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

Abstract

An analytical method for free vibration analysis of arbitrarily shaped plates with simply supported edges is presented using the non-dimensional dynamic influence function (NDIF) method, which was introduced by the author. A major difficulty in the theoretical formulation of the proposed method is to analytically measure the values of diagonal elements of the system matrix that gives the eigenvalues and mode shapes of the plate of interest. In addition, particular attentions are given to remove the spurious eigenvalues of the plate. Various case studies show that the proposed method yields very accurate eigenvalues and mode shapes compared with other analytical and numerical methods in spite of its theoretical simplicity.

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

“…It is explained using equation ( 9) why the determinant of the system matrix is calculated as a meaningless value in the low frequency parameter range of L \ 8.3 for N = 32 in Figure 3. Although N eigenvalues are numerically calculated from equation (10) in the range of L \ 8.3, the higher-order eigenvalues l R(L) + 1 , l R(L) + 2 , ., l N become meaningless values because the rank R(L) is less than N. Thanks to this fact, equation (9) calculated by the product of the N eigenvalues including the higher-order eigenvalues also has a meaningless value. Based on this findings, the determinant of the system matrix is calculated by the product of only the valid eigenvalues l 1 , l 2 , ., l R(L) excluding the invalid eigenvalues l R(L)…”

“…In the section, an appropriate way of obtaining mode shapes of the membrane of interest is proposed after a deep study. The j th mode shape for the j th eigenvalue L j extracted by equation (11) for the rectangular membrane (N = 32) can be obtained by utilizing equation (10) as follows. Inserting L = L j into equation (10) yields…”

“…The j th mode shape for the j th eigenvalue L j extracted by equation (11) for the rectangular membrane (N = 32) can be obtained by utilizing equation (10) as follows. Inserting L = L j into equation (10) yields…”

“…[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.…”