Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions

Abstract The obtaining of exact solutions for free vibration of circular cylindrical shells with classical boundary conditions, except for all edges being shear diaphragm, was known to be difficult or impossible. This work presents those exact solutions based on the Donnell–Mushtari shell theory . They are shown to be in simple and compact form, and not as complex as they have generally been regarded. The computed natural frequencies are in excellent agreement with those in literature and those of the highly accurate semi-analytical differential quadrature finite element method developed in this study. The exact solutions can be used as benchmarks for researchers to check their numerical methods and for engineers to design such shell structures. Two of the four eigenvalues are found to be much larger than the other two for some boundary conditions. High accurate membrane approximate solutions are also presented for the circular cylindrical shells.

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

[2]  H. Kraus,et al.  Literature Review : VIBRATION OF PLATES Arthur W. Leissa NASA SP-160, Scientific and Technical Information Division, National Aeronautics and Space Administration, Washington, D. C. (1969) , 1972 .

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

[4]  Myung-Seog Yang,et al.  A study on the free vibration of the joined cylindrical–spherical shell structures , 2000 .

[5]  M. Naeem,et al.  Prediction of natural frequencies for thin circular cylindrical shells , 2000 .

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

[7]  Mohamad S. Qatu,et al.  Recent research advances in the dynamic behavior of shells: 1989–2000, Part 2: Homogeneous shells , 2002 .

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

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

[10]  Bo Liu,et al.  New exact solutions for free vibrations of rectangular thin plates by symplectic dual method , 2009 .

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

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

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

[14]  Francesco Pellicano,et al.  Vibrations of circular cylindrical shells: Theory and experiments , 2007 .

[15]  Z. Pan,et al.  A study on free vibration of a ring-stiffened thin circular cylindrical shell with arbitrary boundary conditions , 2008 .

[16]  Bo Liu,et al.  Exact solutions for the free in-plane vibrations of rectangular plates , 2009 .

[17]  A. A. Khdeir.  Stability of Antisymmetric Angle‐Ply Laminated Plates , 1989 .

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

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

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

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

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

[23]  E. B. Magrab,et al.  Vibrations of Elastic Structural Members , 1979 .

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

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

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

[27]  J. N. Reddy,et al.  Vibration and stability analyses of cross-ply laminated circular cylindrical shells , 1992 .

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

[29]  Young-Shin Lee,et al.  Free Vibrations of Circular Cylindrical Shells with AN Interior Plate Using the Receptance Method , 2001 .

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