Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Non-linear axisymmetric transient analysis of an orthotropic thin circular plate with an elastically restrained edge

Results are presented for the geometrically non-linear axisymmetric transient elastic stress and deflection responses of a cylindrically orthotropic thin circular plate with an elastically restrained edge, including both rotational and in-plane displacements. In the analysis the dynamic analogue of the von Kárman governing differential equations in terms of the normal displacement w and the stress function ψ are employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain are used. Four types of uniformly distributed transient loadings have been considered: step function, sinusoidal and N-shaped pulses, and exponentially decaying loads. The influence of the orthotropic parameter β and the elastic rotational and in-plane edge restraint parameters (K_b, K_i) on the large amplitude response has been investigated. The effect of a prescribed in-plane displacement on the non-linear transient response has also been studied.     

Non-linear oscillations of elastic orthotropic annular plates of variable thickness

A computational analysis of the non-linear oscillations of elastic orthotropic annular plates of variable thickness is presented. The non-linear boundary value problem is converted into a corresponding eigenvalue problem by using a Kantorovich time-averaging method. Then, by a Newton-Raphson iteration scheme in conjunction with the concept of analytical continuation, the solution to the non-linear oscillations of elastic orthotropic annular plates of variable thickness are obtained.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.     

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.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

The dynamic transient analysis of axisymmetric circular plates by the finite element method

The linear elastic, dynamic transient, analysis of some circular plate bending problems is considered by using axisymmetric, parabolic isoparametric, elements with an explicit time marching scheme. The effects of rotatory inertia and transverse shear deformation are included. A special mass lumping scheme and the use of a reduced integration technique allow the treatment of thin as well as thick plates. Several numerical examples are presented and compared with results from other sources.     

Large amplitude axisymmetric vibrations of orthotropic circular plates elastically restrained against rotation

A finite element formulation is employed to obtain the linear and non-linear frequencies of orthotropic circular plates with elastically restrained edges. Results are presented in the form of linear frequency parameters and ratios of non-linear to linear periods for several values of the spring constants, orthotropy parameter and central deflections.     

Non-linear vibrations of elastic plates subjected to transient pressure loading

The large amplitude vibrations of elastic plates of arbitrary plan form subjected to transient pressure loading are analyzed in a relatively simple fashion by using the Berger method in conjunction with the iso-amplitude contour lines method. The analysis provides for both clamped and simply supported edge conditions. By way of illustration, the large amplitude response of elliptical plates under various types of dynamic loading, namely a step function, a sinusoidal pulse and an N-wave, is investigated and the results are presented graphically. Some comparison is made with previously obtained results for circular plates, as available in the literature.     

The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape

Free flexural vibrations of homogeneous, thin, orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the R-function method. The proposed method is based on the use of the R-function theory and variational methods. In contrast to the widely used methods of the network type (finite differences, finite element, and boundary element methods), in the R-function method all the geometric information given in the boundary value problem statement is represented in an analytical form. This allows one to seek a solution in a form of some formulas called a solution structure. These solution structures contain some indefinite functional components that can be determined by using any variational method. A method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described. Numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits.     

A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates

A finite element is presented to analyze the three-dimensional (3-D) vibration of piezoelectric coupled circular and annular plates. The proposed finite element is a modification of a conventional axisymmetric finite element and is capable of conducting both axisymmetric and nonaxisymmetric vibration analysis of circular and annular laminated plates, with piezoelectric layers therein. The present formulation, a two-dimensional model itself, can investigate 3-D vibration of those plates for a preselected number of nodal diameters, and is therefore more economical than the conventional 3-D finite element analysis, yet still has almost the same accuracy and versatility as the 3-D analysis. In cases such as analysis of stators of traveling wave ultrasonic motors where only vibration modes with particular numbers of nodal diameters are of interest, the proposed approach is very convenient and useful.