Three Dimensional Vibration Analysis of Rectangular Plates With Undamaged and Damaged Boundaries by the Spectral Collocation Method

2011 ◽  
Author(s):  
Ma’en S. Sari ◽  
Eric A. Butcher
Keyword(s):  
Boundary Conditions ◽  
Collocation Method ◽  
Vibration Analysis ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Simply Supported ◽  
Grid Points

This paper presents a new numerical technique for the free vibration analysis of isotropic three dimensional elastic plates with damaged boundaries. In the study, it is assumed that the plates have free lateral surfaces, and two opposite simply supported edges, while the other edges could be clamped, simply supported or free. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of three dimensional plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged three dimensional plates indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for damage detection of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.

Free vibration analysis of rectangular and annular Mindlin plates with undamaged and damaged boundaries by the spectral collocation method

2011 ◽  
Vol 18 (11) ◽  
pp. 1722-1736 ◽  
Author(s):  
Ma’en S Sari ◽  
Eric A Butcher
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Collocation Method ◽  
Vibration Analysis ◽  
Natural Vibration ◽  
Numerical Technique ◽  
Free Vibration Analysis ◽  
Mindlin Plate ◽  
Grid Points ◽  
Matrix Vector

The objective of this paper is the development of a new numerical technique for the free vibration analysis of isotropic rectangular and annular Mindlin plates with damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of Mindlin plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational and torsional springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged Mindlin plate indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

Free-Vibration Analysis of Rectangular Plates With Clamped-Simply Supported Edge Conditions by the Method of Superposition

1977 ◽  
Vol 44 (4) ◽  
pp. 743-749 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Formidable Challenge ◽  
Edge Conditions ◽  
Governing Differential Equation

In this paper attention is focused on the free-vibration analysis of rectangular plates with combinations of clamped and simply supported edge conditions. Plates with at least two opposite edges simply supported are not considered as they have been analyzed in a separate paper. It is well known that the family of problems considered here have presented researchers with a formidable challenge over the years. This is because they are not directly amenable to Le´vy-type solutions. It has been pointed out in the literature that most of the existing solutions are approximate in that they either do not satisfy exactly the governing differential equation or the boundary conditions, or both. In a new approach taken by the author the method of superposition is exploited for handling these dynamic problems. It is found that solutions of any degree of exactitude are easily obtained. The governing differential equation is completely satisfied and the boundary conditions are satisfied to any degree of exactitude by merely increasing the number of terms in the series. Convergence is shown to be remarkably rapid and tabulated results are provided for a large range of parameters. The immediate applicability of the method to problems involving elastic restraint or inertia forces along the plate edges has been discussed in an earlier publication.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

Free Vibration Analysis of a Simply Supported Rectangular Tank Partially Filled With a Liquid

2010 ◽  
Author(s):  
Kyeong-Hoon Jeong ◽  
Jin-Seok Park ◽  
Won-Jae Lee
Keyword(s):  
Boundary Conditions ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Ideal Liquid ◽  
Rectangular Plates ◽  
Element Analysis ◽  
Displacement Potential ◽  
Simply Supported ◽  
Rectangular Tank ◽  
Liquid Displacement

This paper presents a theoretical analysis for the hydroelastic vibration of a rectangular tank partially filled with an ideal liquid. The wet dynamic displacement of the tank is approximated by combining the orthogonal polynomials satisfying the simply supported boundary conditions, since the rectangular tank is composed of four rectangular plates. As the facing rectangular plates are geometrically identical, the vibration modes of the facing plates can be divided into two categories: symmetric modes and asymmetric modes with respect to the vertical centerlines of the plates. The liquid displacement potential satisfying the boundary conditions is derived and the wet dynamic modal functions of the four plates are expanded by the finite Fourier transformation for a compatibility requirement along the contacting surface between the tank and the liquid. The natural frequencies of the rectangular tank in the wet condition are calculated by using the Rayleigh-Ritz method. The proposed analytical method is verified by observing an excellent agreement with three-dimensional finite element analysis results.

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