Improved non-dimensional dynamic influence function method for vibration analysis of arbitrarily shaped plates with clamped edges

Sang-Wook Kang,Satya N Atluri

doi:10.1177/1687814016638586

Sang-Wook Kang, Satya N Atluri

Open Access

https://doi.org/10.1177/1687814016638586

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Abstract

A new formulation for the non-dimensional dynamic influence function method, which was developed by the authors, is proposed to efficiently extract eigenvalues and mode shapes of clamped plates with arbitrary shapes. Compared with the finite element and boundary element methods, the non-dimensional dynamic influence function method yields highly accurate solutions in eigenvalue analysis problems of plates and membranes including acoustic cavities. However, the non-dimensional dynamic influence function method requires the uneconomic procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues because it produces a non-algebraic eigenvalue problem. This article describes a new approach that reduces the problem of free vibrations of clamped plates to an algebraic eigenvalue problem, the solution of which is straightforward. The validity and efficiency of the proposed method are illustrated through several numerical examples.

Highlights

For vibration problems of plates with arbitrary shapes, numerical approximate methods such as the finite element method (FEM) and boundary element method (BEM) are usually applied.[1,2,3,4,5] these methods cannot be expected to give very accurate results because a large number of numerical calculations are required as the number of nodes increases
Many researchers have studied various meshless methods, which discretize the entire region of a plate unlike the non-dimensional dynamic influence function (NDIF) method
Krowiak[14] applied the radial basis function (RBF)-pseudospectral method to the free vibration analysis of two-dimensional structures, which combines the meshless feature of the RBF method and the simplicity of the pseudospectral method

Summary

Introduction

For vibration problems of plates with arbitrary shapes, numerical approximate methods such as the finite element method (FEM) and boundary element method (BEM) are usually applied.[1,2,3,4,5] these methods cannot be expected to give very accurate results because a large number of numerical calculations are required as the number of nodes increases. Misra[13] proposed the radial basis function (RBF) method using multiple linear regression analysis to obtain eigenvalues of supported and clamped plates, which were compared with the eigenvalues presented by the authors’ previous research.[9] Krowiak[14] applied the RBF-pseudospectral method to the free vibration analysis of two-dimensional structures, which combines the meshless feature of the RBF method and the simplicity of the pseudospectral method. He used eigenvalues[6] presented by the authors to show the accuracy of his method. Note that the calculation method for the diagonal elements of the above four system matrices for ri Àrk =0 (i.e. i=k) was presented in the authors’ previous paper.[9]

A B ð15Þ

Conclusion