Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

Abstract

Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

On Fourier series for higher order (partial) derivatives of functions

“…And then the Fourier series in the linear combination is intended for the description of the smoothness behavior of the solution function within the domain, and is uniformly convergent and termwise differentiable. The Fourier series method with supplementary terms has been developed successively by Yan [7] and Li and his collaborators [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] for analyzing vibrations of beams, in-plane vibrations of plates, transverse vibrations of plates, vibrations of cylindrical shells, and vibrations of general structures built up by beams or (and) plates. For example, Li [13] proposed an exact series solution for the transverse vibration of rectangular plates, where the displacement solution was expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series.…”

“…Because of the inherent complexity of higher order differentiation of Fourier series of the functions, only Fourier series of (partial) derivatives up to fourth order have been derived for the one-dimensional or two-dimensional functions with general boundary conditions. As to arbitrary order (partial) derivatives of the functions with general boundary conditions, the general formulas for the Fourier series are heretofore unavailable, which holds up further development of some new types of Fourier series methods for linear elastodynamical systems such as the Fourier series direct-expansion method [1][2][3][4][5][6] and the Fourier series method with supplementary terms [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Take the Fourier series direct-expansion method for example, Chaudhuri [1] investigated a general system that is represented by a set of completely coupled linear 2rth (r is a positive integer) order partial differential equations with constant coefficients.…”

“…But it is a pity that these general formulas are not put forward in time. As a result, the structural decomposition of the solution functions and, accordingly, the Fourier series method with supplementary terms are largely restricted to some specific second or fourth order dynamical problems [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. For example, Li [13] developed this method for the vibration analysis of rectangular plates with elastically restrained edges.…”

Dynamic Buckling Analysis of the Cylindrical Shell with Radial Stiffness and Damping Constraints under Axially Impact

“…w P is the radial force per unit length along the axis, and / w z P c w t kw     . The governing equation of the shell is obtained by the Hamilton variation principle as follows [4,5]:…”

A Hybrid Finite Element‐Fourier Spectral Method for Vibration Analysis of Structures with Elastic Boundary Conditions

“…At the early stage of this method, the generalized Fourier series are first substituted into the governing differential equations with relevant boundary equations which are derived from the force equilibrium relationship, and then the Galerkin discretization procedure is incorporated to transform the overall equations into a standard matrix eigenvalue problem. And recently, one more general formulation based on an energy principle where all the unknown coefficients are regarded as generalized independent coordinates [26] has been introduced, using FSM, to vibration analysis of, for example, single rectangular plates with arbitrary nonuniform elastic edge restraints [27], and more complex structures such as elastically coupled rectangular plates [28] and circular cylindrical shells [29]. FSM is even applied to vibroacoustic analysis of a threedimensional acoustic cavity with both the displacement of the plate and the sound pressure constructed in the form of Fourier series with supplemental terms [30].…”