r a i n r e l a ­
t i o n and from g e o m e try ch an g e s due t o l a r g e d e f l e c t i o n .
I t i s c o n c lu d e d t h a t th e r e s p o n s e o f an a i r p l a n e t o a dy­
nam ic lo a d i s d e f i n i t e l y in f l u e n c e d b y th e e f f e c t s o f r i g i d body
t r a n s l a t i o n and r o t a t i o n .
F or l a r g e lo a d s th e n o n - l i n e a r s o lu ti o n
p r e d i c t s a l a r g e r w in g d e f l e c t i o n and a lo n g e r p e r io d o f w ing o s c i l ­
l a t i o n th a n d o es t h e l i n e a r s o l u t i o n .
CHAPTER I
INTRODUCTION
D esig n lo a d s f o r a c c e l e r a t e d a i r p l a n e s t r u c t u r e s a r e f r e q u e n t
I y b a s e d on th e a ssu m p tio n t h a t th e w ing i s p e r f e c t l y r i g i d . ' T h is
may le a d t o f a i l u r e due t o dynam ic o v e r s t r e s s .
F o r ex am p le, a g u s t
lo a d may p ro d u c e w ing b e n d in g moments a t th e f u s e la g e t h a t a r e 1 5 20% g r e a t e r th a n th o s e c a l c u l a t e d on th e assu m p tio n o f a r i g i d w in g .
Dynamic lo a d s c a u s e t r a n s l a t i o n and r o t a t i o n o f th e a i r p l a n e a s a
w hole and a l s o c a u s e v i b r a t i o n s o f t h e s t r u c t u r e .
Dynamic o v e r s t r e s s
i s p ro d u c e d by th e a d d i t i o n a l i n e r t i a f o r c e s a s s o c i a t e d w ith th e
s tru c tu re v ib ra tio n s .
The lo a d d i s t r i b u t i o n on th e w ing i s a l s o a f f e c t e d by th e
w ing d e f o r m a tio n and v i b r a t i o n .
D e te rm in in g th e lo a d d i s t r i b u t i o n
on th e b a s i s o f a r i g i d w ing may l e a d t o r e s u l t s t h a t a r e to o much
in e r r o r t o be u s e f u l .
T h ere may a l s o be s e r i o u s l o s s o f a i l e r o n ,
e l e v a t o r , and r u d d e r c o n t r o l e f f e c t i v e n e s s due t o d e f o rm a tio n o f
th e s t r u c t u r e .
In t h i s p a p e r o n ly th e a i r p l a n e re s p o n s e t o a dy­
nam ic lo a d w i l l be c o n s id e r e d .
The e f f e c t o f t h i s r e s p o n s e on th e
w ing lo a d d i s t r i b u t i o n and c o n t r o l e f f e c t i v e n e s s w i l l n o t be a n a ly s e d
The re s p o n s e o f an a i r p l a n e t o dynam ic lo a d s h a s b een
fre q u e n tly s tu d ie d .
In some s t u d i e s ( 1 1 ) th e a i r p l a n e was co n -
1 Numbers i n p a r e n t h e s i s r e f e r t o l i t e r a t u r e c o n s u lt e d .
-2 -
s i d e r e d t o be p e r f e c t l y r i g i d .
In o t h e r s t u d i e s
(7 ,1 0 ) th e w ings w ere
c o n s id e r e d t o be e l a s t i c b u t th e e f f e c t s o f r i g i d body t r a n s l a t i o n
and r o t a t i o n o f th e a i r p l a n e w ere n e g l e c te d .
B is p li n g h o f f (4) and
H o u b o lt (5) in c lu d e d th e e f f e c t s o f r i g i d body t r a n s l a t i o n in t h e i r
s tu d ie s .
However, t h e y c o n s id e r e d o n ly dynam ic lo a d s t h a t w ere u n i ­
form a c r o s s th e a i r p l a n e so t h a t no r i g i d body r o t a t i o n o c c u r r e d .
Most o f th e s t u d i e s t h a t have b een made do n o t in c lu d e th e
e f f e c t s o f b o th r i g i d body r o t a t i o n and t r a n s l a t i o n .
The p o s s i b i l ­
i t y o f l a r g e n o n - l i n e a r w ing d e f l e c t i o n s i s c o n s id e r e d i n o n ly v e r y
few s t u d i e s .
The p u rp o s e o f t h i s p a p e r i s t o s tu d y th e re s p o n s e o f an a i r ­
p la n e t o a dynam ic lo a d .
The e f f e c t s o f r i g i d body r o t a t i o n and t r a n s ­
l a t i o n and w in g v i b r a t i o n s w i l l be c o n s id e r e d .
A c o m b in a tio n o f th e m o d if ie d G a le r k in m ethod and Hamming's
m o d ifie d p r e d i c t o r - c o r r e c t o r m ethod w i l l be u s e d t o s o lv e th e gov­
e r n in g p a r t i a l d i f f e r e n t i a l e q u a t i o n s .
CHAPTER I I
FORMULATION OF THE PROBLEM
SYSTEM DESCRIPTION
The sy ste m s e l e c t e d f o r s tu d y i s shown in f i g u r e I .
in te n d e d t o be a s im p le r e p r e s e n t a t i o n o f an a i r p l a n e .
o f a f r e e - f r e e beam w ith a lum ped m ass a t th e c e n t e r .
It is '
I t c o n s is ts
The lumped
m ass s im u l a te s th e f u s e la g e o f th e a i r p l a n e and th e l e f t and r i g h t
p o r t i o n s o f th e beam r e p r e s e n t th e w in g s .
The dynam ic lo a d a c t s
a c r o s s th e e n t i r e w ing c r o s s - s e c t i o n , b u t o n ly th e r e c t a n g u l a r p o r ­
t i o n o f t h e w in g i s assum ed t o c a r r y any lo a d .
The m a t e r i a l p r o p e r ­
t i e s a r e assum ed t o be hom ogeneous th ro u g h o u t each w in g .
T h is sy stem
i s i d e n t i c a l t o one u s e d by B i s p l i n g h o f f (10) in h i s s tu d y o f a e r o e la s tic ity .
T h ree c o o r d in a n t sy ste m s a r e u s e d t o d e s c r ib e t h e m o tio n o f
th e s y s te m .
The f i x e d c o o r d ih a n ts X-Y w ith u n i t v e c t o r s i and j a re
u s e d t o d e s c r i b e th e r i g i d body t r a n s l a t i o n and r o t a t i o n o f th e s y s ­
tem .
The m oving c o o r d in a n ts x ^ -y ^ and x ^ -y ^ w ith u n i t v e c t o r s i_^,
and i ^ , j . r e s p e c t i v e l y a r e u s e d t o m easure th e d e f l e c t i o n o f
th e w in g s r e l a t i v e t o t h e f u s e l a g e .
EQUATIONS OF RIGID BODY MOTION
The resp o n se o f th e system can be s p e c if i e d by fo u r eq u a tio n s
o f m o tio n .
The e q u a tio n s d e s c r ib e th e r i g id body t r a n s l a t i o n , th e
-AO r ig in a l P o s i t i o n
D is p la c e d P o s i t i o n
Wing C r o s s - s e c tio n
F ig u re I . The A irp la n e Model
-5 r '.q i d body r o t a t i o n , th e r o t a t i v e d is p la c e m e n t o f th e l e f t w in g , and
th e r e l a t i v e d is p la c e m e n t o f th e r i g h t w in g .
The two e q u a tio n s o f
r i g i d body m o tio n can be d e te rm in e d b y c o n s id e r in g th e f o r c e s a c t in g
on th e f u s e ] a g e .
The f o r c e s a c t i n g on th e f u s e la g e a r e shown in f i g u r e 2.
It
i s assum ed t h a t th e w e ig h t o f th e a i r p l a n e and th e l i f t on th e w ings
c a n c e l each o th e r.
T h e r e f o r e , t h e s e te rm s a r e n o t in c lu d e d in th e
d e r i v a t i o n o f th e g o v e r n in g e q u a t io n s .
A n o th er s im p l if y in g a p p r o x i­
m a tio n i s t h a t no f o r c i n g f u n c tio n a c t s on th e f u s e l a g e .
X
--------------
V
F igu re 2.
Y
F orces on th e F u sela g e
—
6—
The e q u a tio n f o r r i g i d body r o t a t i o n can be o b ta in e d by sum­
m ing moments a b o u t th e c e n t e r o f th e f u s e l a g e .
r (V1 -
V
-HM2 -
T h is y i e l d s :
(I)
/ 1M
w here TJ/ i s th e r i g i d body r o t a t i o n , r i s th e r a d iu s o f th e f u s e la g e ,
and V2 a r e s h e a r s , Mj and M_ a r e i n t e r n a l mom ents, and
m ass moment o f i n e r t i a o f th e f u s e l a g e .
A dot ,
i s th e
, above a q u a n t i t y
i n d i c a t e s d i f f e r e n t i a t i o n w ith r e s p e c t t o tim e and a p rim e , % i n d i ­
c a t e s d i f f e r e n t i a t i o n w ith r e s p e c t t o p o s i t i o n .
By summing f o r c e s i n th e j d i r e c t i o n th e e q u a tio n f o r r i g i d
body t r a n s l a t i o n c a n be shown t o b e :
Y = -
(V1 + V2 ) c o s TJ/ / Mb
(
w here Mj3 i s th e m ass o f th e f u s e la g e .
LINEAR EQUATION OF WING MOTION
The f u s e l a g e o f th e a i r p l a n e c a n b o th d i s p l a c e and r o t a t e .
S in c e th e w in g s a r e r i g i d l y a t t a c h e d t o th e f u s e la g e , any m o tio n o f
th e f u s e la g e w i l l r e s u l t in a c o r r e s p o n d in g m o tio n o f t h e w in g s.
In a d d i t i o n th e w in g s c a n a l s o move r e l a t i v e t o th e f u s e l a g e .
I f i t i s assum ed t h a t c r o s s - s e c t i o n a l p la n e s rem ain p la n e
and i f s h e a r d e f o r m a tio n and r o t a r y i n e r t i a o f b e n d in g a r e n e g le c te d ,
th e n th e l i n e a r e q u a tio n s f o r t h e r e l a t i v e m o tio n o f th e w ings can
2)
-7 be w r i t t e n a s :
Illl
I Y1 +
-E
- E I
(s e e a p p e n d ix B)
(xl r t )
y''" + w (x , t )
= u
= U p
<•
•I
I CosrY ~ r Y -
•.
X 1Y "
..
+
Y^
(3)
Y1 Y ^
cos Y + r Y + XgY + y
w here Y i s th e r i g i d body t r a n s l a t i o n , y
(4)
i s th e r i g i d body r o t a t i o n ,
x^ and. Xg a r e p o s i t i o n s a lo n g th e w in g s , y^ and y
a re th e r e la tiv e
d is p la c e m e n ts o f th e w in g s , W1 arid w^ a r e th e f o r c i n a f u n c t i o n s , and
u i s t h e m ass p e r u n i t le n g th o f w in g .
NON-LINEAR EQUATIONS OF WING MOTION
The n o n - l i n e a r w ing e q u a tio n s in c lu d e th e e f f e c t s o f geom­
e t r y c h a n g e s due t o l a r g e d e f l e c t i o n s and a l s o a c c o u n t f o r th e p o s­
s i b i l i t y o f s t r a i n s t h a t a r e i n th e n o n - l i n e a r p o r t i o n o f th e s t r e s s s tr a in cu rv e .
C o n s id e r an e le m e n t o f th e r i g h t w ing as shown in f ig u r e 3.
A gain assum e p la n e s e c t i o n s rem ain p la n e and n e g l e c t s h e a r deform ­
a t i o n an d r o t a r y i n e r t i a o f b e n d in g .
The e q u a t io n s o f m o tio n o f t h e w in g s w ith r e s p e c t t o th e
f u s e la g e c a n be shown t o b e : (s e e a p p e n d ix C)
m
IV I sin(2cf>J ) -- IM % ° s 2 ( V I
+ u T2(J)V
sinfE 4\)/
12 +
I
I
(5)
w ^ ( x ^ ,t)
= U
jjf c o s C y ) - r Y - X1Y
+ Y - 'y
—8—
M2 ^ o s i n ( 2 ( p c ) - M.
2 00sf^
-U
,
, u [y
c o s ( T ) + rijjr + x T
"
+ y
^
- y T^
2
(6 )
-
where T i s th e w ing th ic k n e s s , and 0 i s th e r o t a t i o n o f th e
wt no
in th e moving c o o rd in a n t system .
l—— — _____ I
f ig u r e 3.
Eoam Element
-9 The in v e r s e ta n g e n t f u n c t i o n c a n be u se d t o r e p r e s e n t th e
tru e s t r e s s - s t r a i n r e la tio n of a m a te ria l.
The s t r e s s c a n be ex
p re sse d a s:
O' = t a n
w here a~ i s t h e s t r e s s ,
(a C ) / b
(7)
£. i s th e s t r a i n , and a and b a r e c u r v e ­
f i t t i n g c o n s ta n ts .
A ssum ing t h a t th e n e u t r a l p la n e o f a w ing alw ay s p a s s e s
th ro u g h th e c e n t r o i d o f th e a r e a and r e c a l l i n g th e a ssu m p tio n t h a t
p la n e s e c t i o n s re m a in p la n e , th e s t r a i n c a n be w r i t t e n a s :
£ = z Cj)
(8)
w here z i s th e d i s t a n c e from th e n e u t r a l a x is t o th e f i b e r b e in g
s tra in e d .
The i n t e r n a l moment c a n be e x p r e s s e d in te rm s o f th e s t r e s s ,
and th e b e n d in g a n g le , (j) , c a n be e x p r e s s e d in te rm s o f y ' .
T h ere­
f o r e , b y p r o p e r s u b s t i t u t i o n s th e e q u a tio n s o f m o tio n o f th e w ing
e le m e n ts c a n be w r i t t e n a s :
W (X1^ t ) - F1
V
T lx V t '
v t l - F2 W
w here
and
a p p e n d ix C .
1
- U Y c o s (h |f) - r'Xjf - x Tjf + y - y-Y
I
I t J
- U Y c o s ( Y ) + r \ j f + X2Tjf + y
- y^jf
a r e n o n - l i n e a r d i f f e r e n t i a l o p e r a t o r s a s d e f in e d in
(9)
(10)
-1 0 IMITIAL and boundary co n d itio n s
The m o tio n o f th e sy stem i s d e s c r ib e d by f o u r d i f f e r e n t i a l
e q u a tio n s .
Two o f th e e q u a tio n s a r e p a r t i a l d i f f e r e n t i a l e q u a tio n s
t h a t a r e f o u r t h o r d e r w ith r e s p e c t t o p o s i t i o n and se c o n d o r d e r w ith
r e s p e c t t o tim e .
The re m a in in g two e q u a tio n s a r e o r d i n a r y d i f f e r e n ­
t i a l e q u a tio n s t h a t a r e seco n d o r d e r w ith r e s p e c t t o tim e .
T h ere­
f o r e , i t w i l l be n e c e s s a r y t o have a t o t a l o f 16 i n i t i a l and b o u n d ary
c o n d itio n s .
F o u r i n i t i a l c o n d i tio n s a r e n e c e s s a r y t o s p e c i f y th e
r i g i d body m o tio n .
The r e l a t i v e m o tio n o f th e w ings c a n be s p e c i f i e d
b y f o u r " n a t u r a l b o u n d a ry c o n d i t i o n s " , f o u r " f o r c e d b o u n d a ry c o n d i­
t i o n s " , and f o u r i n i t i a l c o n d i t i o n s .
The 16 i n i t i a l and b o u n d ary
c o n d i tio n s a r e :
I n i t i a l C o n d itio n s
R ig id Body T r a n s l a t i o n
R ig id Body R o ta tio n
I n i t i a l R e l a t i v e Wing D isp la c e m e n t
-1 1 -
O
Y1 ( X ^ t )
I n i t i a l R e l a t i v e Wing V e lo c i ty
V
0
v tl
B oundary C o n d itio n s
Y1 ( 0 , t ) = 0
R e l a t i v e Wing D isp la c e m e n t a t th e
F u s e la g e
y ( 0 ,t) = 0
2
f y N o,t ) = o
R e l a t i v e Wing S lo p e a t th e
F u s e la g e
JT g (O ft) = 0
( I ., t j] = 0
Moment a t T ip o f Wing
M[y 2 (L' t ) ]
“ 0
V[y (Lrt ) ] . = 0
S h e a r a t T ip o f Wing
v [ y 2 (L ,t)] = 0
DESCRIPTION OF THE FORCING FUNCTION
The f o r c i n g f u n c ti o n i s an a p p ro x im a tio n o f a bomb b l a s t .
The f o r c e on th e w in g s w i l l be in d e p e n d e n t o f p o s i t i o n i f th e s o u rc e
o f th e b l a s t i s a l a r g e d i s t a n c e from th e a i r p l a n e and th e w ings o f
th e a i r p l a n e a r e p a r a l l e l t o th e b l a s t f r o n t .
U nder th e s e c o n d i tio n s a b l a s t lo a d can be a p p ro x im a te d
v e ry c lo s e ly by:
w (t)
-/S t
P e
o
e
= W
(
11)
-1 2 v;here Pq i s th e maximum p r e s s u r e , Wq i s th e e f f e c t i v e w in g w id th ,
t i s tim e , and
/9
i s a b la s t decay c o n s ta n t.
I f th e w in g s o f th e a i r p l a n e a r e n o t p a r a l l e l t o th e b l a s t
f r o n t th e n th e lo a d on th e a i r p l a n e w i l l be a f u n c tio n o f b o th
p o s i t i o n and tim e .
The lo a d in g f o r su c h a c o n d itio n i s r e p r e s e n te d
B la s t C e n te r
F igure 4.
B la s t P ressu re Decay
-1 3 The b l a s t p r e s s u r e d e c a y s a s th e s q u a re o f th e d is ta n c e
from th e b l a s t c e n t e r ( 1 2 ) .
The p r e s s u r e a t some p o s i t i o n A can
be e x p re sse d a s :
P
(12)
A
w here P q i s th e maximum p r e s s u r e , P a
i s th e d i s t a n c e from th e b l a s t
c e n t e r t o th e en d o f th e l e f t w in g , p
i s th e d i s t a n c e from th e b l a s t
c e n t e r t o th e p o s i t i o n A, and P^ i s th e p r e s s u r e a t A.
The b l a s t p r e s s u r e a l s o d e c a y s w ith tim e .
At some p o s i t i o n
B th e p r e s s u r e i s :
(13)
PB - PA 6
ZL2= Po
where
/3
- / 9 ( t - t« .)
(14)
e
i s th e b la s t decay co n sta n t and t ^ i s th e tim e required
fo r th e b la s t fr o n t to reach p o s it io n A.
R e f e r r i n g t o f i g u r e 5, t,* c a n be e x p r e s s e d a s :
x * s in Y
w here Y
i s th e a n g le b etw een t h e b l a s t f r o n t and th e w in g , c i s
.
th e sp e e d o f t h e b l a s t f r o n t , and X ^ i s th e d i s t a n c e from th e end
o f t h e l e f t w ing t o t h e p o in t u n d e r c o n s i d e r a t i o n on th e a i r p l a n e .
S u b s c r ip t I r e f e r s t o th e r i g h t w ing an d s u b s c r i p t 2 r e f e r s t o th e
l e f t w ing.
-1 4 B la s t C e n te r
B la s t F ro n t a t Time t
B la s t F ro n t a t
tim e t + toe
F ig u r e 5 .
A r r i v a l Time o f B la s t
The e x p r e s s io n f o r t ^ i s v a l i d o n ly when th e r a d i u s o f th e
b la s t fro n t,
, is la rg e .
t h e e x p r e s s io n s f o r t ^
t
=
*2.
t
=
lIi
By m aking th e p r o p e r s u b s t i t u t i o n s
can be shown t o h e :
(L-Xz) c o s [ s i n f s / z M ]
c
(16)
(L + Z r+ x J c o s C s in V s //^ )]
c
(17)
w here s i s th e v e r t i c a l d i s t a n c e from t h e b l a s t c e n t e r t o th e f u s e la g e .
CHAPTER I I I
METHOD OF SOLUTION
MODIFIED GALERKIN'METHOD
One m ethod o f s o l u t i o n o f th e e q u a tio n s d e v e lo p e d i s th e
m o d if ie d G a le r k in m eth o d .
The m o d if ie d G a le r k in m ethod i s a v e r s io n
o f th e m ethod o f w e ig h te d r e s i d u a l s a s d e s c r ib e d b y F in la y s o n and
S c r iv e n ( 9 ) .
To s o lv e a s e t o f e q u a tio n s t h a t a r e f u n c ti o n s o f b o th
tim e and p o s i t i o n th e f i r s t s t e p in t h i s m ethod i s t o assume a com­
p l e t e s o l u t i o n t o th e s e t o f e q u a t io n s .
The co m p le te s o l u t i o n w i l l
c o n s i s t o f i n d i v i d u a l s o l u t i o n s f o r e a c h o f th e d e p e n d e n t v a r i a b l e s .
The in d i v i d u a l s o l u t i o n s w i l l be o f th e fo rm :
(18)
The
© (x) te rm s m ust s a t i s f y th e " f o r c e d b o u n d a ry c o n d i tio n s " b u t
do n o t hav e t o s a t i s f y th e " n a t u r a l b o u n d ary c o n d i t i o n s " .
The n e x t s t e p i s t o s u b s t i t u t e th e assum ed s o l u t i o n s i n t o
th e d i f f e r e n t i a l e q u a tio n s t o d e te rm in e th e e q u a tio n r e s i d u a l s ,
R( x ,t) .
S in c e th e assum ed s o l u t i o n s w ere n o t r e q u i r e d t o s a t i s f y
th e " n a t u r a l b o u n d a ry c o n d i t i o n s " t h e r e w i l l be b o u n d a ry e r r o r s o r
re s id u a ls .
F i n a l l y t h e w e ig h te d i n t e g r a l o f th e r e s i d u a l p lu s th e
w e ig h te d b o u n d a ry e r r o r s a r e s e t e q u a l t o z e ro f o r e a c h o f th e e q u a tio n s
R (x ,t) ©• (x) dx + E 0: (x) + E 0 : (x) = O
J
d o mai n
V J
m
J
(19)
-1 6 E i s th e b o u n d ary s h e a r e r r o r and E i s th e b o u n d ary moment e r r o r ,
v
m
E q u a tio n 19 w i l l y i e l d a s e t o f s im u lta n e o u s d i f f e r e n t i a l e q u a tio n s
in tim e f o r e a c h o f t h e te rm s o f th e co m p le te assum ed s o l u t i o n .
T hese e q u a tio n s c a n th e n be s o lv e d t o d e te rm in e th e q ^ ( t ) te rm s .
M u ltip ly in g th e q^ ( t ) te rm s tim e s th e Q j J .x ) te rm s and summing them
f o r ea c h o f t h e d e p e n d e n t v a r i a b l e s w i l l y i e l d th e c o m p le te s o l u t i o n
f o r th e s e t o f e q u a t io n s .
APPLICATION OF THE MODIFIED GALERKH METHOD
The f o u r e q u a tio n s t h a t d e s c r i b e th e m o tio n o f th e sy stem
c a n be re d u c e d t o t h r e e e q u a tio n s b y s u b s t i t u t i n g th e e x p r e s s io n
f o r Y i n t o th e e q u a tio n s f o r th e r e l a t i v e m o tio n o f t h e w in g s.
The r e s u l t i n g e q u a tio n s a r e :
w^ (X^, t ) - N1 ^y1 (X1 , t )
- u -(V 1TV3 )C o s2Y Z ^ r i j f - X 1Y +Y^y-jV"
O
(20)
w2 (X g ,t) - Nr2 I j 2 ^ g z tT j
- u - (Y1TVg)c o s ^ / M ^ T r Y +X gY + y ^ y g -f
0
(21)
T
-
r [(V1 - Y
H-M2 - M 1 / I1
b
w here N1 and Ng a r e d i f f e r e n t i a l o p e r a t o r s .
By a p p ly in g th e m o d if ie d G a le r k in m ethod th e e q u a tio n s
f o r th e r e l a t i v e m o tio n o f th e w in g s c a n be re d u c e d t o e q u a tio n s
in tim e a l o n e .
The s o l u t i o n o f th e e q u a tio n s c a n be g r e a t l y s im p li­
f i e d b y r e s t r i c t i n g th e assum ed s o l u t i o n s f o r th e m o d if ie d G a le rk in
(
22 )
-1 7 m ethod t o one te rm .
U nder t h i s r e s t r i c t i o n th e sy ste m re d u c e s to
t h r e e c o u p le d e q u a t io n s i n q
q
and UJ/ .
They a r e ; .
. 2.
q
= C/UT q + C Iff + C c o s Tir + C
I
I
I
2r
3
Y
4
= DTJ/'qg + DgY +
(23)
c o s \ +
IjT = Q q V q 2
w here
, Cg, C^, C^, D^, D^, D^, and
d i f f e r e n t i a l o p e r a t o r on q^ and q ^ .
a r e c o n s ta n t s and Q i s a
The t h r e e c o u p le d e q u a tio n s
c a n now be s o lv e d n u m e r ic a lly u s in g th e Hamming's m o d if ie d p r e d i c t o r c o r r e c t o r m eth o d .
The d e f l e c t i o n sh ap e c h o se n f o r th e w ings was th e d e f l e c ­
t i o n sh ap e p ro d u c e d b y a u n ifo rm s t a t i c lo a d on a c a n t i l e v e r beam.
The d e f l e c t i o n sh ap e was n o rm a liz e d t o I a t th e en d o f th e w in g s .
The p h y s i c a l d im e n sio n s o f t h e sy stem w ere s e l e c t e d . t o
a p p ro x im a te a s m a ll a i r p l a n e .
The w in g s a r e 250 in c h e s lo n g , 10
in c h e s w id e , 4 .6 in c h e s t h i c k and a r e made o f 6061 T6 alum inum .
The w in g w id th an d t h i c k n e s s a r e s m a ll so t h a t th e w in g s t i f f n e s s
more c l o s e l y a p p ro x im a te s t h a t o f a r e a l w in g .
The b l a s t lo a d
i s assum ed t o a c t on an e f f e c t i v e w in g w id th o f 100 in c h e s .
The
m ass o f t h e body i s 2 0 .7 l b . s e c . / i n .
The s t r e s s - s t r a i n d ia g ra m f o r th e w ing m a t e r i a l i s shown
i n f i g u r e 6 a lo n g w ith th e f u n c ti o n u s e d t o ap p ro x im a te i t .
S tre s s
(1000 p s i )
C o m p ressio n
- ta n
(a £ )
T en sio n
3 .2 7 2 x 10
F ig u re 6 .
S tra in ( i n . / i n . )
S t r e s s - S t r a i n Curve
CHAPTER IV
RESULTS MD CONCLUSIONS
UNIFORM LOAD
I f t h e b l a s t lo a d i s u n ifo rm a c r o s s th e a i r p l a n e th e d e f l e c ­
t i o n o f th e w in g s w i l l be sy m m e tric a l and t h e r e w i l l be no r i g i d body r o t a t i o n .
T h u s, u n d e r th e r e s t r i c t i o n o f a u n ifo rm lo a d th e
t h r e e g o v e rn in g e q u a t io n s can b e ' re d u c e d t o I e q u a tio n .
F o r th e p u rp o s e o f d i s c u s s i o n a b l a s t lo a d w i l l be c o n s id e r e d
" s m a ll " i f i t p ro d u c e s a s t r e s s w h ich i s i n th e l i n e a r p o r t i o n o f
t h e s t r e s s - s t r a i n d ia g ra m .
F u r th e r i t w i l l be r e f e r r e d t o as u n i­
form i f i t i s a f u n c t i o n o f tim e o n ly and n o n -u n ifo rm i f i t i s a
f u n c t i o n o f p o s i t i o n and tim e .
The l i n e a r and n o n - l i n e a r s o l u t i o n s
g iv e n e a r l y i d e n t i c a l r e s u l t s f o r " s m a ll " lo a d s .
F ig u r e 7 shows
t y p i c a l d is p la c e m e n t, v e l o c i t y , and a c c e l e r a t i o n d ia g ra m s o f th e
w ing t i p f o r " s m a ll " l o a d s .
F ig u r e 7 shows t h a t th e w ing r e s p o n s e f o r b o th t h e l i n e a r
and n o n - l i n e a r c a s e i s s i n u s o i d a l and h a s a p e r io d o f a p p ro x im a te ly
.85 s e c o n d s .
The p e r i o d f o r an e q u i v a l e n t c a n t i l e v e r beam v i b r a ­
t i n g i n i t s fu n d a m e n ta l mode i s
.43 s e c o n d s .
I t i s r e a s o n a b le to
e x p e c t t h a t th e w in g w o u ld have a lo n g e r p e r io d b e c a u s e i t i s n o t
t r u l y f i x e d a t th e f u s e l a g e .
T h at i s ,
i t i s a llo w e d t o t r a n s l a t e .
When th e lo a d s become la r g e th e li n e a r and n o n -lin e a r
s o lu t io n s no lo n g e r g iv e th e same r e s u l t s .
The resp o n se c a lc u la t e d
D isplacem ent
-2 0 -
V e lo c i ty
A c c e le r a tio n
max.
2
135,450 i n . / s e c .
.30 s e c .
F iau re 7 .
T y p ica l L in ear and N on-L inear Wina T ip R esponse fo r
Sm all Uniform Load
-2 1 -
V e lo c i ty
337 i n . / s e c .
N o n -L in ear
.30 s e c .
tim e
L in e a r
A c c e le r a tio n
2 ,6 6 4 i n . / s e c .
F igu re 8 .
Comparison o f L inear and N on-Linear S o lu t io n s f o r a
Large Uniform Load
-2 2 u s in a th e n o n - l i n e a r e q u a tio n i s much g r e a t e r th a n th e re s p o n s e
i n d i c a t e d by th e l i n e a r e q u a t io n .
The m ain r e a s o n f o r th e d i f ­
f e r e n c e i s t h a t th e n o n - l i n e a r s o l u t i o n ta k e s i n t o a c c o u n t th e
f l a t t e n i n g o f f o f t h e s t r e s s - s t r a i n c u rv e f o r l a r g e s t r a i n and
th e l i n e a r s o l u t i o n d o es n o t.
F ig u r e 8 com pares l i n e a r and non­
l i n e a r s o l u t i o n f o r l a r g e u n ifo rm l o a d s .
A n o th e r d i f f e r e n c e i s t h a t f o r l a r g e lo a d s th e n o n - li n e a r
s o l u t i o n p r e d i c t s a lo n g e r p e r i o d th a n d o es th e l i n e a r s o l u t i o n .
T h is i n d i c a t e s t h a t th e n o n - l i n e a r w ing s t i f f n e s s d e c r e a s e s f o r
la rg e s t r a i n .
The p e r io d o f o s c i l l a t i o n i n c r e a s e s due t o th e r e ­
duced w ing s t i f f n e s s .
The re d u c e d s t i f f n e s s o f th e n o n - l i n e a r
w ing i s a l s o due t o th e f l a t t e n i n g o f f o f th e s t r e s s - s t r a i n c u r v e .
NON-UNIFORM LOAD
The r e s p o n s e o f th e a i r p l a n e t o a n o n -u n ifo rm b l a s t lo a d
i s much more c o m p lic a te d th a n i t s re s p o n s e t o a u n ifo rm b l a s t lo a d .
A n o n -u n ifo rm b l a s t lo a d c a u s e s non-sym m e'tric w ing r e s p o n s e as w e ll
as r i g i d body r o t a t i o n .
F o r s m a ll n o n -u n ifo rm lo a d s t h e l i n e a r and n o n - l i n e a r
s o l u t i o n s a g a in , a s i n th e c a s e o f u n ifo rm lo a d s , a r e n e a r l y i d e n t ­
ic a l.
They, h o w ev er, a r e n o t s im p ly s i n u s o i d a l as i n d i c a t e d by
f i g u r e 9.
T h ere i s a " f l u t t e r " s u p e r-im p o s e d on th e w ing d i s p l a c e -
-2 3 -
Y2 (L)
15 I n .
tim e
.30 s e c .
11 i n .
.30 s e c .
F i g u r e 9.
T y p ica l L in ear and N on-L inear System D isp la cem en ts
fo r a Sm all Non-Uniform Load
-
24
-
N o n -L in ear
36 i n .
L in e a r
.30 s e c .
N o n -L in ear
34 i n .
L in e a r
.30 s e c .
.096 r a d .
N o n -L in ea r
L in e a r
1 2 0 ,0 0 0 p s i
tim e
.30 s e c .
F ig u re 1 0 . C om parison o f L in e a r and N o n -L in ea r S o lu ti o n s f o r
L arge N on-U niform Loads
-2 5 m e n t.
The f l u t t e r i s a p p a r e n tly a r e s u l t o f th e r i g i d body r o t a ­
tio n e f f e c t s .
L arg e n o n -u n ifo rm lo a d s have th e same e f f e c t a s d id th e
l a r g e u n ifo rm l o a d s .
T h at i s , f o r l a r g e n o n -u n ifo rm lo a d s th e
n o n - l i n e a r s o l u t i o n i n d i c a t e s a l a r g e r r e s p o n s e and a lo n g e r p e r io d
o f o s c i l l a t i o n th a n d o es th e l i n e a r s o l u t i o n .
T h is can be se e n b y
lo o k in g a t f i g u r e 10.
The i n c l u s i o n o f th e e f f e c t s o f r i g i d body r o t a t i o n and
t r a n s l a t i o n and l a r g e d e f l e c t i o n do a f f e c t th e sy stem r e s p o n s e .
The e f f e c t o f i n c lu d in g r i g i d body t r a n s l a t i o n i s t o in c r e a s e
th e p e r i o d o f w ing v i b r a t i o n and d e c r e a s e th e w ing d is p la c e m e n t.
Wing " f l u t t e r " i s c a u s e d b y th e e f f e c t s o f r i g i d , body r o t a t i o n .
The n o n - l i n e a r o r l a r g e d e f l e c t i o n s o l u t i o n i n d i c a t e s a g r e a t e r
s y s te m 'r e s p o n s e and an in c r e a s e d p e r i o d f o r l a r g e lo a d s .
FURTHER STUDIES
The re s p o n s e o f an a i r p l a n e t o a dynam ic lo a d was d e t e r - .
m ined in t h i s s tu d y .
A l o g i c a l n e x t s t e p w ould be t o d e te rm in e th e
e f f e c t s o f th e re s p o n s e on th e c o n t r o l e f f e c t i v e n e s s o f th e a i r p l a n e .
I t w ould a l s o be o f i n t e r e s t t o i n v e s t i g a t e th e e f f e c t s o f damping
on th e sy ste m r e s p o n s e .
• The s t r e s s - s t r a i n r e l a t i o n u s e d in th e n o n - l i n e a r s e c ti o n
i s v a l i d o n ly f o r lo a d in g o f th e sy ste m .
I f th e s t r a i n i s la r g e
-2 6 th e sy stem w i l l n o t u n lo a d a lo n g th e same p a th a s th e lo a d in g oc­
c u rre d .
The c y c l i c re s p o n s e o f th e sy stem f o r l a r g e lo a d s c o u ld
be s t u d i e d i f th e s t r e s s - s t r a i n r e l a t i o n w ere p r o p e r l y d e f in e d t o
a c c o u n t f o r u n lo a d in g .
The a n a l y s i s p r e s e n te d i n t h i s p a p e r c o u ld be a p p lie d to
a w ide v a r i e t y o f a i r p l a n e s .
I t w ould n o t be v a l i d , how ever, i f
t h e r e w ere a l a r g e v a r i a t i o n in w ing s t i f f n e s s .
A lso a l a r g e lum ped
m ass, su ch a s an e n g in e , on th e w in g s w ould r e n d e r th e a n a l y s i s i n ­
v a lid .
APPENDIX
-28 -
APPENDIX A
ACCELERATION EXPRESSIONS
C o n s id e r th e a c c e l e r a t i o n o f some p o in t a on th e l e f t w ing.
The p o s i t i o n v e c t o r t o p o in t a i s :
r a = Y j - r coshjT I + r sinhJT ^ + x 2 *2 + y2 ^ 2
(I)
The a c c e l e r a t i o n o f p o in t a c a n be d e te rm in e d by d i f f e r e n ­
t i a t i n g th e p o s i t i o n v e c t o r t w i c e .
D oing t h i s y i e l d s th e fo llo w in g
a c c e l e r a t i o n e x p r e s s io n :
—
••
. 2.
—
..
.I
—
r Q = (Y + ry c o s ig - r l) 7 s in h jr ) j + ( r n g s in lg +rig c o s ig ) i +
(
2)
(x2- x 2Y ^ -2 y 2Y - y 2i|> ) I g + ( 2 x ^ g + x ^ g + ^ - y ^ / ) J g
The above e x p r e s s io n i s th e t o t a l a c c e l e r a t i o n o f p o in t a.
In d e r i v i n g th e e q u a tio n s o f m o tio n o n ly th e a c c e l e r a t i o n in th e
J 2 d i r e c t i o n i s c o n s id e r e d .
The a c c e l e r a t i o n in th e j
d ire c tio n
c a n be shown t o b e :
r
= Y c o s ig + r ig + x^\g +
a j2.
- YgljT
In th e same m anner th e a c c e l e r a t i o n o f some p o i n t b on
th e r i g h t w ing can be shown t o b e :
r ^ = Y c o s ig - r ig - x^lg + y
Ji
- y Ig
(3)
-2 9 -
APPENDIX B
LINEAR EQUATIONS OF WING MOTION
From e le m e n ta r y beam th e o r y th e e q u a tio n o f m o tio n o f a
beam c a n be shown t o b e :
-E
I y ' 11+ w ( x ,t ) = u a
w here E i s th e m o d u las o f e l a s t i c i t y o f th e beam m a t e r i a l ,
I i s th e
c r o s s - s e c t i o n a l moment o f i n e r t i a , u i s th e mass p e r u n i t le n g th ,
and a i s th e a c c e l e r a t i o n .
The a c c e l e r a t i o n e x p r e s s io n s f o r th e
l e f t and r i g h t w in g a r e d e te rm in e d i n a p p en d ix A.
By m aking th e
p r o p e r s u b s t i t u t i o n s f o r th e a c c e l e r a t i o n te rm s th e e q u a tio n s o f
m o tio n f o r t h e l e f t and r i g h t w in g s becom e:
R ig h t Wing
- E I y^ + W^( x ^ , t ; = u|_Y c o s Y - r i f f - x^Xjf + Y1 - Y^Y
(I)
L e f t Wing
-E
mi
,
p1
I y_ + w ( x . , t ) = u Y c o s ip +
2
2
L
(2)
I
APPENDIX C
NON-LINEAR EQUATIONS OF WING MOTION
C o n s id e r th e f o r c e s on an ele m e n t o f th e r i g h t w ing as
shown in f i g u r e 3.
By summing th e moments ab o u t th e r i g h t fa c e
o f th e e le m e n t th e s h e a r e x p r e s s io n c a n be shown t o b e ;
V = M cos
+ u T^ KjT c o s (p ,
w here V i s th e s h e a r and M i s th e m om ent.
(I)
D i f f e r e n t i a t i n g th e
s h e a r e x p r e s s io n y i e l d s ;
Vf= M c o s (J)1 - M (J)1S i n (J)1 - u T^Y - (J)1 s i n ^ 1 / 12
N ext sum th e f o r c e s i n t h e
d ire c tio n .
(2)
T h is y i e l d s ;
V (J)1 s i n (J)1 - V c o s (J)1 + w^ ( x ^ , t ) = u a
(3)
w here a i s th e a c c e l e r a t i o n .
Now s u b s t i t u t e th e e x p r e s s io n s f o r
/
V, V, and a i n t o th e above e q u a t io n . T liis g iv e s th e e q u a tio n o f
m o tio n o f t h e r i g h t w ing in t h e fo rm ;
M (J)1 s i n (2 (J)1)+ u T2 s i n (2 (J)) (J)1/ ! 2 - M Cos^((J)1) + w ^ ( x ^ ,t) =
(4)
u [y
c o s (OjT) - r i j f - X j Y
+ Y j - Y-jY '2']
D e fin e a n o n - l i n e a r d i f f e r e n t i a l o p e r a t o r F .
Yf (x I ' t )-
(J)1 s i n (2 (J)) -(J)1U T^ s i n ( 2 ( ^ ) / 1 2 + M C os2 ( ^ i )
(5)
-31Then th e e q u a tio n o f m o tio n o f th e r i g h t w ing can be w r i t t e n a s :
wI lx V t l ' F1
I (* 1 , t
- u Y CosTfr - r'rr - x i f +
(6)
In t h e same m anner th e e q u a tio n o f m o tio n o f th e l e f t w ing
c a n be shown to b e :
r..
- u Y c o sl|f + Tljf + .x l|f + y
wZlx2' 11 - F2 Ly2t x Zft1 .
w here F
i s th e same a s F
"
I
Zl
- y rXjT
e x c e p t i t i s w r i t t e n in te rm s o f x
(7)
2
i n s t e a d o f X^.
Now once th e moment, b e n d in g a n g le , s t r a i n , an d s t r e s s
a r e d e f in e d t h e e q u a t io n o f m o tio n w i l l be c o m p le te ly s p e c i f i e d .
The b e n d in g a n g le c a n be e x p r e s s e d a s :
(b <= t a n SLY
T
dx
(8 )
The s t r a i n c a n be d e f in e d a s :
£=z
9x
(9)
w here z i s t h e d i s t a n c e from th e n e u t r a l a x is t o th e f i b e r b e in g
s tra in e d .
The s t r e s s c a n be w r i t t e n i n te rm s o f th e s t r a i n in th e
fo rm :
O-
I
~'
= T - t a n ( a E.)
■ b
F i n a l l y t h e moment c a n be w r i t t e n in te rm s o f t h e s t r e s s ,
(10)
32_
I
' 1
- t a n (a E.) z d z
b
W i s th e w id th o f th e beam.
-.33,.LICERATURE CONSULTED
1.
T. M. H anson: " J u d g in g th e R e l a t i v e Q u a l i t i e s and M e rits
o f G a le r k i n 7s A pproxim ate S o l u t i o n s t o a D y n am ically Loaded
Beam S y ste m "; M a s te r 's T h e s i s , M ontana S t a t e U n iv e r s it y ,
December 1970.
2.
D. M. K o sz u ta : "Maximum D e f le c tio n s o f a B la s t Loaded
C a n t i l e v e r Beam by th e M o d ifie d G a le rk in M ethod"; M a s te r 's
T h e s is , M ontana S t a t e U n i v e r s i t y , December 1969.
3.
G. T . N o lan : "Maximum D e f le c tio n s o f a B la s t Loaded C i r c u l a r
F l a t P l a t e " ; M a s te r 's T h e s i s , M ontana S t a t e U n iv e r s it y ,
A ugust 1969.
4.
R. L. B is p li n g h o f f and T. F . O 'B rie n : "G ust Loads on R ig id
A ir p la n e s w ith P itc h in g N e g le c te d " ; J o u r n a l o f t h e Aero­
n a u t i c a l S c ie n c e s , J a n u a r y 1956.
5.
J . C. H o u b o lt: "A R e c u rre n c e M a trix S o lu ti o n f o r th e Dynamic
R esponse o f E l a s t i c A i r c r a f t " ; J o u r n a l o f th e A e r o n a u tic a l
S c ie n c e s , S ep te m b er 1950.
6.
G. B. M i t c h e l l : "A ssessm en t o f th e A ccuracy o f G ust R esponse
C a l c u l a t i o n s b y C om parison w ith E x p e rim e n ts " ; J o u r n a l o f
A i r c r a f t , M a rc h -A p ril 1970.
7.
F. ¥ . D i e d e r l i c h : "The R esp o n se o f a Large A ir p la n e t o
C o n tin u o u s Random A tm o sp h eric D is tu r b a n c e s " ; J o u r n a l o f
th e A e r n a u ti c a l S c ie n c e s , O c to b e r 1956.
8.
C . E . J a c k s o n and J . E . W h erry : "A C om parison o f T h e o r e tic a l
and E x p e rim e n ta l Loads on th e B-47 R e s u lt in g from D is c r e te
V e r t i c a l G u s ts " ; J o u r n a l o f A ero -S p ace S c i e n c e s , J a n u a ry 1959.
9.
B. A. F in la y s o n and L. D. S c r i v e n : "The M ethod o f W eighted
R e s id u a ls — A R eview "; A p p lie d M echanics R ev iew s, S eptem ber 1966.
10.
R. L. B i s p l i n g h o f f , H. A sh le y , and R. L. H alfm an: A eroe l a s t i c i t y ; A d d iso n -W esley P u b lis h in g Company, I n c . , 1955.
11.
Y. C. F ung: The T heory o f A e r o e l a s t i c i t y ; Jo h n W iley and
S o n s, I n c . , 1955.
-34LITERATURE CONSULTED (c o n tin u e d )
G. F . K in n ey : E x p lo s iv e S hocks i n A ir ; The M acM illan Comnanv.
1962,
13.
U. S . D ep artm en t o f D e fe n s e : The E f f e c t s o f N u c le a r Weapons;
A p r il 1962.
14 .
M. L. J a m e s , G. M. S m ith , and J . C. W o lfo rd : A p p lie d N um erical
M ethods f o r D i g i t a l C o m p u tatio n w ith F o r t r a n ; I n t e r n a t i o n a l
T ex tb o o k Company, 1967.
15.
A. R a ls to n and H. S . W ilfs M a th e m a tic a l M ethods f o r D i g i t a l
C o m p u ters; Jo h n W iley and S o n s, I n c . , 1960.
16 .
R. A. A n d erso n : F u n d am e n tals o f V i b r a t i o n s ; The M acM illan
Company, F i r s t E d i t i o n , 1967.
17.
S . H. C r a n d a l l : E n g in e e r in g A n a ly s is ; M cG raw -H ill Company.
1956.
MONTANA STATE UNIVERSITY LIBRARIES
-4 .