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Research Papers

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

[+] Author and Article Information
Jeng-Tzong Chen

Department of Harbor and River Engineering,
National Taiwan Ocean University,
Keelung 20224, Taiwan;
Department of Mechanical and
Mechatronic Engineering,
National Taiwan Ocean University,
Keelung 20224, Taiwan
e-mail: jtchen@mail.ntou.edu.tw

Jia-Wei Lee, Ying-Te Lee, Wen-Che Lee

Department of Harbor and River Engineering,
National Taiwan Ocean University,
Keelung 20224, Taiwan

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 11, 2010; final manuscript received December 9, 2013; published online February 5, 2014. Assoc. Editor: Christopher D. Rahn.

J. Vib. Acoust 136(2), 021018 (Feb 05, 2014) (8 pages) Paper No: VIB-10-1293; doi: 10.1115/1.4026354 History: Received December 11, 2010; Revised December 09, 2013

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

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Figures

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Fig. 1

An elliptical membrane subject to (a) Dirichlet B.C. and (b) Neumann B.C.

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Fig. 2

The minimum singular value σ1 versus the wave number k using the single-layer potential approach for the elliptical membrane subject to the Dirichlet boundary condition

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Fig. 3

The minimum singular value σ1 versus the wave number k using the double-layer potential approach for the elliptical membrane subject to the Dirichlet boundary condition

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Fig. 4

The minimum singular value σ1 versus the wave number k using the single-layer potential approach for the elliptical membrane subject to the Neumann boundary condition

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Fig. 5

The minimum singular value σ1 versus the wave number k using the double-layer potential approach for the elliptical membrane subject to the Neumann boundary condition

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Fig. 6

Extraction of true eigenvalues by plotting the minimum singular value σ1 versus the wave number k using the SVD updating document for the matrices of [UT] with respect to the clamped elliptical membrane

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Fig. 7

Extraction of true eigenvalues by plotting the minimum singular value σ1 versus the wave number k using the SVD updating document for the matrices of [LM] with respect to the free elliptical membrane

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Fig. 8

Extraction of spurious eigenvalues by plotting the minimum singular value σ1 versus the wave number k using the SVD updating term with respect to the double-layer matrices of [TM]

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Fig. 9

Extraction of spurious eigenvalues by plotting the minimum singular value σ1 versus the wave number k using the SVD updating term with respect to the single-layer matrices of [UL]

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