Vibration Of Orthotropic Rectangular Plates Research Articles

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

Generalized integral transform solution for free vibration of orthotropic rectangular plates with free edges

Free vibration of orthotropic rectangular thin plates of constant thickness with two opposite edges clamped and one or two edges free is analyzed by generalized integral transform technique. Numerically stable eigenfunctions in exponential function forms of Euler–Bernoulli beams with appropriate boundary conditions are adopted for each direction of the plate. The governing fourth-order partial differential equation for the mode function of free vibration is transformed into a system of linear equations, by integral transform in both directions of the rectangular plate. The boundary conditions at free edges are satisfied exactly by considering the terms generated in the transformed equations by integration by parts, which are absent in the equations by traditional Rayleigh–Ritz method. The natural frequencies of free vibration of orthotropic rectangular thin plates obtained by the proposed integral transform solution are compared with available results in the literature and numerical solutions by finite element analysis, showing excellent agreement and high convergence rate.

Free In-Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

Free Transverse Vibration of Rectangular Orthotropic Plates with Two Opposite Edges Rotationally Restrained and Remaining Others Free

Many types of engineering structures can be effectively modelled as orthotropic plates with opposite free edges such as bridge decks. The other two edges, however, are usually treated as simply supported or fully clamped in current design practice, although the practical boundary conditions are intermediate between these two limiting cases. Frequent applications of orthotropic plates in structures have generated the need for a better understanding of the dynamic behaviour of orthotropic plates with non-classical boundary conditions. In the present study, the transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained with the remaining others free was studied by applying the method of finite integral transforms. A new alternative formulation was developed for vibration analysis, which provides much easier solutions. Exact series solutions were derived, and the excellent accuracy and efficiency of the method are demonstrated through considerable numerical studies and comparisons with existing results. Some new results have been presented. In addition, the effect of different degrees of rotational restraints on the mode shapes was also demonstrated. The present analytical method is straightforward and systematic, and the derived characteristic equation for eigenvalues can be easily adapted for broad applications.

A new method for analytical solution of inplane free vibration of rectangular orthotropic plates based on the analysis of infinite systems

In this paper, a new method for in-plane free vibration analysis of rectangular orthotropic plate is presented. The cases for completely clamped and completely free plates are considered in detail. The boundary value problem is essentially reduced to infinite systems of linear algebraic equations. New mathematical theory and associated results are presented for investigating the asymptotic behavior of non-trivial solution of quasi-regular infinite system. By enhancing the theory, an effective and accurate algorithm for determining the natural frequencies and modes shapes vibrations is developed. The accuracy and computational efficiency of the method are demonstrated by examples.

Vibration Cosine Solutions of Free Orthotropic Rectangular Plates on the Elastic Half-Space

In this paper, the analytic solutions of steady vibration of free orthotropic rectangular plate loaded with vertical steady loading on the elastic half-space was given by combining the general solution of double trigonometrically cosine series with supplementary terms with dynamic integral representations for displacements of the elastic half-space loaded with arbitrary vertical steady loading. This solution not only is four-order derivative, but also has less undetermined coefficients. It can be used to solve the problems of bending and steady vibration of orthotropic rectangular plates on the elastic half-space without be classified and be superimposed. This causes this kind of things, bending and steady vibration of orthotropic rectangular plates with four free edges on the elastic half-space, unionization, simplification and systematization. When the material is isotropic, the solutions are turned into analytic solutions of steady vibration of free rectangular plate on the elastic half-space. At last some computational examples are presented and the results are coincided with those in literatures. Then the method in this paper will be of important practical applications.

Comprehensive exact solutions for free in-plane vibrations of orthotropic rectangular plates

All possible exact solutions are successfully obtained for the first time in terms of 10 sets of distinct eigensolutions for the free in-plane vibration of orthotropic rectangular plates when two opposite plate edges are either type of simple support, the other two edges are any combination of classical edge conditions. The exact solutions are validated through both mathematical proof and comparisons with the solutions of differential quadrature method . Some unusual phenomena are revealed in free in-plane vibrations of rectangular plates due to one of the eigenvalues being zero. This work constitutes a natural extension of very recent corresponding work for isotropic rectangular plates by the same authors. Moreover, this work substantially simplified both solution forms and solving procedure of the early work. It is expected that results tabulated here can serve as the benchmarks for the validation of the numerical methods.

Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells

The Haar wavelet discretization technique for solving the elastic bending problems of orthotropic plates and shells is proposed. Free transverse vibrations of orthotropic rectangular plates with a variable thickness in one direction are considered as a model problem. In the case of constant plate thickness, the numerical results are validated by comparing them with an exact solution. The results obtained are found to be in good agreement with those available in the literature.

Transverse vibration of orthotropic rectangular plates under moving bodies

The use of orthotropic plates is common in all the fields of structural engineering: civil, traffic, aerospace and naval. In this paper the transverse vibration of orthotropic rectangular plates under moving bodies is investigated. The method of substructures is used to derive transverse vibration equations of an orthotropic rectangular plate under the action of moving bodies. For the calculation of dynamic response of orthotropic rectangular plate we use Ritz method and numerical integration method.

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

Spline-based investigation of natural vibrations of orthotropic rectangular plates of variable thickness within classical and refined theories

Spline-based investigation of natural vibrations of orthotropic rectangular plates of variable thickness within classical and refined theories

Calculation of eigenfrequencies for rectangular free orthotropic plates – An overview

AbstractIn this paper, the free transverse vibrations of rectangular orthotropic plates with completely free boundary conditions are considered. Some methodologies for calculating the eigenfrequencies and eigenmodes, based on the classical thin plate theory are presented in a comparative overview. The advantages and disadvantages of these methodologies are discussed for an orthotropic plate application with specific material and geometric properties.

Vibration behavior of thermally stressed composite skew plate

Vibration and stability characteristics of composite laminates under thermo-mechanical loads have attracted the attention of many researchers in recent years as they are being increasingly used in thin walled structural components of aerospace, defense, reactor vessels and other high performance application areas. These studies were reviewed and well documented by Tauchert [1] and Noor and Burton [2]. It is observed from the existing literature that, the static buckling and postbuckling behavior of thermo-mechanically loaded rectangular and circular plates have been studied by many researchers. However, limited work has been done on the dynamic characteristics of thermally stressed/buckled laminated plates. Moreover, similar studies on thermo-mechanical vibration and stability characteristics of laminated composite plates other than rectangular/circular geometry have been sparsely treated in the published literature. Thermal load not only influences the stability characteristics of structures, but also changes its vibration frequencies. The notable recent contributions pertaining to small amplitude vibration behavior of thermally stressed/buckled rectangular composite plates may be found in Refs. [3–8]. Chang and Jen [3] studied influences of temperature change and large amplitude on the period of vibration of orthotropic rectangular plates. Noor and Burton [4] investigated the free vibration behavior of thermally stressed angle-ply composite plates using a three-dimensional thermo-elastic model. Librescu and Lin [5] employed single-term Navier-type double sine function to study the vibration behavior of thermo-mechanically loaded flat and curved composite panels taking into account interlaminar shear traction continuity requirement and considering a higher order shell theory. For accurate solution, the assumed mode shape function in the analytical approach should have more terms, thus leading to more numerical work. Numerical methods, like the finite element method is preferable as there is no need for an a priori assumption of the mode shapes as the solution itself predicts the mode shapes. Lee and Lee [6] used the finite element method to investigate the vibration characteristics of thermally stressed composite plates in the preand postbuckling states. Similar studies on the vibration frequencies of thermally buckled piezolaminated composite plates, and composite plates embedded with shape memory alloy fibers were reported by Oh et al. [7] and Park et al. [8]. Moreover, the authors [6–8] employed

The effect of support locations on the steady-state response of viscoelastically point-supported rectangular orthotropic plates

The vibration of orthotropic rectangular plates having viscoelastic point supports symmetrically located on its diagonals is analyzed. The plate is under the effect of a sinusoidally varying force at its center. The Lagrange equations are used in the solution process. To apply the Lagrange equations, the trial function denoting the deflection of the plate is expressed in polynomial form. By using the Lagrange equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, the damping of the supports and the locations of point supports on the steady-state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various values of damping and location of the supports for a certain stiffness value of the supports. The results are given for the considered frequency range of the external force. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.

Determination of the steady state response of viscoelastically corner point-supported rectangular orthotropic plates

Vibration of orthotropic rectangular plates having viscoelastic point supports at the corners is analyzed. The Lagrange’s equations is used to examine the free vibration characteristics and steady state response to a sinusoidally varying force affecting at the center of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. In the study, for applying the Lagrange’s equations, the trial function denoting the deflection of the plate is expressed in the polynomial form: This procedure may be considered as the classical Rayleigh–Ritz method. By using the Lagrange’s equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports to the mode shapes and to the steady state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various damping and stiffness of the supports. The results are given for the frequencies and mode shapes of the first three symmetrical modes. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.

Sound radiation from forced vibration of rectangular orthotropic plates under moving loads

This paper describes an investigation on the sound radiation from forced vibration of rectangular orthotropic plates under moving loads. The vibrations of a rectangular orthotropic thin plate with general boundary conditions traversed by moving loads are first solved. Based on the Rayleigh integral and the dynamic response of the plate, the acoustic pressure distributions around the plate are obtained in the time domain. The effects of light and heavy moving loads are separately studied. Simplified methods for calculation of the acoustic pressure distributions in far field are developed. The effects of moving mass, damping coefficient, boundary conditions and moving speed on the dynamic responses of the plate and the acoustic pressure distributions are investigated.

Determination of the steady state response of viscoelastically point-supported rectangular specially orthotropic plates with added concentrated masses

Vibration of orthotropic rectangular plates having viscoelastic point supports at the corners with symmetrically added four concentrated masses rigidly mounted on the two diagonals of the plate is analyzed. The Lagrange equations are used to examine the steady state response to a sinusoidally varying force applied at the centre of a viscoelastically point-supported orthotropic elastic plate of rectangular shape with the considered locations of the added masses. In the study, for applying the Lagrange equations, the trial function denoting the deflection of the plate is expressed in polynomial form. By using the Lagrange equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the locations of the added masses, mechanical properties characterizing the orthotropy of the plate material and the damping of the supports to the steady state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at the centre. Because of the symmetry of the supports and loading condition, only the first three symmetrical modes occur in the considered frequency range. Therefore, only the results of the first three symmetrical modes are given in the present study. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.

Determination of the steady state response of viscoelastically corner point-supported rectangular specially orthotropic plates under the effect of sinusoidally varying moment

Vibration of orthotropic rectangular plates having viscoelastic point supports at the corners under the effect of sinusoidally varying concentrated moment is analyzed. The Lagrange equation is used to examine the free vibration characteristics and the steady state response to a sinusoidally varying concentrated moment acting at the centre of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. In the study, for applying the Lagrange equation, the trial function denoting the deflection of the plate is expressed in the polynomial form. By using the Lagrange equation, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports on the mode shapes and the steady state response of the viscoelastically point-supported rectangular plates is investigated numerically, for a concentrated moment at the centre for various values of the mechanical properties which characterize the anisotropy of the plate material and for various damping ratios. The results of the natural frequencies are given for the first three antisymmetrical–symmetrical modes, and the steady state responses to a sinusoidally varying concentrated moment are determined for the frequency ranges of the first two antisymmetrical–symmetrical mode types. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions for free vibration analysis of point-supported or completely free rectangular plates for the first three antisymmetrical–symmetrical vibration modes based on the Kirchhoff–Love plate theory.

A SEMI-ANALYTICAL SOLUTION OF THE FLEXURAL VIBRATION OF ORTHOTROPIC PLATES OF VARIABLE THICKNESS

A semi-analytical method based on the finite strip technique in conjunction with the transition matrix is applied to investigate the flexural vibration of orthotropic rectangular plates of variable thickness in one direction. The effect of orthotropy, aspect ratio, and taper ratio on the natural frequencies of the vibration of such plates is investigated. The efficiency of the proposed technique is illustrated by comparing the results with those available in the literature for several examples. The results show excellent agreement and stable convergence for all plate configurations considered in this paper.

Features of the Free Planar Vibrations of Orthotropic Rectangular Plates

Study is made of the spectrum of the symmetric planar vibrations of an orthotropic rectangular plate as a function of the characteristics of orthotropic materials and its dimensions. The results are compared with the solution for isotropic plates