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Research Papers

Free Vibration Analysis of Arbitrarily Shaped Plates With Smoothly Varying Free Edges Using NDIF Method

[+] Author and Article Information
S. W. Kang1

Department of Mechanical Systems Engineering, Hansung University, 389, 2-ga, Samsun-dong, Sungbuk-gu, Seoul 136-792, Koreaswkang@hansung.ac.kr

I. S. Kim, J. M. Lee

School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1 Shinlim-dong, Kwanak-gu, Seoul 151-742, Korea

1

Corresponding author.

J. Vib. Acoust 130(4), 041010 (Jul 14, 2008) (8 pages) doi:10.1115/1.2730531 History: Received February 19, 2006; Revised November 08, 2006; Published July 14, 2008

The so-called non-dimensional influence function method that was developed by the authors is extended to free vibration analysis of arbitrarily shaped plates with the free boundary condition. A method proposed in this paper can be applied to plates with only smoothly varying boundary shapes. In the proposed method, a local polar coordinate system has been employed at each boundary node to effectively consider the free boundary condition, which is much more complex than the simply supported or clamped boundary condition. The local coordinates system devised allowed us to successfully deal with the radius of curvature involved in the free boundary condition, and, as a result, the accuracy of the proposed method has been reinforced. Finally, verification examples showed that the natural frequency and mode shapes obtained by the proposed method agree excellently with those given by other analytical or numerical methods.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

An arbitrarily shaped plate depicted by the dotted contour along which N boundary nodes are distributed

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Figure 13

First eight mode shapes of the arbitrarily shaped plate obtained by the proposed method (N=16)

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Figure 2

Local polar coordinates system (ri,θi) for considering the free boundary condition of node Pi (the origin Oi of the coordinates system is located at the center of curvature measured at node Pi, and Ri is the radius of curvature measured at node Pi)

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Figure 3

Flow chart illustrating the theoretical development procedure of the present method

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Figure 4

(xi,yi), (xk,yk), and (x0i,y0i) denote coordinates of node Pi, node Pk, and origin Oi, respectively, for the global coordinates system (x,y); this figure also illustrates that node Pk approaches node Pi

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Figure 5

Discrete boundary nodes of the circular plate when (a)N=12 and (b)N=16

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Figure 6

Logarithm curves for det(SMplate)∕det(SMmem) of the circular plate when N=12 and N=16; the resolution of increment of the frequency parameter is 1∕1000

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Figure 7

First eight mode shapes of the circular plate obtained by the proposed method (N=16)

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Figure 8

Discrete boundary nodes of the elliptic plate when (a)N=12 and (b)N=16

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Figure 9

Logarithm curves for det(SMplate)∕det(SMmem) of the elliptic plate when N=12 and N=16; the resolution of increment of the frequency parameter is 1∕1000

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Figure 10

First eight mode shapes of the elliptic plate obtained by the proposed method (N=16)

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Figure 11

Discrete boundary nodes of the arbitrarily shaped plate, composed of a half-circle and a half-ellipse, when (a)N=12 and (b)N=16

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Figure 12

Logarithm curves for det(SMplate)∕det(SMmem) of the arbitrarily shaped plate when N=12 and N=16; the resolution of increment of the frequency parameter is 1∕1000

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