Free vibration analysis of cylindrical shells using the Haar wavelet method

Abstract This paper presents a novel and efficient solution for free vibrations of thin cylindrical shells subjected to various boundary conditions by using the Haar wavelet discretization method. The Goldenveizer–Novozhilov shell theory is adopted to formulate the theoretical model. The displacements and their derivatives in the governing equations are represented by Haar wavelet series and their integrals in the axial direction and the Fourier series in the circumferential direction. The constants appearing from the integrating process are determined by boundary conditions and thus the partial differential equations are transformed into a set of algebraic equations. The frequency parameters of the cylindrical shells are obtained by solving the algebraic equations. The present solution is verified by comparing the numerical results with those previously published in literature. Very good agreement is observed. It is shown that accurate frequency parameters can be obtained by using a small number of collocation points and boundary conditions can be easily achieved. The advantages of this current solution method consist in its simplicity, fast convergence, low computational cost and high precision.

[1]  Ömer Civalek,et al.  Numerical analysis of free vibrations of laminated composite conical and cylindrical shells , 2007 .

[2]  Martin Eerme,et al.  Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells , 2009 .

[3]  Taehyo Park,et al.  Nondestructive damage evaluation of plates using the multi-resolution analysis of two-dimensional Haar wavelet , 2006 .

[4]  Li Hua,et al.  Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method , 1998 .

[5]  Xiaoming Zhang,et al.  Vibration analysis of cross-ply laminated composite cylindrical shells using the wave propagation approach , 2001 .

[6]  Ö. Civalek.  Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory , 2013 .

[7]  Michael J. Brennan,et al.  An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions , 2013 .

[8]  Wen-June Wang,et al.  Haar wavelet approach to nonlinear stiff systems , 2001 .

[9]  Y. Xiang,et al.  Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions , 2006 .

[10]  Zheng Zhong,et al.  Three-dimensional analysis of functionally graded plate based on the Haar wavelet method , 2007 .

[11]  C. Hsiao,et al.  Optimal Control of Linear Time-Varying Systems via Haar Wavelets , 1999 .

[12]  C. F. Chen,et al.  Haar wavelet method for solving lumped and distributed-parameter systems , 1997 .

[13]  Zhengjia He,et al.  The construction of wavelet finite element and its application , 2004 .

[14]  Zhi Shi,et al.  A spectral collocation method based on Haar wavelets for Poisson equations and biharmonic equations , 2011, Math. Comput. Model..

[15]  Clive L. Dym,et al.  Some New Results for the Vibrations of Circular Cylinders , 1973 .

[16]  K. M. Liew,et al.  Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells , 2002 .

[17]  Martin Eerme,et al.  Weak formulation based Haar wavelet method for solving differential equations , 2009, Appl. Math. Comput..

[18]  Ömer Civalek,et al.  An efficient method for free vibration analysis of rotating truncated conical shells , 2006 .

[19]  Chun-Hui Hsiao.  State analysis of linear time delayed systems via Haar wavelets , 1997 .

[20]  Ülo Lepik,et al.  Numerical solution of differential equations using Haar wavelets , 2005, Math. Comput. Simul..

[21]  E. B. Lin,et al.  Connection coefficients on an interval and wavelet solutions of Burgers equation , 2001 .

[22]  Yang Xiang,et al.  Exact solutions for vibration of open cylindrical shells with intermediate ring supports , 2002 .

[23]  Ülo Lepik.  Solving PDEs with the aid of two-dimensional Haar wavelets , 2011, Comput. Math. Appl..

[24]  Helle Hein,et al.  Computationally efficient delamination detection in composite beams using Haar wavelets , 2011 .

[25]  Ülo Lepik,et al.  Haar wavelet method for nonlinear integro-differential equations , 2006, Appl. Math. Comput..

[26]  N. M. Bujurke,et al.  Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets , 2008 .

[27]  Oleg V. Vasilyev,et al.  Second-generation wavelet collocation method for the solution of partial differential equations , 2000 .

[28]  Ö. Civalek,et al.  Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique , 2009 .

[29]  H. Hein,et al.  Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets , 2011 .

[30]  Silvia Bertoluzza,et al.  A high order collocation method for the static and vibration analysis of composite plates using a first-order theory , 2009 .

[31]  Francesco Tornabene,et al.  Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution , 2009 .

[32]  Zhi Shi,et al.  Application of Haar wavelet method to eigenvalue problems of high order differential equations , 2012 .

[33]  Zhengjia He,et al.  A study of the construction and application of a Daubechies wavelet-based beam element , 2003 .

[34]  Silvia Bertoluzza,et al.  A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory , 2010 .

[35]  Chang Shu,et al.  Analysis of Cylindrical Shells Using Generalized Differential Quadrature , 1997 .

[36]  Zhengjia He,et al.  A second-generation wavelet-based finite element method for the solution of partial differential equations , 2012, Appl. Math. Lett..

[37]  K. Y. Lam,et al.  VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS USING WAVE PROPAGATION APPROACH , 2001 .