Vibrations of initially imperfect circular plates including the shear and rotatory inertia effects

An analysis of the free flexural vibrations of elastic circular plates with initial imperfections is presented. The analysis includes the effects of transverse shear and rotatory inertia. The vibration amplitudes are assumed to be large, and two non-linear differential equations are obtained for free vibration of the plate and solved numerically. The period of the plate has been calculated as a function of the initial amplitude for four typical supporting conditions.     

Vibrations of annular plates including the effects of rotatory inertia and transverse shear deformation

An investigation of the natural vibrations of isotropic annular plates of uniform thickness has been made by considering the effects of rotatory inertia and shear deformation. The frequency determinantal equations are derived in explicit form for nine sets of common boundary conditions. Numerical results for the frequency parameters of annular plates having various thickness ratios and inner to outer radii ratios have been obtained. The results are compared with those given by the classical plate theory wherever possible. Among the effects of transverse shear deformation and rotatory inertia, the effect of shear deformation has been found to be more prominent.     

Shear and rotatory inertia effect on Beck's column

In this paper the influence of transverse shear deformation and rotatory inertia upon the flutter load of Beck's column with various support characteristics for a variety of slenderness ratios and cross-sectional shapes is presented. The analysis is based on Cowper's formulae for establishing Timoshenko's shear coefficient K'. From this investigation it is found that the inclusion of these parameters may have an appreciable destabilizing effect in the case of a fully fixed cantilever, and particularly in the case of a partially fixed cantilever with an attached mass at the support. This occurs especially in columns with low critical slenderness ratios and thin cross-sections. Moreover, it is noticed that the flutter frequency— for flutter loads obtained by coalescing either of the first and second or second and third flexural eigenfrequencies-never exceeds the precise value 11·01l… of Beck's column.     

Large amplitude flexural vibration of stiffened plates

The large amplitude free flexural vibration of thin, elastic orthotropic stiffened plates is studied. The boundary conditions considered are either simply supported on all edges or clamped on all edges and the in-plane edge conditions are either immovable or movable. The governing dynamic equations are derived in terms of non-dimensional parameters describing the stiffening achieved, and the solutions are obtained on the basis of an assumed one-term vibration mode shape for various stiffener combinations. In all cases, the non-linearity is found to be of the hardening type (i.e., the period of non-linear vibration decreases with increasing amplitude). Some interesting conclusions are drawn as to the effect of the stiffening parameters on the non-linear behaviour. A simple method of predicting the postbuckling and static large deflection behaviour from the results obtained in this analysis is included.     

On the vibration of skew plates of variable thickness

This paper is concerned with the determination of natural frequencies of a vibrating skew plate with variable thickness. Free and forced vibrations are treated for different ratios of the sides, skew angle and taper constant. The static deflection has also been obtained as a by product of the present solution.     

Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates

Large amplitude flexural vibrations of slender beams, and thin circular and rectangular plates have been studied when a compatible longitudinal or inplane mode is coupled with the fundamental flexural mode. It is shown that the effect of longitudinal or inplane deformation and inertia is to reduce the non-linearity in the flexural frequency-amplitude relationship. Further, for slender beams and thin plates, the effect of longitudinal or inplane inertia is negligible.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

Vibration of simply supported-clamped skew plates at large amplitudest

The large amplitude, free, flexural vibration of orthotropic skew plates simply supported along two opposite edges and clamped along the other two are investigated on the basis of an assumed mode shape. The relationship between the amplitude and period is studied for both isotropic and orthotropic skew plates for various aspect ratios and skew angles under two in-plane edge conditions. It is found that the modal equation reduces to the Dufling type equation from which the period of non-linear vibration is found to decrease with increasing amplitude, exhibiting hardening type of non-linearity. The validity of the Berger approximation is investigated for the problem under consideration and this approximation is shown to give reasonably good results.     

Effect of in-plane inertia on buckling of imperfect plates with large deformations

The influence of in-plane inertia on non-linear dynamic buckling of rectangular initially-imperfect plates is studied. A regular, consistent perturbation technique is used, and both the equations of motion and the boundary conditions are perturbed, yielding a relatively simple procedure for solving an otherwise very involved non-linear problem. Both pulse and vibration buckling during parametric resonance are analyzed. It is shown that, in the former case, the in-plane inertia can be disregarded whereas in the latter it plays an important role for non-slender plates.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

Free vibration and buckling analysis of clamped skew sandwich plates by the Galerkin method

Galerkin's variational method has been used in the past by several investigators [1–3] to solve bending problems of clamped skew plates. In this paper the suitability of the Galerkin method for solution of problems of buckling under the action of in-plane forces and of free vibration of skew plates is studied. The method is first applied to investigate the problems for clamped rectangular sandwich plates. After the validity of the method has been established, the method is then extended to analyze similar problems for clamped skew sandwich plates. The governing differential equations for the skew sandwich plates are obtained by transforming the corresponding differential equations in Cartesian coordinates into skew co-ordinates. The parameters considered herein for the buckling and free vibration behaviour of the skew sandwich plates are the aspect ratio of the plate, Poisson's ratio, skew angle and various shearing stiffnesses of the core. Simplicity and quick convergence is the advantage of the method in comparison with other much more laborious numerical methods requiring extensive computer facilities.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.