Vol. 32, No. 5, pp. 455464, 1990 Printed in Great Britain.
00207403/90 $3.00 + .00 © 1990 Pergamon Press pie
I n t . J. M e c h . Sci.
FREE VIBRATION ANALYSIS OF ISOTROPIC A N D ORTHOTROPIC TRIANGULAR PLATES K. Y. LAM, K. M. LIEWand S. T. Crlow Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511, Republic of Singapore
(Received 10 April 1989; and in revised form 15 September 1989) AbstractA highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of twodimensional orthogonal plate functions is used as an admissible displacement function in the RayleighRitz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the GramSchmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.
NOTATION
a, b aij Ci D DO
the sides of plate coefficient coefficient flexural rigidity = Eh3/12(1  v2) bending and twisting rigidities of orthotropic plate
D11 Elh3/12(1   VI2V21 ) Dr2 v12E2h3/12(1   Y12V21) D22 E2h3/12(1  v12vzl ) D66 Gh3/12
D'11 DI~/x/Dl~D22 D12/~fDllD22 D'22 D22/x/DllD22 D'66 D66/N/D11D22 D'I 2
f(~, ~/) G h m Tmax
Um,x W(~, ~/) x, y ~b(~, ~/)
generating function shear modulus of elasticity thickness of plate number of terms used in the doubly infinite deflection series maximum kinetic energy maximum strain energy deflection function Cartesian coordinates orthogonal plate function
x/a r1 y/b x/~ v, v12, v21 ~0(¢, r/) co
nondimensional frequency parameter Poisson's ratios edge function angular frequency side ratio a/b
1. I N T R O D U C T I O N
Flexural vibration studies on different shapes and configurations of plates are well documented in the Leissa report [1]. A very limited amount of research work on vibration of triangular plates was reported. However, the triangular configuration of plates is commonly encountered in many industries; therefore, further research is needed in this area. A setback in the analysis of this type of problem is the difficulty in formulating a displacement function in two implicitly related variables to describe the triangular domain. 455
456
K.Y. LAMet al.
With the rapid advancement in computer technology, the finite element method has become one of the popular methods used [24]. To avoid using the computationaUy costly finite element method, several numerical and analytical methods were developed and published in the literature. Gorman [5, 6] proposed an analytical technique based on superposition of building blocks for free vibration of triangular plates. A gridwork method was used by Christensen I7] to analyse the free vibrations of cantilevered triangular plates. Bhat [8] adopted the RayleighRitz method to solve cantilevered triangular plate problems with different geometries. A set of twodimensional orthogonal plate functions was recently developed by Liew e t al. 19] to analyse free vibrations of plates with various geometrical shapes. Only natural frequencies of rectangular plates were reported. The present paper extends the earlier work by the authors to study the free vibration of isotropic and orthotropic triangular plates. Various combinations of boundary conditions for the plates are considered. The natural frequencies and mode shapes of the plates are obtained using the twodimensional orthogonal plate functions via the RayleighRitz procedure.
2. ORTHOGONAL PLATE FUNCTION The deflection function for a vibrating triangular plate may be defined by a set of twodimensional orthogonal plate functions
w(~, ,1) = ~ c,¢,(~, ,1), i=1
(1)
where { = x / a , rl = y / b , x and y are the cartesian coordinates, a and b are the sides of the plate and m is the total terms in the set. The starting plate function Cx (¢, q) chosen for the triangular plate at least has to satisfy the geometrical boundary conditions of the plate. Better convergence is achieved if ¢1 (~, ~/) can also satisfy the natural boundary conditions. The function ¢~ (~, r/) in equation (1) is given by 3
~b~(~,q) = I[ ~0.(~,r/), n=l
(2)
where 1I denotes the product of terms, q~. are the edge functions which can be easily formulated if the edge support condition of the plate is known. The function ~0(~, q) for different support conditions are summarized below [8, 9]: (a) for simply supported edge
¢p(~, q) =
(b) for clamped edge
{ f
~c
at e d g e ~ = c
q d
at edge q = d
r /  t~  e
at edge q = t~ + e,
( ~   C) 2
(p(~, 71) ~
(c) for free edge
(3)
at edge ~ = c
(~'
d) 2
at edge r/= d
(r l
t~  e) 2
at edge q = t~ + e,
q,(~, ~) = 1.
(4)
(5)
The higher terms (m > 2) in the set of functions can be generated by the recurrence relation m1
q~,(~, r/) =f,,(~, q)4~1(~, r/)  ~ a,,.i~bi(¢, q). i=1
(6)
Analysis of triangular plates
457
Following the GramSchmidt orthogonalization relationship [812] ~b,(~, r/)q~j(~, r/)d~ d r / =
if/=j'
where the integration is carried out over the plate domain R. The coefficient am,~ in equation (6) is obtained as
ffRfm(~,r/)Cb~(~,tt)4~,(~,r/)d~dt~ ,
ffR
a,,,,,=
(8)
r/)dCdr/
wheref(¢, t/) is the generating function. The generated plate function tkm(~, r/) has to satisfy the geometrical boundary conditions. The generating function f(~, r/) is obtained by determining the parameters r = V"x / ~ 
1 3
(9)
t = ( m  1 )  r 2.
(10)
If t is even, then s = t/2;
(11)
O <<. s <<. r
fm(~, r/) = ~rr/s.
(12)
If t is odd, then s=(t1)/2;
0~s~r1
(13)
fm(~, r/) = eSr/',
(14)
where r ] in equation (9) denotes the greatest integer function. To demonstrate the procedure of forming these orthogonal plate functions, an example is given. For an isosceles right triangular plate with edges r/= ~, r/= 0, ~ = 1 having free, clamped and free boundary conditions (FCF), the first six terms of the orthogonal polynomials of the doubly infinite set are given in Table 1.
3.
METHOD
OF
ANALYSIS
The approximate solutions for the free vibration of a thin triangular plate, shown in Fig. l, may be derived using Rayleigh's principle. The natural frequencies for the plate are determined by equating the strain energy to its kinetic energy Umax = Tmax.
TABLE 1. FIRST SIX ORTHOGONAL POLYNOMIALS FOR F  C  F LAR PLATE Generating functions
(15)
ISOSCELES RIGHT TRIANGU
Plate functions
A (4, ,7) = 1 A(¢, ,1) = ¢ A(~, ,1) = ,I f,(~, ,7) = ~,7 L(~, ,~) = ~2
~b3(~, q) = t / ~ l ( ~ , r/)  a3.1 ~bl (~, q)  a 3 . 2 ~ 2 ( ~ , r/)
A ( ¢ , 7) = '72
q~6(¢,,r) = ,124,~(~,'1)  a~,, 4h(¢, 'I)  a6,24'2(¢, '7)
 as.s~b3(~, r/)  a L 4 ~ 4 ( ~ , q)  a6,3~b3(~, /7)  a6,4t~4(~, r])  a6,st~5(~, r])
K.Y. LAM et al.
458
y,ll
b
0
L
I
_1 I
FIG. 1. Geometry of isosceles right triangular plate.
For smallamplitude vibration, the maximum strain and kinetic energies are given by Uma x :
W~ ~abf;R~DII(~2W~ [~\O~J2 + 2D12 (\ ~ 20IWt~2 2 0?]2 } a2b 2
D22f O2W~2 + 4066( ~2WX~2
+~~q2 ) Tma~ =
a2b2kaiO?]]}dCd?]
1 2ff. ~phabto
wE(I, ?])did?],
(16) (17)
where
Elh3
(18)
DlX = 12(1  1~12v21 )
v12E2h3 O12 =
12(1 
(19)
V121,'21)
E2h3
(20)
D22 = 12(1  v12v21) D66
Gha 12
(21)
Dij is the flexural rigidity, p is the material density, h is the plate thickness, to is the angular frequency of vibration, E1 and E2 are Young's moduli, v12 and v21 are the corresponding Poisson's ratios and G is the shear modulus of elasticity. Substituting equation (1) into equations (16) and (17) leads to 2 m Uma x =
~ab 2D12 Ft~2i__~l
+ aeb2L
~2 Ci~)i(~,")02i=~1 Ci(Pi(~,?])12
~2
~?]2
D22F02~i=c~?]1Ci ~)i(~'?2])]+ 4D66[ 02~i=lC/(~i(~'?])]2 t 2 ~ 0iO?] did?]
+ b4 L
(22)
Analysisof triangular plates
459
and minimizing the Rayleigh quotient with respect to the undetermined coefficients C~
~c(Um~x  Tm~) = 0; i = 1, 2. . . . m
(24)
which leads to the governing eigenvalue equation
~_,[K,j  ).Mij ] C, = O,
(25)
Kij = D'lIPij + ot4D'22Qij + 2a2D'12(Rij + Sii) + 4ct2D~6 Tij
(26)
where
(27) Pij = Qij =
c~2
(28)
c~2
f~Rc~2c~i(~'~l)t~2~J(~'~l)d~drl ~]2 ~fl2
R,j=ffR~2Ck'(~''I) O2C~J(~'FI).. at/2 ~ ag d~/ ~2
~2
agog/
(29)
(30) (31)
2 = phc02a4/~
(33)
D'11 = DI 1 / x / / ~ l1D22
(34)
D'12 = O12/x/~11 D22
(35)
D'22 = D 2 z / ~
(36)
D'66 = D 6 6 / x / ~ ~D22
(37)
in which M~j has to satisfy the orthogonality relationship, and a is the side ratio a/b. Solution of equation (25) yields the natural frequencies and related mode shapes of the plate. 4. RESULTS AND DISCUSSION Numerical work has been done to demonstrate the feasibility of the proposed method and the accuracy of the results. Material properties of the composites are given in Table 2. The solutions for the vibration analysis of orthotropic plate are presented in terms of the nondimensional frequency parameter x / Z defined as c°a2 X / ~ x//~ = 2rt(D11D 22 )i/4,
(38)
TABLE 2. MATERIAL PROPERTIES OF TYPICAL UNIDIRECTIONAL COMPOSITES 113"1
Material
E1/E2
G/E2
vl2
Glass epoxy Boron epoxy Kevlar epoxy Graphite epoxy
4.67 11.03 13.82 17.57
0.50 0.30 0.42 0.70
0.26 0.23 0.34 0.28
K.Y. LAM et al.
460
The orthotropic case can be simplified to isotropic case by setting V12 =
V21 "= V •
(39)
0.30
Eh 3 D l l = D22 = D 366 and the nondimensional
=
½(1

(40)
12(1  v) 2
v)D
frequency parameter ~
(41) becomes
£oa 2
w/Z = ~
x//~/D.
(42)
T h r e e i s o s c e l e s r i g h t t r i a n g u l a r p l a t e s w i t h d i f f e r e n t s u p p o r t c o n d i t i o n s a r e c o n s i d e r e d in the analysis, namely: (a) s i m p l y s u p p o r t e d ( S  S  S ) , (b) o n e e d g e s i m p l y s u p p o r t e d a n d t h e o t h e r t w o e d g e s a r e c l a m p e d ( S  C  C ) , a n d (c) c a n t i l e v e r e d ( F  C  F ) .
TABLE 3. CONVERGENCE PATTERN OF THE NONDIMENSIONAL FREQUENCY PARAMETER x/~ = (coa2/2n)(ph/D) 1/2 OF ISOTROPICTRIANGULARPLATES Boundary condition SSS
S~:~C
FCF
Mode number
4
8
Number of terms 12 14
1 2 3 4
8.94 25.25 31.73 53.56
8.05 19.34 25.97 48.88
7.88 16.48 21.62 35.56
1
2 3 4
12.18 35.73 47.86 78.92
11.74 21.82 28.79 42.77
1 2 3 4
1.00 4.44 7.39 16.22
0.98 3.92 5.39 10.88
16
18
7.86 16.35 21.38 29.96
7.85 15.94 21.08 28.74
7.85 15.66 20.46 27.38
11.68 21.43 27.22 36.15
11.68 21.21 26.78 34.92
11.68 21.16 26.71 34.90
11.68 21.16 26.88 34.86
0.98 3.77 5.24 9.51
0.98 3.73 5.21 9.45
0.98 3.73 5.21 9.43
0.98 3.73 5.21 9.41
TABLE 4. CONVERGENCE PATTERN OF THE NONDIMENSIONAL FREQUENCY PARAMETER
d ) : = (om2/2x)(ph/dDllD22) U2
OF AN ORTHOTROPIC TRIANGULAR PLATE (KEVLAR EPOXY)
Boundary condition
Mode number
4
8
Number of terms 12 14
SSS
1 2 3 4
9.25 24.99 36.86 71.50
8.16 18.22 28.72 42.73
7.76 15.98 23.68 33.34
SC~2
1 2 3 4
13.82 38.79 58.34 90.03
12.20 23.11 31.70 47.09
FCF
1 2 3 4
0.53 2.37 3.91 8.93
0.52 2.16 3.18 6.19
16
18
7.74 14.63 23.07 31.67
7.72 14.63 21.79 29.87
7.71 14.52 21.54 27.04
12.07 20.39 29.43 36.03
12.03 20.33 26.67 29.49
12.03 20.33 26.66 29.47
12.03 20.33 26.66 29.47
0.52 2.14 3.15 5.48
0.52 2.13 3.14 5.47
0.52 2.13 3.12 5.47
0.52 2.13 3.09 5.46
Analysis of triangular plates
461
4.1. Convergence study The energy approach gives an upperbound solution to the exact value. In order to achieve a proper convergence, numerical calculation is performed by varying the number of terms used in the deflection function. Tables 3 and 4 summarize the convergence pattern of the isotropic and orthotropic isosceles right triangular plates. (a) Isotropic case. The convergence patterns of the frequency parameter x / ~ for the three different triangular plates are tabulated in Table 3. Stable and convergent results are obtained when 18 terms are used. (b) Orthotropic case. The fibers of the composite material lie in the xdirection of the plate. Table 4 shows the convergence patterns of the frequency parameter x / ~ for the Kevlar epoxy triangular plate. It is sufficient to take 18 terms for the solutions to reach stable convergence. To minimize computational time and to obtain satisfactory results, 18 terms are used for all cases in the present study. 4.2. Comparison of present results with literature The comparison is carried out only for the isotropic case since no literature is available for the orthotropic case. The computed frequency parameters for the plates are listed in Tables 57. The nodal patterns of the isotropic and graphite epoxy plates are shown in Figs 27. (a) Simply supported plate. Table 5 tabulates the frequency parameters of the simply supported isosceles right triangular plate. The results of the isotropic triangular plate are compared with those obtained by Gorman [51, and good agreement is found. The fundamental frequency parameter exactly coincides with the value presented by Gorman. Moreover, less than 1% of discrepancy is observed between the results of the authors and of Gorman in the higher modes obtained. The numerical results for the orthotropic triangular plate are also included in Table 5. The mode shapes of the simply supported isotropic triangular plate are given in Fig. 2. A comparison of the nodal pattern for the second mode, third mode and fourth mode between the authors and Gorman [5] is made. They agree well. Figure 3 shows the first four mode shapes of the simply supported graphite epoxy plate.
TABLE 5. COMPARISON OF THE NONDIMENSIONAL FREQUENCY PARAMETER ~ / ~ ' OF A SIMPLY SUPPORTED TRIANGULAR PLATE
Material Isotropic Glass epoxy Boron epoxy Kevlar epoxy Graphite epoxy
Second mode
' Reference Present Gorman [5]
1
Mode number 2 3
4
7.85 7.85
15.66 15.64
20.46 20.45
27.38 27.31
7.57 7.47 7.71 8.01
15.01 14.29 14.52 14.11
20.21 20.65 21.54 22.61
27.26 26.37 27.04 27.73
Third mode
Fourth mode
FIG. 2. Nodal line patterns for isotropic SSS isosceles right triangular plate.
462
K. Y. LAMet al.
Sec~d mode
Third mode
Fourth mode
FIG. 3. Nodal line patterns for graphite epoxy SSS isoscelesright triangular plate.
(b) One edge is simply supported and the others clamped. The frequency parameters of the S  C  C right triangular plate are given in Table 6. The results obtained for the isotropic triangular plate agreed well with the values presented by Gorman [6]. The fundamental frequency parameter again compares excellently with the result obtained by Gorman [6]. The higher frequency parameters also agree quite favorably with those presented by Gorman. It is expected since the orthogonal plate functions can be accurately described the mode shapes of the plate. The results obtained for the orthotropic composite plate are tabulated in Table 6. No comparison can be made since the numerical results for such problems are not available in the literature. Figure 4 shows the mode shapes of the S  C  C isosceles right triangular isotropic plate. The nodal patterns of the second mode, third mode and fourth mode are compared with those obtained by Gorman [6], and they agree closely. The nodal patterns of the SC C graphite epoxy plate are given in Fig. 5. (c) Cantilevered plate. The frequency parameters of the cantilevered triangular plate are summarized in Table 7. The present numerical results for the isotropic plate are compared with the available experimental results [14] and the results presented by Bhat [8], Cowper [2] and Mirza [4]. It is evident that the present method yields slightly higher frequencies as compared to the experimental results. This difference may be due to the difficulty in imposing a perfectly clamped condition in the experimental setup. G o o d agreement,
TABLE
6.
/ COMPARISON OF THE NONDIMENSIONAL FREQUENCY PARAMETER ~/,,],' OF A S  C ~ [ ~ TRIANGULAR PLATE
Mode number 2 3
Material
Reference
1
Isotropic
Present Gorman [6]
11.68 11.68
21.16 20.96
26.71 26.26
34.90 33.50
11.61 11.74 12.03 12.41
20.44 19.92 20.33 20.82
26.98 27.13 26.66 26.38
31.62 28.41 29.47 30.77
Glass epoxy Boron epoxy Kevlar epoxy Graphite epoxy
5eco~l mode
Third mode
4
Fourth mode
FIG. 4. Nodal line patterns for the isotropic SCC isoscelesright triangular plate.
Analysis of triangular plates
Second mode
Third mode
463
Fourth mode
FIG. 5. Nodal line patterns for graphite epoxy SCC isosceles right triangular plate.
/_'7TABLE 7. COMPARISON OF THE NONDIMENSIONAL FREQUENCY PARAMETER ~/,'],' OF A CANTILEVERED TRIANGULAR PLATE
Mode number 2 3
Material
Reference
1
lsotropic
Present Experiment [14] Bhat 18] Cowper I2] Mirza I4]
0.98 0.92 0.98 0.98 0.98
3.73 3.64 3.74 3.73 3.67
5.21 5.09 5.21 5.20 5.30
9.41 8.70 8.98 8.94 8.90
0.69 0.54 0.52 0.5l
2.81 2.10 2.13 2.14
4.03 3.12 3.09 3.26
6.99 5.58 5.46 5.69
Glass epoxy Boron epoxy Kevlar epoxy Graphite epoxy
4
however, is found between the present numerical results and those reported by Bhat, C o w p e r and Mirza. Similarly, the present m e t h o d is used to predict the natural frequencies of orthotropic composite plate. The obtained results are given in Table 7. The nodal patterns of the second mode, third m o d e and fourth m o d e of the isotropic and graphite epoxy triangular plate are given in Figs 6 and 7 respectively. Again, the agreement of the m o d e shapes for isotropic plate between the authors and those obtained by Bhat [81, C o w p e r I2] and Mirza 14] is very good.
Second mode
Third mode
Fourth mode
FIG. 6. Nodal line patterns for isotropic FCF isosceles right triangular plate.
Second mode
Third mode
Fourth mode
FIG. 7. Nodal line patterns for graphite epoxy FCF isosceles right triangular plate.
464
K.Y. LAMet al. 5. C O N C L U S I O N
A set of o r t h o g o n a l plate functions is used as a n admissible plate function in the R a y l e i g h  R i t z p r o c e d u r e to predict the n a t u r a l frequency of the t r i a n g u l a r plates. The present m e t h o d serves as a simple alternative to the finite element m e t h o d , finite difference m e t h o d a n d existing a n a l y t i c a l methods. The p r o p o s e d m e t h o d has been illustrated for three t r i a n g u l a r plates with different s u p p o r t conditions. The energy a p p r o a c h gives an u p p e r  b o u n d solution~ to the exact value, therefore, a sufficient n u m b e r of terms in the deflection series m u s t be used to ensure that stable convergence is reached. Stable convergence a n d satisfactory results can be o b t a i n e d when 18 terms are used. T h e p r e d i c t e d results of the i s o t r o p i c t r i a n g u l a r plates by the present m e t h o d in general indicate g o o d a g r e e m e n t with the available literature. This is expected because the present m e t h o d can a c c u r a t e l y describe the deflected shapes of the plates. N a t u r a l frequencies of the o r t h o t r o p i c c o m p o s i t e t r i a n g u l a r plates are also obtained; however, no c o m p a r i s o n can be m a d e since no results are available in the literature. REFERENCES 1. A. W. LEISSA,Vibration of plates. NASA SP160 (1969). 2. G. R. COWPER, E. KOSKO,G. M. LINDaERGand M. D. OLSON, Static and dynamic application of a highprecision triangular plate bending element. AIAA J. 7, 1957 (1969). 3. S. MIRZAand M. BIJLANI,Vibration of triangular plates of variable thickness. Comput. Struct. 21, 1129 (1985). 4. S. MIRZAand M. BIJLANI,Vibration of triangular plates. AIAA J. 21, 1472 (1983). 5. D. J. GORMAN,Highly accurate analytical solution for free vibration analysis of simplysupported right triangular plates. J. Sound Vibration 89, 107 (1983). 6. D. J. GORMAN, Free vibration analysis of right triangular plates with combination of clampedsimply supported boundary conditions. J. Sound Vibration 106, 419 (1986). 7. R.M. CHRISTENSEN,Vibration ofa 45,~right triangular cantilever plate by a gridwork method. AIAA J. 1, 1790 (1963). 8. R. B. BHAT, Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables. J. Sound Vibration 114, 65 (1987). 9. K. M. LIEW,K. Y. LAM and S. T. CHOW,Free vibration analysis of rectangular plates using orthogonal plate function. Comput. Struct. (accepted for publication). 10. T. S. CHIHARA,An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978). 11. D. JACKSON,Series of orthogonal polynomials. Ann. Math. 34, 527 (1933). 12. D. JACKSON,Formal properties of orthogonal polynomials in two variables. Duke Math. J. 2, 423 (1936). 13. S. W. TsAI and H. T. HOIIN,Introduction to Composite Materials, Vols I and II. AFML Technical Report TR78201 (1979). 14. P. N. GUSTAFSON,W. F. STOKEY and C. F. ZOROWSKI,An experimental study of natural vibration of cantilevered triangular plates. J. Aerospace Sci. 20, 331 (1953).