Improved non-dimensional dynamic influence function method based on tow-domain method for vibration analysis of membranes

Abstract

This article introduces an improved non-dimensional dynamic influence function method using a sub-domain method for efficiently extracting the eigenvalues and mode shapes of concave membranes with arbitrary shapes. The nondimensional dynamic influence function method (non-dimensional dynamic influence function method), which was developed by the authors in 1999, gives highly accurate eigenvalues for membranes, plates, and acoustic cavities, compared with the finite element method. However, it needs the inefficient procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues and mode shapes. To overcome the inefficient procedure, this article proposes a practical approach to make the system matrix equation of the concave membrane of interest into a form of algebraic eigenvalue problem. It is shown by several case studies that the proposed method has a good convergence characteristics and yields very accurate eigenvalues, compared with an exact method and finite element method (ANSYS).

Advanced non-dimensional dynamic influence function method considering the singularity of the system matrix for accurate eigenvalue analysis of membranes

“…which is the Bessel function of the first kind of order zero. 1,8 In equation (1), r À r k j j means the distance between P and P k of which the position vectors are r and r k , respectively. Also, L = v= ffiffiffiffiffiffiffiffi ffi T =r p is a frequency parameter where v, T , and r are the angular frequency, the uniform tension per unit length, and the mass per unit area, respectively.…”

“…[3][4][5] Until recently, in-depth studies have been conducted by the author to overcome the frequencydependent problem of the system matrix in the NDIF method. [6][7][8][9][10][11] Although a vast literature exists on analytical and semi-analytical methods for obtaining accurate eigenvalues of membranes having no exact solution and the author has scrutinized the vast literature, only relatively recent studies are introduced in the paper. Gol'dshtein and Ukhov 12 obtained estimates for the first non-trivial eigenvalues of membranes in conformal regular domains.…”

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

“…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.…”

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

“…To overcome this limitation, we have recently developed an improved NDIF method for membranes 17,18 and acoustic cavities 19 with arbitrary shapes. In this article, the basic concept of the improved NDIF method [17][18][19] is extended to arbitrarily shaped plates with clamped edges. The proposed method is validated by comparing its results with those of other methods such as the NDIF method, FEM (ANSYS), and the exact method.…”

“…As a result, the NDIF method needs the inefficient procedure of searching frequency values that make the system matrix singular in the frequency range of interest. To overcome this limitation, we have recently developed an improved NDIF method for membranes 17,18 and acoustic cavities 19 with arbitrary shapes. In this article, the basic concept of the improved NDIF method [17][18][19] is extended to arbitrarily shaped plates with clamped edges.…”