Vibration of Plates
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However, based on the author's knowledge, there are no reports on using the finite integral transform to analyze the rectangular plate on elastic foundation, like in Zhong et al , Li et al , Li et al and Li et al In this paper, the double finite cosine integral transform method is adopted to acquire the theoretical solutions of eigenfrequncies and vibration modes for the rectangular thin plate on foundation with four edges free. In the analysis the elastic foundation was modeled by the Winkler elastic foundation. Because it only uses the basic dynamic elasticity equations of the thin plate on elastic foundation and there is no need to select the deformation function arbitrarily, the developed solution is reasonable.
In order to proof the correction of formulations, the numerical results are presented to compare with those from other references. According to the theory the classical Kirchhoff plate, the governing equation of motion for an unloaded plate on the foundation is. In which E is Young's moduli. Also is Poisson's ratios. Assuming a harmonic vibration, one way write.
Vibration of plate on foundation with four edges free by finite cosine integral transform method
Substitution of Eg. In order to solve the partial differential equation 3 , the double finite cosine integral transform approach  is exploited. The double integral transform of the first partial derivative term appeared in Eg. The second term is split into two parts. The first part considers the partial derivative with respect to y first. Substitution of equations into equation 3 leads to.
The boundary conditions of a free plate are .
Of cause, there is another simplified expression of the boundary conditions for a free plate. Substituting the boundary conditions that are described by equation 11 and equation 14 into equation 10 , one can obtain. Because the right-hand side of the equation 17 is definite integral, it is the constant.
It is clear that the equation 19 can meet the boundary conditions described by equations 11 , 13 , 14 and From the remaining boundary conditions presented by equations 12 and 15 , one can obtain. What follows is a system of homogeneous algebraic equations. Non-trivial solution of those equations requires the coefficient matrix to vanish to any desired degree of accuracy. Non-trivial solution of Eqs. From this determinant the eigenfrequencies of the plate are calculated. The associated vibration modes are given by equation 19 after inserting the eigenfrequencies. The infinite series that occur in the corresponding equations see Eqs.
Using Eq. The dynamic version of principle of virtual work when applied to the plate leads to:. Employing Green's theorem in Eq. The governing differential equations are as follows:. Here the stress resultants appear in the governing equations and boundary conditions are defined as follows:. The inertia terms appeared in the governing equations and boundary conditions are expressed as follows:.
Simply supported orthotropic square and rectangular plates occupying the region given by the Eqn. The governing differential equations 7 and the associated boundary conditions 8 and 9 , in terms of displacement variables, for free vibration of square and rectangular plates under consideration are as follows:.
Vibration of Plates
The governing equations for free flexural vibration of orthotropic plate can be obtained by setting the applied transverse load equal to zero in Eq. A solution to resulting governing equations, when expressed in terms of displacement variables, which satisfies the associated boundary conditions time dependent , is of the following form:.
Substitution of solution form given by Eqn. Equations 29 through 32 can be written in the following matrix form:. The solution of Eqn. The orthotropic plate has following material properties as given by Srinivas et al. In the present paper free vibration analysis of simply supported square and rectangular orthotropic plate for aspect ratio 10 is attempted.
The results obtained using present theory are compared with exact results and those of other higher order theory results available in literature wherever applicable. Following non-dimensional form is used for the purpose of presenting the results in this paper. The percentage error in the results obtained using a particular model with respect to the results of exact elasticity solutions is calculated as follows:.
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Results obtained for frequencies of bending mode, thickness shear mode and thickness stretch mode are compared and discussed with the corresponding results of classical plate theory CPT , first order shear deformation theory, Reddy's theory and exact theory . It can be seen from Table 1 that the present theory yields excellent values of frequencies for all modes of vibration.
The present theory, Reddy's theory  and Mindlin's theory  predicts exact result of bending frequency for fundamental mode i. Maximum percentage error predicted by present theory is 5. The theory of Kirchhoff CPT [9, 10] overestimates the fundamental bending frequency by 4. Because of unavailability of exact results percentage error is not quoted. From Table 2 it can be observed that present theory shows slightly lower values of bending frequency as compared to those of Reddy's  and Mindlin's  theories. In case of rectangular plate, CPT overestimates the results of bending frequency as compared to those of other higher order theories due to the neglect of transverse shear deformation and transverse normal stress effects in the classical theory.
The present theory yield excellent results for the thickness shear mode frequency for all higher modes of vibration. The theories of Reddy  and Mindlin  show higher values for the thickness shear mode frequency for all modes of vibration as compared to those of exact and present theory. The frequencies obtained by present theory and theory of Reddy  of this plate are more or less identical with each other. However, Mindlin's  theory predicts higher values of this frequency.
Dynamic shear correction factor is the most important parameter in the dynamic analysis of plates. The exact value of this factor is given by Lamb . The theories of Reddy  and Mindlin  shows less accuracy of results for higher modes as compared to those of present and exact theories. The results of this frequency by other higher order theories are not available in the literature due to the neglect of transverse normal strain effect in these theories. Following conclusions are drawn from the free vibration analysis thick orthotropic plates using variationally consistent trigonometric shear deformation theory.
The present theory is capable to produce frequencies of thickness stretch mode of vibration. The present theory yields the exact value of dynamic shear correction factor from the thickness shear motion of vibration.
Finite Element Procedures. Prentice Hall of India Pvt.
Vibration of Plates
Bhimaraddi and L. A higher order theory for free vibration of orthotropic, homogenous, and laminated rectangular plates. Die Akustik. Leipzig, Ghugal and R. A review of refined shear deformation theories for isotropic and anisotropic laminated plates. Hanna and A.
A higher order shear deformation theory for the vibration of thick plates. Mechanics of Composite Materials. There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory  and the Mindlin-Reissner theory. Solutions to the governing equations predicted by these theories can give us insight into the behavior of plate-like objects both under free and forced conditions.
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This includes the propagation of waves and the study of standing waves and vibration modes in plates. The derivatives in the governing equations are defined as. Summation over repeated indices is implied. The strain-displacement relations for Kirchhoff-Love plates are. In solid mechanics , a plate is often modeled as a two-dimensional elastic body whose potential energy depends on how it is bent from a planar configuration, rather than how it is stretched which is the instead the case for a membrane such as a drumhead.
In such situations, a vibrating plate can be modeled in a manner analogous to a vibrating drum. However, the resulting partial differential equation for the vertical displacement w of a plate from its equilibrium position is fourth order, involving the square of the Laplacian of w , rather than second order, and its qualitative behavior is fundamentally different from that of the circular membrane drum.
For free vibrations, the external force q is zero, and the governing equation of an isotropic plate reduces to.