Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells

Abstract This paper presents an analytical procedure and closed-form vibration solutions with analytically determined coefficients for orthotropic circular cylindrical shells having classical boundary conditions. This analysis is based upon the Donnell–Mushtari shell theory. This is the simplest thin shell theory and its results for the lowest frequencies of a closed cylinder may not be as accurate. It is known that the exact procedure is complicated for orthotropic shells and this complexity has apparently prevented most researchers from getting results. Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work. They are found in a compact form. Moreover, the characteristics of the eigenvalues are examined. The exact solutions are validated through numerical comparisons with available solutions in literatures and the semi-analytical differential quadrature finite element method (S-DQFEM) solutions calculated by the authors.

[1]  J. B. Greenberg,et al.  Vibrations of Laminated Filament-Wound Cylindrical Shells , 1981 .

[2]  Mohamad S. Qatu,et al.  Vibration of Laminated Shells and Plates , 2004 .

[3]  Bert L. Smith,et al.  Vibration of a circular cylindrical shell closed by an elastic plate. , 1967 .

[4]  Y. C. Das.  Vibrations of orthotropic cylindrical shells , 1964 .

[5]  Mohamad S. Qatu,et al.  Recent research advances on the dynamic analysis of composite shells: 2000-2009 , 2010 .

[6]  K. Ip,et al.  VIBRATION ANALYSIS OF ORTHOTROPIC THIN CYLINDRICAL SHELLS WITH FREE ENDS BY THE RAYLEIGH-RITZ METHOD , 1996 .

[7]  Vibration and stability of orthotropic circular cylindrical shells subjected to axial load , 1984 .

[8]  K. Y. Sze,et al.  APPLICATION OF MATCHED ASYMPTOTIC EXPANSIONS TO THE FREE VIBRATION OF A HERMETIC SHELL , 1998 .

[9]  Joseph Callahan,et al.  A closed-form solution procedure for circular cylindrical shell vibrations , 1999 .

[10]  Yang Xiang,et al.  Vibration of open circular cylindrical shells with intermediate ring supports , 2006 .

[11]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[12]  Mohamad S. Qatu,et al.  Accurate equations for laminated composite deep thick shells , 1999 .

[13]  Joseph C. S. Lai,et al.  Prediction of natural frequencies of finite length circular cylindrical shells , 2000 .

[14]  W. Soedel,et al.  SIMPLIFIED EQUATIONS AND SOLUTIONS FOR THE VIBRATION OF ORTHOTROPIC CYLINDRICAL SHELLS , 1983 .

[15]  M. Darvizeh,et al.  Natural frequencies of laminated orthotropic thin circular cylinders , 1984 .

[16]  Li Xuebin,et al.  A new approach for free vibration analysis of thin circular cylindrical shell , 2006 .

[17]  Mohamad S. Qatu,et al.  Recent research advances in the dynamic behavior of shells: 1989-2000, Part 1: Laminated composite shells , 2002 .

[18]  K. M. Liew,et al.  Buckling and vibration analysis of isotropic and laminated plates by radial basis functions , 2011 .

[19]  C. Wang,et al.  Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio , 2011 .

[20]  K. M. Liew,et al.  Effects of initial twist and thickness variation on the vibration behaviour of shallow conical shells , 1995 .

[21]  W. Soedel.  Vibrations of shells and plates , 1981 .

[22]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[23]  K. R. Sivadas,et al.  Vibration analysis of orthotropic shells with variable thickness , 1990 .

[24]  K. Ip,et al.  Material characterization for orthotropic shells using modal analysis and Rayleigh-Ritz models , 1998 .

[25]  Kevin Forsberg.  Influence of Boundary Conditions on the Modal Characteristics of Thin Cylindrical Shells , 1964 .

[26]  C. B. Sharma,et al.  Free vibration of specially orthotropic, multilayered, thin cylindrical shells with various end conditions , 1987 .

[27]  Guang Liu,et al.  A Differential Quadrature Finite Element Method , 2010, A Differential Quadrature Hierarchical Finite Element Method.

[28]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[29]  Bo Liu,et al.  High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain , 2009 .

[30]  Bert L. Smith,et al.  Natural frequencies of clamped cylindrical shells. , 1968 .

[31]  L. Rayleigh,et al.  The theory of sound , 1894 .

[32]  J. B. Greenberg,et al.  Buckling and vibration of orthotropic composite cylindrical shells , 1980 .

[33]  K. R. Sivadas,et al.  Vibration analysis of orthotropic cantilever cylindrical shells with axial thickness variation , 1992 .

[34]  D. F. Vronay,et al.  Free vibration of circular cylindrical shells of finite length , 1970 .