Free Vibration of Annular Plates by Discrete Singular Convolution and Differential Quadrature Methods (original) (raw)
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Application of a new differential quadrature methodology for free vibration analysis of plates
International Journal for Numerical Methods in Engineering, 2003
A new methodology is introduced in the differential quadrature (DQ) analysis of plate problems. The proposed approach is distinct from other DQ methods by employing the multiple boundary conditions in a different manner. For structural and plate problems, the methodology employs the displacement within the domain as the only degree of freedom, whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co-ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem. Employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently. In order to demonstrate the capability of the new methodology, all cases of free vibration analysis of rectangular isotropic plates, in which the conventional DQ methods have had some sort of difficulty to arrive at a converged or accurate solution, are carried out. Excellent convergence behaviour and accuracy in comparison with exact results and/or results obtained by other approximate methods were obtained. The analogous DQ formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised. It must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods. Copyright © 2002 John Wiley & Sons, Ltd.
Axisymmetric free vibration of thick annular plates
International Journal of Mechanical Sciences, 1999
This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin's first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.
The free vibration analysis of point supported rectangular plates using quadrature element method
Journal of Theoretical and Applied Mechanics, 2017
In this study, the hybrid approach of the Quadrature Element Method (QEM) has been employed to generate solutions for point supported isotropic plates. The Hybrid QEM technique consists of a collocation method with the Galerkin finite element technique to combine the high accurate and rapid converging of Differential Quadrature Method (DQM) for efficient solution of differential equations. To present the validity of the solutions, the results have been compared with other known solutions for point supported rectangular plates. In addition, different solutions are carried out for different type boundary conditions, different locations and number of point supports. Results for the first vibration modes of plates are also tested using a commercial finite element code, and it is shown that they are in good agreement with literature.
Detailed Numerical Study of Free Vibration Analysis of Plates
2019
Plates are widely used in aircraft, ship building, boiler and automobile applications. It is important to study the vibration characteristics of the plates in order to avoid the resonance in their working condition. Because catastrophic failure occurs in resonance condition due violent vibrations. So to avoid these resonance condition vibration analysis of plates are very important. In this article the modal frequencies and mode shapes of the plates were analysed numerically using ANSYS. Structural steel is considered as the plate material. All edges clamped (CCCC) and all edges free (FFFF) boundary conditions were considered for the study. The effect of parameters such as aspect ratio, thickness, area and shape of the plates on natural frequency was studied. The aspect ratios of 1, 1.5, 2 and 2.5 keeping the surface are same, is considered for the study. Plates of thickness 1 mm, 1.5 mm, 2 mm and 3 mm are analysed. The effect of shape of the plate is studied by considering various shapes like square, circular, equilateral triangle and right angular triangle with same area. It is observed that the natural frequency increases with increase of aspect ratio and thickness. As the surface area increases the natural frequency decreases. The values of natural frequencies of the plates descend in the order of circular, equilateral triangular, square and right angled triangular shapes for all edges free. Whereas the values of natural frequencies of the plates ascend in the order of circular, square, equilateral triangular and right angled triangular shapes for all edges fixed conditions.
Vibration Analysis of Annular Circular and Sector Plates with Non-Uniform Thickness
Far East Journal of Dynamical Systems, 2017
In this paper, generalized differential quadrature method (GDQM) is employed and vibration analysis of annular circular and sector plates with non-uniform thickness is investigated numerically. Natural frequencies and corresponding vibration modes are obtained for various boundary conditions. Accuracy and efficiency of the presented GDQM are tested against previous results for free vibration analysis of uniform ones and numerical results are proposed for free vibration analysis of annular circular plates with non-uniform thickness in three cases of linear variable thickness, parabolic variable thickness and exponential variable thickness and also for sector plates with linear variation in thickness. Meanwhile, effects of radii ratio, sector angle and variation of the thickness on the natural frequencies are investigated.
American Journal of Computational and Applied Mathematics, 2012
Vibrat ion characteristics of monoclinic rectangular plate of exponentially varying thickness resting on elastic foundation have been studied on the basis of classical plate theory. Following Lévy approach i.e. t wo parallel edges (y = 0 and b) are assumed to be simp ly-supported while the other two edges (x = 0 and a) may have either of three co mbinations CC , C-S or C-F, where C, S and F stand for clamped, simp ly supported and free edge, respectively. Assuming the transverse displacement w to vary as sin (p y/b), the part ial differential equation wh ich governs the motion of equation is reduced to an ordinary differential equation in x with variab le coefficients. The resulting ordinary differential equation has been solved by Generalised Differential Quadrature Method (GDQM) for all the boundary conditions considered here. The effect of various plate parameters has been studied on the natural frequencies for the first three modes of vibration. Convergence studies have been carried out for four decimal exactitude. Mode shapes for all the three plates have been presented. The efficiency of generalized differential quadrature method for the natural frequencies of vibrat ion of monoclin ic rectangular p lates has been examined.
2019
In this article, using generalized differential quadrature (GDQ) methods, free vibration of a thin annular plate coupled with two open circuit piezoelectric layers, is numerically studied based on the classical plate theory. The governing differential equations with respective boundary conditions are derived and transformed into a set of algebraic equations by implementing the GDQ rule, then solved as an eigenvalue problem to obtain the natural frequencies and mode shapes of the plate. Convergence of the solutions obtained for the natural frequencies is studied. Also, the present numerical model validated by comparing its numerical results with those reported in literature. Finally, parametric studies are carried out and the effects of a number of important parameters on the natural frequencies are investigated.
Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique
International Journal of Pressure Vessels and Piping, 2009
Free vibration analysis of rotating cylindrical shells is presented. Discrete singular convolution (DSC) method has been proposed for numerical solution of vibration problem. The formulations are based on Love's first approximation shell theory, and include the effects of initial hoop tension and centrifugal and Coriolis accelerations due to rotation. Frequencies are obtained for different types of boundary conditions and geometric parameters. In general, close agreement between the obtained results and those of other researchers has been found.
36th Structures, Structural Dynamics and Materials Conference, 1995
In this paper, the analysis of the titled problem is based on classical thin plate theory and its numerical solution is carried out by a semi-analytical differential quadrature method. The thin rectangular plates considered herein arc simply supported on two opposites edges. Thc boundary conditions at the other two edges may be quite general and between these two edges, the plates may have varying thickness. However, the rcsults contained in this paper are for plates which are elastically restrained against rotation at the these edges and have linearly tapered thickness. On the basis of comparison with the available results in the published literature, it is bclicved that this solution method guarantees high numerical accuracy for the problem. Moreover, the computation times involved in the evaluation of free vibration characteristics are sufficiently small indicating that the solution method may possibly be further developed for the real time analysis and design of vibrating plate systems.
Composite Structures, 2018
Free vibration analysis of laminated composite and functionally graded materials (FGM) composite annular plates is investigated. The equations of motion of annular plates have been obtained via conical shell equations. Shear deformation theory is used for shell equation of motion. After the implementation of the Regularized Shannon delta (RSD) kernel and Lagrange delta sequence (LDS) kernel, the method of discrete singular convolution (DSC) is used for numerical solution of the governing equations to obtain the frequency values. To verify the accuracy of this method, comparisons of the present results are made with results available in the open literature. Some parametric results for annular plates and conical panels have depicted for isotropic, laminated composite and functionally graded composite materials. It is found that the convergence and accuracy of the present DSC method is very good for vibration problem of annular plates with functionally graded materials (FMG) and laminated composite cases. Some results about carbon nanotube reinforced (CNTR) composite plate have also been approved.