The free vibrations are studied of laminated anisotropic elliptic plates with clamped edges. The analytical formulation is based on a Mindlin-Reissner type plate theory with the effects of transverse shear deformation, rotary inertia, and bending-extensional coupling included. The frequencies and mode shapes are obtained by using the Rayleigh-Ritz technique in conjunction with Hamilton's principle. A computerized symbolic integration approach is used to develop analytic expressions for the stiffness and mass coefficients and is shown to be particularly useful in evaluating the derivatives of the eigenvalues with respect to certain geometric and material parameters. Numerical results are presented for the case of angle-ply composite plates with skew-symmetric lamination

Andersen, C. M.

Noor, A. K.

English

NASA Technical Reports Server

FREE VIBRATIONS OF LAMINATED COMPOSITE ELLIPTIC PLATES 
C. M. Andersen 
College of William and Mary 
Ahmed K. Noor 
Joint Institute for Advancement of Flight Sciences 
The George Washington University 
SUMMARY 
A study is made of the free vibrations of laminated anisotropic elliptic 
The analytical formulation is based on a Mindlin- plates with clamped edges. 
Reissner type plate theory with the effects of transverse shear deformation, 
rotary inertia, and bending-extensional coupling included. The frequencies 
and mode shapes are obtained by using the Rayleigh-Ritz technique in conjunc- 
tion with Hamilton's principle. A computerized symbolic integration approach 
is used to develop analytic expressions for the stiffness and mass coefficients 
and is shown to be particularly useful in evaluating the derivatives of the 
eigenvalues with respect to certain geometric.and material parameters. 
Numerical results are presented for the case of angle-ply composite plates 
with skew-symmetric lamination. 
INTRODUCTION 
Although a number of studies have been devoted to the free-vibration 
analysis of isotropic elliptic plates (refs. 1 to 41, investigations of 
orthotropic plates are rather limited in extent (refs. 5 and 6), and to the 
authors' knowledge, no publications exist dealing with,the free vibration of 
laminated anisotropic elliptic plates. The present study focuses on this 
problem. More specifically, the objectives of this paper are (1) to present 
a computational procedure based on the use of computerized symbolic integration 
in conjunction with the Rayleigh-Ritz technique for the free-vibration analysis 
of laminated anisotropic elliptic plates and 12) to study the effect of vari- 
ations in the lamination and geometric parameters of the plate on its 
vibration characteristics. 
The analytical formulation is based on a form of the Mindlin-Reissner 
plate theory with the effects of transverse shear deformation, anisotropic 
material behavior, rotary inertia, and bending-extensional coupling included. 
The frequencies and mode shapes are obtained by using the Rayleigh-Ritz 
technique in conjunction with Hamilton's principle. 
coefficients are developed using the symbolic and algebraic manipulation 
language MACSYMA (refs. 7 and 8). Computerized algebraic manipulation, in 
addition to reducing the tedium of the analysis and the likelihood of errors, 
The stiffness and mass 
425 
https://ntrs.nasa.gov/search.jsp?R=19770003328 2020-03-22T12:11:53+00:00Z
was shown to be particularly useful in evaluating the derivatives of the 
eigenvalues with respect to certain geometric and material parameters. 
applications of computerized algebraic manipulation in structural mechdnics 
are reported in references 9 and 10. 
Other 
SYMBOLS 
al'a2 
a$yp %3$3 C 
D 
@SYP 
E~ f E ~  
F 
a 8YP 
G~~ G~~ 
h 
Ki j 
[MI 
ij M 
m ,m m 
0 1' 2 
T 
U 
LT v 
semimajor and semiminor axes of elliptic plate 
extensional and transverse shear stiffnesses of plate, 
respectively 
bending stiffnesses of plate 
elastic moduli in direction of fibers and normal to fibers, 
respectively 
stiffness interaction coefficients of plate 
shear moduli in plane of fibers and normal to plane of 
fibers, respectively 
plate thickness 
element stiffness matrix 
stiffness coefficients 
mass matrix 
mass coefficients 
density parameters of plate 
kinetic energy of plate 
strain energy of plate 
displacement components in coordinate directions 
fiber orientation angle of individual layers 
Poisson's ratio measuring strain in T-direction due to 
uniaxial normal stress in the L-direction 
functional defined in equation (1) 
material density of the plate 
426 
rotation components 
$8 
(9 1 
Jii 
52 plate domain 
w 
vector of undetermined parameters 
ith component of vector (9) 
circular frequency of vibration of the plate 
MATHEMATICAL FORMULATION 
The analytical formulation is based on a form of the Mindlin-Reissner 
plate theory with the effects of transverse shear deformation, anisotropic 
material behavior, rotary inertia, and bending-extensional coupling included. 
A displacement formulation is used with the fundamental unknowns consisting 
of the displacement and rotation components of the middle plane of the plate 
ua, w, and ,$a. (See fig. 1 for sign convention.) Throughout this paper, the 
range of the Greek indices is 1,2 and a term in which any Greek index appears 
twice is to be summed over that index. The fundamental unknowns are assumed 
to have a sinusoidal variation in time with angular velocity 
frequency of vibration of the plate). 
of the stiffness and mass matrices is given by 
w (the circular 
The functional used in the development 
where 
a u  a $  
aBYP 01 B Y P u='J[. a u  a u  + 2 ~  2 asyp 01 B Y P 
T = 1 2 u2j[m 0 (u a a  u + ww) 4 2m l a a  u $ + m 2 a a  $ $ Id52 (3)  
In equations (2) and ( 3 ) ,  C D and F are extensional stiff- 
ClBYP' OlBYP' aBYP 
nesses, bending stiffnesses, and stiffness interaction coefficients of the 
plate ; are transverse shear stiffnesses of the plate; m m and 
%3B3 0' 1' 
427 
m2 are density parameters of the plate; Sl is the plate domain; and 
The displacement and rotation components are approximated by 
of the form 
expressions 
(4) 
where [k] is a matrix of a priori chosen approximation functions and {$I 
is a vector of Undetermined coefficients. 
in the matrix [k] are chosen to be polynomials in x and x 
In the present study the functions 
1 2' 
The stiffness and mass matrices of the plate are obtained by first 
replacing the generalized displacements in equations (2) and (3 )  by their 
expressions in terms of the approximation functions and then applying the 
stationary condition of the functional n, namely, 
6II = 0 (5) 
If the undetermined coefficients {$I  are varied independently and simultan- 
eously, one obtains the following set of equations for the plate: 
where [K] and [MI are the stiffness and mass matrices of the plate, res- 
pectively. The matrix [K] is symmetric and positive definite and the 
matrix [MI is symmetric. The eigenvalues and eigenvectors are obtained 
by using the technique described in reference 11. 
EVALUATION OF STIFFNESS AND MASS COEFFICIENTS 
The stiffness and mass coefficients were evaluated using the computerized 
symbolic and algebraic manipulation system MACSYMA. 
in evaluating these coefficients is given in the appendix. 
performed on MACSYMA are 
The MACSYMA program used 
The major tasks 
(1) Selecting approximation functions for each of the fundamental 
unknowns with undetermined coefficients ($1 in equation (4) and developing 
analytic expressions for the strain and kinetic energies as quadratic 
functions in I$) 
(2) Specifying a pattern-matching technique for evaluating the integrals 
over the elliptic domain (using the function INT(F) (see appendix)) 
4 2% 
(3 )  Forming the stiffness and mass coefficients as second derivatives 
of the strain and kinetic energies with respect to the undetermined coef- 
ficients as 
2 a2T 
w Mij = (7) 
In view of the symmetry of Ki. and Mij, only the upper triangular portions 
are formed in a machine readable (LISP) format. These are subsequently 
converted using the MACSYMA system to-a form which closely resembles FORTRAN 
code (the MACSYMA program used in the conversion is not included in the 
appendix). Finally, a TECO program (DEC's editor for PDP-10 computers) is 
executed to produce the final code. 
The aforementioned computerized algebraic manipulation approach signifi- 
cantly reduced the tedium of the analysis and the lik'elihood of errors. 
Moreover, since analytic exact expressions are obtained for both the stiffness 
and mass coefficients, the- derivatives of the eigenvalues with respect to any 
of the material or geometric parameters can be readily computed by using the 
following formula (ref. 12) : 
2 aM -- ad - {$I;@] - (ai [.]3 {*I, , 
where d refers to any of the material or geometric parameters of the plate and 
subscript i refers to the ith eigenvalue and eigenvector. In equation (8), 
the eigenvectors are assumed to be [MI orthonormal, i.e., 
The two matrices [s] and 
system. 
[E] can be easily evaluated using the MACSYMA 
Equation (8) shows that the derivatives of the eigenvalues with respect 
These derivatives can be used to obtain an approxi- 
to any of the geometric and material parameters of the plate can be calculated 
with little extra work. 
mate estimate for the eigenvalues corresponding to a modified (new) value 
of the parameters without having to resolve the eigenvalue prodem, 
equation (6). To accomplish this, a first-order Taylor's series expansion 
of the eigenvalues in terms of the problem parameter is used (see ref. 12) 
where an asterisk refers to a modified (new) value. 
429 
NUMERICAL STUDIES 
Numerical studies were conducted to investigate the effects of variations 
in the plate geometry and lamination parameters on the vibration characteris- 
tics of elliptic plates with clamped edges. 
antisymmetric lamination with respect to the middle plane are considered. 
The material characteristics of the individual layers were taken to be those 
typical of high-modulus graphite-epoxy composites, namely, 
Angle-ply laminates having 
EL/ET=40 G /E =0.6 GTT/ET=0.5 vLT=0.25 LT T 
where subscript L refers to the direction of the fibers, subscript T refers 
to the transverse direction, and VLT is the major Poisson's ratio. The fiber 
orientation was taken to be +0/-8/+0/-9/..., (0<0<45). All numerical studies 
were obtained using the Rayleigh-Ritz technique wzth 10-term approximation 
functions for each of the fundamental unknowns. The special symmetries ex- 
hibited by the free-vibration modes of antisymmetric laminates were utilized 
in the analysis (see refs. 13 and 14). The four combinations of symmetry and 
antisymmetry with respect to the xi- and x2-axis have been considered. 
results are presented in figures 2 to 4 showing the effects of variations in 
each of the following parameters on the vibration frequencies: (1) the aspect 
ratio of the plate ai/a2, (2) the number of layers of the plate NL, and 
(3) the fiber orientation angle 8 of-the individual layers. 
Typical 
Figure 2 shows that for elliptic plates having the same h/a2, the fre- 
quencies of free vibration decrease with the increase in the aspect ratio 
al/a2. The differences between the frequency curves for thick and thin 
plates in figure 2 are mainly attributed to transverse shear deformation. As 
expected, these differences are more pronounced for the higher modes. Figure 
3 shows that the frequencies increase rapidly as the number of layers increases 
from 2 to 4. Further increase in the number of layers does not have signifi- 
cant effect on the lower frequencies. Figure 4 shows that the minimum 
frequency associated with each of the four basic symmetric-antisymmetric modes 
increases with the increase in the fiber orientation angle 8 from 5O to 45'. 
This is not true, in general, for the higher modes. 
CONCLUDING REMARKS 
The free-vibration response of anisotropic plates with clamped edges is 
studied. The analytical formulation is based on Mindlin-Reissner type 
theory with the effects of transverse shear deformation, rotary inertia, and 
bending-extensional coupling included. 
obtained by using the Rayleigh-Ritz technique in conjunction with Hamilton's 
principle. 
analytic exact expressions for the stiffness and mass coefficients and is 
The frequencies and mode shapes are 
A computerized symbolic integration approach is used to develop 
430 
shown to be particularly useful for evaluating the derivatives of the eigen- 
values with respect to certain geometric and material parameters. Numerical 
results are presented showing the effects of variation in the geometric and 
material parameters on the free-vibration response of composite elliptic 
plates with clamped edges. 
431 
roo roc 
a l a ,  
C k  
w a l  w w  
4 w o .4 roa 
m w  
c o  
-4 oro 
3 0  
rl 4 
m o  
3 
c w  
4 al a a 
m c  
.!j 
+ 
.. 
n 
U 
U 
432  
APPENDIX 
2 
111 
3 
3 
- 
N 
3 s 
Q 
Y z 
111 
h 
h 
a r. 
d 
n 
I 
L 
L 
L 
d U  
I U  
433 
APPENDIX 
5 
-4 
4J u c 
E 
v1 
Y 
E X  
II 
h 
x, 
7 -  Irr 
n 
E 
2 3  
0 c M n 
a, IJ m 
c. 
n 
r.. 
,.-. ..-. 
I- 3 
5 3  
I- I-I- 
I- LZ; 
H U  
w w  
(..‘:I (..!I 
J-I 
W W  
.. 
h-, 
c. c 
.-.. 
n 
iilJ 
-1 
n 
“-I 
I 
4 
U 
- U 0 
u 
I- 
LL 
LL 
u 
I- 
I 1 1  
t . .  
H 
.. 
LL 
LL 
I=I 
U 
.. & 
0 
LL 
.. .. 
d I i j  
-1 -1 
LL 
U 
434 
mFERENCES 
Callahan, W. R.: Flexural Vibrations of Elliptical Plates When Transverse 
Shear and Rotary Inertia Are Considered. J. of the Acoustical Society 
of America, vol. 36, May 1964, pp. 823-829. 
Leissa, A. W.: Vibration of a Simply Supported Elliptical Plate. J. of 
Sound and Vibration, vol. 6, July 1967, pp. 145-148. 
1. 
2. 
3 .  
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
Sato, K.: Free Flexural Vibrations of an Elliptical Plate With Simply 
Supported Edge. J. of the Acoustical Society of America, vol. 52, 
Sept. 1972, pt. 2, pp. 919-922. 
Sato, K.: Free Flexural Vibrations of an Elliptical Plate With Free 
Edge. J. of the Acoustical Society of America, vol. 54, Aug. 1973, 
pp. 547-550. 
Desiderati, F. W.; and Laura, P. A.: Vibrations of Rib-Stiffened Ellipti- 
cal and Circular Plates. J. of the Acoustical Society of America, 
VO~. 48, pt. 1, pp. 6-11. 
Hoppmann, W. H., 11: Flexural Vibration of Orthogonally Stiffened Circu- 
lar and Elliptical Plates. Proceedings 3rd U . S .  National Congress on 
Applied Mechanics, June 1958, pp. 181-187. 
Moses, J.: MACSYMA - The Fifth Year. ACM Sigsam Bull., vol. 8, no. 3, 
Aug. 1974, pp. 105-110. 
Martin, W. A.; and Fateman, R. J.: The MACSYMA System. Proceedings 
Second Symposium on Symbolic and Algebra@ Manipulatbn, S. R. Petrick, 
Ed., Association for Computing Machinery, 1971, pp. 59-75. 
Levi, I. M.: Symbolic Algebra by Computer-Applications to Structural 
Mechanics. AIAA Paper No. 71-363, Presented at the AIAA/ASME 12th 
Structures, Struct. Dynamics and Materials Conference, Anaheim, CA, 
April 19-21, 1971. 
Wilkins, D. J.: Applications of a Symbolic Algebra Manipulation Language 
for Composite Structures Analysis. Computers and Structures, vol. 3 , -  
pp. 801-807. 
Martin, R. S.; and Wilkinson, J. H.: Reduction of the Symmetric Eigen- 
problem, AX=XBX and Related Problems to Standard Form. Numerische 
Mathematik, Bd. 11, 1968, pp. 99-110. 
Fox, R. L.; and Kapoor, M. P.: Rates of Change in Eigenvalues and Eigen- 
vectors. AIAA J., vol. 6, no. 12, 1968, pp. 2426-2429. 
435 
13. Noor, A. K.: Symmetries in Laminated Composite Plates. Developments in 
Theoretical and Applied Mechanics, vol. 8, Proceedings of the Eighth 
Southeastern Conference on Theoretical and Applied Mechanics, 1976, 
pp. 225-246. 
14. Noor, A. K.; Mathers, M. D.; and Anderson, M. D.: Exploiting Symmetries 
for Efficient Post-Buckling Analysis of Composite Plates. 
at the AIAA/ASME/SAE 17th Structures, Structural Dynamics, and 
Materials Conference, Valley Forge, PA, May 5-7, 1976. 
Presented 
436 
t 
1 X 
J 
/ 
Figure  1.- E l l i p t i c  p l a t e  and sign convention. 
437 
2 4  
P W  a2 
ET h2 
-Symmetric - symmetric 
----Symmetric - antisymmetric 
\,\***-Antisymmetric - antisymmetric 
1.0 
2 4  
P W  a2 
ET h2 
al'a2 
(a) h/a2 = 0.1. Cb) h/a2 = 0.01. 
Figure 2.- E f f e c t  of a l / a 2  on t h e  frequencies  of clamped e l l i p t i c  
p l a t e s  with antisymmetric lamination. Eight-layered p l a t e s  
wi th  f i b e r  o r i e n t a t i o n  45'/-45'/ 45°/-450/450/-450/ 4!jo/-45O. 
-Symmetric - symmetric 
----Symmetric - antisymmetric 
--Antisymmetric - symmetric 
****Antisymmetric - antisymmetric l.2[Fj __--  - - - 
1.0 
2 4  
P W  a2 
ET h2 
I I I J 
4 6 8 10 
Number of layers NL 
0 
2 
Figure 3.- E f fec t  of number of 
l a y e r s  on the frequencies  of 
clamped e l l i p t i c  p l a t e s  w i t h  
antisymmetric lamination. 
2 4  
P W  a2 
ETh2 
3 x 10 
-.-. 
5' i5 ;5 3; i 5  
Fiber orientation angle 13 
Figure 4.- E f fec t  of f i b e r  orien- 
t a t i o n  8 on t h e  frequencies  of 
clamped e l l i p t i c  p l a t e s  w i t h  
antisymmetric lamination. Eight- - 
h/a2 = 0.01; a /a 
f i b e r  o r i e n t a t i o n  45O/-45O/.. . = 1.5; layered p l a t e s ;  h/a2 = 0.01; a /a = 1.5; f i b e r  o r i e n t a t i o n  4 - 2  
d-f3h3/-s/e/-s/e/-e 
438 


Free vibrations of laminated composite elliptic plates

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