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

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. KeywordsNon-dimensional dynamic influence function method, free vibration, arbitrarily shaped plate, algebraic eigenvalue problem, natural frequency, mode shape Date

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

“…As in the authors' previous papers, [12][13][14][15] equation (45) may yield not only the eigenvalues of the plate of interest but also the eigenvalues of a similarly shaped membrane, which are called the spurious eigenvalues. To confirm these spurious eigenvalues, the proposed method is applied to a simply supported circular plate of unit radius and Poisson's ratio of 0.3.…”

“…r, r i , and r k represent position vectors for boundary points P, P i , and P k , respectively, and n i denotes the direction of the outward normal of the boundary at point P i . As the same manner as in the authors' previous paper 12,13 that dealt with arbitrarily shaped clamped plates, a general solution of the plate is assumed by linearly superposing the NDIFs as follows W (r) =…”

“…Note that the NDIF is a wave-type solution that exactly satisfies the given governing differential equation and physically describes the displacement response of a point in an infinite plate due to a unit displacement excited at another point. 6,12,13 In order to consider the simply supported boundary conditions (equations ( 4) and ( 5)) at N boundary points P 1 , P 2 , . .…”

“…For more detailed explanation on the function, refer to the author's previous papers. 6 Later, the author improved the NDIF method to effectively extract the eigenvalues of membranes 8,9 In addition, the author extended the NDIF method to arbitrarily shaped acoustic cavities 10,11 and plates with the clamped boundary condition, 12,13 a mixed boundary condition, 14 and the free boundary condition. 15 As known in the author's previous researches, [12][13][14][15] the author developed the NDIF method for arbitrarily shaped plates with various boundary conditions except the simply supported boundary condition.…”

Modification of boundary condition for the optimization of natural frequencies of plate structures with fluid loading

“…The study of plate-like structures can serve as a prerequisite for the analysis of the dynamic performance of more complicated structures. 1,2 A variety of approaches have accordingly been developed for the optimization of the natural frequencies of plate structures. 3 These methods include modification of the dimension of the plate, [4][5][6] addition of masses to the plate, 7 modification of the topology of the structure, 8,9 and so forth.…”