On the Use of Characteristic Orthogonal Polynomials in the Free Vibration Analysis of Rectangular...

7/27/2019 On the Use of Characteristic Orthogonal Polynomials in the Free Vibration Analysis of Rectangular Anisotropic Plate

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Compurrrs c4 Strucrures Vol. 62. No. 4. PP. 699-713. 1997Copyright 8 1996 Elsevier Science Ltd

PII: SOO45-7949(96)00264-7 Printed in Great Bruin. All rights reserved0045-7949/97 $17.00 + 0.00

ON THE USE OF CHARACTERISTICORTHOGONAL POLYNOMIALS IN THEFREE VIBRATION ANALYSIS OF RECTANGULAR

ANISOTROPIC PLATES WITH MIXEDBOUNDARIES AND CONCENTRATED MASSES

T.-P. Changt and M.-H. WuDepartment of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan,Republic of China

(Received 17 May 1995)Abstract-In this paper, the free vibration analysis of mass-loaded rectangular composite laminates platewith mixed boundaries was performed by using the orthogonal polynomial functions and Ritz method.We developed the subdomain method to derive the governing eigenvalue equation. In the solution process,we used the subdomain weighted residual to satisfy the compatibility at the interconnect boundaries fortwo adjacent subdomain and carry out continuity matrices, then we adopted the Gram-Schmidtorthogonalization process to find the orthogonal functions set which satisfy the simply subdomainboundary condition. Finally, we used the continuous matrices to develop the global energy functional andapplied the Ritz method to obtain the governing eigenvalue equation. By solving the governing eigenvalueequation, we can obtain the natural frequencies and mode shapes of the composite laminates.Furthermore, we also investigated the effects of mixed edge ratio, ply orientation and concentratedmasses on the free vibration of the rectangular symmetric composite laminates. Copyright 0 1996 ElsevierScience Ltd.

I. INTRODUCTIONIn recent years, orthogonal polynomials have beenwidely used in solving the problem of the freevibration of plates. The following researchers havealready contributed to the development of this field:in 198 1, Narita (11 applied a series-type method to thefree vibration of an orthotropic rectangular platewith mixed boundary conditions; in 1983, Fan andCheung [2] used the spline finite strip method to studythe flexural free vi bration response of thin rectangu-lar plates with complex support condition. Moreover,some researchers [3-l 11 used characteristic poly-nomials and applied Rayleighs method to solve thenatural frequencies and mode shapes of elliptical,circular, trapezoidal cantilever, rectangular plates,etc.More recently, the method of orthogonal poly-nomials and the Ritz method were extended toanisotropic plates and composite laminates. In 1989,Liew et al. [l2] used the orthogonal polynomialfunction, whch was developed by the Gram-Schmidtorthogonalization process, and adopted the Ray-leigh-Ritz procedure to solve flexural vibration oftriangular composite plates. In 1992, Chow et al. [I31

t To whom correspondence should be addressed.

used the Rayleigh-Ritz method with the two-dimensional orthogonal polynomials to derive thetransverse vibration of symmetrically laminatedrectangular composite plates. In 1993, Liew et al. [141used the method which decomposed a plate withmixed edges into a subdomain of simple boundaryconditions to solve the natural frequencies and modeshapes of the anisotropic plates.In the present study, we used the above methods tosolve the free vibration of the composite laminatewith mixed boundaries and concentrated masses. Weinvestigated the effects of the fiber orientation, mixededge ratio and the concentrated masses on the freevibration of these symmetric composite laminates.

2. EQUATIONOF MOTIONIn most practical applications of thin plates the

magnitude of the stresses acting on surface parallel tothe middle plane is small compared to the bendingand membrane stresses. Since the plate is thin, thisimplies that the tractions on any surface parallel tothe midplane are relative small. In particular, anapproximate state of plane stress exists. A standardx, y, z coordinate system, as shown in Fig. I, is usedto derive the equation of motion of the anisotropiclaminate plate. The displacements in the _v,y, z

699


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T.-P. Chang and M.-H. Wu

Fig. 1. Coordinate system of anisotropic laminates.

directions are denoted by u, u, w respectively. For thecomposite laminates, the following relations aredefined between the stress and moment resultants andthe stains and curvatures.

where

A,, Au ,416.421 A22 A2 aAu A .52 AM-----B , , B12 BL6& I B 2 2 B x861 Be2 8 6 6

BI , B ,2 BMBZ I 8 2 2 B 26& I Be.2 B,-----D, , 012 DmD21 022 D2aD* , D b2 De,

(1)

b,2(A t , , Bu, D;i ) = s $(1,t,z2)dz i,j= 1,2,6.- / I ,2 (2)Qti = (Q,, - Q,2 - 2Q&s

+ (Q12 Q22 + 2Q& *c ,

01, = Q,d + 2(Q,2 + 2Qd s2c2 + Q22s4,

@1 2 (QI I + Q22 4Q& *c2 + Q,2(s + c),

t 7 22 Qus + 2 (Q i 2 + 2Qds2c2 + Q22c4 ,

0~ = (Ql l - Q,2 - 2~4~3~

+ (Q12 Q22 2Qds3c,

02 6 (Q, , - Q,2 2Qd s3c

+ (Q12 Q22 2Q& 3s,

Q66 = (Q,, - 2Q12 + Qu - 2Qdc2s2+ Q& + c4), (3)

in which s=sinO, c=cosO andv 1 2E 2QI I =&, ___u = 1 V IZV21

Q22& 7 Qt i c G,2 , v , 2E , = v12 E2 .(4 )

Using the equation of equilibrium and eqn (I), andconsidering the symmetrical composite laminates, thefollowing equation can be derived:

adwDI 1G + 4016 dx,dy4W + 2(D,2 + 20%) $&$-4&h $$ + D22$ = q(x, y). (5)

Including the dynamic effect and considering theplate without the applied normal loading, the freetransverse vibration of a symmetrical anisotropicplate with several concentrated masses is governed bythe following equation:

D 3 + 4D a4101ax4 ,6 s + Wi2 + 2&d a4wax *ay2

- [ ph + c,6(x - a,Fv_Y BJ,=I 1ahx at=0 (6)

where p is the density of the plate, m denotesthe concentration mass, (a, /I) is the location of theconcentrated mass and NM is the total number ofthe concentrated masses.The undamped natural free vibration is harmonicand thus the displacement is of the following form:

Fig. 2. Geometry of ply laminates with boundaries andconcentrated masses.


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Characteristic orthogonal polynomials 701

X

Fig. 3. Interconnected subdomain of anisotropic laminates.

w(x, y, t) = W(x, y)sin(wf) (7)where o denotes the free vibrational naturalfrequency of the plate. Therefore, the equilibriumcondition for the symmetrical anisotropic compositelaminated plate can be written as:

+4D cwa2w+4D 26ay2 axay

x 6(y - B,)I>dx dy=:tationary value. (8)

To set up the governing eigenvalue equation foreqn (8), we assume the solution as the form of a seriesof products of bea.m characteristic shapes

where 4.,(x) and v.(y) are the characteristic shapesof free vibration satisfying a set of the followingboundary conditions on each end:

Fig. 4. A laminate plate with mixed (simply supported andclamped) boundaries (S-SC-SCS).

W = 0 $y = 0 (for a clamped boundary)

W = 0 $$ = 0 (for a simply-supported edge)

cy = 0 $7 = 0 (for a free edge) (10)where s is the outward normal along the boundary.

3. ORTHOGONAL POLYNOMIALS

In this paper, the beam characteristic shapes 4.,(x)and $.(y) in eqn (9), are sets of orthogonalpolynomials which are selected to satisfy thegeometric boundary conditions of the plate. Somebasic properties of these orthogonal polynomials aredescribed briefly as follows.

Given a polynomial &(x), an orthogonal setpolynomials in the interval a < x < b can begenerated by using Gram-Schmidt process asfollows:

#I(.~) = (x - BlM4X),(p(X) = (x - B,)#b I(X) - C,#i- z(x) (11)

B, = hg(x)q%;_(x) dx g(x)+;- 0) dx,hc, =sg(xM - I(X)+, - Ax) dx g(x)$;_ z(x) dx,,I

(12)

where g(x) is the weighting function and thepolynomials 4,(x) satisfy the following orthogonalitycondition,

Table I. Layer material properties for composite laminate platesMaterials E,/Ez Gl?lEZ VI2 P (kg m-7

E-glass-epoxy 2.45 0.48 0.23 72.0h (ml0.05


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702 T.-P. Chang and M.-H. Wu

Fig. 7.-ig. 5. A laminate plate with mixed (simply supported andclamped) boundaries (C-G-C-SC).A laminate plate with mixed (simply supported andclamped) boundaries (C-CSC-C-CSC).

and applying the boundary conditions (15) gives thehs(x)&(x)+/(x) dx = i,, if; f ; deflection shape as(13)0 X(x) = A4[(3a%2 - a3 - br) + (a + b3 - 3a26

In the present application, the weight function ischosen as unity, that is, - 3&)x + 6&x2 - 2(a + b)x + x]. (17)

Let a = 0 and b = 1, then eqn (17) becomes:s/$(x) dx = 1. (14) X(x) = J&(X - 2x3 + x4), (18)0 where Aq is an arbitrary constant. The normalizedConstruction of the first member &(x) is carried outso as to satisfy all the boundary conditions of thebeam problems accompanying the plate problem. Inthe present paper, laminates having a variety ofboundary conditions are considered. The compli-cated composite laminates domain is to be cut intoseveral small subdomains with same edges for simpleboundary conditions as described in eqn (10). Weused fourth or fifth order polynomials as startingterms, the coefficients of the polynomial being chosento satisfy the equivalent beam end conditions. Thefirst polynomial in the orthogonal set of polynomialsis constructed so as to satisfy at least the boundaryconditions of the subdomain, both geometric andnatural. The method is described below.3.1. Pl ate w i th opposite edges simpl y support ed (S-S)

Both of the beam problems have the sameboundary conditions, namelyX(a) = x(a) = X(b) = X(b) = 0. (15)

mode function is obtained as

9.(x)=(x-2x3+x4)~(~x(x)dx)1.2. (19)

3.2. Pl ate w i th opposite edges clamped (C-C)Both of accompanying beam problems have thesame boundary conditions, namely

X(a) = r(a) = X(b) = x(6) = 0. (20)Assuming the beam deflection as

X(x) = Ao + A,x + A*X2 + A3x3 + /44X4 (21)and applying the boundary conditions (21) gives thedeflection shape as

X(x) = AJ aW - 2ab(a + b)x + (a + 4ab+ b2)X2 2(n + b)x + x4]. (22)

Assuming the beam deflection as Let a = 0 and b = 1, then eqn (22) becomes:X(x) = A4(X2 - 2x + x4), (23)

X(x) = Ao + A,x + AIX + A3x3 + A4x4 (16) where A4 is an arbitrary constant. The normalizedmode function is obtained asS C S IIII e&(x) = (x2 -2x+ x4)/(d X(x)dx)112. (24)I

j sI It might be convenient, at this point, to introduceI the terminology to be used throughout the remainderFig. 6. A laminate plate with mixed (simply supported and of the paper for describing the boundary conditionsclamped) boundaries (S-SCS-S-SCS). of the laminates considered. The types of laminates


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Characteristic orthogonal polynomials 703Table 2. Frequency parameter i, = (~~wW/D#~ of four-layer laminates (0/-0/-13/f?) with S-S-S-Sboundary conditions

Number of modeAl,gorithm 1 2 3 4 5 6

Liew et al. [14] 15.19Leissa el nl. [22] 15.19Chow et 01. [13] 15.19Present method 15.19Leissa et uf. [22] 16.29Chow et al. [I31 16.29Present method 16.28

e = 033.30 44.42 60.78 64.53 90.3033.30 44.42 60.77 64.53 90.2133.31 44.52 60.78 64.55 90.3133.29 44.44 60.76 64.54 90.26

e=45037.71 41.63 63.29 77.56 79.6037.71 41.63 63.29 77.56 79.6037.68 41.60 63.23 77.52 79.56

considered in the present work are S-S and C-C. Thestarting functions used in this study are as follows:

S-S X(x) = C,(x - 2x3 + x4)c-c X(x) = &(x2 - 2x + x);

C, is an arbitrary constant.

4. METHODOF ANALYSISThe Gram-Schmid method which generated

orthogonal polynomials in conjunction with Ray-leigh-Ritz method had been used for the study of freevibration of isotopic rectangular plates. This tech-nique will be adopted to perform the free vibrationof rectangular composite laminates with severalconcentrated masses. It is assumed that the plate liesin the x - y plate, is bounded by x = 0, a andy = 0, b, is of uniform thickness, rectangularlycomposite laminates material. For the free vibrationof the laminates, the deflection w may be representedby the expression

w(x, y, t) = W(x, y)sin wt

= C C A,,,,&,(x)+,(v) sin it (25),I$ ,I

where 4,,,(x), Ii/,,(v) are appropriate polynomialfunctions satisfying at least the geometric boundary

conditions of the laminates, w is the radian naturalfrequency by vibration and t is the time.In the present study, the technique developed byLiew et al. [14] was adopted to establish thegoverning eigenvalue equation for composite Iami-nates wth partial mixed discontinuous edges. Thecomplex composite laminates domain to be cut intosmall subdomains with same edges for simpleboundary conditions, as shown in Fig. 3. Thedisplacement W of each subdomain is chosen tosatisfy the boundary conditions of the subdomain

FW(x, v) = 1 ~4$#$(x)lj/i(y) (26)i n

and

Wi(X, y) = 1 c A;#$;:: (x)$i: (y), (27)n, nwhere t = 2,3, . . . , h4, and M, is the total number ofsubdomains.Consider the two connected subdomains, thedisplacement function Wmust have the same 4(x) or$b) corresponding the same boundary condition inthe s- or y-direction, i.e.

&:(.u) = $,;:-l(x); m = 1,2, 3.. , M (28)or

r,@(y) = I&-~(v); n = 1,2,3.. , N. (29)

Table 3. Frequency parameter /1 = (phwW/Do)~of four-layer laminates (t?/-fl-e/e) with C-C-C-Cboundary conditionsNumber of modeAlgorithm 1 2 3 4 5 6

Liew ef ai. 1141 29.10Chow et al. [13] 29.13Present method 29.12Chow et aI. [13] 28.53Present method 28.53

8=050.83 67.29 85.68 87.14 118.650.82 67.29 85.67 87.14 118.650.84 67.34 85.69 87.17 118.60 =4555.56 60.22 85.25 102.6 105.255.58 60.24 85.29 102.6 105.2


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704 T.-P. Chang and M.-H. WuTable 4. Frequency parameter I = (pho*a4/D#* of four-layer laminates (O/~/O/O) withS-SC-SCS boundary condrtions

Number of modeAlgorithm 1 2 3 4 5 6d=O

Liew et al. [14] 20.43 45.68 47.03 69.52 83.71 95.28Present method 20.43 45.68 47.05 69.50 83.74 95.36d= 0.5Ltew et al. [14] 18.74 40.66 45.64 65.52 74.22 94.45Present method 18.67 40.55 45.66 65.51 74.00 94.51d= 1.0Liew et al. [14] 15.19 33.30 44.42 60.78 64.53 90.30Present method 15.19 33.29 44.44 60.76 64.54 90.26

Because the interconnecting boundary is continu-ous, the deflection, slope and higher derivatives of theWI and WI - I f must be equal to the exact solution.Therefore, the following expression is defined byusing the weighted residual method.

R = L[ WI: _ WC I (30)where L is a linear differential operator which denotesthe nth partial derivative of the function inside thebracket with respect to x or y. Now, we will dividethe interconnecting boundary into I subdomains thenwe use the method of subdomain weighted residual tointegrate the residual R over each subdomain.sds=O; i = 1,2,3 ,..., I, (31)0therefore

WI,_, as as (33)

where i=l,2,3 ,..., Ie; t=2,3,4 ,..., M,; r= l ,2,3, . , N - 1 and s = x or s = y depending on thedirection of connecting boundary. The deflections Wof each subdomain have been defined by eqns (26)and (27).A set of continuity matrices for the eigenvectors A,of the adjacent subdomains can be obtained fromthe above method of subdomain weighted residual.The derivation process is described here with forsubdomains 1 and 2. These two subdomains have thesame &(x) for the same boundary condition in thex-direction, i.e.

@I = &f(x); form = 1,2, 3,. . . , M . (35)Also we know

Wll(x, y) = 11 ,gJf#l~(x)$~(y) (36)n, n

Table 5. Frequency parameter ,I = (phc~*a~/D~)~ of four-layer laminates (0/00/00/00) withSSCS-SSCS boundary conditionsNumber of modeAlgorithm 1 2 3 4 5 6

d=OLiew et al. [14] 15.19 33.30 44.42 60.78 64.53 90.30Present method 15.19 33.29 44.44 60.76 64.54 90.26d = 0.5Liew et al. [14] 20.11 44.72 46.36 67.02 81.67 94.65Present method 20.12 44.79 46.41 67.17 81.79 94.75

(37)

d= 1.0Liew et al. [14] 20.43 45.68 47.03 69.52 83.71 95.28Present method 20.43 45.68 47.05 69.50 83.74 95.36


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Characteristic orthogonal polynomials 705Table 6. Freqeuncy parameters I = (uAr4/D~)~*f four-layer laminates for case 4 (C-CSC-C-CSC) with concentrated mass

Number of modeM/phab 0 1 2 3 4 5 6

0.0 (0/00/00/00) 13.91 22.24 35.04 37.96 43.41 58.71(lS/ -45/ -45/45) 12.90 24.32 30.22 40.89 46.37 54.70

(0/90/90/00) 13.67 22.86 34.00 40.10 43.14 59.931.0 (~/OO/OO/oy 7.662 15.97 25.73 36.49 39.26 52.13

(4s0/- 45/ - 45/45) 7.420 15.89 25.17 35.76 42.40 49.54(0/90/90/00) 7.663 15.88 26.04 36.21 40.63 52.28

2.0 (oO/oO/o~/oq 5.671 15.72 25.53 36.47 39.24 52.01(45=/-45/-45/45) 5.481 15.47 25.14 35.62 42.36 49.50

(0/90/90/00) 5.627 15.59 25.87 36.16 40.62 52.153.0 (0/00/00/00) 4.639 15.65 25.47 36.46 39.23 51.98

(45/-45/-45/45) 4.537 15.35 25.13 35.57 42.35 49.48(0/90/90/00) 4.650 15.51 25.82 36.14 40.62 52.11

4.0 (~/0/0/00) 4.040 15.61 25.44 36.46 39.22 51.96(uO/-45/ -45/45) 3.956 15.29 25.12 35.55 42.35 49.48

(0/90/90/00) 4.051 15.47 25.79 36.14 40.62 52.095.0 (W/OO/OO/OO) 3.626 15.59 25.42 36.46 39.22 51.95

(45O/ - 45/ - 45/45) 3.552 15.25 25.12 35.53 42.34 49.47(0/90/90/00) 3.636 15.45 25.77 36.13 40.62 52.08

Substituting eqns (36) (37) into eqns (32), (33) and From eqns (38)-(41), the following equations can be(34), we can obtain the following equations: derived readily:

+ + H& k; = 0, (42)agi_+&!@+..+@!

+. . + H !?i!k= 0 (43)Max t(39)

_ zi w:;(x)T t,+;(y) dy = 0. (40)3Now if we define

Fig. 8. Frequency parameter A of case 1 (S-SC-S-CS)without concentrated mass for various simply supportedmixed ratio.CAS 6214-D


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Fig. 9. Frequency parameter 1 of case 2 (C-CS-C-SC)without concentrated mass for vartous clamped mtxed ratio.

In order to satisfy eqns (42)


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Characteristic orthogonal polynomials 707

Fig 13. Frequency parameter i of case 2 (C-CS-CSC)with concentrated mass for various clamped mixed ratio. Fig. 15. Frequency parameter /I of case 4 (C-CSC-C-CSC)with concentrated mass for various simply supported mixedratio.?( -g,. - si(y) dy. (53)?IIFrom the above process, the derivation of thecontinuity matrix can be easily generalized to three ormore subdomains for two interconnecting bound-aries.

Now we are ready to derive the eigenvalueequation by using the continuity matrix. Let thestrain energy W and the kinetic energy TiCi for arectangular subdomain be expressed as follows:

and

+20 a*w 2*wa*w I2ax* ay* D 22 ay2( >+40 d2Wa*w+4D Ea*wI6ax* axay 26ay* axay

dx dy (54)

T:) = !a22 ph + y mJ(x - 51),=I

x SOJ - ni)> Wdx dy. (55)

Substituting eqns (26) or (27) into eqns (54) and (55)yields:

Vi: = c c 1 1 A&;,mn I,

Fig. 14. Frequency parameter I of case 3 (S-SCS-SSCS) Fig. 16. Frequency parameter ,I of case 1 (S-SC-S-G) withwith concentrated mass for various clamped mixed ratio. concentrated mass at point (0.25,0.25).


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Frg. 17. Frequency parameter 1 of case 1 (S-SC-S-CS) withconcentrated mass at point (0.5,0.5). Frg. 19. Frequency parameter ibf case I (S-SC-SCS),with concentrated mass located along the y-axts (5 = 0.5).

(56) in which r, s = 0, 1, 2. Substituting eqns (58) and (59)into eqns (56) and (57) yields:and p: = sAJT[K:f]{A) dR (60)R

TEci {A}T[M~~]{,4} dRs (61)I wherex [~.,(5)~,(5)11/.(11)~,(~)15 drl (57)

K,\:i, = $- { DI I %I$ + y4DzeF:;0where 5 = x/a, q = y/b and mk are some masses atpoints (&, Q). Now denote EL;,, FL; as follows: + Y~DIz(G%?; + E:ljti;)

and

Fig. 18. Frequency parameter I of case 1 (S-SC-S-CS),with concentrated mass located along the x-axis (q = 0.5).

+ 4yD&!,F!; ) (62)and

(63)where m,i=1,2,3 ,_.., M, n,j=1,2,3 ,..., Nand Do = E,h/[12(1 - v,~v~,)], 7 = b/a. Now, wetransform [K::] and [MC] for subdomain {t) usingcontinuity matrix [Pfl] of eqn (50), we can then derivethe following expression:

[RkI] = [SII:][KIG][$I] (64)and

(65)


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Characteristic orthogonal polynomials 709T = T + 3 Tk!, (70)


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A = 35.21Mode 1

x = 17.41 A = 45.21Mode 2 Mode 5

X = 27.38 Jt 51.76Mode 3 Mode G

Fig. 21. The first six mode shapes of case 2 (45/ -45/-45/45) with concentrated mass.

[K] = [KX] + % [p] (75)r=2and

[M] = [MI:] + 2 [&::I. (76)


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Characteristic orthogonal polynomials 711works. The non-dimensional frequency parameter isexpressed as l(pli~~~~/&)~~. It can be seen fromTables 2-5 that the comparisons reveal goodagreement as far as the first six eigenvalues areconcerned. For simplicity, the concentrated mass isassumed to be only one which is located at point(0.25,0.25), and the non-dimensional frequencyparameter is expressed as l(w2a4/D0)2. The first sixfrequency parameters for the case 4 are given inTable 6. A study on the frequency parameters 1 dueto the variation of concentrated mass from zero tofive times the laminate mass combined with plyoritentation 0 is carried out for the compositessymmetric laminates.

In the following discussion, the concentrated massis assumed to be equal to half of the laminated massand the non-dimensional frequency is expressed as1 = (wa/D#2 with concentrated mass orI = (ph&4/Do)~2 without concentrated mass. InFigs 8-l 1, the eigenvalues of cases 1-4 without theconcentrated mass have been presented for varioussimply supported or clamped ratio of mixedboundaries, respectively. In Figs 12-15, the par-ameters and boundary conditions are the same asthose in Figs 8-11, except that the compositelaminate is loaded with a concentrated mass locatedat point (0.25,0.25). As it can be seen fromFigs 12-15, the results are quite similar to those inFigs 8-l 1. In Figs 16-l 9, the boundary condition as

in case 1 with mixed ratio equal to 0.5 is considered.The concentrated mass which is varied from zero tofive times the laminate mass, is located at point(0.25,0.25), as shown in Fig. 16 and at point(0.5,0.5), as shown in Fig. 17. It can be seen fromFigs 16 and 17 that the first eigenvalue of thecomposite laminates is larger when the concentratedmass is located at point (0.25,0.25), also the firsteigenvalue is more strongly affected by the concen-trated mass located at point (0.5,0.5) rather than thatlocated at point (0.25, 0.25). We loaded theconcentrated mass which is located at TV 0.5 alongthe x-axis, as shown in Fig. 18 and at 5 = 0.5 alongthe y-axis, as shown in Fig. 19. We presented thevariation of the first eigenvalue due to the variouslocation of the concentrated mass in Figs 18 and 19.Finally in Figs 20-23, the first six mode shapes of thecomposite laminates (45/ -45/ - 45/45) with orwithout concentrated mass are shown for cases 2and 4 individually. It can be concluded that themode shapes of the composite laminates arestrongly affected by the boundary conditions andconcentrated masses.

6. CONCLUSIONSA simple, efficient and accurate approximate

method was introduced to study the free vibrationalbehavior of rectangular symmetrically composite

A = 12.90Mode 1 Mode 4

Mode 2X = 46.37

Mode 5

Mode 3 Mode GFig. 22. The first six mode shapes of case 4(45/-45/-45/45) without concentrated mass.


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Mode 2X = 42.46

Mode 5

X = 49.64Mode 3 Mode GFig. 23. The first six mode shapes of case 4 (45/-45/ -45/45) with concentrated mass.

laminates with mixed edge boundary conditions andconcentrated masses. The orthogonal polynomialfunctions set was constructed by using Gram-Schmidt orthogonalization process. The energyfunctional was obtained by using subdomain methodwith a set of orthogonal polynomials. Based on theRayleigh-Ritz procedure, the governing eigenvalueequation for the composite laminates was thenderived by minimizing the energy functional withrespect to each unknown coefficient. By solving thisgoverning eigenvalue equation, we can obtain thenatural frequencies and the mode shapes for thelaminates composite plate with concentrated mass.

In the previous study, various mixed edgeboundary conditions and concentrated masses wereconsidered, and different ply angles of the compositelaminates were investigated as well. It can beconcluded that the eigenvalues of the compositelaminates are strongly influenced by the concentratedmasses and mixed boundaries, also the mode shapesof the composite laminates with the concentratedmasses are quite different from those of the compositelaminates without the concentrated masses. Inpractice, we believe that the present work is quiteimportant, since it provides additional useful designinformation for engineers who are engaged in thearea of thin composite laminates plate with mixedboundaries and concentrated masses.

Acknowledgement-This work was partially supported bythe National Science Council of the Republic -of Chinaunder grant NSC 84-2212-E005-008. The authors aregrateful for this support.

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REFERENCESY. Narita, Application of a series-type method tovibration or orthotropic rectangular plates with mixedboundary conditions. J. Sound Vibr. 71, 345-355(1981).S. C. Fan and Y. K. Cheung, Flexural free vibrationsof rectangular plates with complex support conditions.J. Sound Vibr. 93(l), 81-94 (1984).C. Rajalingham, R. B. Bhat and G. D. Xistris, Naturalfrequencies and mode shapes of elliptic plates withboundary characteristic orthogonal polynomials asassumed shape function. J. Vibr. Acousr. 115, 353-358(1993).R. B. Bhat, Natural frequencies of rectangular platesusing characteristic orthogonal polynomials in Ray-leigh-Ritz method. J. Sound Vibr. 102, 493499(1985).B. Singh and S. Chakraverty, Transverse vibration ofsimply supported elliptical and circular using boundarycharacteristic orthogonal polynomials in two variables.J. Sound Vibr. 152(l), 149-155 (1992).P. A. A. Laura, R. H. Gutierrezder and R. B. Bhat,Transverse vibration of a trapezoidal cantilever plate ofvariable thickness. AIAA 27(l), 921-922 (1989).T. Mizusawa, Natural frequencies of rectangularplates with free edges. J. Sound Vibr. 105(3), 451459(1986).


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Characteristic orthogonal polynomials 7138.

9.

IO.

11.

12.

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On the Use of Characteristic Orthogonal Polynomials in the Free Vibration Analysis of Rectangular...

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