The fundamental frequency of vibration Maximum deflection for the different values vibration is computed for a simply supported-free-simply supported-free plate non-homogeneity constant and aspect ratios, for various values of taper constant. Graphical presentation of present work is shown in this paper.

Two of these take modal solutions of the three-dimensional equations just mentioned; these solutions satisfy the differential equations and boundary conditions on the major surfaces of the plate exactly, and a number of such solutions are summed so as to satisfy the remaining boundary conditions on the minor surfaces approximately, using either the variation formulation or the method of least squares. Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory.

Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, i fully clamped, ii two opposite edges are clamped and other two are simply supported. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration.

### Transverse Vibration of Plates

The effect of various plate parameters on the vibration characteristics has been analyzed. Three dimensional mode shapes have been plotted. The WPA was found to be a very popular tool to compute the vibrational properties of plates. Recently, the strong formulation of WPA has been applied for investigations fundamental frequencies of single-walled carbon nanotubes and detailed discussion is given in our earlier published work [ 30 , 31 , 32 ].

The main objective of the present work is to generalize a modified model based on WPA first time and is determine how to calculate the frequencies of plates under various boundary conditions. In our case, the WPA is applied to solve the presented dynamical equations. The frequencies of the plates increase by increasing the modal number and CC-CC frequencies are greater than the frequencies of other boundary conditions. Then, a , b , and h are called its geometrical parameters. Suppose that w x,y,t designates the deformation displacement out of the plane of motion in the transverse direction.

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The strain energy, U, of this rectangular plate when it is vibrating, is expressed as:. The Lagrangian energy variational functional is formulated by considering the expressions for the strain and kinetic energies of the vibrating rectangular pate and is written as:. For deriving, pate governing equation is obtained by applying the Hamiltonian variational principle [ 33 ]. This principle states that during very short interval of time, the change in the Lagrange functional is minimized. So using this principle to the expression 3 , we get the following:.

This process furnishes the governing equation that states the flexural vibration for the rectangular plates as:. Usually, energy variational methods are applied to investigate the vibration characteristics of structural elements namely: beams, plates, and shells. These methods consist of the Rayleigh-Ritz method [ 19 ] and the Galerkin method [ 16 , 21 ]. When a vibrating problem is written in the integral form, the Rayleigh-Ritz technique is applied.

When this problem is in the form of differential equations, the Galerkin procedure is applied. In both these techniques, axial modal dependence is assumed by different of mathematical functions which meet the boundary conditions described on the ends of structural elements. Frequently, beam functions are exploited for this purpose. These functions are obtained from the solutions of the beam differential equation for various end conditions. In the solution 8 , there are four terms which are functions of the axial variable, x.

They represent the negatively decaying evanescent wave, the positively wave, the negatively propagating wave, and the positively propagating wave.

## Free In-Plane Vibration of Rectangular Plates

Determination of values of A i ' s and the axial wave mode k m are associated with edge conditions. Moreover, they are related to the eigenvalues of characteristic beam functions. Expression 8 may further be written in the following:. This is illustrated by the example of boundary conditions viz. Here, a and b are the dimensions, that is, length and width of the rectangular plate.

Area of determination of solutions of plate equation describing vibration phenomenon has been remained an attractive interests of mathematicians and engineers for their applied aspects. A comprehensive material on plate solutions has been compiled by Leissa [ 2 ]. The present era has been said to be the era of computer and its applications.

The invention of computer has made the mathematical computational process very simple, and complicated expressions are simplified by applying computer software packages like Mathematica, Matlab, Maple, etc. Various numerical techniques are available to solve the differential equations found in engineering fields.

With new developments in the world of computer science, complicated problems encountered in the areas of engineering and technology have been solved very easily and in efficient way. Linear and nonlinear differential equations have solved by the finite difference method, Rayleigh-Ritz method, the Galerkin method, finite element method, Fourier series method, and boundary element method. Ultimately, there are numerous commercially developed software packages.

It is the basic interest of a researcher to apply a method which implicate less time and labor. This concept leads to develop a new technique which is more efficient and simple and provides accurate results. It has been seen that in the recent years, the wave propagation approach has been employed successfully to solve a number of shell and tube problems [ 28 , 30 , 31 , 32 ].

Application of this approach reduces the differential equations in simple algebraic equations. For the present plate problem, this procedure is used to get the plate eigenvalue equation. Modal displacement functions. For classical solutions of partial differential equations, method of separation of variables is employed to split the independent variables. In the governing differential equation of motion for rectangular plates, three independent variables are involved viz. For splitting variables, the following modal displacement function forms are adopted:.

They are taken from algebraic functions and assumed to meet boundary conditions. A trigonometric function or an exponential complex function represents harmonic response. When modal form 7 or 8 or 9 is substituted in the equation of motion of plates,. For wave propagation approach, the space modal functions X x and Y y are supposed to be the following forms:.

## Free In-Plane Vibration of Rectangular Plates | AIAA Journal

Making substitutions of expressions 14 and 15 in Eq. Most of the vibration analysis of rotating functionally graded cylindrical shell with ring supports has been performed using the simply supported boundary conditions. In this case, the axial deformation displacement is estimated by the trigonometric functions, that is:. Differential equations represent a physical problem and involve unknown functions. These functions are determined by applying some constraints on the boundary of solutions.

These conditions are called boundary conditions. Plate vibration is an initial-boundary value problem and is transformed into the boundary value problem and four boundary conditions are described at four ends of a rectangular plate. For a rectangular plate with edges a and b , there are eight physical boundary conditions: Fully simply supported end conditions. Using axial wave numbers, various frequency formulas can be formed for a number of boundary conditions.

The obtained results are discussed and compared with earlier theoretical results and simulation methods using same sets of material and geometrical parameters. Here, a number of results are presented for vibrating rectangular isotropic plates. The vibration frequency equation for the plate has been obtained in terms of vibration, geometrical, and material parameters.

The wave propagation approach has been applied for various boundary conditions. For the accuracy and stability of the present method, the findings are in good agreement with the existing results. Tables 1 and 2 show the comparison of natural frequencies of simply supported square plate with FEM [ 23 ] and SEM [ 29 ]. As the number of modes increases, the frequencies also increase. This comparison shows that present approach is efficient to find the vibration of plates.

In Tables 3 — 5 , the frequencies for a vibrating rectangular plate have been evaluated for modal parameters m , n. It is observed that as m is kept fixed, n is allowed to vary, the frequency for the square plate is increased. Figures 3 and 4 show the variation of frequencies versus modal wave number. As the constraints in the end conditions are applied more, the frequencies increase. Figures 5 and 6 show the variation of natural frequencies Hz of rectangular plates versus the vibration modal wave number m , n for the boundary conditions on four edges viz.

It is observed that frequency increases by increasing the value of n.

It is also concluded that the frequency curves with SS-SS boundary condition are the lowest for varying the modal wave number. For these boundary conditions, on increasing the length a and width b , the frequencies also increase m , n. Figures 5 and 6 also show that the trend of frequency is same for the symmetry of m , n.

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In this study, vibrations of isotropic square and rectangular plates have been investigated for modal parameters. Wave propagation approach has been engaged to solve this problem. The axial deformations along axial variables are approximated by the complex exponential functions. As the modal wave numbers are enhanced, the frequencies for the plates indefinitely. Moreover, the influence of the boundary conditions has been studied for by changing the axial wave modes. In the present study, vibrations of square and rectangular plates are analyzed by applying the wave propagation approach.

This is an approximate technique related to the axial wave modes obtained characteristic beam functions. These axial wave modes represent boundary conditions specified at four ends of a rectangular plate.

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Natural frequencies for vibrating square and rectangular plates are obtained for various boundary conditions. The natural frequencies of plates are investigated versus modal numbers by varying the length and width of the plates with simply supported- simply supported SS-SS , clamped-clamped CC-CC , and simply supported-clamped SS-CC boundary conditions. If we change the nature of material of plate or other physical parameters applied to maintain motion in radial direction, then a new problem can be formed.

These problems can be solved for different set of boundary conditions. This analysis can be applied to examine the vibrations of functionally graded material plates.