FREE VIBRATION OF LINE SUPPORTED RECTANGULAR PLATES USING A SET OF STATIC BEAM FUNCTIONS

doi:10.1006/jsvi.1998.2043

Journal of Sound and Vibration

Volume 223, Issue 2, 3 June 1999, Pages 231-245

https://doi.org/10.1006/jsvi.1998.2043 Get rights and content

Abstract

The free vibration of thin orthotropic rectangular plates, which may be continuous over a number of intermediate line supports in one or two directions, is analyzed by the Rayleigh-Ritz method. A new set of admissible functions which are the static solutions of a point supported beam under a series of sine loads is developed. The eigenfrequency equation for the plate is derived by minimizing the potential energy. A very simple and general computer programme has been compiled. The basic concept to form the set of static beam functions is very clear and requires no complicated mathematical knowledge. Some numerical results presented are compared with those obtained by other numerical methods in the literature. It is shown that this set of static beam functions has some advantages in terms of computational cost, application versatility and numerical accuracy, especially for the plate problem with a large number of intermediate line supports and/or when higher vibrating modes need to be calculated.

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Cited by (22)

Linear and nonlinear vibrations of isotropic rectangular plates resting on full or partial line supports
2022, Materials Today: Proceedings
Citation Excerpt :
The early research on the subject was concerned with rectangular plates having intermediate line supports in one direction. From Venetos and Newmark [1] (1959), until Topal [2] (2012), different methods have been adopted to investigate multi-span plate vibrations, such as the receptance method By Azimi [3], the Rayleigh-Ritz method (RRM) with trial functions calculated as products of multispan beam functions [4], and the discrete method applied in [5]. In this article, the trial plate functions used in the RRM are obtained as products of x- and y-beam functions satisfying the appropriate boundary conditions only at the two ends and omitting the intermediate line supports which are taken into account via distributions of translational springs contributing to the whole system strain energy, with a stiffness tending to infinity at the limit case of rigid supports, which ensures solutions with zero deflections at the intermediate support locations.
Linear and geometrically nonlinear free vibration analyses of isotropic rectangular plates with intermediate full or partial line supports in one or two directions are performed by a semi-analytical method. The Rayleigh-Ritz method is used first to calculate the plates examined linear frequency parameters and associated mode shapes which are then used as trial functions in the geometrically non-linear analysis. The line supports are modelled by distributions of translational springs contributing to the whole system strain energy with a stiffness tending to infinity at the limit case of rigid supports. The trial plate functions used in the RRM are obtained as products of x- and y- beam functions satisfying the appropriate boundary conditions only at the two ends and omitting the intermediate line supports. The calculation concept is classical, clear and requires no complicated mathematical tools for all aspect ratios, various edge conditions and any number of full or partial line supports in one or two directions. The mode shapes obtained from linear analysis are used in the non-linear theory to estimate the non-linear frequencies and mode shapes amplitude dependence at large vibration amplitudes of many multi-span plates. The problem of partial supports which are smaller than the size of the plate is also treated.
Natural vibrations of anisotropic plates with an internal curve with hinges
2017, International Journal of Mechanical Sciences
Citation Excerpt :
Several complicating effects have been considered such as: elastically restrained boundaries, presence of elastically or rigidly connected masses, variable thickness, anisotropic material, presence of holes, etc. In references [8–11] general studies on vibration of plates with point supports have been presented and vibration of plates with line supports have been developed in references [12–14]. A review of the literature has shown that there is only a limited amount of information for the vibration of plates with line hinges.
The objective of this paper is to propose a general algorithm to obtain approximate analytical solutions for the study of the free vibrations of a rectangular anisotropic thin plate with an internal curved line hinge and general restraints. In this system, there exists an intermediate condition that requires the continuity of the transverse displacements. It is well known that the difficulty in choosing admissible functions has been the most significant drawback of the Ritz method. To relax the admissibility requirement the Ritz method, with polynomials as coordinate functions, in conjunction with the Penalty Function method is proposed. This study is focus on different problems related the curved hinge and a natural parametrization is used to treat the mentioned curves. The accuracy of the formulation is ensured by comparing some numerical examples with those available in the literature. Cases not previously treated are particularly analyzed. Frequencies parameters and several sets of vibration mode shapes are included, to provide a better understanding of the dynamical behavior of these plates.
Free vibrations of a generally restrained rectangular plate with an internal line hinge
2012, Applied Acoustics
Citation Excerpt :
Several complicating effects have been considered such as: elastically restrained boundaries, presence of elastically or rigidly connected masses, variable thickness, anisotropic material, and presence of holes. In Refs. [8–11] general studies on vibration of plates with point supports have been presented and vibration of plates with line supports have been developed in Refs. [12–14]. A review of the literature has shown that there is only a limited amount of information for the vibration of plates with line hinges.
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 vibration analysis of continuous rectangular plates
2010, Journal of Sound and Vibration
Citation Excerpt :
Numerical results were presented for four-equal-span, four-unequal-span, three-equal-span and three-unequal-span plates. The other methods, such as the Rayleigh–Ritz method [4,5], the finite strip method [6], the finite layer approach [7] and the Lévy method [8] were also used. Lin and his cooperators [9,10] analyzed free vibration of a finite row of continuous skin-stringer panels and column-supported cooling towers.
Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.
Semi-analytical determination of natural frequencies and mode shapes of multi-span bridge decks
2009, Journal of Sound and Vibration
The determination of the natural frequencies and mode shapes of structures requires an analytical, semi-analytical or numerical method. This paper presents a new semi-analytical approach to determine natural frequencies and mode shapes of a multi-span, continuous, orthotropic bridge deck. The suggested approach is based on the modal method, which differs from other approaches in its decomposition of the admissible functions defining the mode shapes. Implementation of this technique is simple and enables avoidance of cumbersome mathematical calculations. In this paper, application of the semi-analytic approach to a three-span, orthotropic roadway bridge deck is compared in the first 16 modes of previously published fully analytical results and to a finite element method analysis. The simplified implementation matches within 2 percent in all cases, with the additional benefit of including intermodal coupling. The approach can be extended to similar bridges with more than three spans.
Vibration analysis of rectangular Mindlin plates with internal line supports using static Timoshenko beam functions
2002, International Journal of Mechanical Sciences
In this paper, the vibration characteristics of rectangular Mindlin plates with internal line supports in one or two directions are studied by using the Rayleigh–Ritz method. The static Timoshenko beam functions are employed as admissible functions, which are composed of a set of transverse deflection functions and a set of rotation-angle functions due to bending. The static Timoshenko beam functions are derived from a point-supported strip of unit width taken from the particular plate under consideration in a direction parallel to the edge of the plate and acted upon by a series of static sinusoidal loads distributed along the length of the strip. It can be seen that the suggested approach is very simple mathematically, and each of the beam functions is only a sine or cosine function plus a polynomial function of not more than the third order. A unified program can be easily prepared, because the changes in boundary conditions, number and locations of internal line supports and thickness ratio of the plate will result in a corresponding change of only the coefficients of the polynomials. Both high accuracy and low computational cost have been verified by convergence and comparison studies. In addition, it can be seen that the admissible functions presented in this paper can also properly describe the discontinuity conditions of the shear forces at the line supports. Therefore, accurate results can be expected for the analysis of dynamic response and internal force distribution of the plate.

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Regular articleFREE VIBRATION OF LINE SUPPORTED RECTANGULAR PLATES USING A SET OF STATIC BEAM FUNCTIONS

Abstract

Regular article
FREE VIBRATION OF LINE SUPPORTED RECTANGULAR PLATES USING A SET OF STATIC BEAM FUNCTIONS