Free vibration analysis of shells of revolution using GDQ method

In this paper, the Generalized Differential Quadrature (GDQ) Method is applied to analyze the dynamical behaviour of double curved shells of revolution. The GDQ method was developed to improve the differential quadrature (DQ) method for the computation of weighting coefficients [1]. It has been found that the GDQ technique can obtain accurate numerical solutions using just a few grid points and requiring very small computational resources. The First-order Shear Deformation Theory (FSDT) [2-4] is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. By using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing equations of motion [5]. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. The results are obtained taking the meridional and the circumferential co-ordinates into account, without using the Fourier modal expansion methodology [4]. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. The results presented illustrate the validity and the accuracy of GDQ method. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. GDQ results, which are based upon the FSDT, are compared with those results obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Mechanica.