In this paper, the transverse vibrations of rectangular plate with circular central hole have been investigated and the natural frequencies of the mentioned plate with point supported by Rayleigh-Ritz Method have been obtained. In this research, the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate. To determine the kinetic and potential energies of plate, admissible functions for rectangular plate are considered as beam functions and it has been tried that the functions of the deflection of plate, in the form of polynomial functionsproportionate with finite degrees, to be replaced by Bessel function, which is used in the analysis of the vibrations of a circular plate. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. In this study, the effects of increasing the diameter of the hole and the effects of number of point supported on the natural frequencies were investigated and the optimum radius of the circular hole for different boundary conditions are obtained. The method has been verified with many known solutions. Furthermore, the convergence is very fast with any desirable accuracy to exact known natural frequencies. پرونده مقاله

In this study, the analysis of transverse vibrations of rectangular plate with circular central hole with different boundary conditions is studied and the natural frequencies and natural modes of a rectangular plate with circular hole have been obtained. To solve the problem, it is necessary to use both Cartesian and polar coordinate system. The complexity of the method is to apply an appropriate model, which can solve the problem of transverse vibrations of a plate. So, it has been tried that the functions of the deflection of plate, in the form of polynomial functionsproportionate with finite degrees, to be replaced by Bessel function, which is used in the analysis of the vibrations of a circular plate. Then with the help of a semi-analytical method and orthogonality properties of the eliminated position angle, without any need to analyze so many points on the edges of the rectangular plate, we can prevent the coefficients matrix from becoming so much large as well as the equations from becoming complicated. The above mentioned functions will lead to reducing the calculation time and simplifying the equations as well as speeding up the convergence. پرونده مقاله

This paper presents a numerical solution for vibration analysis of a cantilever trapezoidal thick plate. The material of the plate is considered to be graded through the thickness from a metal surface to a ceramic one according to a power law function. Kinetic and strain energies are derived based on the Reissner-Mindlin theory for thick plates and using Hamilton's principle, the governing equations and boundary conditions are derived in the Cartesian coordinates. A transformation of coordinates is used to convert the equations and boundary conditions from the original coordinate into a new computational coordinates. Generalized differential quadrature method (GDQM) is selected as a strong method and natural frequencies and corresponding modes are derived. The accuracy and convergence of the proposed solution are confirmed using results presented by other authors. Finally, the effect of the power law index, angles and thickness of the plate on the natural frequencies are investigated. پرونده مقاله

In this paper, vibration analysis of multiple-stepped Bernoulli-Euler and Timoshenko beams carrying point masses is presented analytically for various boundary conditions. Each attached element is considered to have both translational and rotational inertias. The method of solution is “transfer matrix method” which is based on the changes in the vibration modes at the vicinity of any discontinuity in geometrical and natural parameters; these changes are shown by transfer matrices depended on the geometry of each step or value of the translational and rotational inertias of each attached element. First, natural frequencies and corresponding normal mode shapes are obtained by implementation of the compatibility conditions and external boundary conditions; Then, the precision of the proposed method is checked by comparison of the results with other exact solutions; Finally, the effect of the translational and rotational inertias and position of the attached elements on the natural frequencies of multi-stepped beams are investigated for various boundary conditions. پرونده مقاله