Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

Free vibration analysis of continuous rectangular plates

Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes 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.     

Free vibration of rectangular plates of arbitrary thickness

Free vibration of thick rectangular plates is investigated by using the “method of initial functions” proposed by Vlasov. The governing equations are derived from the three-dimensional elastodynamic equations. They are obtained in the form of series and theories of any desired order can be constructed by deleting higher terms in the infinite order differential equations. The numerical results are compared with those of classical, Mindlin, and Lee and Reismann solutions.     

Rayleigh's contributions to modern vibration analysis

Although Rayleigh performed his work on vibrations approximately one hundred years ago, his contributions with their emphasis on energy principles, approximate methods of solution and use of physical insight to assess problems lead directly to some of the most powerful methods of vibration analysis, which are in use today. His contributions and their relationship to modern analytical methods are outlined and illustrated by simple examples.     

Free vibration of a shearwall-frame building on an elastic foundation

The frequencies and associated mode shapes of a shearwall-frame building interacting with an elastic foundation are determined by using an iterative technique iith the Rayleigh-Ritz method. The eigenfunctions of a clamped-free prismatic shear-flexure beam are suitably modified to generate the required shape functions. The influence of the soil on the frequency spectrum of a shearwall-frame building is investigated.     

Free vibration of some circular plates of arbitrary thickness

In this paper, in order to be self-contained, a brief account of the construction technique of the method of initial functions which has been developed for circular plates by the author is given. Then the new method is applied to investigate the free vibration of two circular plates: i.e., simply supported and completely free plates. Numerical results are obtained and compared with those of the classical, Reissner and Mindlin theories.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

Free vibration of continuous polar orthotropic annular and circular plates

The free vibration of a polar orthotropic annular plate supported on concentric circles is analyzed by the Ritz method with use of Lagrange multipliers. A trial function for the deflection of the plate is expressed in terms of simple power series, and a frequency equation for the plate is derived by the condition for minimizing the total potential energy with the constraint equations included. In the numerical examples it is also shown that the method can directly yield quite accurate frequency values for a solid circular plate. Natural frequencies of annular and circular plates are calculated for wide ranges of the support location and orthotropic parameters.     

Free vibration analysis of Mindlin plates with linearly varying thickness

A method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness. Natural frequencies and mode shapes of Mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant.     

Free vibration analysis of cantilever plates by the method of superposition

The method of superposition is employed to analyze the first five symmetric and antisymmetric free vibration modes of a cantilever plate for a wide range of aspect ratios. It is shown that this method provides a simple, straightforward and highly accurate means of solution for this family of problems. Convergence to exact values is shown to be remarkably rapid. The first two symmetric and antisymmetric modal shapes for a square plate are accurately described by means of contour line drawings. The numerous advantages of this method over previously used methods are discussed. It is shown that it lends itself readily to the entire family of rectangular plates with classical edge conditions: i.e., clamped, simply supported, and free. Its applicability to a wide family of rectangular plates with boundary conditions other than the classical type is also discussed.