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Research Papers

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

[+] Author and Article Information
Roshan Lal

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Indiarlatmfma@iitr.ernet.in

Dhanpati

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

J. Vib. Acoust 131(1), 011007 (Jan 06, 2009) (9 pages) doi:10.1115/1.2980399 History: Received September 12, 2007; Revised May 12, 2008; Published January 06, 2009

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Rectangular plate resting on Pasternak foundation

Grahic Jump Location
Figure 2

Convergence graphs: (a) C-C plate and (b) C-S plate for μ=0.5, α=0.5, β=−0.5, K=0.02, G=0.002, and a∕b=1. —, first mode; ------, second mode; – – – –, third mode. Percentage error=∣Ωm−Ω18∣∕Ω18×100, m=9(1)22.

Grahic Jump Location
Figure 3

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for β=0.5, a∕b=1. —, C-C; ------, C-S; △, α=−0.5, K=0.0, G=0.00; ▲, α=0.5, K=0.0, G=0.00; ◻, α=−0.5, K=0.02, G=0.00; ◼, α=0.5, K=0.02, G=0.00; ○, α=−0.5, K=0.02, G=0.002; ●, α=0.5, K=0.02, G=0.002.

Grahic Jump Location
Figure 4

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for α=0.5, a∕b=1. —, C-C;------, C-S; △, μ=−0.5, K=0.0, G=0.00; ▲, μ=0.5, K=0.0, G=0.00; □, μ=−0.5, K=0.02, G=0.00; ◼, μ=0.5, K=0.02, G=0.00; ○, μ=−0.5, K=0.02, G=0.002; ●, μ=0.5, K=0.02, G=0.002.

Grahic Jump Location
Figure 5

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for β=0.5, a∕b=1. —, C-C; ------, C-S; △, μ=−0.5, K=0.0, G=0.00; ▲, μ=0.5, K=0.0, G=0.00; □, μ=−0.5, K=0.02, G=0.00; ◼, μ=0.5, K=0.02, G=0.00; ○, μ=−0.5, K=0.02, G=0.002; ●, μ=0.5, K=0.02, G=0.002.

Grahic Jump Location
Figure 6

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for α=0.5, β=0.5. —, C-C; ------, C-S; △, μ=−0.5, K=0.0, G=0.00; ▲, μ=0.5, K=0.0, G=0.00; □, μ=−0.5, K=0.02, G=0.00; ◼, μ=0.5, K=0.02, G=0.00; ○, μ=−0.5, K=0.02, G=0.002; ●, μ=0.5, K=0.02, G=0.002.

Grahic Jump Location
Figure 7

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for α=0.5, a∕b=1. —, C-C; ------, C-S; △, μ=−0.5, β=−0.5, G=0.00; ▲, μ=0.5, β=−0.5, G=0.00; ◻, μ=−0.5, β=0.5, G=0.00; ◼, μ=0.5, β=0.5, G=0.00; ○, μ=−0.5, β=0.5, G=0.002; ●, μ=0.5, β=0.5, G=0.002.

Grahic Jump Location
Figure 8

Frequency parameter for C-C and C-S plates: (a) first mode, (b) second mode, and (c) third mode for α=0.5, a∕b=1. —, C-C; ------, C-S; △, μ=−0.5, β=−0.5, K=0.0; ▲, μ=0.5, β=−0.5, K=0.0; ◻, μ=−0.5, β=0.5, K=0.0; ◼, μ=0.5, β=0.5, K=0.0; ○, μ=−0.5, β=0.5, K=0.02; ●, μ=0.5, β=0.5, K=0.02.

Grahic Jump Location
Figure 9

Normalized displacements: (a) C-C plate and (b) C-S plate for β=0.5, K=0.02, G=0.002, and a∕b=1. —, first mode; ------, second mode; – – – –, third mode; △, μ=−0.5, α=−0.5; ▲, μ=0.5, α=−0.5; ◻, μ=−0.5, α=0.05; ◼, μ=0.05, α=0.05.

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