Vibration Problem Of Thin Plates Research Articles

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

Solving biharmonic problems on irregular domains by stably evaluated Gaussian kernel

Abstract The paper extends recently developed idea of stable evaluation of the Gaussian kernel. Owing to this, the Gaussian radial basis function that is sensitive to the shape parameter can be stably evaluated and applied to interpolation problems as well as to solve differential equations, giving highly accurate results. But it can be done only with grids being the Cartesian product of sets of points, what limits the use of this idea to rectangular domains. In the present paper, by the association of an appropriate transformation with the mentioned method, the latter is applied to solve biharmonic problems on quadrilateral irregular domains. As an example, in the present work this approach is applied to solve bending as well as free vibration problem of thin plates. In the paper some strategies for the implementation of the boundary conditions are also presented and examined due to the accuracy. The numerical tests show high accuracy and usefulness of the method.

Conformable solution of fractional vibration problem of plate subjected to in-plane loads

This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.

A virtual element method for the vibration problem of Kirchhoff plates

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose anH2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

Vibration analysis of stepped rectangular plates by the discrete singular convolution algorithm

A novel scheme in using discrete singular convolution (DSC) algorithm is proposed for analyzing the challenging vibration problem of thin plates with geometric discontinuity . An n th-order interpolation polynomial is employed on each side of a step. Then two jump conditions together with two continuous conditions at the step are established to correlate the two polynomials. Detailed formulations and solution procedures are given. Several numerical examples are analyzed to demonstrate the applicability of the proposed method to the free vibration analysis of the stepped rectangular plates . Results obtained by the modified DSC are compared with existing solutions to verify the proposed method. For lower order mode frequencies, DSC results agree well with existing analytical and numerical solutions or data obtained by the finite element method (FEM). Besides, the DSC can also yield relatively accurate frequencies for higher order modes. This research extends the application range of the DSC method. • A modified DSC is presented for analyzing vibration of stepped rectangular plates. • Jumped conditions at the step are enforced using the interpolation method. • The ability of the modified DSC to obtain accurate higher order frequencies is demonstrated.

Wavelet Finite Element Method Analysis of Bending Plate Based on Hermite Interpolation

In this paper, a new kind of finite element method (FEM) is proposed, which use the two-dimensional Hermite interpolation scaling function constructed by tensor product as the basis interpolation function of field function, and then combine with the energy functional with related mechanics and variational principle, the wavelet finite element equations for solving elastic thin plate unit that constructed in this paper are derived. Then the bending problem of thin plate is solved very quickly and availably through the matlab program. The numerical example in this paper indicates the correctness and validity of this method, and has high calculation precision and convergence speed. Moreover, it also provides a reliable method to solve the free vibration problem of thin plate and the pipe crack problems.

Analogy of TE waveguide and vibrating plate with sliding edge condition and exact solutions

Presented herein is a proof that the TE (transverse electric) waveguide problem is analogous to the vibration problem of thin plates with zero slope and zero twisting moment at the freely sliding edges in the vertical (transverse) direction. This analogy is confirmed because of the mathematical similarity of the governing equations and the boundary conditions of the two problems. We also present exact solutions for these two analogous problems for various domain shapes. The exact solutions are very useful for checking the validity, convergence and accuracy of numerical methods for solving the Helmholtz equation.

A piecewise linear finite element method for the buckling and the vibration problems of thin plates

The aim of this paper is to analyze a piecewise linear finite element method to approximate the buckling and the vibration problems of a thin plate. The method is based on a conforming discretization of a bending moment formulation for the Kirchhoff-Love model. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. The method is proved to converge with optimal order for both spectral problems, including an improved order for the eigenvalues. Numerical experiments are reported to assess its performance and to compare it with other low-order finite element methods.

Solving vibration problem of thin plates using integral equation method

This paper deals with reducing differential equations of vibration problem of plates with concentrated masses, elastic supports and elastically mounted masses into eigenvalue problem of integral equations. By applying the general function theory and the integral equation theory, the frequency equation is derived in terms of standard eigenvalue problem of a matrix with infinite order. So that, the natural frequencies and mode shapes can be determined. Convergence of this method has also been discussed at the end of this paper.

Vibration of plates with straight-line, clamped and free edges

This paper deals with vibration problems of thin plates having straight-line, mutually perpendicular, clamped and free edges and subjected to a load consisting of a set of transverse, arbitrarily located random forces. It is assumed that the number of edges of a plate forming recurring figures is optional but each of these edges is either clamped or free along its entire length. The procedure for solving the boundary problem based upon the R-functions method and for estimation of transverse displacements based upon the correlation analysis is presented. Numerical calculations are carried out for two example plates.