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Journal of Sound and Vibration (1988) 124(2), 269-283

B Y

D E T E R M I N A T I O N O F E L A S T I C C O N S T A N T S

O F O R T H O T R O P I C P L A T E S

A M O D A L A N A L u T E C H N I Q U E

L. R .

DEOB LDt ND

R . F . G m S O N

Department o f Mechan ical Engineering, University o f Idaho, M oscow, Idaho

83843,

U.S.A.

Received 26 February 1987, and in revised form 21 October 1987)

Th e f ir st par t of th is pa pe r descr ibes a com puter program me in w hich equat ions der ived

by t he Ray l e i gh - Ri t z t e chn i que a r e u s ed t o m ode l t he v i b r a ti ons o f r ec t angu l ar o r t ho t r op i c

pla tes. The charac te r is t ic func t ions of v ibra ting beams w ere used as the assumed func t ions

for p la tes wi th bo un da ry condi t ions consi st ing of c lamped and f ree edges . Natura l

f requencies and m ode sh apes f rom the prog ramm e w ere veri f ied by f in it e e lement ana lys i s

and mo dal analys is for square a lum inium and g raph i te /ep ox y p la tes . Th e p la te v ibra t ion

m ode l w as t hen i nco r po r a t ed i n t o a s econd com p u t e r p rog r am m e wh i ch was de s igned t o

us e t he m eas u r ed na t u r a l f r equenc ie s o f o r t ho t r op i c p la t es t o de t e r m i ne t he f ou r ap pa r en t

e las t i c cons tant s . Natura l f requencies measured by an impulse t echnique were used to

de te rmine two Yo ung 's m odul i , the in-p lane shear modu lus , and a P oi sson ra t io for each

plate.

", 1. I N T R O D U C T I O N

M o d e r n e n g i n e e r i n g d e s i g n r e q u i r e s th e u s e o f m a t e r i a l s in a w a y w h i c h o p t i m i z e s t h e i r

i n h e r e n t p r o p e r t i e s . T h e g e n e r a l c l as s o f m a t e r i a l s w h i c h i s m o s t s u i ta b l e f o r o p t i m u m

d e s i g n i s c o m p o s i t e s . C o m p o s i t e m a t e r ia l s a r e u s e d i n e n g i n e e r i n g s t ru c t u r e s f o r a v a r i e t y

o f r e a s o n s , s u c h a s h i g h s p e c i f i c s ti ff n es s . T h e f o c u s o f t h is p a p e r i s o n t h e e l a s ti c c o n s t a n t s

w h i c h c o n t r i b u t e t o t h e s t if f ne s s o f s t r u c t u r a l c o m p o s i t e s .

S t r u c tu r e s m a d e o f a d v a n c e d c o m p o s i t e s s u c h a s f i b r e - re i n f o r c e d p l as ti cs o f t e n c o n s i s t

o f a n u m b e r o f l a y e r s h a v i n g u n i d i r e c t i o n a l f i b re s . T h e s e l a y e r s , o r l a m i n a e , w i ll g e n e r a l l y

b e o r t h o t r o p i c . W h e n o r t h o t r o p i c l a m i n a e a r e s t a c k e d t o f o r m a l a m i n a t e , t he r e s u lt in g

s t r u c t u r e is g e n e r a l l y a n i s o t r o p i c . T h e l a m i n a t e w i ll b e o r t h o t r o p i c o n l y f o r c e r t a i n s ta c k i n g

s e q u e n c e s . H e n c e f o r t h , i n t h i s p a p e r , t h e p l a t e s c o n s i d e r e d a r e s p e c i a l l y o r t h o t r o p i c , s u c h

t h a t a l l t h e f i b re s a r e o r i e n t e d p a r a l l e l t o t w o o f t h e p l a t e ' s e d g e s . A c o o r d i n a t e s y s t e m

is d e f i n e d w h i c h h a s t h e x a x i s o r i e n t e d a l o n g t h e f i b re d i r e c t i o n a n d t h e y a x is o r i e n t e d

p e r p e n d i c u l a r t o t h e f i b r e d i r e c t i o n . T h e f l e x u r a l d e f o r m a t i o n s a r e d e p e n d e n t o n t h e

m a t e r i a l e l a s t i c c o n s t a n t s i n a c c o r d a n c e w i t h c l a s si c a l l a m i n a t i o n t h e o r y [ 1 ]. T h e s e e l a s t ic

c o n s t a n ts i n c l u d e t h e l o n g i tu d i n a l Y o u n g ' s m o d u l u s , E x , t r a n s v e rs e Y o u n g ' s m o d u l u s ,

Ey ,

m a j o r P o i s s o n r a t i o ,

V,,y,

m i n o r P o i s s o n r a ti o , V y,,, a n d t h e i n - p l a n e s h e a r m o d u l u s ,

G~y. O f t h e f iv e c o n s t a n ts n a m e d , o n l y f o u r a re i n d e p e n d e n t s i n ce v~y /Ex = vyx /E y d u e

t o s y m m e t r y o f th e c o m p l i a n c e m a t r ix [ 1 ] .

D e t e r m i n a t i o n o f t h e o r t h o t r o p i c e l a st ic c o n s t an t s o f fi br e c o m p o s i t e s is i m p o r t a n t f o r

o p t i m u m d e s ig n , q u a l it y c o n t r o l , a n d d a m a g e d e t e c t io n . A S T M S t a n d a r d s D 3 0 3 9 -7 6 a n d

D 4 2 5 5 - 8 3 p r e s e n t s t a t ic t e n s il e a n d s h e a r te s ts f o r d e t e r m i n i n g t h e e l a s ti c c o n s t a n t s o f

t Present address: D epartmen t o f M echanical Eng ineerinz. Un iversity of W ashington, Seattle, W ashington

98195, U.S.A.

269

0022-460X/88/140269+ 15 S03.00/0 9 1988 Acade mic Press Limited

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270 L R DEOBAL D AND R F GIBSON

fibre-reinforced composites. To obtain one set of constants, two tensile test specimens

and one shear test specimen must be fabricated and tested. One disadvantage of these

tests is that resulting strains measured by strain gauges occur at localized areas.

Three of the elastic constants can be measured dynamically by measuring the frequency

response of beams. Vibration tests of beams with fibre orientations of 0, 45 and 90 degrees

will produce two Young s moduli and a shear modulus [2]. The Poisson ratio must be

determined by an alternative method, since it is used to calculate the shear modulus. The

disadvantage of all of the methods mentioned thus far is that they are slow, expensive

and destructive tests.

If vibrations are induced in an orthotropic plate, then its dynamic response will be a

function of plate geometry, density, boundary conditions and the elastic constants. This

implies lhe possibility of using plate vibration theory to develop a non-destructive test

to determine the dynamic elastic constants of an orthotropic plate. References [3] and

[4] each presented a unique method for determining the elastic constants of an isotropic

plate. Reference [5] reported on a technique for determining the Poisson ratio of an

orthotropic plate. DeWilde e t a l have been developing a technique very similar to the

work being presented here [6, 7].

In DeWilde s method a model developed by Galerkin s method based on Lagrangian

polynomials as assumed shape functions is used to extract the six elastic rigidities of

rectangular anisotropie plates. Our work and DeWilde s work was carried out concurrently

and independently. Differences in the two approaches include variances in modal analysis

procedures and difference in assumed mode shapes (in the work reported here the

characteristic functions of vibratir~g beams were used). Also, in DeWilde s technique a

least squares method is used in an over-determined matrix in order to obtain one set of

constants, whereas in the present work the resulting elastic constants from various

combinations of measured natural frequencies are averaged.

The first objective of this research was to develop and verify a computer programme

which models a vibrating orthotropic plate. The second and primary objective was to

develop software consisting of the plate vibration model to use measured natural frequen-

cies for obtaining the apparent elastic constants. When four natural frequencies are known,

the possibility exists for determining one set of four independent constants. The third

objective was to develop a fast, accurate and portable technique suitable for quality

control testing, which meant that the computer programmes should be based in a micro-

computer/modal analysis system.

The fourth objective was to develop modal analysis techniques for the accurate measure-

ment of natural frequencies from rectangular plates with various boundary conditions.

The final goal was to apply the technique for the measurement of elastic constants of

aluminium and graphite/epoxy plates.

2. PLATE VIBRATION MODEL

Three methods available for modelling the dynamic behaviour of a rectangular plate

include finite element analysis, Rayleigh s method, and the Rayleigh-Ritz technique.

Finite element analysis was rejected because it would have to be based in a large computer

and would consume considerable computer time. Rayleigh s method was easily imple-

mented, but was found to be f~lr too inaccurate in this particular application. The

Rayleigh-Ritz technique was chosen to model plate vibrations, since it provided good

accuracy, yet could be based in a microcomputer.

A wealth of information is present in the literature on the modeling of vibrating

rectangular plates. A limited number o f rof~erences are cited here in which vibrating plates

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O R T H o T R O P I c P L A T E E L A S T I C C O N S T A N T S 2 7

o f v a ry i n g d e g r e e s o f a n i s o t ro p y a r e m o d e l e d a n d a v a r ie t y o f m o d e l i n g m e t h o d s i s u se d .

Y o u n g ' s ap p l i c a t i o n o f t h e R a y le ig h - R i t z t e ch n iq u e t o a c an t i lev e r ed i s o t r o p i c p l a t e w as

u s e d a s a g u i d e f o r th e m o d e l d e v e i o p e d i n t h e p r e se n t w o r k [ 8 ]. R a y l e i g h - R i tz m o d e l s

f o r i s o t r o p i c p l a t e s w e r e a l s o p r e s en t ed i n r e f e r en ces [ 9 , 1 0] . Le i s sa co n s o l id a t ed a g r ea t

d ea l o f i n f o r m a t io n o n t h e m o d e l in g o f v ib r a t i n g i s o t r o p i c p l a t e s [ 1 1, 1 2] . M an y r e s ea r ch e r s

h a v e s u c c e s s f u ll y m o d e l e d v i b ra t in g o r t h o t r o p i c - p l a t e s [ 1 3 - i 5 ] . T h e R a y l ei g h m o d e l

p resen ted in r e fe rence [14 ] was o r ig ina l ly u t i l i zed fo r th i s work , bu t lacked su f f ic ien t

accu r acy . A m o r e accu r a t e v a r i a t i o n o f R ay le ig h ' s t e ch n iq u e w as p r e s en t ed i n r e f e r en ce

[ 1 5] a n d w a r r a n ts c o n s i d e r a t io n f o r s u b s e q u e n t w o r k . O r t h o t r o p i c p l a t e s m a y b e c o n -

s id e r ed a s p ec i a l c a s e o f an i s o t r o p i c p l a t e s ; t h u s , s o m e an i s o t r o p i c p l a t e v ib r a t io n m o d e l s

a r e a l s 0 m en t io n ed [ 1 6 , 1 7 , 1 8 ] . A G a le r k in ap p r o ach w as u s ed i n r e f e r en ce [ 1 8 ] an d th e

R a y le ig h - R i t z t e c h n iq u e w as u t i l i z ed i n r e fe r en ces [ 1 6 , 1 7]. I n a l l t h r ee o f t h e s e p ap e r s

c h a r a c te r is t ic b e a m f u n c t io n s w e r e u s e d t o r e p r es e n t t h e a s s u m e d m o d e s h a p e s .

Th e d e r iv a t i o n o f t h e R a y le ig h - R i t z m o d e l f o r o r th o t r o p i c p l a t e v ib r a ti o n s i s b r i e f ly

s u m m ar i zed h e r e . R e f e r en ce [ 1 9 ] p r e s en t ed t h e n eces s a r y d e r iv a t i o n s f o r ap p ly in g t h e

R a y l e i g h - R i t z m e t h o d t o o r t h o t r o p i c p l a t e s . A n e x p r e s s i o n f o r t h e m a x i m u m p o t e n t i a l

en e r g y , V m ax , w as d e r iv ed f o r a r e c t an g u la r p l a t e i n a s t a t e o f t r an s v e rs e v ib r a t i o n , b y

f o l lo w in g th e u s u a l a s s u m p t io n s f o r c l a s si ca l l am in a t io n t h eo r y [1 9]. Th e b en d in g

s t i f f n e s s e s D l t D l 2

and D22 , and to r s iona l s t i f fness ,

0 6 6

w h ich co n t r i b u t e t o t h e

p o ten t i a l en e r g y a r e d e f in ed i n t e r m s o f t h e p ly t h i ck n es s e s an d s t if f n es s e s [ 1 ]. H en ce f o r th ,

a l l f o u r t e r m s a r e r e f e r r ed t o a s b en d in g s t i t t n e s s e s , w i th t h e u n d e r s t an d in g t h a t 9 6 6 is

ac tua l ly a to r s iona l s t i f fness .

Th e f i r s t s t ep i s t o f o r m R ay le ig h ' s q u o t i en t f o r a r e c t an g u la r o r th o t r o p i c p l a t e . F o r a

c o n s e r v a ti v e s y st e m , th e m a x i m u m p o t e n t ia l e n e r g y e q u a l s t h e m a x i m u m k i n et ic e n e r g y .

Th i s r e l a t i o n s h ip m ay b e r e - a r r an g ed to fo r m th e R ay le ig h ' s q u o t i en t a s

t o 2 = 2 ( V . , ~ x / f

I o ' w 2 d y d x ) i ( 1 )

ph \

/ J o

R ay le g h ' s q u o t i en t p r o v id es an e s t im a te o f t h e f u n d am en ta l f r eq u en cy , to , i n t e r m s o f

th e m ax im u m p o ten t i a l en e r g y , V ,,ax, m a te r i a l d en s i t y , p , a s s u m ed m o d e s h ap e , w (x, y) ,

p la t e t h ick n es s , h , an d p l a t e d im en s io n s , a an d b . Th e R a y le ig h - R i t z m e th o d i s ap p l i ed

b y a s s u m i n g t h e m a x i m u m d e f le c t io n , w(x , y ) , t o b e r ep r e s en t ed a s a l i n ea r s e r i e s o f

a s s u m e d f u n c ti o n s :

p q

w (x ,y )= Z Z A , ,, .~ , ,, (x )O. (y ). (2)

m = l n = l

Th e a s s u m ed f u n c tio n , q ~,, an d O , m u s t b e ad m is s ib le , s u ch t h a t t h ey s a t is f y t h e e s s en t i a l

b o u n d a r y c o n d i t i o n s o f t h e p l at e . In o t h e r w o r d s , t h e a s s u m e d f u n c t io n s ' b o u n d a r y

c o n d i t i o n s m u s t c o r r e s p o n d t o th e p r e s c r i b e d d i s p l a c e m e n t s a n d r o ta t io n s o f t h e p l a t e ' s

e d g e s .

Th e s t a t i o n a r y v a lu e o f t h e f r eq u en cy , to , i s f o u n d b y t ak in g t h e p a r t ia l d e r iv a t iv e o f

eq u a t io n ( 1 ) w i th r e s p ec t t o e ach co e f f ic i en t, A ,,n , an d eq u a t in g t o ze ro . Th e ap p l i ca t i o n

o f th e p r o d u c t r u le f o r d e r iv a t iv e s t o eq u a t io n ( 1 ) re s u l ts i n

OV,.~x to2ph O. fa fbw2dxdy=O

3 )

o o

A i k 9 2 O A i k

S in ce eq u a t io n ( 2 ) i s a f i n i t e s e r i e s , t h e a s s u m ed m o d e s h ap e , w(x , y ) , w i l l o n iy

ap p r o x im a te t h e t r u e m o d e s h ap e . Th u s , w h en th e ,$ o e f fi c ien t s, A , . . , a r e s e lec t ed acco r d in g

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272 L R DEOBALD AND R F GIBSON

t o eq u a t i o n s ( 3 ), th e R ay l e i g h - R i t z t e ch n i q u e w i ll p r o v i d e a d i s c re t e n u m b er o f s ta t i o n a r y

v a l u es , w s , w h i ch a r e t h e l o w es t u p p e r b o u n d s o f t h e ac t u a l n a t u r a l f r eq u en c i e s [ 1 1 , 2 0 ].

O n ce t h e ex p r e s s i o n f o r t h e max i mu m p o t en t i a l en e r g y i s s u b s t i t u t ed i n t o eq u a t i o n s

( 3 ), a l g eb r a i c ma n i p u l a t i o n p r o d u ces a se t o f l in ea r h o m o g e n eo u s eq u a t i o n s a s f o ll o w s

[19] :

{ I f L ] [ f o L ]

n ~ ~ r O n O k d y + D 2 2 r 1 6 2 O ~ O ~ d y

m = i n = l

I f

o

o

o ]

9 q - D l 2 t J b ~ t ~ i d x 0 , ,0 ~ d y + r d x

~ ) : ( ~ k

d y

[ I o L ]

! I l

+ 4 D 6 6 ~ m ~ ) i d x OnO k dy

- P h o : [ f : r

i = 1 , 2 , 3 , . . . , p ,

k = 1 , 2 , 3 , . . . , q.

4)

f o

Y.k = b O.O ~ d y

o

~ k = b 0 0 ~ d y .

(6)

O n ce t h e a s s u med f u n c t i o n s h av e b een ch o s en an d t h e i n t eg r a l s ev a l u a t ed , eq u a t i o n s

( 4) r e d u c e t o a n e i g e n v a l ue p r o b l e m k t h e b e a m e i g e n v a lu e s a p p e a r d u e t o th e o r t h o g o n a l i ty

o f the i n t eg ra ls i n equ a t ion s (5 ) ) :

P q

E E [C,k.m,,--Ar,,,,S,,k]Am,, =0 . (7)

m = l n = l

~ 0 ~ 9

F~ i = a ~b , ,qb id x

o

I t

H , , / = a

~ q b i

dx ,

Th e g en e r a l a p p r o ach t o t h e s o l u t io n o f eq u a t i o n s ( 4 ) is fi rs t t o ch o o s e a s s u m ed f u n c t i o n s

w h i ch s a t i s f y t h e ap p r o p r i a t e b o u n d a r y c o n d i t i o n s a n d t h en s o l v e t h e s p ec if i ed i n teg r a ls .

F o l l o w i n g t h e p r o c ed u r e o f Y o u n g [ 8 ] t h e ch a r ac t e r is t i c eq u a t i o n s o f v i b r a t in g b ea m s

w er e u s ed a s t h e a s s u m ed f u n c t i o n s , ~ , , an d O , . Th e ch a r ac t e r i s t ic eq u a t i o n s o f v i b r a ti n g

b e a m s w e r e c h o s e n s o th a t t h e b o u n d a r y c o n d i t i o n s o f t h e b e a m m a t c h e d t h o s e o f t h e

p l a t e , g u a r an t ee i n g s a t i s f ac t io n o f t h e e ss en t ia l b o u n d a r y co n d i t io n s . T h e ap p r o p r i a t e

c h a r a ct e ri s ti c e q u a t i o n s o f v ib r a ti n g b e a m s w h i c h w e r e u s e d f o r t h e a s s u m e d ' m o d e s h a p e s

w er e p r e s en t ed b y B l ev in s [2 0 ] . B eam e i g en v a l u es , e r , an d f r eq u en cy p a r am e t e r s , a , ,

w h i ch ap p ea r i n t h e ch a r ac t e r i s t i c eq u a t i o n s a r e a l s o l i s t ed i n r e f e r en ce [ 2 0 ] .

Th e i n t eg r a ls w h i ch ap p ea r i n eq u a t i o n s ( 4 ) h av e b een e v a l u a t ed b y u s i n g o r t h o g o n a l i ty

r e l a t io n s h i p s [ 2 1] . Th e r e s u lt in g eq u a t i o n s a r e l is t ed i n re f e r en ce [ 2 0] an d t ab u l a t ed v a l u es

a r e p r e s en t ed i n r e f e r en ce [ 8 ] . Th e i n teg r a ls i n t h e f o r m o f eq u a t i o n s ( 5 ) a r e n o n - ze r o

o n l y f o r eq u a l i n d i ce s m an d i o r eq u a l in d i ce s n an d k

f o ~ / o

o ~ o ~

b qb'. ' d x an d 0 0 ~ dy, qbr~qb~ d x an d O , OR dy. (5)

Th e f o u r r ema i n i n g d i s t in c t i n teg r a ls w r i t ten a s eq u a t i o n s ( 6 ) f o l lo w i n g a r e n o w n a m ed

F~ i, F~ k, H ~i, HY~k; h e i n t eg r a l s w e r e n o n - d i men s i o n a l i zed , s o t h a r o n l y o n e ca l cu l a t i o n

w as n eces s a r y :

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O R T H O T R O P I C P L A T E E L A S T I C C O N S T A N T S

73

Here

2 2 2

A = to s p h a b ,

Q , , i k . , . . = ( e ' [ ) 4 ( b 2 / a 2 ) 8 , k S , . . , Q 2 ., k. ., ~ = ( e ~ , ) 4 ( a 2 / b 2 ) C S i k S . , .,

F , , . F k . + F i . , F ~ k ,

3 i k m n ~ x y x y

C ~k.m,, = D lfQ l .~ k. , , , , + D22Q2.1k.mn q- D1 2Q a.ik .~n 1-D66Q4.ik,m,,

x u

Q 4 .i k.m n - - 4 H m ~ H n k ,

i= 1, 2, 3, .. ., p, k= 1, 2, 3, .. ., q, 8,~ = {10

for r=~}

for r~

No integrat ion is performed during the modeling process. Once the appropriate integral

values and bean eigenvalues are read from computer memory, the eigenvalue problem

may immediately be formulated according to equations (7).

The solution of an eigenvalue problem of practical size is not a trivial task. A variety

of methods exists for such a solution. For this research, the determinant search method

was chosen. In this technique secant iteration is used on the characteristic polynomial to

approximate an eigenvalue, which is then used in the inverse power method to determine

an accurate eigenpair [22, 23]. The determinant search method calculates eigenpairs in

ascending order, beginning with the first eigenpair.

Plate vibration modes are identified by a pair of indices representing the number of

flexural half waves along the x and y axes. If the eigenvalue problem results in a maximum

vector entry, A,,, then r and s correspond to the plate modal indices. The Rayleigh-Ri tz

techniques models plates which have an infinite number of modes as a system with a

discrete number of degrees of freedom. This means that certain modes are excluded in

th~ model. The modes which will be excluded have their first and second modal indices

greater than the target values p and q, respectively, in equations (7).

T~e mathematical plate vibration model was incorporated in a computer programme.

The results of this programme were compared with values from finite element analysis

and experimental modal analysis (see Tables 2 and 3).

3. ELASTIC CONSTANTS FROM PLATE VIBRATION MEASUREMENTS

The primary source of the information presented in this paper is reference [24]. Before

presenting the technique for obtaining the elastic constants, the eigenvalue problem is

written in a more favourable form, Thus, equations (7) are now written as

p •

[ C o - A ~ o ] A ~ = O , C o = D I Q I u + D 2 Q 2 0 + D 3 Q 3 u + D 4 Q 4o . (8)

The bending stiffnesses, Dl l, D22, Dr2 and D66, are renamed DI, D2, D3 and /)4,

respectively, and the double indices i k and m n are changed to single indices i and j,

respectively. The four integral matrices, Qlo, Q2u,

Q 3 j

and Q40, result when the four

bending stiffnesses are factored out of the stiffness matrix, C o . T h e integral matrices are

so named because they are exclusively functions of the integrals which appear in equations

(5) and (6).

If four natural frequencies of the orthotropic plate are measured, the possibility exists

for determining one set of elastic constants. The complexity of the problem suggests the

use o f an iterative technique. The :first preliminary step would be to form the four integral

matrices and store them to allow repeated use. The second preliminary step would be to

use an initial guess of the elastic constants to calculate four bending stiffnesses. The

bending stiffnesses would be calculated as i f the plate consisted of unidirectional aligned

laminae and the laminae properties equaled the,initial guess of the elastic constants [1].

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274

L R DE OBA L D AND R F GI BSON

To b eg i n t h e i t e r a t i o n p r o ces s , t h e e i g en v a l u e p r o b l em s h o w n a s eq u a t i o n s ( 8 ) mu s t

b e f o r m u l a t ed an d s o l v ed w i th u s e o f t h e f o u r i n i ti a l b en d i n g s t if fn e s s e s. Th e f o u r

exp er im en ta l na tu ra l f r eque nci es , f~ ' , conv er t s imply t o t he fou r e igenvalues A~, A~, A~,

and A~, as g iven by

A [ = p h (27 rfeab) 2.

(9)

I f th e i n i ti a l e s t i ma t e s o f t h e e l a s t ic co n s t an t s a r e r e a s o n ab l e an d t h e e i g en v a l u es o f

i n te r e s t f r o m t h e p l a t e v i b r a t i o n m o d e l a r e f a i rl y w e ll is o l a t ed , t h en t h e f o u r e i g en v a l u es

f r o m t h e ex p e r i men t a l n a t u r a l f r eq u en c i e s may b e u s ed a s s h i f t s f o r t h e i n v e r s e p o w er

m e t h o d . O t h e r w i s e , t h e m o d a l i n d i ce s o f th e e i g en v a l u es m u s t b e id en t if i ed an d t h e

c o m p l e t e d e t e rm i n a n t s e a r c h m e t h o d p e r f o rm e d . A s e a c h e i g e n p a ir i s f o u n d , i ts m o d a l

i n d i ce s w i l l b e co mp ar ed w i t h t h e s p ec i f i ed i n d i ce s . Th e s ea r ch co n t i n u es u n t i l f o u r

e i g en p a i rs h av e b ee n i d en t i fi ed an d s t o r ed . Th e r e s u l t o f t h e e i g en f u n c t i o n s o l u t i o n i s

f o u r ap p r o x i ma t e e i g en p a i r s , A I , A ~ ~), A2,

A~2 , A3,

A~3) and

A4, A~ ,

c o r r e s p o n d i n g t o

the e igenvalue s A~, A~, A~ and A,~, r espec t ive ly . N o te t ha t t he fou r de t e rm ined e igenp ai r s

d o n o t n eed t o b e t h e fi rs t f o u r o f t h e e i g en v a l u e p r o b l em , n o r d o t h ey n eed t o b e i n an y

par t i cu l a r o rder .

Th e n ex t s t ep i s t o u s e t h e ap p r o x i ma t e e i g en v ec t o r s t o f o r mu l a t e t h e f o u r b y f o u r

s o l u t i o n ma t r i x r e l a t i n g t h e f o u r e i g en v a l u es t o t h e f o u r b en d i n g s t i f f n e s s e s . F o r each

eige npa i r , As an d A~~ t h e h t h r o w i n eq u a t i o n s ( 8 ) i s i s o l a t ed :

A,A(h =

A~')( D~Q lh, + D 2Q2h, + D3Q 3h, + D4Q 4h,)

+ At20(Di Q I h2

d r -

D2Q282 + D3 Q382 + D4Q 4h2)

+ ' ' § A t~ Q~h , + D2Q2h, + DaQ3h,, + D4Q 4h, ), .

i = 1 , 2 , 3 ,4 , l ~ h < ~ n n = p x q .

l o )

I n t h eo r y , th e s e l ec t i o n o f t h e f o u r eq u a t i o n s f r o m t h e s e t o f n eq u a t i o n s m ay b e

a r b i t r a r y . F o r t h i s w o r k , t h e h t h eq u a t i o n w as s e l ec t ed f o r each e i g en p a i r s u ch t h a t t h e

eigenvector ent ry Ath~ w a s t h e m a x i m u m v a l u e . T h e f o u r a p p r o x i m a t e e i g e n v a l u e s i n

eq u a t i o n s ( 1 0 ) a r e rep l aced b y t h e e ig en v a l u es ca l cu l a t ed f r o m t h e ex p e r i men t a l n a t u r a l

f r eq u en c i e s . O n e ca n r ed u ce e q u a t i o n s ( 1 0 ) t o th e f o l l o w i n g eq u a t i o n s b y f ac t o r in g o u t

t h e f o u r b en d i n g s t it tn e s s e s :

/ - .

~ . [J L H ~ H . 2 / -/ 4 3 H 4 . J D 4

( l l )

H e r e

n O = E A ~ --~Q j~ i : = 1 2 3 4 = p • l< ~ h < . ._

Th e en t ri e s i n t h e s o l u t i o n m a t ri x , / ~ j , a r e f u n c t i o n s o f t h e ap p r o x i ma t e e i g en v ec t o r s ,

A~~ and the i n t eg ra l m at r i ces Q~m, . Eq uat ion s (11) r e l a t e t he e igenva lues f rom the

ex p e r i men t a l n a t u r a l f r eq u e n c i e s t o n ew e s t ima t e s o f t h e f o u r b en d i n g s t i ff n e ss e s . Th e

s o l u t i o n ma t r ix id en t i fi ed i n t hi s p ap e r is an a l o g o u s t o D e W i l d e ' s s en s i t i v i ty ma t r i x

[ 7 ] .

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ORTHOT ROPIC PLATE ELASTIC CONSTANTS 75

E v e n w h e n t h e e s t i m a t e d e l a s t ic c o n s t a n t s a r e c o n s i d e r a b l y d i f f e re n t f r o m t h e t r u e

v a l u e s, th e a p p r o x i m a t e e i g e n v e c to r s f r o m t h e a p p r o x i m a t e e i g e n v a lu e p r o b l e m a r e

g e n e r a l l y a g o o d r e p r e s e n t a t i o n o f t h e tr u e e i g e n v e c t o r s . E q u a t i o n s ( 1 1 ) a re s o l v e d w i t h

t h e u s e o f e i g e n v a l u e s f r o m t h e e x p e r i m e n t a l n a t u r a l f r e q u e n c i e s to o b t a i n n e w e s t i m a t e s

o f t h e b e n d i n g s ti ff ne ss e s. T h e c y c l e is t h e n r e p e a t e d , c o n t i n u a l l y u p g r a d i n g t h e a c c u r a c y

o f t h e b e n d i n g s ti ff n e ss e s. F i n a l l y , t h e f o u r e l a st ic c o n s t a n t s a r e t h e n d e t e r m i n e d b y

V xy = D 3 / D 2 , V yx = r x y D 2 / D l ,

E x = ( 1 2 /h 3 )( 1 - ~'xy~'~.x)D ~ , E y = ( 1 2 / h 3 ) ( 1 - vxyv~,~)D2, Gxy = 12 /h3 )D 4.

. . (12)

4 . M O D A L A N A L Y S I S O F P L A T E S

F o r th i s r e s e a r c h , p r o p e r m o d a l a n a l y s i s o f t h e p l a t e s is v i ta l . T o d e t e r m i n e a c c u r a t e l y

t h e n a t u r a l f r e q u e n c i e s , m o d a l a n a ly s is w a s p e r f o r m e d o n s i x p la t es . R e f e re n c e s [ 2 5 - 2 8 ]

w e r e u s e d a s g u i d e s f o r p r o p e r m o d a l a n a ly s is . T h r e e p l a te s w e r e m a d e o f a l u m i n i u m

a n d t h re e w e r e m a d e f r o m g r a p h i t e / e p o x y ( se e T a b l e 1 ). T h e b o u n d a r y c o n d i t io n s

i nc lu d e d C - C - F - C , C - F - F - F a n d F - F - F - F , w h e re C = c l a m p e d a n d F = f r ee , a s in

r e f e re n c e [2 0 ]. T h e b o u n d a r y c o n d i t i o n f o r t h e C - C - F - C p l a te w a s o b t a in e d b y a d h e s i v el y

b o n d i n g s te e l s h o u l d e r s t o b o t h s u r f a c e s o f e a c h p l a te . T h e h o r s e s h o e s h a p e d s h o u l d e r s

i s o la t e d a 2 5 . 4 c m ( 1 0 in c h e s ) s q u a r e a r e a o n a n a l u m i n i u m o r a g r a p h i t e / e p o x y p l a te .

T h e a s s e m b l y w a s t h e n c l a m p e d t o th e s u r f a c e o f a n i s o l a ti o n t a b l e w h e r e m o d a l a n a l y si s

w a s p e r f o r m e d . T o o b t a i n t h e o n e e d g e c l a m p e d b o u n d a r y c o n d i t i o n , a lu m i n i u m s h o u l d e r s

w e r e b o n d e d a l o n g o n e e d g e o f a l u m i n i u m a n d g r a p h i t e / e p o x y p l at e s. F o r t e st in g , th e

p l a t e s w e r e c l a m p e d i n a v is e lo c a t e d o n a n i s o l a t i o n t a b l e . F i n a l ly , to o b t a i n t h e b o u n d a r y

c o n d i t i o n o f a ll ed g e s f r e e , t h e p l a t e s w e r e p l a c e d o n a s o f t c o t t o n p a d .

T o p e r f o r m m o d a l a n a l y s is , th e e q u i p m e n t s h o w n i n F i g u r e 1 w a s u s e d . ' T h e i m p u l s e

t e c h n i q u e w a s c h o s e n f o r th i s r e se a r c h b e c a u s e o f th e e a s e o f im p l e m e n t a t i o n a n d t h e

q u i c k n e s s o f th e t es t. A f o r c e t r a n s d u c e r w a s p o s i t i o n e d o n a n i m p u l s e h a m m e r w h i c h

w a s u s e d t b in d u c e e x c i ta t io n . A n o n - c o n t a c t i n g e d d y c u r r e n t p r o x i m i t y p r o b e w a s u s e d

t o d e t e c t th e r e s p o n s e i n o r d e r t o a v o i d n a t u r a l f r e q u e n c y sh i ft s d u e t o a d d e d m a s s . T h e

r e s p o n s e w a s tr a n s f o r m e d i n t o t h e f r e q u e n c y d o m a i n b y a s p e c t r u m a n a ly z e r . O n l y th e

r e s p o n s e w a s u s e d , s o a s to d e c r e a s e n o i s e a n d i n c re a s e f r e q u e n c y r e s o lu t io n . T h e r e s p o n s e

w i l l n o t h a v e t h e s a m e a m p l i t u d e a s t h e t r a n s f e r f u n c t i o n , b u t r e s o n a n c e w i l l s t i l l o c c u r

a t th e s a m e f r e q u e n c y . T h e z o o m f e a t u re o f th e s p e c t r u m a n a l y z e r w a s u s e d , a ll o w i n g

b e t t e r f r e q u e n c y r e s o l u ti o n . T r y i n g to d e c r e a s e th e s p a n b y t o o m u c h i n t r o d u c e d m o r e

n o i se . T h e o p t i m u m s p a n f o r th e s e te s ts w a s 2 5 0 H z . M e a s u r e d d a m p i n g w a s u s e d t o

e s t im a t e t h e f r e q u e n c y s h i ft b a s e d o n a s i n g le d e g r e e o f f r e e d o m s y s t em . T h e d a m p i n g

r a t i o w a s l e ss t h a n 0 . 0 1 , s o t h a t t h e f r e q u e n c y s h i ft s d u e t o d a m p i n g w e r e n e g l ig i b le . T h e

d a m p i n g r a ti o w a s o b t a i n e d b y a p p l y i n g th e h a l f p o w e r b a n d w i d t h t e c h n i q u e to i n d i v id u a l

p e a k s [ 2 1 ] . T h e h a l f p o w e r b a n d w i d t h t e c h n i q u e i s b a s e d o n a l i g ht ly d a m p e d s in g l e

d e g r e e o f f re e d o m s y s te m . T h e C R T o n t h e s p e c t ru m a n a l y z e r is d iv i d e d u p i n to a n u m b e r

o f d i s c r e t e v a lu e s . T h e s p e c t r u m is p l o t t e d b y t r a c i n g s t r a ig h t l i n e s b e t w e e n t h e d i s c r e t e

v a l u e s . S h o u l d a n a t u r a l f r e q u e n c y f a ll b e t w e e n t h e d i s c r e t e v a lu e s , e r r o r w o u l d b e

i n t r o d u c e d b y t ry i n g t o r e a d t h e n a t u r a l f r e q u e n c y d i r e c tl y . In s t e a d , s o f t w a r e u s e d i n a

d e s k t o p c o m p u t e r r e a d t h e p l o t f r o m t h o - sp e c t r u m a n a l y z e r a n d d e t e r m i n e d t h e n a t u r a l

f r e q u e n c y a s t h e l o c a t io n h a l f w a y b e t w e e n t h e h a l f p o w e r p o i n t s o f th e r e s o n a n t p e a k .

T h e s p e c t r u m a n a l y z e r u s e d i n t h is p r o j e c t h a s a v e r a g i n g c a p a b i l it i e s. T h e s p e c t r a f o r

m e a s u r e m e n t s u s e d w e r e t h e r e s u lt s o f a t le a s t f o u r r .m . s , a v e r a g e s . T h e m e a s u r e d n a t u r a l

f r e q u e n c i e s w e r e g e n e r a l l y c o n s i s t e n t w i t h i n 1 , ,,

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2 7 6

L . R . D E O B A L D A N D R . F . G I B S O N

-~- Added mass

9 \ ~. Force transducer

I m p u l s e ~ " \ / PCB 2 08 A O2

h a m m e ~ ~ , , A ~ V

J J ~ " ~ Eddy current

J . T es t p la te ~ ~ ] ' p , - M e t a l l i c f oi l

- Co tton pad \ Tabletop

I I C ~ 1 7 6m n lifie r Ii [ ~ 1isplacemen.~=., ,= t i x

9 | E x c i t a t i o n

| s igna l

/ Response

J FFT I s / s ignal

I a n a l y z e r I

J H P 3 5 8 2 A j ~

Desk

computer

HP 9836

Printer

HP 2671G

Figure 1 . B lock diagram for m odal analysis equipment9

A p p r o x i m a t e m o d e s h a p e s p l o t t e d fr o m th e R a y l e i g h - R i t z s o l u ti o n w e r e u s e d t o

c o r r e l a te t h e r e s o n a n t p e a k s w i t h th e a p p r o p r i a t e m o d a l i n d ic e s . A g i ve n r e s o n a n t p e a k

w o u l d n o t a p p e a r w h e n t h e p l a t e w a s i m p a c t e d o n t h e c o r r e s p o n d i n g n o d a l l i n e s .

5 . R E S U L T S

T h e m a t e r ia l p r o p e r t ie s a n d d i m e n s i o n s o f s ix s q u a r e a l u m i n i u m a n d g r a p h i t e / e p o x y

p l a te s a r e li st ed in T a b l e I . T h e s e v a lu e s w e r e u s e d i n a F O R T R A N p r o g r a m m e i n w h i c h

e q u a t i o n s d e r i v e d b y t h e R a y l e i g h - R i t z t e c h n i q u e a r e u s e d t o m o d e l v i b r a t i n g p l a t e s .

A S A P I V fi n it e e l e m e n t m o d e l [ 2 9 ] c o n s i s t in g o f 1 4 4 p l a t e e l e m e n t s w a s s o l v e d a s a

c o m p a r i s o n . N e i t h e r t h e R a y l e i g h - R i t z m o d e l n o r t h e f in it e e l e m e n t m o d e l i n c l u d e d

t r a n s v e r s e s h e a r o r r o t a r y i n e r t ia e f f ec ts . T a b l e s 2 a n d 3 li st t h e m o d a l i n d i c e s a n d n a t u r a l

f r e q u e n c i e s o f t h e s ix p l a t e s a s d e t e r m i n e d b y e x p e r i m e n t a l m o d a l a n a ly s is , 3 6 - te r m

R a y l e i g h - R i t z m o d e l , a n d S A P I V F .E . M . V a lu e s in s q u a r e b r a c k e t s g iv e t h e p e r c e n t

d i f f e r e n c e s b e t w e e n t h e a n a l y t i c a l a n d e x p e r i m e n t a l n a t u r a l f r e q u e n c i e s .

T h e a c t u a l n a t u r a l f r e q u e n c i e s w e r e e x p e c t e d t o li e b e l o w t h e R a y l e i g h - R . it z s o l u t i o n

a n d a b o v e t h e f i n i t e e l e m e n t r e s u l t S . I n t h e S A P I V f i n i t e e l e m e n t p r o g r a m m e a l u m p e d

m a s s m a t ri x is u se d , w h i c h w o u l d h a v e a t e n d e n c y t o p r o d u c e f r e q u e n c ie s w h i c h a r e a

l o w e r b o u n d o n t h e a c t u a l f r e q u e n c i e s [3 0 ]. T h e R a y l e i g h - R i t z a n d f in it e e l e m e n t n a t u r a l

f r e q u e n c i e s c o n s is t e n tl y a g r e e d w i t h in 3 % . W h e n c o m p a r i n g t h e e x p e r i m e n t a l n a t u r a l

f r e q u e n c i e s t o th e t w o s e t s o f a n a l y ti c a l n a t u r a l f r e q u e n c i e s , t h e e r r o r w a s v e r y i n c o n s i s t e n t

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O R T H O T R O P I C P L A T E E L A S T IC C O N S T A N T S

TAaLE 1

D i m e n s i o n s a n d m a t e r i a l p ro pe rtie s o f p l a t e s

277

a b

Boundary (cm

condition (inches))

h p E~ E~ G~y

(mm (g/cm 3 (GPa (GPa (GPa

(inches)) (Ibm/in3)) (MPs i) ) (M ps i )) (Mpsi)) v~ v.,,

Aluminium p la tes t

F-F-F-F$

C-F-F-F- 2 5 . 4 3.160 2.77 72.4 72.4 28.0 0.33 0-33

(10.0) (0 .1244) (0-100) (10 5) (10.5) (4.06)

C-C-F-C

Graphite expoxy platesw

F-F-F-F 1-483

(0.05839)

C-F-F-F 25,4 1.688

(10,0) (0.06646)

C-C-F-C 1.379

(0.05428)

1.584 127.9 10.27 7.312 0.22 0.0177

(0.05723) (1 8. 55 ) (1 .4 89 ) (1.061)

t 2 0 2 4 A l u m i n u m T 6 ; p r o p e r t i e s t a k e n f r o m r e f e r e n c e [ 3 5 ].

w G r a p h i t e / e p o x y c o n s i s t i n g o f 1 2 p l i e s o r i e n t e d a t 0 d e g r e e s a n d f a b r i c a t e d f r o m F i b e ri t e H y - E 1 0 3 4 C p r e p r e g ;

p r o p e r t i e s t a k e n f r o m r e f e r e n c e [ 2 ] .

:~ B o u n d a r y c o n d i t i o n c o n v e n t i o n x = 0 ) - y = b ) - x = a ) - y = 0 ) ; C , c l a m p e d ; F , fr e e .

ranging from 0.1 to 24%. The natural frequency errors of the isotropic aluminium plates

are,generally less than those of the highly orthotropic graphite/epoxy plates, and the

plates having all edges free produced natural frequencies with less error than those with

clamped edges. One should notice that the mode order of the orthotropic plate is different

from that of the isotropic plate.

The difference between the analytical and experimental natural frequencies may be

~ittributed to a variety of factors. One source of possible error is the lack of ideal boundary

conditions. Shoulders for the plates with clamped edges were fabricated from metal and

adhered to the plate surface. The actual boundary structure would exhibit a degree of

elasticity which would tend to lower the natural frequencies, slightly. The F -F-F-F plate

boundary condition was established by testing the plates on soft foam rubber and cotton

pads. The foam rubber substantially increased the natural frequencies of the first three

modes when compared with the results from the cotton pad. The experimental natural

frequencies of the plates on the cotton pad were utilized for this project. The stiffness of

the cotton pad would tend to increase the natural frequencies.

Another source Of error is the inability of the analytical models to predict the actual

natural frequencies. Neglecting the transverse shear for the isotropic aluminum plate

probably had little effect on the natural frequencies [11]. When the longitudinal Young's

modulus is much greater than the in-plane shear modulus, which is the case for the

graphite /epoxy plates used here, neglecting the through-the-thickness shear deformations

can cause substantial error in the natural frequencies [31, 32]. Although vibration ampli-

tudes were very small here, when. the vibrational amplitude is greater than one half the

plate thickness, membrane forces become important and the system becomes non-linear

[33]. Another problem which must be addressed is the frequency dependence of the

elastic constants [2, 34]. The epoxy matrix tends to be frequency dependent, which in

turn will cause the apparent elastic constants to be,different for various natural frequencies.

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278 L. R. DEOBALD AND R. F. GIBSON

TABLE 2

atural frequencies o f aluminium plates

Boundary Modal

condition indices

r

Experimental

Natural frequencies (Hz)

SAP IV F.E.M.

36-Term

Rayleigh-Ritz

F-F-F-F 22

13

31

23

32

14

41

33

42

24

C-F-F-F 11

12

21

13

22

23

31

14

32

24

C-C-F-C 11

21

12

31

22

32

13

41

23

42

156.7 161-4 [3 .0]t

232-5 230.0 [- 1. 1]

300.4 292.3 [- 2.7]

411.7 412.9 [0.3]

411-7 412.9 [0-3]

744.9 719.3 [- 3.4]

744.9 719.3 [-3. 4]

755.7 755.9 [0.1]

821.8 803.2 [- 2. 3]

936.5 917.6 [-2. 0]

41.03 42.10 [2.6]

99.04 102.6 [3.6]

247.0 254.6 [3.1]

330-1 325.8 [- 1. 3]

359.8 371.8 [3.3]

638.9 648.5 [1-5]

730.5 735.9 [0.7]

769.0 752.9 [-2. 1]

830.1 851.6 [2.6]

-- 1102.0

246.9 292.0 [18.0]

481.0 487.4 [1.3]

660.2 774.8 [17:0]

884.8 929.2 [5.0]

926.4 989.5 [6.8]

1428.0

-- 1505.0

1621 . 0

1733 0

-- 2105.0

163-2 [4.2]

237.6 [2-2]

299-9 [-0 .2 ]

424.3 [3 1]

424.3 [3 1]

749.4 [0-6.]

749.4 [0 6]

780.5 [3.3]

843.4 [2.6]

949.8 [1.4]

42.50 [3.6]

103.3 [4.3]

259.7 [5.1]

333.3 [1.0]

377.6 [5.0]

661.8 [3.6]

750.7 [2.8]

783.6 [1.9]

870-6 [4.9]

1136.0

292.4 [18.0]

488-3 [1.5]

773.4 [17.0]

936.7 [5.9]

984.5 [6.3]

1427.0

1496.0

1641.0

1715.0

2115.0

t Percent error when compared with experimental values.

The proce dure for obt ining the elastic constants presented in the previous section was

incorporated in a BASIC computer programme called MA RT . The programm e was

based in a desktop computer which was interfaced with a specti'um analyzer (see Figure

1). Before discussing the apparent elastic constants which resulted from MART, the

solution characteristics are described. The first characteristic to be considered is the

inherent tendency towards ill conditioned solution matrices. The programme will always

form 4 x 4 soluti on matrices in order to solve for four new be ndin g stiffnesses. The pro ble m

is that one o r more of the stitinesses often cont ribu te little or noth ing to any given

eigenvalue. If one of th e stiffnesses does not contribute muc h to all four natural frequencies

used for the solution, then a singular or ill conditioned solution matrix will occur. The

stiffness D~2, which has the Poisson ratio as the primary contributor, was usually the

cause of ill conditioned matrices when they occurred. The ill conditioned matrices were

detected and bypassed by limiting the maximum condition number, which was based on

the Euclidean norms.

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ORTHOTROPIC PLATE ELASTIC CONSTANTS

TABLE 3

atu ral frequencies o f graphite epoxy plates

279

Natural frequencies (Hz)

Boundary Modal 36-Term

condition indices Experimental SAP IV F.E.M. Rayleigh-Ritz

F-F- F-F 22 49.37 51.33 [4.0 ]t 51.81 [4.9]

13 78.89 60.54 [-23 .0 ] 60.19 [-24.0]

23 113.2 120.7 [6.6] 122.0 [7.8]

14 188.3 161.8 [-14.0 ] 165.9 [-12 .0 ]

31 210-5 208.6 [-0.9] 212.7 [1-1]

24 222.4 225.2 [1 .3] 229.0 [3.0]

32 231.6 231.6 [0.0] 236.7 [2-2]

33 295.2 299.2 [1.4] 306.5 [3.8]

15 350.0 316.2 [- 9. 7] 326.1 [- 6.8]

25 381-3 378.8 [-0.7] 386-2 [1-3]

C-F-F-F I1 33.94 37.93 [12.0] 38.05 [12.0]

12 51.01 52.23 [2.4] 52.61 [3.1]

13 102.4 103.9 [1.5] 105.5 [3.0]

14 210.9 212.3 [0.7] 217.3 [3.0]

21 234.1 235.9 [0.8] 238.5 [1.9]

22 283.1 254.3 [-10. 0] 257.5 [- 9.0]

23 314.7 310.9 [- 1. 2] 316.9 [0.7]

15 350.3 384.3 [9.7] 394.8 [13.0]

24 380.4 415.9 [9-3] 423.6 [11.0]

25 526.5 577.0 [9-6] 589.9 [12.0]

C- C- F- C 11 71-29 71.90 [0.9] 70.97 [- 0. 5]

'~ 12 170.9 171.9 [0.6] ~_67~5 [-2.0]

21 199.8 219.5 [10.0] 219.8 [10.0]

22 307.9 305.6 [-0. 8] " 298.2 [-3-2]

13 315.3 328.1 [4.1] 314.9 [-0 .1 ]

23 388.0 457.0 [18.0] 432.5 [12.0]

9 14 511.4 544.5 [6.5] 511.9 [0.1]

31 - - 566.9 564.0

24 - - 624.0 618.0

32 - - 676.2 624.7

t Percent error when comp ared with experimental values

The convergence characteristics of the solution can now be discussed. The converg ence

characteristics of the elastic constants are inconsistent. If the solution is nearly singular,

then the elastic constants may diverge. As mentioned in the previous paragraph, such a

solution can be bypassed. Some combinations of natural frequencies will form solutions

which conver ge smoothly to the actual elastic constants. The third type of convergence

which is apparent is when one or more of the bending stiffnesses oscillate about the

actual value during the iteration process. Since each iteration consumes considerable

computer time, a subroutine was written to detect oscillating stittnesses and to artificially

increase the convergence rate.

Tables 4 and 5 list the elastic const~,nts of aluminium and graphite/epoxy plates,

respectively. The acquisition of good results for the actual exper imental natural frequencies

was found to be difficult. A slight error in the natural frequencies greatly magnifies the

error in the elastic constants. The first five natural frequencies listed in Table 2 were used

to calculate the 9 constants of the F- F- F~ F aluminium plate. The two Young' s

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2 8 0

L R DE OBAL D AND R F GI BSON

TABLE 4

Elast ic constants from aluminium plates

Bo un dar y M od e E, E>. G.~y

cond i t ion com bina t ions (GP a) (G Pa ) (GP a) V.~y

F -F - F - F 22 , 13, 31 , 14 69 .8 69 .9 25 .7 0 .361

22, 13, 23, 14 69.3 70.1 25.7 0.355

22, 31, 23, 14 69-0 69.6 25.7 0.366

13, 31, 23, 14 69.8 69-6 25.5 0.363

Av erage 69.5 69.9 25.6 0.361

Stan dard devia t ion 0 .40 0 .24 0 .09 0 .005

C - F - F - F 11, 12, 21, 13 62.6 66.2 25.4 0.418

11, 21, 13, 22 62.8 67.3 25.5 0-412

12, 21, 13, 22 66.5 77 .6 26-1 0-244

A verag e 64.0 70.4 25.3 0.358

Stan dard devia t ion 2 .20 6 .34 0-82 0 .099

TABLE 5

Elast ic constants from graphi te epoxy plates

Bo und ary M ode E~ Ey Gx).

cond i t ion com bina t i ons (GP a ) (G Pa ) (GPa ) vxy

F -F -F - F 22 , 31 , 24 , 32 125.1 10-3 6 .6 -0 .26 7

22, 31, 24, 33 125.2 10.3 6.6 -0 .2 43

22, 24, 32, 33 125.2 10.3 6-6 -0 9

31, 24, 32, 33 125.2 10.3 6.6 -0 9

Ave rage 125.2 10 .3 " 6 .6 -0 .2 49

Stan dard devia t ion 0-05 0 .00 0 .00 0 .012

C - F - F - F 12, 13, 21, 23 105.1 8.9 6.6 1.39

C - C - F - C 11, 12, 21, 22 68.1 8.6 11-1 1-18

m o d u l i , t h e s h e a r m o d u l u s a n d t h e P o i s s o n r a t io a r e w i t h in 4 , 4 , 8 .5 a n d 1 0 % , r e s p e c t i v e l y ,

o f t h e v a l u e s l is t e d in T a b l e 1 . O n e s h o u l d n o t e t h a t t h e e l a s t i c c o n s t a n t s i n T a b l e 1 a r e

t y p i c a l v a l u e s p r e s e n t e d i n r e f e re n c e [ 3 5 ] a n d m a y v a r y f r o m t h e a c tu a l p r o p e r t i e s o f t h e

p l a t e . F o r e x a m p l e , a f t e r t h e c a l c u la t i o n s w e r e ' c o m p l e t e d , i t w a s f o u n d t h a t t h e v a l u e s

o f E , G , a n d v in T a b l e 1 d i d n o t s a ti s f y t h e r e l a t io n s h i p a m o n g e l a s ti c c o n s t a n t s

G = E / 2 ( i + v ) . ( 1 3)

V a l u e s o f E = 1 0.1 M p s i , 0 = 3 . 8 M p s i a n d v = 0 . 3 3 f r o m r e f e r e n c e [ 3 6] d o s a ti s f y t h is

e q u a t i o n a n d a r e a l s o i n b e t t e r a g r e e m e n t w i t h t h e m e a s u r e m e n t s r e p o r t e d i n T a b l e 4 .

T h e f i rs t f iv e n a t u r a l f r e q u e n c i e s o f t h e C - F - F - F a l u m i n i u m p l a t e t a k e n f o u r a t a t i m e

p r o d u c e d t h r e e s u c c e s s f u l so l u ti o n s 9 T h e e r r o r in t h e tw o Y o u n g ' s m o d u l i , s h e a r m o d u l u s

a n d t h e P o i s s o n r a t i o a v e r a g e s . a b o u t 3,;5 , 1 2, 10 a n d 1 6 % , r e s p e c t i v e l y . O n e s h o u l d n o t i c e

t h a t t h e tw o Y o u n g ' s m o d u l i a r e d if f e re n t d u e t o a b i a s c a u s e d b y t h e l a c k o f d o u b l e

s y m m e t r y in t h e b o u n d a r y c o n d it io n 9 A n a t t e m p t w a s m a d e t o o b t a i n e l a st ic c o n s t a n t s

f o r t he C - C - F - C a l u m i n i u m p l at e . E it h e r t h e m o d e c o m b i n a t i o n s p r o d u c e d i ll c o n d i t i o n e d

s o l u t io n m a t r i c e s o r th e e l a s t ic c o n s t a n t s w e r e c o m o l e t e l v u n r e a s o n a b l e .

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ORTHOTROPi C PLATE ELASTIC CONSTANTS 28

The natural frequencies used to obtain elastic constants for the F -F-F -F graphite/epoxy

plate had modal indices 22, 31, 24,.32 and 33. The two Young's moduli are very close to

the values listed in Table 1 [2]. The shear modulus differs by about 10 . The resulting

Poisson ratio is obviously incorrect. As explained earlier, the problem is that the small

Poisson ratio has a negligible effect on the natural frequencies of the graphite/epoxy plate.

For the C-F-F-F and C-C-F-C graphite plates, only the first four natural frequencies

were used to obtain one set of elastic constants each. The three moduli of the C-F-F-F

plate have an estimated error range from 11 to 22 percent. The error of the moduli from

the C-C-F-C plate ranges from 19 to 88 percent. For both plates, the Poisson ratios were

completely unreasonable.

To conclude the discussion of the resulting elastic constants, some general observations

are made. The best results were from the F -F-F-F plates and the worst results were from

the C-C-F-C plates. The F-F-F-F plates provided the largest percentage of well condi-

tioned solution matrices. The F-F-F-F plates also appeared to be the least sensitive to

error in the natural frequencies. Mention must be made of the validity of the elastic

constants presented in Tables 4 and 5. The error in the two Young's moduli was

inconsistent. The differences between the calculated shear moduli and the shear moduli

from Table 1 were consistently about 10 for the F-F-F -F and C-F -F -F plates. A

reasonable Poisson ratio was obtained only from the F-F-F-F aluminium plate. The

authors have confidence only in the results from the F -F-F-F aluminium plate, because

its experimental natural frequencies consistently differ from the Rayleigh-Ritz predictions

by a small amount.

6. CONCLUSIONS AND RECOMMENDATIONS

As previously discussed, individual orthotropic elastic constants have been calculated

by vibration tests, but this research was one of only two attempts to obtain all four elastic

constants in a single test. Although this is a good start, the technique needsmore ref inement

and verification before it may be widely used.

A major problem in this research was that the experimental natural frequencies did

not always closely match the predicted values. The key to this technique is to model

accurately the plates' natural frequencies while maintaining relatively small eigenfunction

matrices so that they may be repetitively solved in a reasonable amount of computer

time. Before a researcher proceeds with a method to obtain the elastic constants, the

authors suggest that a Rayleigh-Ritz model be developed which produces natural frequen-

cies which match experimental values consistently within 3 percent. A variety of assumed

mode shapes may be used in the Rayleigh-Ritz model. The effects of through-the-thickness

shear deformations may not be neglected when the ratio of longitudinal Young's modulus

to the in-plane shear modulus is very large [31, 32].

The technique used for experimental modal analysis seemed to be very effective. Since

a major problem was the difficulty in establishing near ideal boundary conditions, the

authors suggest that subsequent research use plates which have all edges free. A variety

of materials should be tested, and orthotropic ply stacking Sequences, such as cross-ply

plates, could also be considered.

The method for obtaining the elastic constants from the natural frequencies also caused

some difficulties. The original objective.of developing a test suitable for quality control

was not realized due to long computer run times. Computational time may be reduced

by using a compiled language and more efficient algorithms. Since the Poisson ratio often

causes an ill conditioned solution matrix , three elastic moduli may be found by using

three natural frequencies and an independently,,determined Poisson ratio. Likewise, an

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2 8 2

L R DE OBAL D AND R F GI BSON

o p t i o n m a y b e p r o v i d e d a l l o w i n g t h e d e t e r m i n a t i o n o f t h e t w o i s o t r o p i c e l a st i c m o d u l i

f r o m t w o n a t u r a l f r e q u e n c i e s . T h e m e t h o d u s e d f o r a v e r a g i n g th e e la s ti c c o n s t a n t s w a s

i n e f fi c ie n t , a n d t h u s i n f u t u r e w o r k o n t h is p r o j e c t a l t e r n a t i v e t e c h n i q u e s s h o u l d b e

e x p l o r e d .

A l t h o u g h a ll o f t h e o b j e c t i v e s w e r e n o t r e a l i z ed , t h i s r e s e a r c h m a y b e c o n s i d e r e d

s u c c e s s fu l . T h e p o t e n t i a l t i m e s a v i n g s a n d v e r s a t i li t y o f th e t e c h n i q u e c o m b i n e d w i t h

s e v e r a l p r o m i s i n g r e s u lt s in d i c a t e s th e m e t h o d w a r r a n t s f u r t h e r c o n s i d e r a t i o n . W i t h

c o n t i n u e d d e v e l o p m e n t , th is m o d a l a n a l y s i s / R a y l e i g h - R i t z t e c h n i q u e f o r o b t a i n in g

o r t h o t r o p i c e l a s t i c c o n s t a n t s w i l l b e c o m e a v a l u a b l e n o n - d e s t r u c t i v e t e s t .

7 . A C K N O W L E D G M E N T S

W e w i s h to t h a n k H e w l e t t - P a c k a r d C o r p o r a t i o n f o r t h e e q u i p m e n t d o n a t i o n s t o th e

U n i v e r s i t y o f I d a h o . T h i s p r o j e c t w o u l d n o t h a v e b e e n p o s s i b l e w i t h o u t t h e c o m p u t e r ,

s p e c t r u m a n a l y z e r a n d p r i n t e r w h i c h w e r e d o n a t e d . W e e x t e n d o u r a p p r e c i a t i o n t o C a r o l y n

A h e r n , u n d e r g r a d u a t e a s s i st a n t, a n d D a r r e l B r o w n , m a c h i n i s t , f o r t h e ir v a l u a b l e c o n t r i b u -

t i o n s t o th i s p r o j e c t. T h e c a r e f u l t y p i n g o f t h e m a n u s c r i p t b y V a l er ie S m i t h is a ls o g r a t e f u l l y

a c k n o w l e d g e d .

R E F E R E N C E S

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9 See pp. 3 4-46 and 147-187.

2. S. A. SUAREZ 1984

Ph.D. Dissertation, Unit,ersity o f Idaho.

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" D e t e r m i n a ti o n o f t h e d y n a m i c p a r a m e t e r s o f p o l y m e r p l a te s .

4 . J . A. WOLF, JR. and T. G. CARNE 1979 Meeting of Society for Exper nental Stress Analysis,

San Francisco, California, P aper No . A-48. Ident i f ica t ion of the e las t ic cons tan ts for com posi tes

us ing modal ana lys i s .

5 . G . CALDERSMITH and T. D. ROSSING 1984 Applied Acoustics 17 , 33 -44 . De t e rmina i i on o f

modal coupl ing in v ibra t ing rec tangular p la tes .

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International M oda l Analysis Conference, Orlando, FL , I , 44 -49 . De t e rmina t i on o f t he m a te r i a l

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1058-1063. C oupl ing o f Lag range in te rpola-

t ion , mo dal ana lys i s and sens i t iv i ty ana lys i s in the de te rm ina t ion of an iso t ropic p la te r ig id it ies .

8. D. YOU NG 1950 Journal o f Applied Mechan ics 17, 448-453. V ibra t ions o f rec tang ular p la tes by

the Ri tz method.

9 . S . M. DICK INSO N an d E. K. H. LI 1982 Journal of Sou nd and Vibrat ion SO, 292-297. O n the

use o f s imp ly suppor t ed p l a t e f unc t i ons i n t he Ray l e igh -R i t z me thod app l i ed t o t he f l exu ra l

v ibra t ion of rec tangular p la tes .

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Journal o f Soun dan d Vibration

102,493-499. Na tura l f requencies of r ec t an gu lar

p l a t e s u s ing cha rac t e ri s t ic o r t hogon a l po lynom ia ls i n Ray l e igh -R i t z me thod .

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N A S A S P - 1 6 0 ,

43-115. Vibra t ion of p la tes .

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o f r ec t angu l a r o r t ho t rop i c p l a te s , w

14. S . M. DICKINSON 1978 Journal of Sou nd a nd Vibrat ion 61, 1-8 . The buckl ing and f requency

of f l exu ra l v ib r a ti on o f r e c t angu l a r i so t rop i c and o r t ho t rop i c p l a t e s u s ing Ray l e igh ' s me thod .

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approx ima te exp re s s ions fo r t he na tu r a l f r equenc i e s o f i so t rop i c and o r t ho t rop i c r ec t angu l a r

plates .

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O R T H O T R O P I C P L T E E L S T I C C O N S T N T S 2 8 3

1 6. J . E . A S H T O N a n d M . E . W A D D O U P S 1 96 9 Journal of Com posite M aterials 3 , 1 4 8 - 1 6 5 . A n a l y s i s

o f a n i s o t r o p i c p l a t e s .

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v i b r a t io n s o f u n s y m m e t r i c a l l y l a m i n a t e d a n i s o t r o p i c p la t e s w i t h c l a m p e d e d g e s.

1 9. J . E . A S H T O N a n d J . M . W H I T N E Y 1 9 70 Theory of Lam inated Plates . S t a m f o r d C T : T e c h n o m i e

P u b l i s h i n g C o . S e e p p . 1 1 - 1 2 a n d 3 1 - 3 5 .

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R e i n h o l d . S e e p p . 1 0 1 - 1 0 9 , 2 5 3 -2 6 1 a n d 4 6 1 - 4 6 3 .

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