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.

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.

scholarly journals Analysis of Laminated Shells by Murakami’s Zig-Zag Theory and Radial Basis Functions Collocation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
D. A. Maturi ◽  
A. J. M. Ferreira ◽  
A. M. Zenkour ◽  
D. S. Mashat
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Basis Functions ◽  
Transverse Displacement ◽  
Laminated Shells ◽  
Radial Basis

The static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to Murakami’s zig-zag (ZZ) function (MZZF) theory . The MZZF theory accounts for through-the-thickness deformation, by considering a ZZ evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions.

scholarly journals Free Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions by the Method of Reverberation-Ray Matrix

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Dong Tang ◽  
Guoxun Wu ◽  
Xiongliang Yao ◽  
Chuanlong Wang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Traveling Wave ◽  
Vibration Analysis ◽  
Cylindrical Shells ◽  
Free Vibration Analysis ◽  
Wave Form ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Circular Cylindrical Shells

An analytical procedure for free vibration analysis of circular cylindrical shells with arbitrary boundary conditions is developed with the employment of the method of reverberation-ray matrix. Based on the Flügge thin shell theory, the equations of motion are solved and exact solutions of the traveling wave form along the axial direction and the standing wave form along the circumferential direction are obtained. With such a unidirectional traveling wave form solution, the method of reverberation-ray matrix is introduced to derive a unified and compact form of equation for natural frequencies of circular cylindrical shells with arbitrary boundary conditions. The exact frequency parameters obtained in this paper are validated by comparing with those given by other researchers. The effects of the elastic restraints on the frequency parameters are examined in detail and some novel and useful conclusions are achieved.

Free vibration analysis of Euler–Bernoulli nanobeam using differential transform method

2018 ◽  
Vol 07 (03) ◽  
pp. 1850020 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Differential Transform Method ◽  
Inverse Transformation ◽  
Algebraic Equations ◽  
Transform Method ◽  
Differential Transform

In this paper, a semi analytical-numerical technique called differential transform method (DTM) is applied to investigate free vibration of nanobeams based on non-local Euler–Bernoulli beam theory. The essential steps of the DTM application include transforming the governing equations of motion into algebraic equations, solving the transformed equations and then applying a process of inverse transformation to obtain accurate mode frequency. All the steps of the DTM are very straightforward, and the application of the DTM to both the equations of motion and the boundary conditions seems to be very involved computationally. Besides all these, the analysis of the convergence of the results shows that DTM solutions converge fast. In this paper, a detailed investigation has been reported and MATLAB code has been developed to analyze the numerical results for different scaling parameters as well as for four types of boundary conditions. Present results are compared with other available results and are found to be in good agreement.

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