Ocean Engineering 26 (1999) 565–573

Fundamental frequency of tapered plates by themethod of eigensensitivity analysis

Oscar Barton, Jr*

Mechanical Engineering Department, US Naval Academy, Annapolis, MD, USA

Received 9 July 1997; received in revised form 29 August 1997; accepted 2 September 1997

Abstract

The fundamental frequency of a rectangular isotropic plate having a linear thickness vari-ation is computed using the method of eigensensitivity analysis. The approach incorporateseigen derivatives to evaluate a Maclaurin series representation of the desired eigenvalue, herethe fundamental frequency. Comparison with published results, for various taper ratios, aspectratios, and support conditions, demonstrates the accuracy and utility of the expression andmethodology. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:Fundamental frequency; Tapered plates; Eigensensitivity analysis

1. Introduction

Structures with variable geometric parameters are commonplace in many aspectsof engineering, and provide the means for optimal performance and cost savings. Anotable drawback which occurs when using these structures is the increased difficultyencountered in analysis. Addition effort is required to solve the governing differentialequation, having variable coefficients, for free vibration of plates and beam structureswith a variable thickness. Therefore, research efforts have sought to provide method-ologies that accurately and efficiently address this problem. Of particular interest isthe problem of computing the fundamental frequency of isotropic plates having alinear thickness variation.

* Tel.: 410 293 6510; Fax: 410 293 2591; E-mail: obarton@nadn.navy.mil

0029-8018/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0029 -8018(97)10003-8

566 O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

The vibration of isotropic plates having a linear thickness variation has been stud-ied by many investigators. Early work includes that of Gumeniuk (1955) who com-puted the fundamental frequency of simply supported plates using a finite differenceapproach and Appl and Byers (1965) also studied the vibration of simply supportedplates using the work of Collatz as the basis of the analysis procedure. More recently,Ng and Araar (1989) computed the vibration of clamped rectangular plates usingthe Galerkin method. Kukreti et al. (1992) have computed the fundamental frequencyof rectangular plates using differential quadratures for simply supported, clampedand mixed boundary conditions. Kukreti compares results of this method with thosecomputed from the finite element approach and the approach presented by Appl.Later Gutierrez and Laura (1994, 1995) incorporated the method of differential quad-ratures to study vibration of rectangular and circular plates. In addition, Laura etal. (1995) used an optimized Rayleigh–Ritz approach to determine the fundamentalfrequency of rectangular plates with a bilinear thickness variation in one direction.Comparative results for this type of thickness variation are also generated using afinite element approach. Gutierrez et al. (1995) studied the fundamental frequencyof rectangular plates that have a discontinuous thickness variation over a segmentof the plate. The authors also incorporated an optimized Rayleigh–Ritz method usingorthogonal polynomials as the selected basis functions. Rossi et al. (1996) provideda numerical finite element solution for the transverse vibration of rectangular cantil-evered plates having a discontinuous thickness variation. These authors presentedexperimental results to assess the accuracy of the numerical procedure.

In this article, the method of eigensensitivity analysis is used to compute the funda-mental frequency of rectangular isotropic plates and compared with the results pro-vided by Kukreti.

2. Problem statement

The equation governing the free vibration of a rectangular, isotropic plates withvariable thickness is given by

D(x,y)F∂4w∂x4 1 2

∂4w∂x2∂y2 1

∂4w∂y4G1

2∂D(x,y)∂x F∂3w

∂x3 1∂3w∂xy2G

12∂D(x,y)

∂y F∂3w∂x2y

1∂3w∂y3G

1∂2D(x,y)

∂x2 F∂2w∂x2 1 n

∂2w∂y2G

1 2(1 2 n)∂2D(x,y)

∂x∂y∂2w∂x∂y

1∂2D(x,y)

∂2y

Fn∂2w∂x2 1

∂2w∂y2G 5 r0h(x,y)v2w (1)

567O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

Boundary condition to be considered are clamped given by

w 5∂w∂x

5 0 at x 5 0,a

and

w 5∂w∂y

5 0 at y 5 0,b

and simply supported given by

w 5∂2w∂x2 5 0 at x 5 0,a

and

w 5∂2w∂y2 5 0 at y 5 0,b

In Eq. (1),D(x,y) is the position dependent flexural stiffness,w(x,y) is the displace-ment,r0 is the mass density,h(x,y) is the variable thickness andv is the frequency.In this paper, the thickness is assumed to vary in thex-direction only (Fig. 1) and sat-isfies

h(x) 5 h0G(x/a) 5 h0F1 1 bxaG

G(0) 5 1 (2)

G(1) 5 1 1 b

Fig. 1. Geometry of tapered plate used in current study.

568 O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

As a result of Eq. (2), the flexural stiffness becomes a cubic function of the spatialvariable and can be represented as

D 5 D[G(x/a)]3 (3)

whereD are the constant flexural stiffnesses corresponding to the uniform plate atx 5 0. An exact solution to Eq. (1) is impeded by the non-uniform thickness. There-fore, an approximate closed-form solution to Eq. (1) will be determined using themethod of eigensensitivity analysis.

3. Problem formulation

The discrete Ritz equations, corresponding to Eq. (1), are obtained by expandingw(x,y) in a complete, kinematically admissible basis

w(x,y) 5 ONm

ONn

amnXmSxaDYnSy

bD (4)

and substituting this representation into Eq. (1). Next, taking the inner product ofthis result with Xp ,Yq provides the discrete Ritz equations

[K(b)]apq 5 lpq[M(b)]apq (5)

whereapq is interpreted as components of the eigenvector whose corresponding eig-envalue islpq. The elements of the stiffness matrix [K(b)] and mass matrix [M(b)]may be expressed as

a4

abKpqmn(b) 5 D[Apm(b)bqn 1 n(Cpm(b)cnq 1 Cmp(b)cq)R2 (6)

1 Bpm(b)aqnR4 1 2(1 2 n)Epm(b)eqnR2]

In Eq. (6) and Eq. (7),R is the plate aspect ratioa/b and

1ab

Mpqmn(b) 5 r0h0Bpm(b)bqn (7)

Apm 5 (G3X0p,X0m) Bpm 5 (G3Xp,Xm) Cpm 5 (G3X0p,Xm)

Epm 5 (G3X9p,X9m) aqn 5 (Y0q,Y0n) bqn 5 (Yq,Yn)

cqn 5 (Y0q,Yn) eqn 5 (Y9q,Y9n) Bpm 5 (GXp,Xm)

lpq 5 v2

(8)

represent matrix elements corresponding to the inner product of the basis functions.Also ()9 denotes differentiation with respect to the indicated argument and the symbol(•,•) represents the L2 inner product on [0,1]. The matrices defined by Eq. (8) obvi-ously depend upon the basis functions selected. In this paper, beam shape functionssatisfying similar support conditions for a uniform beam have been selected as the

569O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

basis functions. Boundary conditions are implicitly included through these basis. Foreach boundary condition evaluated, the matrices are computed once and stored forlater use.

4. Sensitivity analysis

An approximate expression for the eigenvaluelpq can be determined by introduc-ing parametersS1 andS2 into Eq. (5) and considering

[K (S1)]{ apq(S1,S2)} 5 lpq(S1,S2)[M (S2)]{ apq(S1,S2)} (9)

where

[K (S1)] 5 [KD] 1 S1[DK] (10)

[M(S2)] 5 [MD] 1 S2[DM]

Here [KD] and [MD] are diagonal matrices obtained from [K] and [M], respectively,by deleting all off diagonal elements; [DK] and [DM] are matrices which have zeroson the diagonal and contain only the off-diagonal elements of [K] and [M]. TheparametersS1 and S2 range from 0 to 1. IfS1 5 S2 5 0, the solution to Eq. (9)becomes the ratio of the diagonal elements of the stiffness matrix [KD] and massmatrix [MD]. If S1 5 S2 5 1, the original eigenvalue problem, Eq. (5) is recovered.The desired eigenvaluelpq is obtained by expandinglpq in a Maclaurin series aboutS1 5 S2 5 0 and evaluating atS1 5 S2 5 1. Thus

lpq 5 lpq(1,1) > lpq(0,0) 1 dlpq(0,0) 112

d 2lpq(0,0) (11)

The desired results appearing on the right hand side of Eq. (11) can be shown to be

lpq(0,0) 5Kpqpq

Mpqpq

dlpq(0,0) 5 0 (12)

and

d 2lpq 5 22

M2pqpq

OmÞp

OnÞq

H[KpqpqDMmnpq 2 MpqpqDKmnpq]2

KmnmnMpqpq 2 KpqpqMmnmnJ (13)

Indeed, Barton and Reiss (1995) have provided a complete derivation for these termsby considering the buckling of a uniform symmetric angle-ply laminate. SubstitutingEqs. (12) and (13) into Eq. (11) provides the required quadratic approximate closed-form expression

lpq 5Kpqpq

Mpqpq

21

M2pqpq

OmÞp

OnÞq

H[KpqpqDMmnpq 2 MpqpqDKmnpq]2

KmnmnMpqpq 2 KpqpqMmnmnJ (14)

570 O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

5. Discussion and results

Eq. (14) provides a general quadratic approximate closed-form quadraticexpression for the eigenvaluelpq. Evaluating Eq. (14) with the stiffness and massmatrices given through Eqs. (6) and (7) provides the eigenvalue corresponding tothe tapered, isotropic plate. The fundamental eigenvalue will generally occur whenp 5 q 5 1. However, some plate configurations may require the evaluation of severalvalues ofp and q in order to identify the lowest eigenvalue. Therefore, care mustbe taken when evaluatinglpq since there is no a priori guarantee thatl11 is the leastfrequency, although this will often be the case. To facilitate numerical computationof the required frequency, introduce the normalized frequency corresponding to Eq.(14) as

kpq 5 √lpq 5 vpqa2!r0h0

D

Therefore, the normalized fundamental frequency corresponding to Eq. (14) isk11.Before any calculations can be made, the basis functions must be selected. If thebasis functions are indeed the exact eigenfunctions, the off-diagonal elements of the[K] and [M] vanish identically, that is [DK] and [DM] 5 [0] and the approximationof Eq. (14) is exact. If the basis functions provide a good approximation to the exacteigenfunctions, then the elements [DK] and [DM] are small compared the those of[KD] and [MD] respectively, and Eq. (14) can be expected to provide an excellentapproximation to the desired eigenvalues. Accordingly, the importance of selectingan appropriate basis when using the approximation of Eq. (14) cannot be overem-phasized.

The problems investigated consisted of rectangular plates with simply supported,clamped supported and a combination of simply supported and clamped boundaryconditions. For the mixed support case, the plate’s taper will be on the simply sup-ported side which is in thex-direction. The aspect ratiosR for each plate consideredwere 0.5, 1 and 2 and the taper parameterb varied from 0 to 1. For an accuratecomparison, the same number of terms was taken in the quadratic expression as thatused in the differential quadrature method. Therefore nine terms,N 5 9, were usedin the displacement expansion. Finally, the value used for Poisson ration was n5 0.3333.

Table 1 presents results for the plate simply supported on all sides. Columns 1and 2 provide the values used for the aspect ratioR and taper ratiob, columns 3and 4 provide the normalized fundamental frequency as computed by the differentialquadrature method (DQM) and the approximate closed-form expression, respectively,and the last column provides the percentage difference between the two approaches.For this plate, the largest discrepancy occurs forR 5 2, b 5 0.8. Here the DQMpredicts a value of 67.6021 and the approximate expression, Eq. (14), predicts avalue of 67.7339, resulting in a 0.195% difference. For a smaller taper ratio ofb 50.1 the difference is 0.079%. For the square plate,R 5 1, the difference varies froma 0.043% atb 5 0.1 to a maximum of 0.074% atb 5 0.8. ForR 5 0.5, the differencefor all taper ratios considered is under 0.070%.

571O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

Table 1Comparison of normalized frequency for a simply-supported plate

R b DQM Quadratic % Difference

0.5 0.1 12.9518 12.9482 0.030.5 0.2 13.5539 13.5490 0.0360.5 0.3 14.1473 14.1412 0.0430.5 0.4 14.7332 14.7264 0.0460.5 0.5 15.3123 15.3060 0.0410.5 0.6 15.8850 15.8812 0.0240.5 0.7 16.4510 16.4529 20.0120.5 0.8 17.0101 17.0220 20.0701.0 0.1 20.7296 20.7206 0.0431.0 0.2 21.7025 21.6915 0.0511.0 0.3 22.6669 22.6541 0.0561.0 0.4 23.6239 23.6105 0.0571.0 0.5 24.5740 24.5624 0.0471.0 0.6 25.5177 25.5113 0.0251.0 0.7 26.4547 26.4584 20.0141.0 0.8 27.3845 27.4048 20.0742.0 0.1 51.8244 51.7834 0.0792.0 0.2 54.2056 54.1611 0.0822.0 0.3 56.5370 56.4916 0.0802.0 0.4 58.8237 58.7847 0.0662.0 0.5 61.0699 61.0489 0.0342.0 0.6 63.2796 63.2915 20.0192.0 0.7 65.4562 65.5182 20.0952.0 0.8 67.6021 67.7339 20.195

Table 2 contains the results for the clamped plate on all sides. In general, thelargest percentage difference occurs when the taper parameter becomes 1.0. The onlyexception is when the aspect ratio is 0.5. For this case, the largest discrepancy of0.337% occurs whenb 5 0.8. For this geometry, the DQM computes 33.7888 andthe approximate closed-form expression computes 33.6789. For the square plate, thelargest difference is 1.40% with the DQM predicting 52.6374 and the approximateclosed-form expression predicting 53.3772. Recall that beam shape functions wereselected as the basis functions. Using orthogonal polynomials as the basis for theapproximate closed-form expression provides a slightly better comparisons of52.9218 yielding a 0.540% difference. For the larger aspect ratioR 5 2.0, the percentdifference is 3.41% when using the beam shape functions with the DQM predicting140.5029 and Eq. (14) providing 145.2929. An improvement of 1.93% results if oneuses the orthogonal polynomials as basis functions.

Finally, Table 3 presents the results for a plate with mixed simply supported andclamped boundary conditions with the simply supported boundary taken in the taperdirection. For this configuration, largest percentage difference of 3.49 occurs forR5 0.5 andb 5 1.0.

572 O. Barton, Jr /Ocean Engineering 26 (1999) 565–573

Table 2Comparison of normalized frequency for a plate clamped on all sides

R b DQM Quadratic % Difference

0.5 0.0 24.5877 24.5789 0.040.5 0.2 26.9971 26.9708 0.0970.5 0.4 29.3233 29.2602 0.2150.5 0.6 31.5836 31.4856 0.3100.5 0.8 33.7888 33.6749 0.3370.5 1.0 35.9465 35.8460 0.2801.0 0.0 36.0056 35.9812 0.0681.0 0.2 39.5485 39.5396 0.0231.0 0.4 42.9408 43.0339 20.2171.0 0.6 46.2504 46.4923 20.5231.0 0.8 49.4776 49.9357 20.9261.0 1.0 52.6374 53.3772 21.4052.0 0.0 98.3475 98.3155 0.0332.0 0.2 107.8149 108.0055 20.1772.0 0.4 116.6358 117.4573 20.7042.0 0.6 124.9588 126.7766 21.4552.0 0.8 132.8875 136.0392 22.3722.0 1.0 140.5029 145.2929 23.409

Table 3Comparison of normalized frequency for simply supported and clamped plate

R b DQM Quadratic % Difference

0.5 0.0 23.8113 23.8157 20.0180.5 0.2 26.1427 26.0570 0.3280.5 0.4 28.3959 28.0829 1.1020.5 0.6 30.5831 29.9778 1.9790.5 0.8 32.7166 31.8006 2.8000.5 1.0 34.8036 33.5891 3.4901.0 0.0 28.9572 28.9515 0.0201.0 0.2 31.7956 31.8030 20.0231.0 0.4 34.5371 34.5873 20.1451.0 0.6 37.2012 37.3319 20.3511.0 0.8 39.8010 40.0568 20.6431.0 1.0 42.3453 42.7752 21.0152.0 0.0 54.8235 54.7551 0.1252.0 0.2 60.1797 60.1385 0.0682.0 0.4 65.3223 65.3754 20.0812.0 0.6 70.2943 70.5170 20.3172.0 0.8 75.1258 75.6006 20.6322.0 1.0 79.8387 80.6513 21.018

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6. Conclusion

The method of eigensensitivity analysis has been employed to determine a quad-ratic expression to compute the fundamental frequency of an rectangular, isotropicplate with a linear thickness variation in one direction. Various support conditionswere analyzed including simply supported and clamped boundary conditions. For allplates configurations considered, the maximum discrepancy for any was 3.5%, whencompared with the differential quadrature results with most well under 1.0%.

References

Appl, F.C., Byers, N.R., 1965. Fundamental frequency of simply supported rectangular plates with linearlyvarying thickness. Journal of Applied Mechanics Transactions of the ASME, 32, 163–168.

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