Differential quadrature analysis of free vibration of rhombic plates with free edges

Abstract

An accurate free vibration analysis of rhombic plates with free edges is presented by using the modified differential quadrature method (DQM). Fourteen combinations of boundary conditions with free edges and five skew angles for each plate are analyzed. It is found for the first time that the problem is extremely sensitive to grid spacing and the method to applying the multiple boundary conditions when the skew angle is very large. Various existing non-uniform grid spacing and methods for applying the multiple boundary conditions are tested. It is surprising to see that DQM with only one grid spacing can yield accurate and reliable solutions, besides applying boundary conditions correctly at free corners is also important for success. For verifications, the DQ results are compared with either existing accurate solutions or data obtained by finite element method. Due to bending stress singularities occurred at corners having obtuse angles, the rate of convergence is low for a few cases and quite different from rhombic plates without free edges. Some new results are presented and waiting for comparison with future data obtained by other investigators.

Introduction

The behavior of skew or rhombic plates, one of the common structural elements in many kinds of high-performance surface and air vehicles, is important to structural engineers. Due to the complicated mathematical structure of the governing equations, it is not an easy task to obtain an accurate fundamental frequency by either approximate or numerical methods for plates having a large skew angle [1], [2], [3], [4], not to mention to obtain closed-form solutions. Therefore, the free vibration problems of thin and thick skew or rhombic plates have caused a lot of attention and various approximate and numerical methods, such as the Rayleigh–Ritz method [1], [2], [3], [4], finite difference and differential transformation method [5], discrete singular convolution (DSC) [6], the differential quadrature method (DQM) [7], [8], [9], and the moving least square Ritz (MLS-Ritz) method [10], have been used for obtaining frequencies. Due to the stress singularity at the obtuse angles of the rhombic plates, many approximate and numerical methods have encountered serious convergence problems when the skew angle is large. A vast body of literature exists and is well documented in a paper written recently by Zhou and Zheng [10].

The free vibration problem of skew plates has been successfully analyzed by MLS-Ritz method [10]. Different number of MLS-Ritz points has been tried and most results are very close to the accurate upper bound solutions reported in [1], [2], [3], [4]. It is noted that, however, for a few cases, namely, the CSCS, CSSS, and SSFF rhombic plates with large skew angle ($75°$), some mode frequencies are quite different to the data reported in [1], [2], [3], [4]. Here symbols S, C, and F stand for simply supported, clamped, and free boundary conditions. Since the convergence of the MLS-Ritz is not monotonic with the increase of the number of points, the question may arise which frequencies are more accurate. Besides, accurate frequencies of rhombic plates with many other combinations of boundary conditions have not been reported thus far.

The differential quadrature method (DQM) is a relatively new and efficient method for solving static and free vibration problems of isotropic and anisotropic rectangular plates [11], [12], [13], [14]. Very accurate results can be obtained by the DQM with relatively much less computational effort and the DQM is a very promising approach for the vibration analysis of plates [15]. Previous limited researches show that accurate results can be obtained by the DSC [6] and DQM [7], [8], [9], [16] for skew plates without free edges, even when the skew angle is very large ($75°$) [17]. Although the DSC and DQM are capable to solve free vibration problems of straight-sided or curved quadrilateral thin and thick plates [6], [7], [8], [9], [16], the information on their applications to thin isotropic skew plates with free edges is limited.

The free vibration of isotropic thin rhombic plates shows mathematically analogy to the free vibration of anisotropic thin rectangular plates. In the DQ analysis, non-uniform grid spacing should be used since uniform grid spacing cannot yield reliable solutions for anisotropic rectangular plates [18]. Besides, appropriately applying the multiple boundary conditions is also important for success in using the DQM. If the boundary conditions at free corners are not appropriately applied, the rate of convergence of the DQM may become quite low even for isotropic thin rectangular plates [19]. Various non-uniform grid points have been summarized in [20] and available methods for applying the multiple boundary conditions can be found in [21].

Since appropriately applying the multiple boundary conditions is important, the modified DQM [17], [21], [22] is to be employed in the present investigations. Although the formulation of the weighting coefficient of the first order derivative for the conventional DQM and the modified DQM is exactly the same, however, the formulations of the weighting coefficients of higher order derivatives are quite different. In the modified DQM, the weighting coefficients of the second order derivative at two end points are slightly modified thus two additional degrees of freedom, the first order derivatives at two end points, are introduced. The weighting coefficients of the third and fourth order derivatives are formulated based on the modified weighting coefficient of the second order derivative. Since the number of degrees of freedom at boundary points in the modified DQM is exactly the same as the number of boundary conditions, the difficulty of implementation of boundary conditions for plate problems in conventional DQM has been completely removed. Although the form of weighting coefficients is symbolically the same as the differential quadrature element method (DQEM) [23], [24], the modified DQM uses Lagrange interpolation but the DQEM uses the Hermit interpolation to determine the weighting coefficients. Besides, the modified DQM has only three degrees of freedom (w, $\partial w/\partial \xi$, $\partial w/\partial \eta$) at the plate corners, but the DQEM has four degrees of freedom (w, $\partial w/\partial \xi$, $\partial w/\partial \eta$, ${\partial }^{2}w/\partial \xi \partial \eta$) at the plate corners. The $\xi \text{,}\eta$ are the oblique coordinate system for a skew plate. The introduce of the degree of freedom ${\partial }^{2}w/\partial \xi \partial \eta$ causes the difficulty in applying the boundary conditions at the corner point, since only three boundary conditions are available.

Our previous numerical experience shows that even with the modified DQM, the solution is also very sensitive to grid spacing for the case of skew plate with free edges but without free corners [17]. It is even astonished to find that the DQM with the most widely used non-uniform grid spacing in open literature, the CGL (Chebyshev–Gauss–Lobatto) grid distribution, cannot yield reliable solutions for rhombic plate with free edges. Except one non-uniform grid distribution listed in [20], i.e., (N − 2) Gaussian quadrature points plus $\pm 1$, the DQM with all other non-uniform grid points cannot give reliable solutions. Here N is the number of grid points. Therefore, the suitability of the modified DQM to analyze isotropic thin skew plates with free edges and free corners deserves further investigated.

The objective of the present investigation is to analyze the free vibration of thin rhombic plates with free edges by the modified DQM or by the combination of the modified DQM with DQEM. Various non-uniform grid points are tried. Fourteen combinations of simply supported, clamped, and free boundary conditions and five skew angles for each plate are analyzed. Formulations and solution procedures are given. For verification and investigation of the solution accuracy, the DQ results are compared to the accurate upper bound solutions [1], [2], [3], [4], data obtained by the MLS-Ritz method [10], or results obtained by NASTRAN with very fine meshes. New results are provided and waiting for comparison with future data obtained by other investigators.