Flexural Vibration Of Rectangular Plates Research Articles

Rigorous implementation of the Galerkin method for stepped structures needs generalized functions

In this paper we study free vibrations of stepped structures, specifically for longitudinal vibrations of bars and flexural vibrations of rectangular plates, providing two versions of the Galerkin method. Specifically, we first apply the straightforward version of the Galerkin method which stipulates the employment of the Galerkin procedure to be conducted in each subdomain, or step, of the structure. Second, the rigorous realization of the Galerkin method is presented where the structural parameters, like rigidity and mass, are treated as generalized functions over the entire domain. This latter implementation utilizes unit step functions, as well as the Dirac's delta function, and its derivative to treat the changes of the structural parameters across the steps.It turns out that this rigorous implementation leads to additional terms that do not appear in the straightforward (or “naïve”) realization of the Galerkin method. Both versions of Galerkin methods are compared with the exact solutions of the considered problems. It turns out that with increase of number of terms in the expansion, the rigorous, generalized-functions based Galerkin method tends to exact solution. In contrast the naïve realization of Galerkin's method, which is usually utilized in literature, does not tend to exact solution. This study demonstrates that extreme care must be taken when implementing the Galerkin's method for stepped structures, and only the rigorous version should be employed.

Mathematical analysis of flexural vibrations of inhomogeneous rectangular plates and beams subjected to cyclic loadings of temperature change and external force

Flexural vibrations of rectangular plates and beams which consist of functionally inhomogeneous materials due to cyclic loadings of external force and temperature change are analyzed mathematically. Effects of aspect ratio of rectangular plates and loading frequency on the interference between the flexural vibration due to cyclic loading and that due to cyclic heating are discussed. The amplification factor in the dynamic behavior to the quasi-static one is also studied. Especially, comparing transient responses in flexural vibration of rectangular plates with those of beams, the difference in the amplifications of out-of-plane deflection and stresses between plates and beams is discussed.

Flexural vibration of rectangular plates subjected to sinusoidally distributed compressive loading on two opposite sides

Natural frequencies and mode shapes of rectangular plates subjected to sinusoidally distributed in-plane compressive loading on two opposite edges were considered in this work. Although vibration analysis of rectangular plates was investigated by numerous authors, flexural vibration of plates with nonlinearly distributed in-plane loading received much less attention. However, certain problems involving thermal stresses may have such nonlinear loading conditions in the plane of the plate. The analysis procedure for the title problem involves first finding a plane elasticity solution for the in-plane problem satisfying all boundary conditions. Then using this in- plane solution, flexural vibration analysis has to be carried out. In a related work investigated recently by the present authors, an in-plane elasticity solution was developed and was compared with existing analytical solutions in the literature and with the finite element method. One aspect of this study was the stress diffusion (i.e., reduction of stress) phenomenon along the length of the plate as well as the presence of all three in-plane stress components. Using this in-plane solution, vibration analysis of rectangular plates was carried out for various plate aspect ratios and the results were compared with corresponding values from finite element analysis.

Measuring the Bending Stiffnesses of Orthotropic and Symmetric Laminates from Flexural Vibrations

The paper develops an extensive analysis of the problem of determining the bending stiffnesses of orthotropic and symmetric laminates from flexural vibrations of rectangular plates with different boundary conditions consisting of clamped or free edges. Rayleigh’s approximation and Ritz method are considered for evaluating the flexural vibrations of plates. The experimental results and an analysis of the sensibility show clearly the tendency for the evaluation of the bending stiffnesses from the natural frequencies of the flexural vibrations of an orthotropic or symmetric plate to be an ill conditioned problem. It results that iterative procedures associated with Ritz method or with numerical method as finite element analysis cannot be used for evaluating the bending stiffnesses from the natural frequencies of a single plate in industrial applications. For orthotropic laminates, the paper shows how the Rayleigh’s approximation allows to overcome the problem of the divergence of the iterative processes. This approximation suggests another methods for evaluating the bending stiffnesses and their variations with the frequency. Next for symmetric laminates, the evaluation of the stiffnesses from the flexural vibrations of plates is implemented using the measurement of the mode shapes deduced from experimental modal analysis.

On The Free Flexural Vibration Of Rectangular Plates With Straight Or Curved Internal Line Supports

The Rayleigh-Ritz method is used, with simple polynomials as admissible functions, to obtain the eigenvalue equation for the free vibration of a rectangular plate which has an internal line support lying along a line describable in terms of a general polynomial. Numerical and graphical results are presented for plates with internal supports lying along straight lines, for which comparison results also exist, and for plates with the supports lying along different types of curves, including a central circular support.

On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method

Sets of simple polynomials are proposed for use as admissible functions in the Rayleigh-Ritz method, for the study of the flexural vibration of rectangular plates. The plates may be subject to a number of different complicating factors, including the presence of in-plane forces, concentrated masses, point and line spring supports and special orthotropy of the plate material. The sets of polynomials are shown to be equivalent to the orthogonally generated polynomials used very successfully in earlier studies but are more easily generated and the subsequent evaluation of the integrals in the resulting eigenvalue problem are more simply performed. Numerical results are presented for several problems, illustrating the applicability of the method and its accuracy.

ANALYSIS OF FREE FLEXURAL VIBRATION OF RECTANGULAR PLATES BY THE FINITE DIFFERENCE TRANSFER MATRIX METHOD

This paper is concerned with the analysis of free flexural vibration of rectangular plates by the finite difference transfer matrix method.In this method, the algebraic computations are conducted by the transfer matrices which are formulated with the use of the ordinary finite difference equations for the differential equation of plate vibration.The proposed method is efficient to reduce the size of the coefficient matrix of simultaneous equation in the ordinary finite difference method and finite element method. Some numerical results are also presented to prove that the proposed method has the wide applicability and possibility of future development of structural analysis.

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.

Nonlinear flexural vibrations of rectangular plates subjected to in-plane forces using a new shear deformation theory

Recently developed shear deformation theory is used to analyse large amplitude, flexural vibrations of laminated rectangular plates subject to in-plane forces. Single mode approach, in conjunction with the Galerkin method, is used to reduce the five coupled equilibrium equations to a single second order ordinary differential equation in time variable by ignoring certain inertia terms. This reduced equation involves quadratic and cubic nonlinearities and the solution of which is obtained by the method of Multiple Scales. Numerical results are presented in tabular form for various parameters of the rectangular plate considered.

Flexural vibrations of rectangular plates with free edges

Flexural vibrations of rectangular plates with free edges

Large amplitude flexural vibration of layered composite plates with cutouts

The large amplitude flexural vibration of rectangular plates in investigated. A finite element based on a Reissner-Mindlin type of shear deformable theory governing laminated anisotropic composite plates, and the non-linear (large rotations) strain-displacement relations of the von Karman theory are employed. Numerical results are presented for rectangular plates with rectangular cutouts. The parametric effects of side to thickness ratio (shear deformation), aspect ratio, plate side to cutout side ratio, anisotropy, and lamination on linear and non-linear frequencies are investigated.

The vibration of rectangular plates treated using simply supported plate functions in the Rayleigh—Ritz method

The problem of the flexural vibration of rectangular plates has been treated extensively, with good result, using appropriate beam vibration mode shapes as admissible functions in both the Rayleigh and Rayleigh—Ritz methods [D. Young, J. Appl. Mech. 17, 448–453 (1950); G. B. Warburton, Proc. I. Mech. E. 168, 371–384 (1954); A. W. Leissa, J. Sound Vib. 31(3), 257–293 (1973); S. M. Dickinson, J. Sound Vib. 61(1), 1–8 (1978)]. An alternative set of admissible functions, derived from the mode shapes of vibration of plates having two parallel edges simply supported and boundary conditions on the other two edges appropriate for the plate under consideration, was suggested by Dickinson [J. Sound Vib. 59(1), 143–146 (1978)] and their use in Rayleigh's method demonstrated. The present paper describes the use of these functions in the Rayleigh—Ritz method and illustrates the improvement (or otherwise) in the natural frequencies so calculated over those obtained using beam functions for several example plates. It is found that for plates supported in some manner on all four edges, the simply supported plate functions yield superior results.

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.

Nonlinear Flexural Vibrations of Rectangular Plates Subjected to Uniformly Distributed Periodic Forces

Nonlinear Flexural Vibrations of Rectangular Plates Subjected to Uniformly Distributed Periodic Forces

Flexural Vibration of Rectangular Plates with Stiffeners Parallel to the Edges

SummaryThe basis of the procedure described in the paper is the replacement of the stiffeners by an approximately equivalent system of line springs. One of two methods may then be used to determine the natural frequencies. A rectangular plate with edge stiffeners, point-supported at the four corners, is used to demonstrate the application of the Rayleigh-Ritz method. Numerical results obtained are compared with known approximate solutions based on finite difference equations. A Holzer-type iteration is employed in the case of a plate with parallel stiffeners, where the two edges perpendicular to the stiffeners are simply supported, the other two edges having any combination of conditions.

Vibration of Stiffened Rectangular Plates

The free flexural vibrations of rectangular plates with various boundary conditions have been considered by Warburton. The natural frequencies were calculated by the Rayleigh method, the mode assumed being the product of the characteristic beam functions for the given boundary conditions. Comparison with experimental results shows that the method gives reasonably good approximations. The present note describes a method of obtaining the approximately equivalent characteristic beam functions to enable Warburton's method to be extended to plates having one or more stiffeners parallel to an edge. As a numerical example expressions for the frequencies are derived for a plate, simply supported along two opposite edges, and having a central stiffener parallel to the other two free edges. The results are compared with those given in a recent note by Kirk, who solved the same problem by the Rayleigh-Ritz method, using a mode with one arbitrary parameter. In the case of the fundamental frequency of the unstiffened plate, the characteristic beam function in a direction perpendicular to the free edges is simply a constant, and the solution is less accurate than that given by the Rayleigh-Ritz method. However, numerical analysis of a square plate shows that above a certain stiffener depth the characteristic beam function method is more accurate than the Rayleigh-Ritz method. The two methods are also compared for the 2/2 mode.

Vibration of Centrally Stiffened Rectangular Plate

Natural frequencies of free flexural vibration of rectangular plates may, in many cases, be considerably increased by attaching to the plate one or more elastic stiffening ribs parallel to one edge, or by casting or machining the plate and stiffeners integrally.Hoppmann has determined by a semi-empirical method the natural frequencies of an integrally stiffened simply-supported square plate, using the concept of a homogeneous orthotropic plate of uniform thickness having elastic compliances which are equivalent to those of the stiffened plate. Filippov has obtained the exact solution for the fundamental frequency of a simply-supported square plate having a number of equally spaced stiffeners and has considered the effect of point loads applied to the stiffeners in a direction perpendicular to the plane of the plate.

Vibration Characteristics of Stiffened Plates

This paper describes methods of calculating natural frequencies of free flexural vibration of rectangular plates which have stiffeners parallel to one edge. An approximate frequency equation is derived for a clamped plate with closely spaced stiffeners by extending the Rayleigh method as applied by Warburton (1)† to rectangular isotropic plates. For a plate containing a single central stiffener, an exact frequency equation is derived for simply supported edges and an approximate fundamental frequency equation is derived for clamped edges. Comparison between observed and calculated frequencies is made and the practical effect of stiffening upon natural frequencies is discussed. The application of the Rayleigh method to the calculation of natural frequencies of integrally stiffened plates is illustrated by calculating frequencies determined by Hoppmann and Thorkildsen (2).

On the Transverse Vibrations of Rectangular Orthotropic Plates

Abstract Frequency equations and modal eigenfunctions have been determined for the flexural vibrations of rectangular plates of orthotropic material, for those cases which may be treated by the method of M. Lévy (1). In addition, for a broader class of boundary-value problems, an orthogonality criterion for the eigenfunctions has been established and relations for the kinetic and potential energies derived. The value of these energy functions in dealing with forced vibrations is demonstrated.