Modal Analysis: the International Journal of Analytical and Experimental Modal Analysis v 8 n 1 Jan 1993 p15-33

by T.C. Lai and T.C. Lau, Hong Kong Polytechnic

ABSTRACT This paper describes a method of finding the elastic constants of a generally orthotroplc composite thin plate

through modal analysis based on a Rayleigh·Ritz formulation. The natural frequencies and mode shapes for a plate with free-free boundary conditions are obtained with chirp excitation. The characteristic functions of vibrating beams have been assumed for the plate deflection in developing the eigenvalue problem. Based on the eigenvalue equation and the constitutive equations of the plate, an iteration scheme is derived using the experimentally determined natural frequencies to arrive at a set of converged values for the elastic constants. Four sets of experimental data are required for the four independent constants: namely the two Young's moduli £1 and E2, the in-plane shear modulus G12,and one Poisson's ratio v12. The other Poisson's ratio v21 can then be determined from the relationship among the constants. Comparison whh static test results indicate good agreement. Choo,sing the right combinations of natural modes together with a set of reasonable initial estimates for the constants to start the iteration has been found to be crucial in achieving convergence.

A 111, eigenvectors and deflection coefficients dimensions and thickness of plate a,b,h

-

D,D

Gl2 [K]

-

Q.Q

bending stiffness of plate

Young's modulus along and perpen­dicular to the fiber direction shear modulus transfonnation matrix

lamina stiffness components integrals involving characteristic beam functions maximum potential energy

W( x ,y) max irnum transverse deflection of plate w transverse deflection of plate

x,y rectangular coordinates

Greek 8 angle between fiber and x -axis

l eigenvalues

V12, V21 major and minor Poisson's ratio

p density of plate material cpm(x),8,(y) characteristic beam functions

m vibration frequency of plate

T.C. Lai (SEM memher), Senior Lecturer, Department of Mechanical EnRineering, Hong Kong Polytechnic. Kowloon, Hong Kong.

T.C. Lau, Comulrant, Parsons Brinckerhoff(Asia), Ltd . . Kornhill. Hong Kong.

Final manuscript received: May 26, 1992

15

; ' ' . .

Composite structures such as glass fiber reinforced plastics are normally made of a number of layers

with orthotropic properties. These laminae are stacked to fonn a laminate plate. The plate is said to be specially orthotropic if the fiber directions are parallel to the edges of the plate and generally orthotropic

if they are at an angle to them. The static and dynamic behaviour of these plates depend on the elastic

constants of the material and have been investigated in detail [1]. If the plate being studied is not truly

anisotropic and effects of transverse shear and rotary inertia are ignored, then four independent constants

are required to describe its behaviour.

There are standard tests for finding these constants but they are destructive and require separate

specimens [2,3]. A non-destructive test is thus called for so that orthotropic plates can be tested in the

laminate fonn. To avoid difficulties in supporting the edges when obtaining the required boundary

conditions, the plate should preferably be tested with all its edges free. Modal testing which yields natural frequencies of a vibrating plate provides such a test [4].

Analysis of a vibrating specially orthotropic plate with free edges as a means to ascertain the material

elastic constants has been accomplished [5]. The Rayleigh-Ritz technique was used to study the plate

vibration with an assumed function based on the characteristic functions of vibrating beams. This approach

is well established for analyzing isotropic plates and the integrals resulting from the analysis have been evaluated [6,7]. It was then extended to investigate the influence of anisotropy along with the effect of

varying stacking sequences and orientation on the dynamic behaviour of generally orthotropic plates [8]. A s imilar study was carried out using power series as admissible functions [9] . Determination of anisotropic

plate rigidities using sensitivity analysis has also been undertaken { 10,11].

The present work for a generally orthotropic plate is an extension to that by Deobald and Gibson [5]. In order to have dose control of the test procedure a plate is excited with a periodic chirp signal. The natural

frequencies and mode shapes obtained experimentally relate to the eigenpairs in the eigenvalue problem

resulting from the Rayleigh-Ritz technique. Together with the constitutive equations of the composite plate, an iterative procedure can be established to obtain converged values of the elastic constants.

. .. -. .. ,... .. .... t.•l :� .. •• w

For free transverse vibration of a thin rectangular generally orthotropic plate with mid-plane symmetry,

the equation of motion is shown to be [ 1]

Representing the plate deflection by

the Rayleigh's quotient is

16 January 1993

w(x,y,t) = W(x,y)sinox (2)

(3)

I . , .

.

: .'.

. . . .

.. ' • · ..

.

.

-:

..

.

,

. .... . , .o· ·, ..

:

•. .. ::

To apply the Rayleigh- Ritz technique, the maximum deflection is assumed to be

p q W(x,y) = _L_LAmn<I>m(x)8n(Y)

m=l 11=l (4)

where <I> m(x) and 911(y) are the free-free characteristic beam functions in Ref. [7]. Substituting Eq. (4) into Eq. (3) and minimize a1- with respect to the deflection coefficients results in

a set of linear homogeneous equations in the fonn of

m=l n=I (5)

i = 1,2,3 . . . p k = l,2,3 . . . q

where the beam function integrals QI,im,kn etc. have been evaluated numerically and are tabulated in the Appendix.

Equation (5) represents an eigenvalue problem

p q L L [ C;k,mn - AOimokn )Amn = 0 m=I 11=l

where

and

i = 1,2,3 ... p

= 1 for r = s

= 0 for r :F. s

k = 1,2,3 ... q

By rearranging the suffices of the D's and Q's, Cik,mn is rewritten as

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

Substitute Eq. (7) into Eq. (6) will give

(6)

(7)

Modal Analysis 17

'I • .. '

.

. 0: !

!

. . �

i: ' ' .. > ·• ' ::

.I ,

. ,

.� .

' ·> �· . ·' .> ' ·.

' ' ..

. � ·�­·'· . . · . . . . ,

t "

: .� . . . ' ..

· . ' .. .. i ::

.I'; . , > ·; .

,

·� ,. " '

· ' � .. '�� ·� ··: .ll · :�: l ·.�

, '

. • ' . ... ' . .i · . • •

Mik = C;k. 1 1 A11 + c,.k,l2Al2 + ....... + c,.k ,pqApq = [ Q,,ik. 1 1 A11 + Ql,ik .12 A12 + ....... + Qt,ik,pqApq ]v. +[ Ql.ik.ll A,, + Qz,ik,t2 A12 + ....... + Q2,;k.pqApq ]n2 + [ Q3.ik, 11 A11 + Q3,;k.t2At2 + ....... + Q3,ik,pqApq ]v3 +[ Q4,ik.tt A,, + Q4,ik,l2Al2 + ······· + Q4,tk,pqApq ]D4 + [ Q5,ik,J 1 Att + Q.s.tk,I2At2 + ....... + Q5,ik.pqApq ]Ds

or

where

+[ Q6,ik.tt Au + Q6,ik,t2AJ2 + .. .. .. . + Q6.ik,pqApq ]v6

p q H-��Amn Q r-�� A r,ik.mn

m=l n=l ik

(8)

r = 1,2,3,4,5,6

i=1,2.3 ....... p k = 1,2,3 ....... q

From the constitutive equations of a generally orthotropic laminate [12], the lamina stiffness can be expressed in terms of the four basic elastic constants as

-

Ql Ku -

Q2 K21 -

Q3 K31 Q4 K4t

-

Q5 Kst -

Q6 K61 m4 n4

m2n2 -

m2n2 m3n mn3

where m = cos 9 and n = sin a and

18 January 1993

Dt D2 D3 D4 Ds D6

--

Kt2 K13 Kt4 K22 K23 K24 Qll K32 K33 K34 Q22 K42 K43 K44 Ql2 Ks2 Ks3 Ks4 Q66 K62 K63 K64

n4 2m2n2 4m2n2 m4 2m2n2 4m2n2 Qll

m2n2 m4 + n4 -4m2n2 Q22 m2n2 -2m2n2 m4 + n4 -2m2n2 Q12 -mn3 mn3 -m3n 2mn3 -2m3n Q66 -m3n m3n-mn3 2m3n-2mn3

Ql Ku K12 K13 K14 - -

Q2 K21 K22 KzJ Kz4 Dr h3 K3t K32 K33

-

QJ K34 D2 --- -

12 Q4 K41 K42 K43 K44 D3 (9) Q5 Kst Ksz K.s3 Ks4 D4 Q6 K6I K62 K63 K64

' .

.

where

Dt Qll - 3 Q22 D2 h -- -D3 1 2 Q12 -

Q66 D4

Substituting Eq. (9) into Eq. (8) gives

Ku K12 Kt3 Kt4 K21 K22 K23 K24 v.

K33 -

A.= [H1H2H3H4HsH6] K31 K32 K34 D2

K42 K43 K44 -K41 D3 (10) Kst Ksz K53 Ks4 D4 K6t K62 K63 K64

-Four experimental natural frequencies are chosen to evaluate four A. 's which are used with Eq. ( 1 0) to form

the final equation

Kn K12 K13 Kl4 -A.l Hu H12 H,3 H14 Hl5 H,6 K21 K22 K23 K24 Dt - -A.2 H21 H22 H23 H24 H2s H26 K3l K32 K K34 D2 33 -- - -A.3 H31 H32 H33 H34 H3s H36 K41 K42 K43 K44 D3 . -.:1.4 H41 H42 H43 H44 H4s .H46 Kst Ks2 Ks3 K54 D4

K6t K62 K63 K64 -

(l l) Hu H,2 H13 H14 Dl - -- H21 H22 H23 H24 D2 - - - - . -H31 H32 H33 H34 D3 - - - -

H41 H42 H43 H44 D4

Equation ( l l) is the basic equation with which the iteration of the elastic constants is carried out. Figure 1 is a flow chart showing the entire procedure. The required accuracy is indicated by the percentage difference between two successive iterations being less than 0.03%.

The plate specimen was a Syncoglass E-glass/epoxy(Araldite MY750 and hardener HY956 at I 00:23 by weight) square plate with unidirectional fiber orientation at 60 degrees to one of the plate edges. This

Modal Analysis 19

. ..

. · . . . : : . . . · . . . . ·.·

' ·:: .

•. : • • . . . . . . . . . . . ;.· . , . . ·O.

'>:

'· '

, ; ' ' . , . . :: . . ; :·· . . . . .. . . . . .

•: . . , . . . . ' ·• . , :: • . · .. . , .

: ; ; . . . ;;.J.: ,.r . �.;:1 . ;·o� ,. ·� . .

. :• . . ,; .

F � • ·; . • . :;. '( .,

·1 .

_, .I ; I

. " ' q :j � I j ol ., . , ·I' �

i . . _, ., .•. -1 •I 1 ·)

t(;

. !

J ! . { •• " '·

Rule of mixture to estimate

Q11, 022, 012. 066 and

D 1 to 04

01 to 06 by equation (9)

Form and solve eigenvalue

problem to obtain

eigenpairs of equation (6)

Mode identification and find Hr by equation (8)

Form solution matrix LHJ and solve equation ( 11)

Elastic constants converge?

Yes

Fig. 1 Flow chart for the Iteration procedure

20 January 1993

edge is along the x -axis of the coordinate system. Thirty layers of fiber were prepared and dimensions and physical properties of the plate are listed in Table 1.

The plate under test was then suspended by soft cords in a vertical position to simulate a free-free boundary condition. Excitation was provided by a suspended shaker via a force transducer and a stinger rod. Acceleration signals were measured by attaching an accelerometer to thirty-six locations on the plate surface. The natural modes were excited individually with periodic chirp signals centered around the natural frequencies [ 13]. This type of signal was chosen because it gives good signal to noise ratio for the test. They were generated by the analyzer with a frequency span of I 00 Hz [ 14]. This corresponds to as weep time of 8 seconds and once the transient response of the plate subsided, periodicity was achieved. The measurement was leakage free and required no averaging. Modal analysis was then carried out and four modes were chosen for the iteration. Figure 2 is a schematic diagram of the experimental set-up.

{,.'0" • -0 ·-'.'''"( , ;r- "'· ,._ ·-x� , , -

I

• p ·a;."wy:.::A .. .. . /1 :',.;.·

TABLE 1 PHYSICAL PROPERTIES OF GLASS EPOXY PLATE

fiber direction p(kg!Jri3) vol. fraction (%) length a(m) 1319.9 9.4 .2003

-

thickness h(m) 9.4 E-3

.------t Power Amplifier f4----t Dynamic Signal 1---• HP 9000/238 B&K 2708 Analyzer HP 3562A Computer

t Shaker

B&K 4809 !charge Amplifiel

��ger

I Force Transducer B&K 8200

Specimen

1 1 Accelerometer B&K 4344

B&K 2635 ....,_____.

•

_... �harge Amplifie 1------...J '---- --..._ B&K 2635

Flg.2 Schematic diagram of the experimental set-up

Operating Program Entek structural Analy111 V3.58

Modal Analysis 21

The initial estimate of the elastic constants from the rule of mixtures results in

-

Dt 732.84

D2 - 333.88 -

D3 -

147.91 -

102.50 D4

Substituting into Eq. (9) gives

Dt 365.95 Dz 565.43 D3 215.58 --D4 170.17 D5 47.31 D6 125.45

These values of D1 to D6 are then substituted into Eq. (7) with the values of the integral Q's from the Appendix to form the [C) matrix. This 16 x 16 matrix has the 1 , 1; 1 ,2 and 2,1 rows/columns equal to zero and thus reduces to an order of 13 x 13. Further inspection reveals that it can be divided into two sub­matrices of order 7 x 7 and 6 x 6 respectively. Thus, instead of having p x q = 16 for Eq. (6), the split results in a reduced value of either 7 or 6. The frrst sub-matrix [ C 1] consists entirely of doubly symmetric and doubly antisymmetric modes and the other [C2] of the symmetric-antisymmetric and antisymmetric­symmetric modes. These matrices are listed below.

[c.] =

22 January 1993

--

C13,13 c22,13 c24,l3 c31,13 c33.l3 c42,t3 c44,13

2.8£3 5.6£4

-3.9£4

7.4£4

-4.9£4

6.5£4

-4.4£4

c,3.22 c22.22 c24,22 c31.22

0

c42,22 c44,22

5.6£4

9.8£4

1.1£5 2.1£4

0 1.1Ej 1.3£5

c13.24 c,3.31 c,3.33 c.3.42 CJ3,44 Cz2.24 Czz.3t 0 c22.42 C22 44 •

c24,24 c24,31 c24,33 c24,42 c24,44 c31,24 c31.31 c31,33 c3l.42 c31,44 c33.24 c33.3l c33.33 C33.42 c33.44 c42,24 c42,3t c42,33 c42,42 c42.44 c44.24 c44,3l c 44,33 c44,42 c44,44

.

-3.9£4 7.4£4 --4.9£4 6.5£4 -4.4£4

1.1£5 2.1£4 0 1.1E5 1.3£5 3£6 2.4£4 6.4£5 4.9£5 6.2£5 2.4£4 1.8£5 --4.9£4 -1.5£4 -1.7£4

6.4£5 -4.9£4 2.2£6 3.7£5 -2.8£5 4.9£5 -1.5£4 3.7£5 2.3£6 6.2£5 6.2£5 -1.7£4 -2.8£5 6.2£ 5 1.3£7

c,4,14 c,4,23 c,4,32 c,4,34 0 cl4,43 c23,14 c23,23 c 23,32 c23,34 C23,41 c23,43

[c2 ) = c32,14 c32,23 c32,32 c32,34 C32,41 c32,43 c34.14 c34,23 c c34,34 C34,41 c34,43 34,32

0 c4,,23 c4,,32 c4,,34 c4,,41 c4,,43 c43,14 c43,23 c43,32 c 43,34 C43,41 c43,43 2.1£6 2.9£5 1.6£5 -1.8£5 0 3.4£5

2.9£5 6.9£5 1.5£5 -8.4£4 1.6£5 3.6£5

1.6£5 1.5£5 5.9£5 3.6£5 1.1£5 -4.9£4 ---1.8£5 -8.4£4 3.6£5 6.2£6 1.3£5 1.1£6

0 1.6£5 1.1£5 1.3£5 1.4£6 -1.8£5

3.4£5 3.6£5 --4.9£4 1.1£6 -1.8£5 5.6£6

The eigenpairs of these two matrices are then determined with each eigenvector norma1ized to its largest

element. The location of this element indicates the indices of the corresponding natural mode. The

following eigenpairs are chosen for illustration. Since modes 3,2 and 2,3 each have two dominant elements

in the eigenvector, their mode shapes should also be used to ascertain the modal indices.

mode 'A ill -2,2 74202 { -0.2597 1 -0.0362 -0.0096 0.0098 -0.0352 -0.0076 }T 3,1 142226 { -0.4959 -0.1190 -0.0182 1 0.0125 0.0299 0.0005 }T 3,2 433914 { 0.0 1 71 -0.8129 1 -0.0902 0.0515 0.0859}T 2,3 660325 { -0.2794 1 0.8379 -0.0297 -0.3567 -0.0522 }T

These eigenvectors with Eq. (8) will enable H 1 to H 6 for each mode to be found and the matrix [H) is given

as

0

500.56 [H]=

0

0

0

0

-174.21

57.97

522.63

0

--4.26 -115.86

500.56

-71.82 0

2127.61 -658.11 -738.77

0 500.56 -243.08 2231.83 752.25 114.39

- -Four eigenvalues A 1 to A 4 are calculated from the experimental natural frequencies corresponding to the

chosen modes. They will be used in Eq. ( 11) with [ Jl] to evaluate a new set of values for l5 's such that

-D1 -D2 ---D3 -D4

The process now repeats until convergence occurs.

768.30

317.19

150.50

83.61

Modal Analysis 23

' ' ' ' ' ' I

' I ' !

The first eight natural modes of the vibrating plate have been extracted by modal analysis [ 15]. Their

particulars are shown in Table 2. Natural frequencies calculated by the Rayleigh-Ritz method using rule

of mixture estimates are also listed for comparison. The mode shapes are illustrated in Fig. 3. Of the eight

natural modes five of them belong to the sub� matrix C 1 of doubly antisymmetric and doubly symmetric

groups and the other three are from the sub-matrix C2 of symmetric-antisymmetric and antisymmetric­

symmetric groups. There are altogether seventy possible combinations belonging to four categories.

Category I consists of three modes from C 1 and one mode from C 2 while category II has three from C 2 and

one from C 1. Category Ill will contain two modes from C 1 and the other two from C 2• Finally, category

IV have all the modes taken from either C1 or C2. Concentrating initially on the first six modes it is seen that calculation of natural frequencies by the

Rayleigh-Ritz method are in good agreement with experimental values. The largest discrepancy is found

to be around 5%. Fifteen mode combinations were used to iterate the elastic constants using initial values

obtained by the rule of mixtures. Table 3 shows the converged values and Figs. 4 to 8 show the convergence

rate of these constants. Eleven of the fifteen combinations converge. The other four either diverge or

converge to obviously incorrect values.

A review of all the results shows that the two elastic constants E1 and £2 have been estimated with a

good degree of consistence. Comparing the values obtained through modal testing to that obtained by the

static tensile test described in Ref. [12], the maximum deviation registered for these two quantities are

11.4% and 17.4% respectively. The same cannot be said fortheother three constants. Largest discrepancies

between values for the shear modulus, major and minor Poisson's ratios amount to 67.3%, 64.1% and

49.1 %. This may be explained by the inherent weakness of the static test which does not determine the shear

modulus accurately.

For an accuracy of0.03%, four to twenty iteration steps were required to achieve convergence. When

the accuracy was relaxed to 1% convergence was then obtained within ten iterations for all eleven mode

combinations. This approach thus represents a quick and convenient non-destructive way of detennining

the elastic constants of a generally orthotropic plate.

The study is now extended to include the higher order modes 3,3 and 4,2; both from the C 1 sub� matrix.

Their natural frequencies were not predicted by the Rayleigh-Ritz method to the same degree of accuracy

;a ;' . .

TABLE 2 MODAL PARAMETERS OF GLASS EXPOXY PLATE

modal indices

24 January 1993

2,2 3,1 1,3 3,2 2,3 4,1 3,3 4,2

natural frequency (Hz)

experimental Rayleigh-Ritz

298.9 306.4 403.3 424.2 616.5 646.9 714.7 741.0 912.0 914.1

1281.0 1342.5 1313.3 1500.1 1519.4 1612.4

s ub-matrix mode shape

c, D-A

c1 D�S

c, D-S

c2 S·A

c2 A-S

c2 A-S c1 D-S c, 0-A

.. . . ,. , . . :�:·

' . ' . ,

. . . . ;: ·: .. , · . . �-: ' • •.:

? .

·. �:: .. :··

•' .. . ' �:

. . . ' , . . , ! •

·'· I . . . :

• . ..

.... \j :.: tj ..

,. . . . .

' ' . :: • • •.

� · . . . . . , •' •'

:I .. . . ::1 ' • ' • :-1 'I n

:j · � . . • . ' .. . , . . . ·< }: . , . " ·: l . .

:�. lj-

'' j ., . · . .. . ·<>

19 II -11!1 -18

I

v / J ,_./ / '-...,- - --·

IS -15

9

1\ / � --v v /

-9 2

-13 15

31 -9 I 1 2 21 y

Lx $TAATING CGMT�r 8

YOCE: U Fll!tl. •11111

•• y 17 18 7 11

L�: STARTING :ONTOUR. e

UOO£: 2,3 FREQ· 11ZHa CCNfOIJUII'VIV>.i.: I

-14 1 -26

-13 -14

-11!1

-1!1 19

7 Ul

a y 3 -8 -15 -I?

LY. !TAAT HG COtmiURo II

-z• -4

-17 -e

lZ

J6 34

-!I

-'Z2 •

y

Lx ST�RT1N6 CONTOUR: 9

Fig. 3 Modes of the free-free generally orthotroplc plate

y

Lx IIOOE U

17 y

Lx -- (,1

8 2

43

29

37

58

s;r y

Lx MODE; (,2

-9

-91

-45

:7

H

2-4 y

Lx

-29 -39 -II 42

FRED: 1lll t Ill

Ill I 11

-12 -21 31 28

�

..... -21!1 -9!5 32 2 1

67 46 -53 -67

1 -38 4!1 5&

25

-59

-53

-44

-&6

-95

:16

7-4

35

-38

-se

-25

Modal Analysis

TCIMh e

• Ql

25

-

MODULUS E1 OPIC PLATE

:'c'c10:"c'c1 ______________________________________________ --, .. ,

8 7 8L----L---�--�--�--�--�---L--­

o 2 4 6 8 10 12 14 18 NO. OF ITERATIONS

-a- 1,3;3,2;2,ll;4, 1

2,2;3,1;1,3:2.3

lillie t.n1111 tell

-41--- 3,1;3,2;2,3;4,1

:•c':10:"'cc1 ______________________________________________ � 12 r

8 0 2 • • 8 10 12 14 18 18 20

NO. OF ITERATIONS

- 2,2,3;1!8,2,2,8 � t.ll,a, t,a,z,4. t � t,t,u,a.z,z.a � 2,1:!1,8,8.2!4.1 � 2.1;1,8;1,ll;4,1 -+- 8.1;\318.1:2,8 � 3,111.818.2:4,1 - 111tlo teneue e .. t

Fig. 4 Convergence of Young's modulus E1 for generally orthotropic plate

012

012(0Pal -------------------------------·------------� •,-----

2.0

1 L__�--�--�--�--�--���� Q 2 4 6 8 10 12 14 18

NO. Off ITERATION&

2

2,2;8,1(t,8!1,2 .......... 2.213,1;1,8;2,8 ......... 8,1;3,2J2.S.-<f.1

-e- 1,3;3,2;2,8:4,1 llltiO Mnllll Mil

012 (OPI)

CONVERaENCE OF SHEAR NODULUS Q12 ,OR GENERALLY ORTHOTAOPIC PLATE

o1t.gory Ill

1L_�--���--��--�--=��� o 2 4 a a 10 12 14 1s m 20 NO. OF ITERATIONS

� a,aJt.1J3.2,2,t -+- a,a1a,1,t,2,4,1 ....... 2,21t,3,a,a,a,a -e- a,a;t,a,s,.a,4,1

......... 2.2:1.8;2,8;4,1 -+- 3,11t,a1a,a,a.s _... 8,1!1,8;8,2;4,1 - 11111o t.n•ll• t.1t

Fig. 6 Convergence of shear modulus G12 for generally orthotropic plate

26 January 1993

E2 IGPa) •

•••

• ·� •• •

3 . .

2. • 0 2

CONVERGENCE OF YOUNG'S MODULUS E2 FOR GI!NI!AALLY OATHOTROPIC PLATE

Oltetorlll I I II

"*

•

=>-

e e to NO. OF ITERATIONS

12

2,2;3,1;'f.8;3,2 -+- 2,2;8,1:1.3:2,3

ttatlc .. n1U1 IHI

CONVERGENCE OF YOUNG'S MODULUS E2 FOR GII!!NI!RALLY OfiTHOTROPIC PUT!

caltogory Ill

•• ,.

0 2 • 8 8 "' 12 ,. 18 20 NO. OF ITI!AATION8

- t,z,a,t,&.2J2,3 -+- 2,.2,3, 1!8,2;4,1 � 2,2,1,8,3,2,2,3 -e- 2,2rt,8,8,.2,4, t

� 1,2;1,8;2,8;4,1 � 8,1;1,8;3,2;.2,8 � 8,1!1.8;8,2;"'·' - ••• ,,., t.llelll felt

Fig. 5 Convergence of Young's modulus E2 for generally orthotropic plate

"' 0.7 0.8 0.0

CONVERGENCE OF MAJOR POISSON'S RATIO FOR GENERALLY ORTHOTROPIC PLATE

o1tegorl•• 1 a II

...

·� = 0.4 0.3 0.2 0. 1

0 0

'

2 • 8 8 10 NO. OF ITERATIONS

12

2.2;8,1;1,8,3,2 -+- 2.2;8,1;1,3;2,8

CONVERGENCE OF MAJOR POISSON'S RATIO FOR GENERALLY ORTHOTROPIC PLATE

Clt.gory m

:'':'-------------------------------------------------l 0.7 1 0.0

0.3 �� ==''='== 0.2

18

0.1 O L__-L----��----�----L---=---����---0 2 4 8 8 10 12 14 � 18 20

NO. OF ITERATIONS

2,2,3, 1!8.212,3 -+-2.2:1.8;2.3=4.1 .......

2,2,8,t,a,a,4,t ..._ 2,2,t,a,s,2,a,a -e- 2,.ll,t,a,a,2,4, 1

3,1;1,8,3,2;2,3 -6- 8,1;1,3;8,2!4,1 - 1t1llo Mn1ll1 Mit

Fig. 7 Convergence of Poisson's ratio v12 for generally orthotropic plate

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TABLE 3 ELASTIC CONSTANTS OF GLASS EPOXY PLATE

mode combination

rule of mixture

static tensile test

2,2;3,1;1,3;3,2

2,2;3,1 ;1,3;2,3

2,2;3, 1 :1 ,3;4, 1

2,2;3,2;2,3;4, 1 311 ;3.2;2,3;4, 1

1 ,3;3,2;2,3 ;4,1

2,2;3,1;3,2;2,3

2,2;3, 1 ;3,2;4, 1

2,2;3,1 ;2,3;4, 1

2,2;1,3;3,2;2,3

2,2;1 ,3 ;3,2;4,1

2,2;1 ,3 ;2,3;4, 1

3,1 ;1 ,3 ;3,2;2,3

3,1 ;1 ,3 ;3,2;4, 1

3,1 ;1 ,3;2,3;4,1

average

E1 (GPa)

9.57

9.49

9.49

9.79

10.29

9.12

10.19

10.57

8.57

8.93

9.02

9.83

8.83

9.51

E2 (GPa)

4.36

4.21

4.02

3.51

4.07

3.67

4.03

4.04

3.51

3.93

3.48

3.90

4.23

3.85

G12 (GPa)

1.47

1.25

1.33

1.57

0.440

0.284

0.347

0.339

no convergence

no convergence

1.24 0.407

1.80 0.400

1.28 0.385

1.25 0.405

no convergence

2.09 0.466

1.50 0.413

1.90 0.428

1.41 0.305

1.20 0.408

no convergence

1.51 0.391

0.200

0.131

0.147

0.121

0.161

0.161

0.152

0.155

0.191

0.182

0.165

0.121

0.195

0.159

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Fig. 8 Convergence of Poisson's ratio v21 for generally orthotroplc plate

iteration category

14 I 12 I

I

II

7 II

5 II

20 Ill

16 Ill

Ill

7 IU

8 Ill

7 Ill

8 Ill

4 Ill

Ill

Modal Analysis 27

:0 � :i � :; � ·' 'I . 1 ... .. ·i

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as the first six modes. Forty trials were undertaken, half of them include mode 3,3 while the other half

include mode 4,2 but none containing both in the same set. Only eight sets of converged data were obtained from the forty combinations. Table 4 shows the values

for the elastic constants. They are significantly inferior compared to previous results. The maximum discrepancies between the converged values and the static test values for E1, E2, G12, v12, v21 are now respectively 22.0 %, 37.5 %, 88.3 %, 146.3% and 111.0 %. These observations deliver a very useful message, i.e., modes with low order indices should be selected in order to attain a better chance of convergence and a reliable set of constants. This is particularly true if the functions in the mathematical model are to not have an excessive number of terms since the modal indices must not be larger than the number of terms in the corresponding assumed function. Furthermore, measurement of modal parameters at low frequencies can also be made more accurate.

The next test brings together five combinations of category IV containing modes exclusive! y from sub�

matrix C 1• Only one set converged after eight iterations and the final values obtained were unacceptable. Such bias selection of modes does not seem to draw comparable contribution from the various elastic constants and therefore should be avoided.

TABLE 4 ELASTIC CONSTANTS OF GLASS EPOXY PLATE

mode combination

3,1 ;1 ,3;2,3;3,3

3,1 ;1,3;4,1 ;3,3

3,1 ;2,3 ;4, 1 ;3,3

1,3 ;2,3 ;4,1 ;3,3

2,2;3,1;3,2;4,2

3,1;1,3;3,2;4,2

3,1 ;1 ,3;4,1 ;4,2

3,1 ;3,2 ;4,1 ;4,2

average

28 January 1993

E1 (GPa)

9.99

7.40

10.62

8.50

8.11

8.23

7.87

8.33

8.63

-

E2 (GPa)

2.70

3.07

2.75

2.63

3.98

4.37

3.37

4.26

3.39

G12 (GPa)

1.82

1.52

1.61

2.35

1.45

1.12

1.47

1.21

1.57

0.376 0.102

0.666 0.277

0.699 0.181

0.590 0.183

0.270 0.131

0.447 0.238

0.586 0.251

0.405 0.207

0.505 0.196

TABLE 5 EFFECT OF INITIAL ESTIMATES ON CONVERGENCE

initial values

rule of mixture

£2=9.57 GPa

G12=1.47 Pa

E1= 9.57 Pa

E2= 4.36 Pa

E1=9.57 Pa, E2=4.36 Pa

number of convergent sets

11

10

9

6

3

3

iteration category

12 I

15 I

14 Ill

8 Ill

12 I

28 I

15 I

6 Ill

., .. , . ..• . . . ·

. ' 'I·

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' . .. : · . . . .. �· � :.

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•• I

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Finally, the effect of initial estimates on the iteration process is examined. Changes are introduced to

the starting values of individual constants and the iteration repeated for the fifteen mode combinations used in Table 3. The other constants are still the values obtained by the rule of mixture. Details of these changes are shown in the. left-hand column of Table 5. The right-hand column indicates the number of convergent sets out of the original eleven. No new convergence was discovered with these changes.

It is seen that the Young's moduli have a more profound influence on convergence than the shear modulus. When really poor initial estimates of E 1 and £2 were used only three combinations converged. Give the same order of deviation to G12 and there are still nine convergent sets. While initial values may affect convergence, nevertheless, as long as convergence is achieved the final values are always the same.

Four elastic constants E1• E2, G12 and v12 of a thin generally orthotropic glass/epoxy plate have been determined by modal testing based on a Rayleigh-Ritz approach. The fifth constant, v 21 is calculated from

'

the reciprocality relation of the compliance matrix. Four natural modes of vibration are required to implement the iteration leading to the four independent constants. These modes should preferably b e of the lowest natural frequencies otherwise an excessive number of terms in the assumed function of the mathematical model would be required. In this work, four terms of the free-free characteristic beam

functions were employed and were found to give satisfactory results when used in conjunction with the first six natural modes.

In order to avoid lack of contribution from any of the elastic constants in the iteration the four modes should not be taken out from just one category but evenly spread out in the four categories of modes. It is believed that this will prevent any one of the constants from being unduly inaccurate. The two Young's moduli have been found to have the most prominent effect on the iterative process. They were also

•

calculated more accurately than the others. The success of the method depends critically on a good modal test and acceptable initial estimates of the Young's moduli.

For the specimen used it was found that periodic chirp excitation has the advantage of providing a well­controlled experimental procedure, giving leakage free measurements, and producing very clean frequency response cmves for mode extraction. The mode shapes are required to substantiate mode index identification . This is useful in situations where the corresponding eigenvector has more than one dominant element.

Static tensile test results indicate that this method produces reasonable data for the elastic constants and shows itself as a promising non-destructive test. It also has the advantage of being able to find all the constants simultaneously. As there is no well established test for finding the shear modulus of composite laminates this technique should prove to be particularly useful in that application.

[I] Ashton, J. E.; Whitney, J. M. Theory of Laminated Plate .... Stamford, CT: Technomic Publishing Co.; 1970.

L2J ·Test Method for Tensile Properties of Oriented Fiber Composites." ASTM D 3039- 76. Phila­delphia, PA: American Society for Testing and Materials; 1976.

[3] "Guide for Testing lnplaneShearProperties of Composites Laminates." ASTM D4255- 83. Phila­

delphia� PA: American Society for Testing and Materials; 1983.

l4l Came, T. G.; Wolf, J. A. "Identification of r.he Elastic Constants for Composites Using Modal

Modal Analysis 29

Analysis." SAND79-0527. Albuquerque, NM: Sandia National Laboratories; 1979.

l5 J Deobald, L. R.; Gibson, R. F. "Detennination of Elastic Constants of Orthotropic Plates by a Modal

Analysis/Rayleigh-Ritz Technique." .I Sound Vib v 124 n 2 1988 p 209-283

f 61 Young, D. "Vibration of Rectangular Plates by the Ritz Method." 1 Appl M ech Trans ASME v 17

1 950 p 448-453

f71 Blevins. R.D. Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand

Reinhold Co.; 1979. (8] Ashton, J.E. Natural Modes of Free-Free Anisotropic Plates. Shock Vih Bull n 39 pt 4 1969

p 93-99

[9) Sivakumaran, K.S. "Frequency Analysis of Symmetrically Laminated Plates with Free Edges."

J Sound Vih v 1 25 n 2 1988 p 2 1 1-225

[ 1 0] DeWilde, W .P.; Narmon, B.; Sol, H.; Roovers, M. "Determination of the Material Constants of an

Anisotropic Lamina by Free Vibration Analysis." Proceedings of the 2nd International Modal

Analysis Conference, Orlando, FL, Feb 6-9, 1984. v 1 p 44-49 [ I I ] De Wilde, W .P.; Sol, H.; Van Overmeire, M. "Coupling of Lagrange Interpolation, Modal Analysis

and Sensitivity Analysis in the Determination of Anisotropic Plate Rigidities." Proceedin};S of the

4'h International Modal AruJ/ysis Conference, Los Angeles, CA, Feb 3-6, 1986. v 2 p 1058-1063 [ l 2J Jones, R. M. Mechanics o.f Composite Materials. Washington, D. C.: Scripta Book Co.; 1975.

[ l 3] Olson, N. Excitation functions for structural frequency response measurements. Proceedings of

the 2"d International Modal Analysis Conference, Orlando. FL, Feb 6-9, 1984. v 2 p 894�902 [ 141 "Hewlett-Packard Model 3562A Dynamic Signal Analyzer Operating Manual., Hewlett�Packard

Co; J 985.

[ 1 5 j "Structural Analysis Instruction Manual-Version 3.56" Entek Scientific Corporation; 1988.

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Modal Analysis 31

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Modal Analysis 33

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