An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges

In this investigation, the free in-plane vibration analysis of orthotropic rectangular plates with non-uniform boundary conditions and internal line supports is performed with a modified Fourier solution. The exact solution for the problem is obtained using improved Fourier series method, in which both two in-plane displacements of the orthotropic rectangular plates are represented by a double Fourier cosine series and four supplementary functions, in the form of the product of a polynomial function and a single cosine series expansion, introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined over the entire x–y plane. The unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the two sets of the boundary springs at the all boundary of the orthotropic rectangular plates without the need of making any change to the solutions. The excellent accuracy of the current result is validated by comparison with those obtained from other analytical approach as well as the finite element method (FEM). Numerical results are presented to illustrate the current method that is applied not only to the homogeneous boundary conditions but also to other interesting and practically important boundary restraints on free in-plane vibrations of the orthotropic rectangular plates, including varying stiffness of boundary springs, point supported, partially supported boundary conditions and internal line supports. In addition to this, the effects of locations of line supports are also investigated and reported. New results for free vibration of moderately orthotropic rectangular plates with various edge conditions and internal line supports are presented, which may be used for benchmarking of researchers in the field.

A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports

Frequency equation for the in-plane vibration of a clamped circular plate

Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev–Lagrangian method

BasedontheKantorovich ‐Krylovvariationalmethod (Kantorovich,L.V.,andKrylov,V.I., ApproximateMethods of Higher Analysis , Noordhoff International, Groningen, The Netherlands, 1964, pp. 241 ‐357), we analytically solve for modal frequency and displacement mode shapes of in-plane vibration of rectangular plates with free and clamped boundary conditions. Free and clamped boundary conditions do not allow closed-form solutions from partial differential equations. We develop analytical expressions of plate mode shapes consisting of a linear combination of progressive waves with unknown wave amplitudes. We then use these to calculate the modal frequencies and the wave amplitudes with analytical mode shape expressions. An iteration scheme is presented to calculate efe ciently the natural frequency and corresponding mode shape functions based on the Kantorovich ‐ Krylov method. Three cone gurations are considered: a plate with four edges clamped, a plate with three edges clamped and one edge free, and a plate with two parallel edges clamped and the other two edges free. The e rst six natural frequencies predicted by our approach are validated using NASTRAN and other analyses from the literature. Improvements in accuracy of the e rst six predicted natural frequencies were achieved using the Kantorovich ‐Krylov method when compared against other results in the literature. a = length of plate in x direction b = length of plate in y direction E = Young’ s modulus of plate material h = plate thickness k1;2 = two wave numbers in the solution of mode shape functions m;n = number of half-wavelengths for mode shapes

Free In-Plane Vibration of Rectangular Plates

The equations of in-plane vibration of thin plates are solved for rectangular panels with two parallel edges clamped, while the other two edges are either both clamped, both free, or one clamped and one free. The propagation characteristics of in-plane waves are investigated and the frequency bands of attenuation and propagation of the waves are identified. Simple expressions for the cutoff frequencies of different wave components are derived. The nature of the coupling between in-plane longitudinal and in-plane shear waves is mathematically and physically illustrated. Although all the modes in a plate panel are coupled through Poisson’s effect, it is shown that the coupling is very weak in plate panels with all edges clamped as compared to the cases where one or two parallel edges are free. As a first order approximation, the resonant modes of in-plane waves are classified into uncoupled and coupled mode pairs. By using the coupled mode pairs approximation, it is possible to predict the dynamic response of the plate panel with a reasonable accuracy. Simple expressions are given for approximate estimation of the resonance frequencies of coupled and uncoupled modes. Mode shapes are given, for each case of edge conditions, which satisfy both the displacement and force conditions at the plate edges. A simple procedure is given for the determination of resonance frequencies and mode shapes without excessive computations. The predicted resonance frequencies and mode shapes are compared to the finite element results and good agreement is found. The mode shapes of the in-plane vibration are depicted for the first eight resonant modes for each case of edge conditions.

Jie Pan

Papers

Modal characteristics of in-plane vibration of rectangular plates