Free vibration analysis of composite laminated cylindrical shells using the Haar wavelet method

Free vibration analysis of truncated circular conical shells with variable thickness using the Haar wavelet method

“…The elastic bending problems of orthotropic plates and shells were solved by Majak et al [20] using this method. Recently, Xie et al [21,22] applied the Haar wavelet method to investigate the free vibrations of isotropic and composite cylindrical shells. However, to the knowledge of the authors, it appears that the Haar wavelet method has not been developed to solve the vibration of conical shells with variable thickness.…”

“…In the Refs. [21,22], the integration constants are solved by the matrix form, which makes it hard to solve the problems of the simply or free boundary conditions for conical shells. In this paper, the integration constants are obtained directly from the boundary equations, making it universal to cope with all types of boundary conditions.…”

“…The mode shapes of ( ), ( ) and ( ) which corresponds to a certain frequency can be simultaneously obtained by substituting the corresponding eigenvector back into the Eq. (22). The detailed expressions of K and M can be expressed as follows:…”

Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature

“…The CSTs are based on the first approximation of Love-Kirchhoff hypothesis, in which the effects of the transverse shear deformation are neglected. Many studies investigated the vibration characteristics of laminated shells on the basis of the CSTs [4][5][6][7][8][9][10][11]. The CSTs are however invalid for moderately thick shells, which are investigated in this paper.…”

“…Tornabene and his coauthors [2,[34][35][36][37][38][39][40] provide a general numerical method for the stress recovery and vibration analysis of moderately thick laminated doubly-curved shells and panels based on the Generalized Differential Quadrature (GDQ) technique. In this method, the partial derivative of each displacement function of a doubly-curved 5 shell with respect to the coordinate variable is approximated by a weighted sum of displacement function values at all the discrete points in shell domain. The GDQ results are compared well with those presented in literature and numerical solutions obtained using commercial programs.…”