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Research Papers

Application of the Nondimensional Dynamic Influence Function Method for Free Vibration Analysis of Arbitrarily Shaped Membranes

[+] Author and Article Information
Sang Wook Kang1

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

Satya N. Atluri

Department of Mechanical and Aerospace Engineering,  University of California, Irvine, California 92697satluri@uci.edu

Sang-Hyun Kim

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

1

Corresponding author.

J. Vib. Acoust 134(4), 041008 (May 31, 2012) (8 pages) doi:10.1115/1.4006414 History: Received April 22, 2011; Revised February 15, 2012; Published May 29, 2012; Online May 31, 2012

A new formulation for the NDIF method (the nondimensional dynamic influence function method) is introduced to efficiently extract eigenvalues and mode shapes of arbitrarily shaped, homogeneous membranes with the fixed boundary. The NDIF method, which was developed by the authors for the accurate free vibration analysis of arbitrarily shaped membranes and plates including acoustic cavities, has the feature that it yields highly accurate solutions compared with other analytical methods or numerical methods (the finite element method and the boundary element method). However, the NDIF method has the weak point that the system matrix of the method is not independent of the frequency parameter and as a result the method needs the inefficient procedure of searching eigenvalues by plotting the values of the determinant of the system matrix in the frequency parameter range of interest. An improved formulation presented in the paper does not require the above-mentioned inefficient procedure because a newly developed system matrix is independent of the frequency parameter. Finally, the validity of the proposed method is shown in several case studies, which indicate that eigenvalues and mode shapes obtained by the proposed method are very accurate compared to those calculated by exact, analytica, or numerical methods.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Infinite membrane with harmonic excitation points that are distributed along the fictitious contour (dotted line) with the same shape as the finite-sized membrane of interest

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Figure 2

Circular membrane discretized by 16 nodes

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Figure 3

Mode shapes of the circular membrane obtained by the proposed method when N=16 and M=20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode

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Figure 4

Rectangular membrane discretized by 24 nodes

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Figure 5

Mode shapes of the rectangular membrane obtained by the proposed method when N=24 and M=20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode

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Figure 6

Arbitrarily shaped membrane discretized by 20 nodes

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Figure 7

Mode shapes of the arbitrarily shaped membrane obtained by the proposed method when N=20 and M=20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode

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