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Research Papers

# Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Annular Plates of Varying Thickness

[+] Author and Article Information
Seema Sharma

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, Uttarakhand, Indiadikshitseema@yahoo.com

U. S. Gupta

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, Uttarakhand, Indiausgpt_10@yahoo.co.in

R. Lal

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, Uttarakhand, Indiarlatmfma@iitr.ernet.in

J. Vib. Acoust 132(4), 041001 (May 20, 2010) (13 pages) doi:10.1115/1.4001495 History: Received June 04, 2006; Revised March 16, 2010; Published May 20, 2010; Online May 20, 2010

## Abstract

Free axisymmetric vibrations of polar orthotropic annular plates of variable thickness resting on a Pasternak-type elastic foundation have been studied based on the classical plate theory. Hamilton’s energy principle has been used to derive the governing differential equation of motion. Frequency equations for an annular plate for two different combinations of edge conditions have been obtained employing Chebyshev collocation technique. Numerical results thus obtained have been presented in the form of tables and graphs. The effect of foundation parameter and thickness variation together with various plate parameters such as rigidity ratio, radius ratio, and taper parameter on natural frequencies has been investigated for the first three modes of vibration. Mode shapes for specified plates have been presented. A close agreement of results with those available in the literature shows the versatility of the present technique.

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## Figures

Figure 6

Frequency parameter for fundamental mode for α=0.5, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) K=0, G=0; (△) K=200, G=0; (○) K=200, G=20. (◻, △, ○) n=1; (◼, ▲, ●) n=2.

Figure 1

Convergence of frequency parameter for first three modes of vibrations for (a) C-C and (b) C-S plates for n=1; (c) C-C and (d) C-S plates for n=2 for ε=0.3, α=−0.5, p=2.0, K=200, G=25. (◆) fundamental mode; (◼) second mode; (▲) third mode. Percentage error=(|Ωm−Ω18|/Ω18)×100.

Figure 2

Frequency parameter for fundamental mode for ε=0.3, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) K=0, G=0; (△) K=200, G=0; (○) K=200, G=20. (◻, △, ○) n=1; (◼, ▲, ●) n=2.

Figure 7

Frequency parameter for (a) fundamental, (b) second, and (c) third modes for α=0.5, ε=0.3, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) G=0; (△) G=20. (◻, △) n=1; (◼, ▲) n=2.

Figure 8

Frequency parameter for (a) fundamental, (b) second, and (c) third modes for α=0.5, ε=0.3, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) K=0; (△) K=200. (◻, △) n=1; (◼, ▲) n=2.

Figure 9

Normalized displacements for (a) C-C plate and (b) C-S plate for p=5.0, K=200, G=25, ε=0.3. (————) n=1; (----------) n=2. (◻) α=−0.5; (◼) α=0.5.

Figure 3

Frequency parameter for second mode for ε=0.3, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) K=0, G=0; (△) K=200, G=0; (○) K=200, G=20. (◻, △, ○) n=1; (◼, ▲, ●) n=2.

Figure 4

Frequency parameter for third mode for ε=0.3, p=5.0. (————) C-C plate; (-------------) C-S plate. (◻) K=0, G=0; (△) K=200, G=0; (○) K=200, G=20. (◻, △, ○) n=1; (◼, ▲, ●) n=2.

Figure 5

Frequency parameter for fundamental mode for ε=0.3, α=0.5. (————) C-C plate; (-------------) C-S plate. (◻) K=0, G=0; (△) K=200, G=0; (○) K=200, G=20. (◻, △, ○) n=1; (◼, ▲, ●) n=2.

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