Vibration Of Orthotropic Rectangular Plates Research Articles

Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints

A set of static beam functions, which are the solutions of an elastically point-supported beam under a Fourier series of static sinusoidal loads distributed along the length of the beam, are developed as the admissible functions to analyze the vibrations of orthotropic rectangular plates with elastic intermediate line-supports using the Rayleigh–Ritz method. Both the elastic rotational and the elastic translational constraints along the edges of the plate are also considered simultaneously. Unlike conventional admissible functions, this set of static beam functions not only can automatically adjust to the stiffnesses of the intermediate line-supports but also can properly describe the discontinuity of shear forces at the line-supports so that higher accuracy and faster convergence can be expected for the dynamic analysis of such plates. The suggested approach is effective even for various limiting cases by letting the corresponding stiffnesses approach their natural limits of zero or infinity. The present method is theoretically sound and mathematically simple, with each of the static beam functions being only a third-order polynomial plus a sine function. A common and efficient computational program can be compiled because of the fact that a change of the line-support parameters (locations, number and stiffnesses) and the boundary conditions of the plate only results in a corresponding change of the coefficients of the polynomial in the static beam functions. Several numerical examples are presented and the results obtained, where possible, are compared with the known solutions in literature. The present method has proved to be extremely effective for solving the aforementioned problems.

QUINTIC SPLINES IN THE STUDY OF TRANSVERSE VIBRATIONS OF NON-UNIFORM ORTHOTROPIC RECTANGULAR PLATES

An analysis and numerical results are presented for free transverse vibrations of orthotropic rectangular plates of linearly varying thickness along one direction and resting on an elastic foundation of the Winkler type on the basis of classical plate theory. Following the Lévy approach i.e., two parallel edges being simply supported, the fourth order differential equation governing the motion of such plates has been solved by using the quintic splines interpolation technique for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of the elastic foundation together with the orthotropy, aspect ratio and thickness variation on the natural frequencies of vibration is illustrated for the first three modes of vibration. Normalized displacements are presented for two different values of the taper constant keeping other plate parameters fixed for all the three boundary conditions. A comparison of the results with those available in literature is presented.

Generic Free Vibration of Orthotropic Rectangular Plates with Clamped and Simply Supported Edges

The generic free vibration of specially orthotropic rectangular plates with combinations of clamped and simply supported edges is investigated by the superposition technique. The introduction of affine transformation reduces the two rigidity parameters required for the free vibration analysis to one, making complete tabulation of computed eigenvalues much easier. Numerical calculations are performed on clamped plates, plates with one edge simply supported and plates with two adjacent edges simply supported, along with a number of representative values of the generalized rigidity parameter and the affine aspect ratio.

Nonlinear analysis of rectangular orthotropic plates

The present study deals with the large deflections and nonlinear flexural vibrations of orthotropic rectangular plates. In this paper, generalized double Fourier series in terms of generalized beam eigenfunctions for deflection and force functions are presented. The influences of aspect ratio, material properties, transverse load, rotational edge restraint coefficients, and two types of in-plane conditions are studied. The results of deflection and frequency ratio for special cases are compared with those values given in the literature and good agreement is obtained.

Free vibrations of rectangular orthotropic plates with a pair of parallel edges simply supported

Natural frequency parameters for rectangular orthotropic plates with a pair of parallel edges simply supported are studied numerically for the effects of a variety of boundary conditions, material orthotropy, Winkler foundation modulus, and aspect ratio for the first several modes. The results indicate that the material orthotropy, foundation modulus and aspect ratio have more influence on the frequency parameter at the lower modes whereas the influence of the boundary condition on the frequency parameter depends on the stiffness of the boundary condition and is the same for all modes.

Nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales

The method of multiple scales (MMS) in conjunction with the Galerkin method is used to analyze the nonlinear forced and damped response of a rectangular Orthotropic plate subjected to a uniformly distributed harmonic transverse loading. The effects of damping and in-plane loads are considered. The analysis considers simply supported as well as clamped plates. For each case, both movable and immovable edge conditions are considered. By using MMS, all possible resonances such as primary, subharmonic, and superharmonic reso- nances are studied. For the undamped response without in-plane loading, comparisons of the MMS results with those obtained by the finite-element method show excellent agreement. EVELOPMENT of composite materials comprising lam- inates of Orthotropic or multilayered anisotropic materi- als Recently has been receiving substantially growing research efforts. Due to the increasing demands for energy-efficient, high strength, minimum weight aircraft designs, many re- searchers believe that the use of composite materials offers promising alternatives for aircraft designs. Thin, laminated, composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher. Responses of this kind cannot be predicted by using the linear theory. Consequently, the need to study large- amplitude-deflection vibrations of composite structures is of paramount importance. The literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von Karman geometrical nonlinearity were first considered by Whitney and Leissa.1 Based on these equa- tions, different methods of analysis have been developed by several researchers. An excellent number of collections on nonlinear free and forced vibrations of composite plates cov- ering the work through 1979 can be found in the comprehen- sive book by Chia. 2 Bert3 has conducted a survey on the dynamics of composite plates for the period 1979-81. A re- view of the literature on nonlinear vibrations of plates can be found in the review paper by Sathyamoorthy4 and the book by Nayfeh and Mook.5 Large deflection analysis of symmetrically laminated rec- equations of motion are presented in terms of the lateral displacement and stress function. The equations are nondi- mensionalized following the transformation introduced by Brunelle and Oyibo.9 Though multimode analysis can be treated, the present study is focused on single-mode analysis. A deflection function representing the first mode and satisfy- ing the boundary conditions is assumed, and subsequently the stress function is found. Next, the Galerkin method is applied to obtain the modal equation, which is solved analytically by using the method of multiple scales (MMS).10 The MMS also provides solutions for subharmonic and superharmonic reso- nances. The effects of damping ratio, plate aspect ratio, and in-plane loading then are studied.

Vibration of Orthotropic Rectangular Plates With Variable Thickness

The problem vibration of rectangular orthotropic plates with variable thickness and mixed boundary conditions are solved by a modified energy method. A general expression is written for the deflection of the plate without aiming at any particular combination of boundary conditions. Boundary conditions are satisfied approximately by adjusting a set of so-called fixity factors. A computer program has been developed to solve for natural frequencies of plates with variable thicknesses and having different orthotropic properties.

Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh-Ritz technique

The first part of this paper describes a computer programme in which equations derived by the Rayleigh-Ritz technique are used to model the vibrations of rectangular orthotropic plates. The characteristic functions of vibrating beams were used as the assumed functions for plates with boundary conditions consisting of clamped and free edges. Natural frequencies and mode shapes from the programme were verified by finite element analysis and modal analysis for square aluminium and graphite/epoxy plates. The plate vibration model was then incorporated into a second computer programme which was desgned to use the measured natural frequencies of orthotropic plates to determine the four apparent elastic constants. Natural frequencies measured by an impulse technique were used to determine two Young's moduli, the in-plane shear modulus, and a Poisson ratio for each plate.

Transverse vibrations of rectangular orthotropic plates with the edges elastically restrained against rotation and a free, straight corner cut-out

A survey of the literature reveals that the title problem has not been previously solved. Frequent applications of composite materials in many fields of modern technology—from ocean and aerospace engineering to printed circuit boards—have generated the need for a sound understanding of the dynamic behaviour of orthotropic and isotropic plates. The fundamental frequency of transverse vibration of the orthotropic structural element is determined by expanding the displacement amplitude function in terms of simply polynomial coordinate functions which satisfy identically the essential boundary conditions and approximately the natural requirements. The Ritz method is used to generate the frequency equation.

Application of a series-type method to vibration of orthotropic rectangular plates with mixed boundary conditions

The object of this paper is to show an application of a series-type method to the free vibration of an orthotropic rectangular plate with mixed boundary conditions. The rectangular plate considered is elastically constrained or clamped along a few parts of its edge and simply supported on the remainder. A frequency equation is derived to accommodate rectangular orthotropy of the plate by the extension of a previously developed method, and subsequent algebraic manipulation leads to the improved equation, yielding natural frequencies of the plate with very good accuracy. The method is demonstrated for numerical examples where one, two or three parts are constrained along the boundary of the square plate, and the effects of varying constraint and orthotropic parameters on the natural frequencies and mode shapes are studied.

Transverse vibrations of orthotropic rectangular plates with thickness varying in two directions and with edges elastically restrained against rotation

Apparently the theme of the present paper has not been considered in the literature. Problems of this type are of practical importance in view of the continuously increasing use of fibre reinforced materials. The title problem is solved in the case of linear variation of the thickness in the x and y directions using a simple coordinate function, which partially satisfies the boundary conditions. The approximate value of the fundamental frequency coefficient is then determined using the Ritz method.

A perturbation solution of non-linear vibration of rectangular orthotropic plates

Classical perturbation theory is applied to the non-linear dynamic response of orthotropic plates. Expressions are derived for the ratio of non-linear to linear frequency, membrane stress, and the ratio of the maximum total stress to the maximum bending stress. Where possible the analysis is compared to other available numerical solutions, and excellent agreement is shown.

Effects of large amplitude, shear and rotatory inertia on vibration of rectangular plates

In this paper, the governing equations applicable for the large amplitude free, flexural vibration of orthotropic rectangular plates are formulated in terms of the displacement components u, v and w. The formulation and the solutions presented for the cases of isotropic and orthotropic simply supported rectangular plates incorporate the effects of the transverse shear deformation and the rotatory inertia on the large amplitude vibration behaviour. The influences of shear and rotatory inertia are significant in the case of moderately thick plates undergoing large amplitude vibration. The method suggested here does not require the use of the Berger approximation for plates with immovable in-plane edge conditions. A comparison with the results available in the literature indicates the inadequacy of the approach adopted earlier for the study of the large amplitude free flexural vibration of orthotropic rectangular plates.

Transverse vibrations of rectangular orthotropic plates with one or two free edges while the remaining are elastically restrained against rotation

A survey of the literature reveals that the title problem has not been previously solved. The fundamentals frequency of transverse vibration of the orthotropic structural element is determined by expanding the displacement amplitude function in terms of simple polynomial coordinate functions which satisfy identically the essential boundary conditions and approximately some of the natural edge requirements. The Ritz method is then used to generate the frequency equations.

Vibrations of orthotropic rectangular plates with edges possessing different rotational flexibility and subjected to in-plane forces

The title problem is solved using polynomial approximations and a weighted residuals approach. It is shown that a very simple, yet accurate, approximate fundamental frequency equation can be generated. A basic, forced vibrations problem is also analyzed. The procedure is quite convenient for design purposes since all the resulting expressions for frequency coefficients, displacements and stress resultants can be easily implemented for numerical evaluation in a programmable, pocket calculator.

Shear and Rotatory Inertia Effects on Plates

The purpose of this note is to present the governing nonlinear equation applicable for the large amplitude free, flexural vibration of orthotropic rectangular plates influenced by the transverse shear and rotatory inertia effects and to present numerical solutions to isotropic and orthotropic rectangular plates clamped along all the edges. The method of analysis suggested here requires the assumption of only the mode shape for w and is based on the stress function approach and Galerkin's method.

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.

Thermally induced vibrations of orthotropic rectangular plate resting on elastic foundation

Thermally induced vibration of orthotropic rectangular plate resting on elastic foundation has been investigated. The problem is solved in terms of double trignometric series. Complete solution is derived from the sum of two deflections—quasi-static and dynamic. The dynamic solution is obtained by the method of Laplace transform.

Vibrations of orthotropic rectangular plates under inplane forces

Transverse vibrations of orthotropic rectangular plates in the presence of inplane forces have been studied on the basis of classical theory of plates. The two parallel edges ( y = 0 and y = b), have been considered to be simply supported. The displacement function has been taken as a product of two functions; one satisfying boundary conditions at the simply supported edges and the other a general function. From the resulting differential equation of motion, an exact solution is obtained for the general function. Frequencies, mode shapes and critical buckling loads have been computed for four different combinations of boundary conditions at the other two edges.

The Frequency of Flexural Vibration of Rectangular Orthotropic Plates With Clamped or Supported Edges

Abstract The Rayleigh method, using characteristic beam functions appropriate to the boundary conditions, is applied to derive closed formulas for the frequencies of vibration of orthotropic plates under any combination of clamped or supported edges.