Fundamental frequency of tapered plates by the method of eigensensitivity analysis

Pergamon Ocean Engineering 26 (1999) 565-573

Fundamental frequency of tapered plates by the

method of eigensensitivity analysis

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

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

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

Keywords: Fundamental frequency; Tapered plates; Eigensensitivity analysis

1. Introduction

Structures with variable geometric parameters are commonplace i n many aspects of engineering, and provide the means for optimal performance and cost savings. A notable drawback which occurs when using these structures is the increased difficulty encountered i n analysis. Addition effort is required to solve the governing differential equation, having variable coefficients, for free vibration of plates and beam structures with a variable thickness. Therefore, research efforts have sought to provide method-ologies that accurately and efficiently address this problem. O f particular interest is the problem of computing the fundamental frequency of isotropic plates having a linear thickness variation.

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

Oscar Barton, Jr

Abstract

0029-8018/99/$—see front matter © 1998 Elsevier Science Ltd. AU rights reserved. PII: 8 0 0 2 9 - 8 0 1 8 ( 9 7 ) 1 0 0 0 3 - 8

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 difference approach and Appl and Byers (1965) also studied the vibration of simply supported plates using the work of Collatz as the basis of the analysis procedure. More recentiy, N g and Araar (1989) computed the vibration of clamped rectangular plates using the Galerkin method. Kukreti et al. (1992) have computed the fundamental frequency of rectangular plates using differential quadratures for simply supported, clamped and mixed boundary conditions. Kukreti compares results of this method with those computed f r o m 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. I n addition, Laura et al. (1995) used an optimized Rayleigh-Ritz approach to determine the fundamental frequency of rectangular plates with a bilinear thickness variation i n one direction. Comparative results for this type of thickness variation are also generated using a finite element approach. Gutierrez et al. (1995) studied the fundamental frequency of rectangular plates that have a discontinuous thickness variation over a segment of the plate. The authors also incorporated an optimized Rayleigh-Ritz method using orthogonal polynomials as the selected basis functions. Rossi et al. (1996) provided a numerical finite element solution for the transverse vibration of rectangular cantil-evered plates having a discontinuous thickness variation. These authors presented experimental results to assess the accuracy of the numerical procedure.

I n 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 with variable thickness is given by

D{x,y) a v + 2 • a v +

+

+

+

+

Boundary condition to be considered are clamped given by dw ^ w = ^ = 0 at X ^ 0,a ox and dw w = ^ = 0 a.ty ^ 0,b dy

and simply supported given by w = -^T^ = 0 at X = 0,a and w = dx a v dy^ 0 at y = 0,b

In Eq. (1), D(x,y) is the position dependent flexural stiffness, w(x,y) is the displace-ment, Po is the mass density, h(x,y) is the variable thickness and co is the frequency. In this paper, the thickness is assumed to vary in the x-direction only (Fig. 1) and sat-isfies h(x) = hoG(x/a) = Hq G(0) = 1 Gil) = 1+ (3 1 + (3 (2)

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

D = D[G(x/a)]^ (3) where D are the constant flexural stiffnesses corresponding to the uniform plate at

X = 0. A n exact solution to Eq. (1) is impeded by the non-uniform thickness. There-fore, an approximate closed-form solution to Eq. (1) w i l l be determined using the method of eigensensitivity analysis.

3. Problem formulation

The discrete Ritz equations, corresponding to Eq. (1), are obtained by expanding w(x,y) i n a complete, kinematically admissible basis

N N / \ / \

y^(x,y) = 2 I.oi,„,Xn(-]Y,il] (4)

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

[Kmc^P, = A^,[M(/3)]a^, ' (5), where ct^^ is interpreted as components of the eigenvector whose corresponding

eig-envalue is Xpq. The elements of the stiffness matrix [K(jS)] and mass matrix [M(jS)] may be expressed as

^p,nM = D[Apnmb,n + v(Cp,Mc„^ + C,„/P)c^)R^ (6) ab

+ Bp,Mci,X + 2(1 - v)EpMe^X]

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

ab M^,„,„(iS) = PohoBpMb,,, (7)

^pm ~^ (.G^X"p,X"„j) Bp,„ — (G^Xp,X,„) Cp,„ — (G^X"p,X,„) Ep,„ = (G^X'pXJ a^n = i r ^ X n ) K„ = (7„F„) Can = iX'qJ,d ^qn = (Y'qX n) Bp,,, = (GXp,X„,)

(8)

represent matrix elements corresponding to the inner product of the basis functions. Also 0 ' denotes differentiation with respect to the indicated argument and the symbol (•,•) represents the L 2 inner product on [0,1]. The matrices defined by Eq. (8) obvi-ously depend upon the basis functions selected. In this paper, beam shape functions satisfying similar support conditions for a uniform beam have been selected as the

basis functions. Boundary conditions are implicitly included through these basis. For each boundary condition evaluated, the matrices are computed once and stored for later use.

4. Sensitivity analysis

A n approximate expression for the eigenvalue A^^ can be determined by introduc-ing parameters and ^'2 into Eq. (5) and considerintroduc-ing

[t(S,)]{ap^{S„S2)} = K,(SuS2m(SM&p,iSuS2)} (9) where

[K(S,)] = [ K J + S,m] (10) [M(S2)] = [Mr,] + 5 2 [ A M ]

Here [Kp] and [ M D ] are diagonal matrices obtained f r o m [ K ] and [ M ] , respectively, by deleting all o f f diagonal elements; [ A K ] and [ A M ] are matrices which have zeros on the diagonal and contain only the off-diagonal elements of [ K ] and [ M ] . The parameters and S2 range f r o m 0 to 1. I f = 52 = 0, the solution to Eq. (9) becomes the ratio of the diagonal elements of the stiffness matrix [ K Q ] and mass matrix [ M Q ] . I f = 5'2 = 1, the original eigenvalue problem, Eq. (5) is recovered. The desired eigenvalue A^^ is obtained by expanding A^^ i n a Maclaurin series about

= ^2 ^ 0 and evaluating at = 5*2 = 1. Thus

= V ( l ' l ) ^ ^P.(0,0) + 5A^,(0,0) + ^ 8%^(0,0) (11)

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

U0>0)

Indeed, Barton and Reiss (1995) have provided a complete derivation for these terms by considering the buckling of a uniform symmetric angle-ply laminate. Substituting Eqs. (12) and (13) into Eq. (11) provides the required quadratic approximate closed-f o r m expression

\ _ ^PgPI _ ^ V* l-^pqpq^M„j„pg — Mp^pgAK„,„pg] 1

M M ^ ^ ^ K M - K M

5. Discussion and results

Eq. (14) pro vides a general quadratic approximate closed-form quadratic expression f o r the eigenvalue A^^. Evaluating Eq. (14) with the stiffness and mass matrices given through Eqs. (6) and (7) provides the eigenvalue corresponding to the tapered, isotropic plate. The fundamental eigenvalue w i l l generally occur when p = q = 1. However, some plate configurations may require the evaluation of several

values of p and q i n order to identify the lowest eigenvalue. Therefore, care must be taken when evaluating A^^ since there is no a priori guarantee that An is the least frequency, although this w i l l often be the case. To facilitate numerical computation of the required frequency, introduce the normalized frequency corresponding to Eq.

Therefore, the normalized fundamental frequency corresponding to Eq. (14) is kn-Before any calculations can be made, the basis functions must be selected. I f the basis functions are indeed the exact eigenfunctions, the off-diagonal elements of the [ K ] and [ M ] vanish identically, that is [ A K ] and [ A M ] = [0] and the approximation of Eq. (14) is exact. I f the basis functions provide a good approximation to the exact eigenfunctions, then the elements [ A K ] and [ A M ] are small compared the those of [ K D ] and [ M Q ] respectively, and Eq. (14) can be expected to provide an excellent ' approximation to the desired eigenvalues. Accordingly, the importance of selecting an 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 boundary conditions. For the mixed support case, the plate's taper w i l l be on the simply sup-ported side which is i n the x-direction. The aspect ratios R for each plate considered were 0.5, 1 and 2 and the taper parameter /3 varied f r o m 0 to 1. For an accurate comparison, the same number o f terms was taken i n the quadratic expression as that used i n the differential quadrature method. Therefore nine terms, N = 9, were used in the displacement expansion. Finally, the value used for Poisson ratio v was v

Table 1 presents results for the plate simply supported on all sides. Columns 1 and 2 provide the values used for the aspect ratio R and taper ratio j8, columns 3 and 4 provide the normalized fundamental frequency as computed by the differential quadrature method ( D Q M ) 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 for = 2, /3 = 0.8. Here the D Q M predicts a value of 67.6021 and the approximate expression, Eq. (14), predicts a value of 67.7339, resulting i n a 0.195% difference. For a smaller taper ratio of j3 = 0.1 the difference is 0.079%. For the square plate, /? = 1, the difference varies f r o m a 0.043% at iS = 0.1 to a maximum of 0.074% at /3 = 0.8. For R = 0.5, the difference for all taper ratios considered is under 0.070%.

(14) as

Table 1

Comparison of normalized frequency for a simply- supported plate

R iS D Q M Quadratic % Difference 0.5 0.1 12.9518 12.9482 0.03 0.5 0.2 13.5539 13.5490 0.036 0.5 0.3 14.1473 14.1412 0.043 0.5 0.4 14.7332 14.7264 0.046 0.5 0.5 15.3123 15.3060 0.041 0.5 0.6 15.8850 15.8812 0.024 0.5 0.7 16.4510 16.4529 -0.012 0.5 0.8 17.0101 17.0220 -0.070 1.0 0.1 20.7296 20.7206 0.043 1.0 0.2 21.7025 21.6915 0.051 1.0 0.3 22.6669 22.6541 0.056 1.0 0.4 23.6239 23.6105 0.057 1.0 0.5 24.5740 24.5624 0.047 1.0 0.6 25.5177 25.5113 0.025 1.0 0.7 26.4547 26.4584 -0.014 1.0 0.8 27.3845 27.4048 -0.074 2.0 . 0.1 51.8244 51.7834 0.079 2.0 0.2 54.2056 54.1611 0.082 2.0 0.3 56.5370 56.4916 0.080 2.0 0.4 58.8237 58.7847 0.066 2.0 0.5 61.0699 61.0489 0.034 2.0 0.6 63.2796 63.2915 -0.019 2.0 0.7 65.4562 65.5182 -0.095 2.0 0.8 67.6021 67.7339 -0.195

Table 2 contains the results for the clamped plate on all sides. In general, the largest percentage difference occurs when the taper parameter becomes 1.0. The only exception is when the aspect ratio is 0.5. For this case, the largest discrepancy of 0.337% occurs when /3 = 0.8. For this geometry, the D Q M computes 33.7888 and the approximate closed-form expression computes 33.6789. For the square plate, the largest difference is 1.40% with the D Q M predicting 52.6374 and the approximate closed-form expression predicting 53.3772. Recall that beam shape functions were selected as the basis functions. Using orthogonal polynomials as the basis for the approximate closed-form expression provides a slightly better comparisons of 52.9218 yielding a 0.540% difference. For the larger aspect ratio R = 2.0, the percent difference is 3.41% when using the beam shape functions with the D Q M predicting

140.5029 and Eq. (14) providing 145.2929. A n improvement of 1.93% results i f one uses the orthogonal polynomials as basis functions.

Finally, Table 3 presents the results for a plate with mixed simply supported and clamped boundary conditions with the simply supported boundary taken i n the taper direction. For this configuration, largest percentage difference o f 3.49 occurs for R = 0.5 and (3 = 1.0.

Table 2

Comparison of normalized frequency for a plate clamped on all sides

R D Q M Quadratic % Difference 0.5 0.0 24.5877 24.5789 0.04 0.5 0.2 26.9971 26.9708 0.097 0.5 0.4 29.3233 29.2602 0.215 0.5 0.6 31.5836 31.4856 0.310 0.5 0.8 33.7888 33.6749 0.337 0.5 1.0 35.9465 35.8460 0.280 1.0 0.0 36.0056 35.9812 0.068 1,0 0.2 39.5485 39.5396 0.023 1.0 0.4 42.9408 43.0339 -0.217 1.0 0.6 46.2504 46.4923 -0.523 1.0 0.8 49.4776 49.9357 -0.926 1.0 1.0 52.6374 53.3772 -1.405 2.0 0.0 98.3475 98.3155 0.033 2.0 0.2 107.8149 108.0055 -0.177 2.0 0.4 116.6358 117.4573 - 0 . 7 0 4 2.0 0.6 124.9588 126.7766 -1.455 2.0 0.8 132.8875 136.0392 -2.372 2.0 1.0 140.5029 145.2929 -3.409 Table 3

Comparison of normalized frequency for simply supported and clamped plate

R D Q M Quadratic % Difference 0.5 0.0 23.8113 23.8157 -0.018 0.5 0.2 26.1427 26.0570 0.328 0.5 0.4 28.3959 28.0829 1.102 0.5 0.6 30.5831 29.9778 1.979 0.5 0.8 32.7166 31.8006 2.800 0.5 1.0 34.8036 33.5891 3.490 1.0 0.0 28.9572 28.9515 0.020 1.0 0.2 31.7956 31.8030 -0.023 1.0 0.4 34.5371 34.5873 -0.145 1.0 0.6 37.2012 37.3319 - 0 . 3 5 1 1.0 0.8 39.8010 40.0568 -0.643 1.0 1.0 42.3453 42.7752 -1.015 2.0 0.0 54.8235 54.7551 0.125 2.0 0.2 60.1797 60.1385 0.068 2.0 0.4 65.3223 65.3754 -0.081 2.0 0.6 70.2943 70.5170 -0.317 2.0 0.8 75.1258 75.6006 -0.632 2.0 1.0 79.8387 80.6513 -1.018

6. Conclusion

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

References

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Barton, O., Reiss, R., 1995. Buckling of rectangular symmetric angle-ply laminated plates determined by Eigensensitivity analysis. A I A A Journal 33, 2406-2413.

Gumeniuk, V.S., 1955. Determination of the free vibrations frequencies of variable thickness plates. Dopov, A N UkrSSR 2, 130-133.

Gutierrez, R., Laura, P., 1994. Vibrations of rectangular plates with linearly varying thickness and non-uniform boundary conditions. Journal of Sound arid Vibrations 178, 563-566.

Gutierrez, R., Rossi, R., Laura, P., 1995. Determination of the fundamental frequency of transverse vibration of rectangular plates when the thickness varies in a discontinuous fashion. Ocean Engineering 22, 663-668.

Gutierrez, R., Laura, P., 1995. Analysis of vibrating circular plates of nonuniform thickness by the method of differential quadrature. Ocean Engineering 22, 97-100.

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Ng, S.F., Araar, Y., 1989. Free vibration and buckling analysis of clamped rectangular plates of variable thickness by the Galerkin method. Journal of Sound and Vibration 60, 263-274.

Rossi, R., Belles, P., Laura, P., 1996. Transverse vibration of a rectangular cantilever plates with thickness varying in a discontinuous fashion. Ocean Engineering 23, 271-276.