Vibration Problem Of Rectangular Plates Research Articles

Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Abstract Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.

A simple finite element procedure for free vibration of rectangular thin and thick plates

A simple finite element procedure is introduced and developed to solve the free vibration problem of rectangular plates. The proposed method first reduces the original plate problem to two simple beam problems. Each beam problem is then discretized separately with the help of one-dimensional finite element method (FEM). Since only the one-dimensional FEMs are dealt with in the proposed approach, the implementation of the proposed approach is much easier than the case where the two-dimensional FEM is applied to the problem. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the vibration problems of thin and thick rectangular plates. The simulations have been done for plates with different boundary conditions including the simply supported, clamped, and free boundary conditions. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

Hencky Bar-Net Model for Vibration of Rectangular Plates with Mixed Boundary Conditions and Point Supports

This paper is concerned with the development of the Hencky bar-net model (HBM) for free vibration analyses of rectangular plates with mixed boundary conditions and point supports. The HBM is a two-dimensional discrete net system composed of rigid segments connected by frictionless hinges and rotational springs. In the model, bending is accommodated by rotational springs at each joint while the twisting by a diagonal spring system in each grid cell. The total mass of the plate is distributed as lumped mass at each joint and the continuous boundary stiffness of plate is simulated by springs located at the edge joints. Owing to the discrete property of HBM, it is able to readily handle any boundary conditions of plates including mixed boundary conditions and point supports. The HBM is herein used to solve some vibration problems of rectangular plates with mixed boundary conditions and point supports to demonstrate its accuracy and convenience for plate analyses.

Exact solutions for the free in-plane vibrations of rectangular plates with arbitrary boundary conditions

Though the free in-plane vibration problem of rectangular plates with classical boundary conditions have been studied using various numerical methods like direct separation of variables, the elastic boundary restraint is more general in the practical structural applications compared with the perfect classical boundary condition. Thus, in this paper, an innovative exact solution on the basis of the method of reverberation ray matrix (MRRM) and golden section search (GSS) algorithm is presented to study the exact free in-plane vibration characteristics of rectangular plates with arbitrary boundary conditions. In the present method, the exact natural frequencies and mode shapes can be very easily obtained when two opposite plate edges have either type of the simply supported boundary conditions and the resting plate edges have the arbitrary boundary conditions including the classical case and general elastic restraints. By the comparison with other published exact solutions, it is shown that the present exact solutions have some obvious advantages which are performed in the excellent accuracy, reliability and efficiency. Performing the study, some of the first-known exact results for the free in-plane vibrations of rectangular plates with elastic boundary restraints are also obtained, which can be worked as the benchmark data for the future approximate methods.

Free vibration analysis of stepped rectangular plates resting on non-homogeneous elastic foundations

Abstract A Half Boundary Method (HBM) is proposed for analyzing the free vibration problem of rectangular plates with stepped thickness resting on non-homogeneous elastic foundations. The unknown quantities of the method exist only on half of the boundary. The non-homogeneous elastic foundation discussed here consists of two-segment elastic Winkler foundation. The fundamental differential equations are established for the bending problem of the plate on elastic foundations. The Green function, which is obtained by transforming these differential equations into integral equations and using numerical integration, is used to establish the characteristic equation of the free vibration. The effects of the modulus of the foundation, the stepped thickness and aspect ratio on the frequency parameters are considered. By comparing the present numerical results with those previously published, the efficiency and accuracy of the present method are investigated.

Vibration of an initially stressed rectangular plate due to an accelerated traveling mass

This paper presents a combined application of the Ritz method, the Differential Quadrature (DQ) method, and the Integral Quadrature (IQ) method to vibration problem of rectangular plates subjected to accelerated traveling masses. In this study, the Ritz method with beam eigenfunctions is first used to discretize the spatial partial derivatives with respect to a co-ordinate direction of the plate. The DQ and IQ methods are then employed to analogize the resultant system of partial differential equations. The resulting system of ordinary differential equations is then solved by using the Newmark time integration scheme. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those of the existing literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQM sampling points. Finally, the effects of following parameters having something to do with the title problem are investigated: moving load speed and acceleration, and transverse inertia of the moving load. Numerical results show that all the above-mentioned parameters have significant effects on the transient response of such structures under traveling dynamic loads.

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.

Thermo-microstretch elastic bodies and plane waves

In the present work, vibration problems of rectangular plates are considered for the determination of upper bounds to the unknown microstretch material properties. The frequencies are obtained by extending the Ritz method to this case. The analysis shows that some additional frequencies characterizing the microstretch effects appear among the classical frequencies. Furthermore, by the increasing values of the microstretch constants, the additional frequencies disappear and only the classical frequencies remain in the spectrum. Considering this phenomenon, an optimization problem is established for the identification of the upper bounds of microstretch elastic constants. In the second part of the work, thermal effects are considered and several theories are discussed. Finally, propagation of the plane waves is investigated.

DSC ANALYSIS FOR BUCKLING AND VIBRATION OF RECTANGULAR PLATES WITH ELASTICALLY RESTRAINED EDGES AND LINEARLY VARYING IN-PLANE LOADING

This paper presents the discrete singular convolution (DSC) method for solving buckling and vibration problems of rectangular plates with all edges transversely supported and restrained by uniform elastic rotational springs. The opposite plate edges are subjected to a linearly varying uni-axial in-plane loading. The rationale for using DSC method stems from its numerical stability and flexible implementation for structural analysis. To verify the present approach, convergence and comparison studies for rectangular plates with different combinations of elastically restrained and classical edges are carried out. Accurate buckling and vibration solutions of plates having two opposite edges elastically restrained and the other two sides clamped, or all edges elastically restrained are presented.

Natural vibration study on rectangular plates with a line hinge and various boundary conditions

A discrete method is proposed for analyzing the natural vibration problem of rectangular plates with a line hinge. The fundamental differential equations and the solutions of these equations are derived for two parts of the plate, which are obtained by dividing the plate along the line hinge. By transforming these equations into integral equations, and using numerical integration and the continuous conditions along the line hinge, the solutions of the whole plate can be expressed by the unknown quantities on the boundary and the quantities of the rotation along the hinge. The Green function of the deflection problem is used to obtain the characteristic equation of the free vibration. The effects of the position of the line hinge, the aspect ratio, the thickness ratio and the boundary condition on the natural frequency parameters are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.

On the identification of microstretch elastic moduli of materials by using vibration data of plates

In the present work, the vibration problems of rectangular plates modeled by Eringen’s microstretch theory are investigated for the identification of the upper bounds of the microstretch moduli of the plate material. The calculated frequencies of the plates are obtained by extending the Ritz method to the microstretch plates. The three dimensional (3D) vibration analysis of the plates shows that some additional frequencies occur among the classical frequencies as characterizing the microstretch effects. Then it is also observed that these additional frequencies disappear and only the classical frequencies remain with the increasing values of microstretch constants. The inverse problem is established for the identification of the upper bounds of the microstretch elastic constants as an optimization problem where an error function is minimized.

Free vibration analysis of rectangular plates with variable thickness and point supports

A discrete method is developed for analyzing the free vibration problem of rectangular plates with point supports. The fundamental differential equations involving Dirac's delta function are established for the bending problem of the plate with point supports. By transforming these differential equations into integral equations and using numerical integration, the solution of these equations is obtained and used as Green function to obtain the characteristic equation of the free vibration. The effects of the point support, the boundary condition, the variable thickness and aspect ratio on the frequencies are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.

IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER-ORDER PDEs: APPLICATION TO FREE VIBRATION OF PLATES

In the context of the free vibration problem of rectangular plates, this paper develops a detailed methodology for implementing multiple boundary conditions in the differential quadrature solutions of higher-order differential equations. It is explained that a certain type of boundary conditions can be built into the differential quadrature weighting coefficients themselves. This simplifies the programming of the differential quadrature solution algorithms. More importantly, however, as shown by the results of fundamental frequencies of a wide spectrum of rectangular plate configurations, the methodology results in strikingly accurate differential quadrature solutions.

IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER‐ORDER PDEs: APPLICATION TO FREE VIBRATION OF PLATES

In the context of the free vibration problem of rectangular plates, this paper develops a detailed methodology for implementing multiple boundary conditions in the differential quadrature solutions of higher-order differential equations. It is explained that a certain type of boundary conditions can be built into the differential quadrature weighting coefficients themselves. This simplifies the programming of the differential quadrature solution algorithms. More importantly, however, as shown by the results of fundamental frequencies of a wide spectrum of rectangular plate configurations, the methodology results in strikingly accurate differential quadrature solutions.

Free vibration of rectangular plates with an arbitrary straight line support

The free vibration problem of rectangular plates with an arbitrary straight line support is treated by using the Rayleigh-Ritz method with products of polynomials as the admissible functions. The numerical results are presented for various diagonally supported plates, cantilever plates with an oblique support and diagonally supported orthotropic cantilever plates. For isotropic square plates, nodal patterns are also presented.

Analytic solution to free vibration of rectangular plates

The Analytical Strip Method (ASM) is presented for the free flexural vibration problem of rectangular plates with intermediate flexible supports and different edge conditions. Orthotropic plates and plates having single or multiple step variation in thickness are considered. The plate is divided into strips, the number of which depends on the geometric discontinuities. The solution employs the classical method of separation of variables, and the deflection for each plate strip is expressed in a single trigonometric series. Results are presented and, when possible, compared with previously published ones.

A general analytical solution for elastic vibration of rectangular thin plates

A general solution of differential equation for lateral displacement lunction of rectangular elastic thin plates in free vibration is established in this paper. It can be used to solve the vibration problem of rectangular plate with arbitrary boundaries. As an example, the frequency and its vibration mode of a rectangular plate with four edges free have been solved.

Vibrations of Rectangular Plates With Nonuniform Elastic Edge Supports

Two methods are introduced for the solution of free vibration problems of rectangular plates having nonuniform, elastic edge constraints, a class of problems having no previous solutions in the literature. One method uses exact solutions to the governing differential equation of motion, and the other is an extension of the Ritz method. Numerical results are presented for problems having parabolically varying rotational constraints.