PREDICTION AND MEASUREMENT OF THE NATURAL FREQUENCIES AND DAMPING CAPACITY OF CARBON FIBRE-REINFORCED PLASTICS PLATES

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PREDICTION AND MEASUREMENT OF THE NATURAL FREQUENCIES AND DAMPING CAPACITY OF CARBON FIBRE-REINFORCED

PLASTICS PLATES

R. Adams, D. Lin, R. Ni

To cite this version:

R. Adams, D. Lin, R. Ni. PREDICTION AND MEASUREMENT OF THE NATURAL FREQUEN- CIES AND DAMPING CAPACITY OF CARBON FIBRE-REINFORCED PLASTICS PLATES.

Journal de Physique Colloques, 1983, 44 (C9), pp.C9-525-C9-530. �10.1051/jphyscol:1983977�. �jpa-

00223427�

J O U R N A L D E PHYSIQUE

Colloque C9, suppl6ment a u n012, Tome 44, dgcembre 1983 page C9-525

P R E D I C T I O N AND MEASUREMENT OF THE NATURAL FREQUENCIES AND DAMPING C A P A C I T Y OF CARBON F I B R E - R E I N F O R C E D P L A S T I C S P L A T E S

R.D. Adams, D.X.

in*

and R . G . ~ i * *

Reader i n Mechanicat Engineering, University

of

BristoZ, BristoZ,

U . K .

haan an xi MechanicaZ Engineering I n s t i t u t e , Xian, Shaanzi, China

* * ~ n s t i t u t e

o f

AeronauticaZ Materiazs, Peking, China

Rdsumd - L ' o b j e t de c e t t e i n v e s t i g a t i o n e s t de p r & d i r e l a frdquence n a t u r e l l e e t l e c o e f f i c i e n t d'amortissement de plaques s t r a t i f i E e s pour d i f f d r e n t s modes de v i b r a t i o n p a r l a technique des 6lEments f i n i s . La d s f i n i t i o n du c o e f f i c i e n t d'amortissement e s t

Y' = A

U / U , oii U e s t l t & n e r g i e maximale de ddformation mise en j e u p a r c y c l e , e t A U , l t E n e r g i e d i s s i p d e pendant l e c y c l e . I c i , U e s t c a l c u l d p a r l a technique des dldments f i n i s p r e n a n t en compte l a dgforma- t i o n en c i s a i l l e m e n t t r a n s v e r s a l e (LT);AU e s t ddtermind

B

p a r t i r d ' u n modsle d'dl6ments a m o r t i s . Les amortissements, l e s modes, e t l e s frdquences n a t u r e l - l e s de quelques plaques r e n f o r c d e s avec des f i b r e s de carbones ou d e v e r r e s o n t p r 6 d i s e t mesurgs.

A b s t r a c t

-

The o b j e c t i v e of t h i s i n v e s t i g a t i o n is t o p r e d i c t t h e n a t u r a l frequency and s p e c i f i c damping c a p a c i t y of laminated composite p l a t e s i n v a r i o u s modes of v i b r a t i o n by u s i n g t h e f i n i t e element method. The simple d e f i n i t i o n o f s p e c i f i c damping c a p a c i t y i s J,

=

AU/U, where U i s t h e maximum s t r a i n energy s t o r e d p e r c y c l e and U i s t h e energy d i s s i p a t e d p e r c y c l e . I n t h i s work, U i s c a l c u l a t e d by a f i n i t e element method which i n c l u d e s t r a n s v e r s e s h e a r deformation; AU i s determined from a damped element model. The s p e c i f i c damping c a p a c i t i e s , mode s h a p e s , and n a t u r a l f r e q u e n c i e s of v a r i o u s f r e e - f r e e carbon and g l a s s f i b r e - r e i n f o r c e d p l a s t i c s p l a t e s have been p r e d i c t e d and measured.

I. INTRODUCTION

F l a t p a n e l s a r e used i n many s t r u c t u r a l a p p l i c a t i o n s i n l a n d and o c e a n - g o i n g v e h i c l e s , a i r c r a f t and s p a c e c r a f t . Both sandwich c o n s t r u c t i o n and f i l a m e n t a r y composite m a t e r i a l s a r e used i n p l a t e t y p e s t r u c t u r e s because t h e s t r u c t u r a l s t i f f n e s s can be i n c r e a s e d w i t h o u t adding e x c e s s weight. For t h e o p t i m a l d e s i g n of p a n e l s made from composite m a t e r i a l s , s t r u c t u r a l d e s i g n e r s r e q u i r e more u s e f u l and p r a c t i c a l methods f o r o b t a i n i n g t h e c o r r e c t numerical r e s u l t s of t h e s t i f f n e s s and damping of lamina- t e d composites. During t h e l a s t decade, s e v e r a l a u t h o r s have t r i e d t o p r e d i c t t h e s t i f f n e s s and damping of p l a t e s , b u t most of them have been r e s t r i c t e d t o t h e n a t u r a l f r e q u e n c i e s of a n i s o t r o p i c laminated composite p l a t e s , and t h e r e have been few on damping.

I n t h i s work we have i n v e s t i g a t e d t h e p r e d i c t i o n of t h e n a t u r a l modes and s p e c i f i c damping c a p a c i t y o f a n i s o t r o p i c laminated p l a t e s u s i n g , and e x t e n d i n g , t h e f i n i t e element method d e s c r i b e d by Cawley and Adams /1/ and t h e damped element model proposed by Adams and Bacon / 2 / .

To i n v e s t i g a t e t h e accuracy o f t h e t h e o r e t i c a l p r e d i c t i o n , a comparison w i l l b e made w i t h t h e e x p e r i m e n t a l r e s u l t s from v a r i o u s laminated CFRP and GFRP r e c t a n g u l a r p l a t e s i n t h e f r e e - f r e e c o n d i t i o n f o r t h e f i r s t s i x modes of v i b r a t i o n . Some g r a p h i c a l methods f o r s i m p l i f y i n g t h e p r e d i c t i o n a r e p r e s e n t e d .

A l l t h e p l a t e s used i n t h i s i n v e s t i g a t i o n were mid-plane symmetric s o a s t o e l i m i n a t e b e n d i n g - s t r e t c h i n g coupling. It i s , however, p o s s i b l e t o i n c l u d e t h i s e f f e c t i n t h e a n a l y s i s if asymmetrical l a m i n a t e s were t o be used.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983977

JOURNAL DE PHYSIQUE

11. THEORY

The s p e c i f i c damping c a p o c i t y i s d e f i n e d a s

where AU i s t h e energy d i s s i p a t e d d u r i n g a s t r e s s c y c l e and U i s t h e maximum s t r a i n energy.

U i s o b t a i n e d a s f o r an undamped system a s f o l l o w s :

where s i j and o a r e t h e s t r a i n s and s t r e s s e s r e l a t e d t o t h e f i b r e d i r e c t i o n . i j

Equation (2) may b e reduced t o a s t a n d a r d form a s

U

= 3

^{61T

[K]

^{I61 (3)}

where I6 i s t h e n o d a l p o i n t displacement m a t r i x . Here, f i v e degrees of freedom f o r each n o d a l p o i n t and 8 n o d a l p o i n t s f o r each element a r e used, and

[K]

i s t h e s t i f f - n e s s m a t r i x . I n t h e e v a l u a t i o n o f t h e maximum s t r a i n energy U, t h e Young's modulus of

oO,

90° u n i d i r e c t i o n a l f i b r e r e i n f o r c e d beams, EL, ET, and t h e s h e a r modulus of a

o0

u n i d i r e c t i o n a l rod GLT a r e used.

A s g i v e n i n Ref. 2

where 6(AU) i s t h e energy d i s s i p a t e d i n each element, and i s d e f i n e d a s 6(AU)

=

&(Lull

+

6(AU2)

+

^{6(AU23) f 6(AU13)}

+

6(AU12)

and 6(AU1)

= 2 JIL

€11 011, 6(AU2)

=

$

JIT

922 022

S u f f i x 1 denotes t h e f i b r e d i r e c t i o n , while 2 and 3 a r e t h e two d i r e c t i o n s t r a n s - v e r s e t o t h e d i r e c t i o n of t h e , f i b r e s . JIL, $ T , e t c . a r e t h e a s s o c i a t e d damping c a p a c i t i e s i n each d i r e c t i o n , and t h e y a r e o b t a i n e d from t e s t s on u n i d i r e c t i o n a l beams.

Equation (4) may be reduced t o m a t r i x form a s :

where

[$I =

- -

$ , O 0 0 0

0 J I T 0 0 0

0 0

JI,,

0 0

0 0 0

JILT

0

0 0 0 0 JILT

- -

Using t h e same method a s with Eqn. ^{( 2 ) ,} Eqn. ( 5 ) may be reduced t o

where C6) i s t h e same m a t r i x a s i n Eqn. ( 2 ) and was o b t a i n e d from t h e f i n i t e element r e s u l t s . [ K ~ ] is t h e s t i f f n e s s m a t r i x o f t h e damped system, and i t may be e v a l u a t e d s e p a r a t e l y .

111. RESULTS AND DISCUSSION

The composite p l a t e s used i n t h i s i n v e s t i g a t i o n c o n s i s t e d of e i t h e r g l a s s o r HM-S carbon f i b r e i n DX-210 epoxy r e s i n . The p l a t e s were made of 8 o r 12 l a y e r s of pre- impregnated f i b r e i n a h o t p r e s s , such t h a t d i f f e r e n t l a m i n a t e o r i e n t a t i o n s could be o b t a i n e d ; d e t a i l s of t h e p l a t e s used a r e given i n Table 1. The m a t e r i a l proper- t i e s used i n t h e t h e o r e t i c a l p r e d i c t i o n a r e given i n Table 2. A l l t h e v a l u e s i n

Table 1 P l a t e Data

P l a t e Density

No. of

---

number M a t e r i a l l a y e r s kg m-3 vf P l y o r i e n t a t i o n

1 CFRP 8 1446.2 0.342 (oO, go0,

oO,

9 0 ' ) ~

2 CFRP 12 1636.4 0.618 ( 0 ° , - 6 0 0 , 6 0 0 , 0 0 , - 6 0 0 , 6 0 0 ) s

3 GFRP 8 1813.9 0.451 ( 0 ° , 9 0 0 , 0 0 , 9 0 0 ) s

4 GFRP 1 2 2003.5 0.592 ( o O , -60°, 60°,

oO,

-60°, 6 0 ' ) ~

N.B. S u f f i x s means mid-plane symmetric

-

Table 2 Moduli and damping v a l u e s f o r m a t e r i a l s used i n t h e p l a t e s

E 1

- -

^{E 2} - ^G12

- ^UJ

¹

- -

^{2 $12}

M a t e r i a l GPa GPa GPa 6 % % ^{Vl ,v2 vf}

HMS/DX-210 1 7 2 . 1 7.20 3.76 0.45 4.22 7.05 0.3 0.50

Glass/DX-210 37.87 10.90 4 . 9 1 0.87 5.05 6.91 0.3 0.50

DX-210/BF3400 3.21 3.21 1.20 6.54 6.54 6.68 0.34 0

-

t h i s t a b l e were e s t a b l i s h e d e i t h e r by u s i n g beam specimens c u t from a u n i d i r e c t i o n a l p l a t e ( l o n g i t u d i n a l and t r a n s v e r s e damping and Young's moduli) o r c y l i n d r i c a l s p e c i - mens ( f o r measuring t h e s h e a r modulus and damping i n t o r s i o n ) . I t should be noted t h a t t h e v a l u e of t h e t o r s i o n a l damping of a 90° f i b r e o r i e n t a t i o n r o d , Q 2 3 , i s n o t important i n t h e p r e d i c t i o n , s i n c e changing it from 6% t o 15% gave no d i f f e r e n c e t o t h e t h e o r e t i c a l r e s u l t s . I n t h e p r e d i c t i o n , Q23 i s t a k e n as t h e same v a l u e a s $12 which i s t h e v a l u e of t o r s i o n a l damping of a O0 f i b r e o r i e n t a t i o n rod ( i n l o n g i - t u d i n a l s h e a r ) . Because of v a r i a t i o n s i n t h e f i b r e volume f r a c t i o n of t h e p l a t e s , t h e m a t e r i a l p r o p e r t i e s used i n t h e t h e o r e t i c a l p r e d i c t i o n were c o r r e c t e d from a s t a n d a r d s e t g i v e n f o r a 50% f i b r e volume f r a c t i o n

.

3.2 Comparison of t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s

Tables 3 and 4 show f o r t h e first s i x modes t h e t h e o r e t i c a l p r e d i c t i o n and t h e experimental r e s u l t s of CFRP p l a t e s f o r v a r i o u s f i b r e o r i e n t a t i o n s . On t h e whole, t h e r e i s good agreement between t h e p r e d i c t e d and measured v a l u e s . The d i s c r e p a n - c i e s i n n a t u r a l f r e q u e n c i e s a r e l e s s t h a n

l o % ,

and t h e v a l u e s of s p e c i f i c damping c a p a c i t y are very c l o s e . Mode 6 i n p l a t e 3 could n o t be o b t a i n e d e x p e r i m e n t a l l y because t h e i n p u t energy from t h e t r a n s i e n t t e c h n i q u e was i n s u f f i c i e n t . Tables 5

C9-528 J O U R N A L

DE

PHYSIQUE

Outer l a y e r

----+

F i b r e d i r e c t i o n

SDC

(%)

(6.65 1

(1.051 0 * 9 1

(2.6)

(0.92) 0 6 0

1 :51 (1.7) No.

L

Outer l a y e r

-

F i b r e d i r e c t i o n

Table 3 N a t u r a l f r e q u e n c i e s and damping Table 4 N a t u r a l frequency and damping o f v a r i o u s modes of an 8 - l a y e r of v a r i o u s modes o f a 12- ( o O , go0,

oO,

go0, go0,

oO,

l a y e r (oO -60°, 60°,

oO,

go0, oO) carbon FRP p l a t e -60°, 60°: 60°, -60°,

oO,

( P l a t e no. 1 ) . Experimental 60°, -60°, 0') carbon FRP v a l u e s i n b r a c k e t s . p l a t e ( P l a t e no. 2 ) . Experi-

mental v a l u e s i n b r a c k e t s .

1

441.62 (452.5) Frtq. (Hz)

58 . l o

(68.88)

213.31 (218.9)

243.47 (251 2 )

302.51 (305.4)

324.16 (323.5)

and 6 g i v e t h e r e s u l t s f o r GFRP p l a t e s . A l l show good agreement between p r e d i c t i o n and measurement.

( 3 . 0 ) Mode shape

H El

r l

H un

SDC(%)

(1.40)

( 0 88) O-

~-

(0.65) 0 . 6 3 .

(1 261

(0.99) 0.98 .

0 92

-

No

5

The e f f e c t o f a i r damping and t h e a d d i t i o n a l energy d i s s i p a t i o n a s s o c i a t e d with t h e s u p p o r t s and t h e s m a l l p i e c e o f m e t a l r e q u i r e d (which i s connected t o e a r t h ) f o r t h e t r a n s d u c e r a f f e c t t h e r e s u l t s o f t h e very low damping modes such a s t h e 4 t h mode of p l a t e 1, t h e 4 t h mode of p l a t e 3 and s o on. These a r e e s s e n t i a l l y beam modes i n which t h e l a r g e m a j o r i t y of t h e s t r a i n energy i s s t o r e d i n tension/compression i n t h e

6

Freq. [Hz)

165 17 (156.6)

279.14 (272.0)

387.8 (372.3)

432.57 (407.81

51 1-43 (486.1)

Mode shape

E3 m

Ell

EZl

Outer l a y e r

---*

F i b r e d i r e c t i o n

Outer l a y e r F i b r e d i r e c t i o n

-

SDC(%)

(6.7) 7*'6 -

(2.8)

"'

(1.9)

"'

4.87

L .9

(3.2 No.

1

*

Table 5 N a t u r a l frequency and damping Table 6 N a t u r a l frequency and damping

gf

o f v a r i o u s modes o f an 8 - l a y e r v a r i o u s modes of a 1 2 - l a g e r ( 0

,

( o O , go0,

oO,

90°, 90°,

oO,

-60°, +60°,

oO,

-60°, 60 ), g l a s s go0, oO) g l a s s FRP p l a t e FRP p l a t e ( P l a t e no.4). Experi- ( P l a t e no. 3 ) . Experimental mental v a l u e s i n b r a c k e t s . v a l u e s i n b r a c k e t s .

f i b r e s and n o t i n m a t r i x t e n s i o n o r s h e a r . However, t h e r e s u l t s f o r a l l t h e p l a t e s used a r e s a t i s f a c t o r y , even when t h e specimens have i m p e r f e c t i o n s such

ag

s l i g h t v a r i a t i o n s i n t h i c k n e s s and t h e nominal a n g l e o f t h e f i b r e s ( 2 2' t o 5 3 e r r o r ) .

(159.2)

rn

189.79

ul

(180.5)

208.87 (200.051 Freq. (Hz)

66.42 (62.2)

131.62 (131.G 1

I t can be s a i d t h a t t h e morg t h g t w i s t i n g , t h e h i g h e r t h e damping. For i n s t a n c e , f o r an 8 l a y e r cross-ply (0 /90 ) GFRP p l a t e ( s e e Table 5 ) t h e two beam-type modes, numbers 2 and 3, appear s i m i l a r , b u t t h e r e l a t i o n s h i p of t h e n o d a l l i n e s t o t h e o u t e r f i b r e d i r e c t i o n means t h a t t h e h i g h e r mode has much l e s s damping t h a n t h e lower one. The o t h e r modes o f v i b r a t i o n of t h i s p l a t e a l l i n v o l v e much more p l a t e

H

Mode shape

El3

6

3.73

^~

164.46

3L7.16

(326.71 (5.8 1

C9-530 JOURNAL

DE

PHYSIQUE

t w i s t i n g and hence m a t r i x s h e a r t h a n do modes 2 and 3 , and s o t h e damping is higher.

It is important for designers to realise the significance of these results, which show that for at2 the plates the damping values are different for each mode.

V. CONCLUSIONS

In t h i s paper a method f o r p r e d i c t i n g t h e n a t u r a l f r e q u e n c i e s , mode shapes and v i b r a t i o n damping parameters of laminated composite p l a t e s has been d e s c r i b e d . The method i s based on t h e f i n i t e element technique u s i n g t h e damped element model i n which t h e e f f e c t s o f t r a n s v e r s e s h e a r deformation and r o t a r y i n e r t i a were c o n s i d e r e d . The s i g n i f i c a n c e of t h e d i f f e r e n c e o f damping v a l u e s from d i f f e r e n t mode shapes and f i b r e o r i e n t a t i o n s must b e emphasized. The damping v a l u e s f o r mode shapes i n which t h e r e is a l o t of t w i s t i n g a r e g r e a t e r t h a n f o r t h o s e i n which t h e l a r g e m a j o r i t y of t h e s t r a i n energy i s s t o r e d i n tension/compression i n t h e f i b r e and n o t i n matrix t e n s i o n or s h e a r .

REFERENCES

1. Cawley, P. and Adams, R. D., J. Composite M a t e r i a l s ,

12

(1978), 336.

2. Adams, R. D. and Bacon, D. G . C . , J. Composite M a t e r i a l s ,

2

(1973), 402.