Transverse Vibrations Of Rectangular Plates Research Articles

Modified Fourier-Ritz Method to Analyze Transverse Vibration of Rectangular Plates with Additional Mass

In practical engineering, rectangular thin plate structures with concentrated mass points or equivalent to concentrated mass points have a wide range of applications in mechanical engineering, electronic engineering, and vehicle engineering, such as supporting workbenches, ship decks, PCB boards and so on. The modified Fourier Ritz method is used to establish the numerical analysis model of rectangular thin plate with lumped mass under general boundary conditions, which can avoid the problems of non-derivability or discontinuity at the boundary of thin plates in traditional methods. In addition, the Fourier expansion in the form of cosine function plus polynomial has better convergence than that in the form of sine function. The calculation method of mass matrix and stiffness matrix of rectangular thin plate with concentrated mass is given, the parameters of different boundary constraints are analyzed, and the influence of the size, position and quantity of concentrated mass on the modal of rectangular plate is discussed. The method and its results can be applied to the vibration analysis and vibration control of rectangular thin plates.

Vibration of rectangular plates: fundamental mode and integer multiple of the fundamental period of vibration

This paper is concerned with a vibration analysis of the transverse vibrations of rectangular plates with freely-supported boundaries and corners fixed, in an imagistic way, to describe the deformation state for the specific natural frequency and mode shape for each degree-of-freedom. Two materials, i.e. aluminum and steel, showing the homogeneous properties in the direction of plate thickness were chosen. The exact solutions for free vibrations of the homogeneous rectangular plate were obtained and the fundamental mode and integer multiple of the fundamental period of vibration were provided. The comparisons show that the analytical model predicts natural frequencies reasonably well for a rectangular plate.

Free vibrations of a generally restrained rectangular plate with an internal line hinge

This paper deals with the study of free transverse vibrations of rectangular plates with an internal line hinge and elastically restrained boundaries. The equations of motion and its associated boundary and transition conditions are rigorously derived using Hamilton’s principle. The governing eigenvalue equation is solved employing a combination of the Ritz method and the Lagrange multipliers method. The deflections of the plate and the Lagrange multipliers are approximated by polynomials as coordinate functions. The developed algorithm allows obtaining approximate solutions for plates with different aspect ratios, boundary conditions, including edges elastically restrained by both translational and rotational springs, and arbitrary locations of the line hinge. Therefore, a unified algorithm has been implemented. Sets of parametric studies are performed and the results are given in graphical and tabular form.

Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire x-y plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions

Extending the previous work on isotropic beams and plates by the second author [Li WL, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J Sound Vib 2009;321:254–69], this paper describes an accurate analytical method for calculating the static deflections and modal characteristics of orthotropic plates with general elastic boundary supports. The displacement function is expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. The series expansions for all the relevant derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, a classical solution can be derived by letting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively. Several numerical examples are presented to demonstrate the excellent accuracy and convergence of the current solutions.

An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports

An analytical method is developed for the vibration analysis of rectangular plates with elastically restrained edges. The displacement solution is expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series. Mathematically, such a series expansion is capable of representing any function (including the exact displacement solution) whose third-order partial derivatives are (required to be) continuous over the area of the plate. Since the discontinuities (or jumps) potentially related to the partial derivatives at the edges (when they are periodically extended onto the entire x– y plane as implied by a two-dimensional Fourier series expansion) have been explicitly “absorbed” by the supplementary terms, all the series expansions for up to the fourth-order derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, an exact solution can be obtained by letting the series simultaneously satisfy the governing differential equation and the boundary conditions on a point-wise basis. Because the series solution has to be truncated numerically, the “exact solution” should be understood as a solution with arbitrary precision. Several numerical examples are presented to illustrate the excellent accuracy of the current solution. The proposed method can be directly extended to other more complicated boundary conditions involving non-uniform elastic restraints, point supports, partial supports, and their combinations.

A multiple scales method solution for the free and forced nonlinear transverse vibrations of rectangular plates

In this paper, first, the equations of motion for a rectangular isotropic plate have been derived. This derivation is based on the Von Karmann theory and the effects of shear deformation have been considered. Introducing an Airy stress function, the equations of motion have been transformed to a nonlinear coupled equation. Using Galerkin method, this equation has been separated into position and time functions. By means of the dimensional analysis, it is shown that the orders of magnitude for nonlinear terms are small with respect to linear terms. The Multiple Scales Method has been applied to the equation of motion in the forced vibration and free vibration cases and closed-form relations for the nonlinear natural frequencies, displacement and frequency response of the plate have been derived. The obtained results in comparison with numerical methods are in good agreements. Using the obtained relation, the effects of initial displacement, thickness and dimensions of the plate on the nonlinear natural frequencies and displacements have been investigated. These results are valid for a special range of the ratio of thickness to dimensions of the plate, which is a characteristic of the Multiple Scales Method. In the forced vibration case, the frequency response equation for the primary resonance condition is calculated and the effects of various parameters on the frequency response of system have been studied.

An analytical solution of transverse vibration of rectangular plates simply supported at two opposite edges with arbitrary number of elastic line supports in one way

This paper presents a new analytical solution of transverse vibration of rectangular plates simply supported at two opposite edges with arbitrary number of elastic line supports in one way. The reaction forces of the elastic line supports are regarded as the unknown external forces acted on the plate. The analytical solution of the differential equation of motion of the rectangular plate, which includes the unknown reaction forces, is gained. The frequency equation is derived by using the linear relutionships between the reaction forces of the elastic line supports and the transverse displacements of the plate along the elastic line supports. The representations of the frequency equation and the mode shape functions are different from those obtained by other methods.

An approximate solution of eigen-frequencies of transverse vibration of rectangular plates with elastical restraints

This paper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple and easily programmed but also have high accuracy. Finally, some numerical results are given and compared with other results obtained.

Natural frequencies of elastically restrained rectangular plates using a set of static beam functions in the Rayleigh-Ritz method

A set of static beam functions is developed. The Rayleigh-Ritz method is used to determine natural frequencies in transverse vibration of rectangular plates with elastic translational and/or rotational edges. By using the static beam functions as basis functions, a relatively small number of functions is required; consequently only a modest quantity of computation is necessary. The reasonable accuracy of the method is demonstrated by solving test problems and comparing with accurate results. The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes in addition to the fundamental one. The approach is simple and straightforward. Complicating factors (orthotropic properties, in-plane forces, etc.) can also be taken into account without great difficulties.

Transverse vibrations of rectangular plates of bilinearly varying thickness

First Page

Determination of the fundamental frequency of transverse vibration of rectangular plates when the thickness varies in a discontinuous fashion

The problem outlined in the title is solved, in an approximate fashion, for the situation where a discontinuous variation of the thickness takes place over a concentric, rectangular subdomain of the plate. Numerical values of the fundamental frequency coefficient are obtained for several combinations of clamped and simply supported edges and the governing geometric parameters. Two approaches have been used: (a) the optimized Rayleigh-Ritz method employing polynomial coordinate functions, and (b) the finite element method. An important practical conclusion of the present investigation is the fact that dynamic stiffening of the plate is achieved for all the situations considered in this paper when the plate possesses a free, concentric, rectangular hole.

Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness

A comparison of natural frequencies of transverse vibration of thin, rectangular plates of non-uniform thickness with different combinations of boundary conditions is presented in this paper, as obtained by using the following methodologies: (1) the Rayleigh-Ritz method with characteristic orthogonal polynomial shape functions; (2) the Rayleigh-Ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient; (3) the optimized Kantorovich method which has been proposed rather recently; (4) the finite element method. The results are presented in tabular form and discussed.

Transverse vibration of rectangular plates subjected to inplane forces by a difference based variational approach

Free vibration characteristics of rectangular plates subjected to inplane loads have been studied using the variational finite difference method. The total energy of free vibration of the system is discretized by replacing the derivative terms by their finite difference equivalents and energy minimization technique is used to obtain a typical eigenvalue problem. Vibration frequencies for various modes for plates subjected to inplane normal loads, pure shear and their combination have been determined for different aspect ratios and edge conditions. It has been observed that the effect of inplane loads on vibration frequencies is more pronounced in the case of plates having similar modes for vibration and buckling.

Transverse vibration of rectangular plates elastically supported at points on edges

This paper studies transverse vibration of rectangular plates with two opposite edges simply supported, other two edges arbitrarily supported and free edges elastically supported at points. A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular plates elastically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourier series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given. Inherent frequencies and mode shape with any accuracy can be gained.

On the effect of free, rectangular cut-outs along the edge on the transverse vibrations of rectangular plates

The title problem is tackled using two alternative approaches: a simple Rayleigh-Ritz methodology and a finite element algorithmic procedure. Numerical results are obtained in the case of simply supported and clamped plates with free, square holes located at the middle of a plate edge. Experimental values are obtained in the case of clamped plates. It is concluded that the agreement is very good from an engineering viewpoint as long as the hole size is moderate. Since this constitutes a valid restriction in practice when ducts, pipes, etc., traverse plates or slabs, one can say that the Rayleigh-Ritz procedure yields fundamental frequency coefficients which possess sufficient accuracy from a designer's viewpoint.

Transverse vibrations of rectangular plates on inhomogeneous foundations, part I: Rayleigh-ritz method

The title problem is solved for the case of Winkler-type inhomogeneous foundation and when the plate edges are elastically restrained against rotation. Fundamental frequency coefficients are determined for several combinations of length to width ratios and flexibility coefficients. The solution given in Part I is based on the Rayleigh-Ritz method and polynomial co-ordinate functions while the problem is solved in Part II [1] by means of the modal constraint method.

Transverse vibrations of rectangular plates on inhomogeneous foundations, part II: Modal constraint method

A method is described for establishing the lowest natural frequencies of simply supported and clamped plates on inhomogeneous elastic foundations. The method is based upon the modal constraint technique described in references [1,2]. The merits of this method lie in the fact that the eigenvalues and eigenfunctions of a simply supported plate and scalar springs are used to calculate the eigenvalues of the modified structures. These modifications consist of coupling these structures and/or adding rotational constraints. A number of configurations of inhomogeneous elastic foundations have been considered. Excellent agreement is shown to exist between the eigenvalues determined in the present paper and those calculated using the Rayleigh-Ritz method by Laura and Gutierrez in Part I [3] of this joint project.

Vibrations of rectangular plates with elastically restrained edges

The Rayleigh-Ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges. By treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions, a relatively small number of functions is required; consequently only a modest quantity of computation is necessary. The good accuracy of the method is demonstrated by solving test problems. The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes in addition to the fundamental. The usefulness and accuracy of existing simplified approaches to the problem are assessed. The effect of in-plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also.

Transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation

The fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation is calculated by using polynomial coordinate functions and the Rayleigh-Ritz method. The approach is simple and straightforward and allows the solution of a rather difficult elastodynamics problem. Complicating factors (orthotropic properties, in-plane forces, concentrated masses, etc.) can also be taken into account without formal difficulties.