An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions

Fundamental frequency of a cantilever composite cylindrical shell

“…Effects of boundary conditions on the free vibration characteristics for a multi-layered cylindrical shell using the Ritz method where beam functions were used as the axial modal functions were studied by Lam and Loy [17]. Dai et al developed a method to study the free vibrations of an isotropic circular shell with various boundary conditions in which the displacements were represented as Fourier series, such that both the governing differential equations based on Flugge's theory and the boundary conditions were satisfied [18]. Semi-analytical approaches to the free vibration analyses of axisymmetric laminated shells with various combinations of boundary conditions were developed by Pinto Correia et al [19,20] and Santos et al [21].…”

Thermal gradient effect on the free vibration of a cardioid cross-section cylindrical shell

“…Many pioneering studies have been carried out on the vibration analysis of the elastic cylindrical shells and structural systems, and some of them can be found, for instance, in (Smith, 1968;Chung, 1981;Joseph et al, 1987;Zhang et al, 2001;Toorani and Lakis, 2004;Gokhan and Burak, 2007;Xuebin, 2008;Liu et al, 2012;Xin et al, 2011;Khalifa, 2011a;El-Kaavazi and David, 2012;Lu et al, 2013;Yuehua et al, 2013;Tomabene et al, 2014a). Unlike the circular cylindrical shells, the study of vibration behavior for the noncircular cross-section shells has been received little attention in the past decade, but only a few important studies are concerned with the vibration analysis of the shells with three-lobed, four-lobed, oval, panel and elliptical cross-section (see for example (Culberson and Boyd, 1971;Sewel and Pusey, 1971;Chen and Kempner, 1978;Koumousis and Armenakas, 1983;Yamada et al, 1984Yamada et al, , 1985Sodatos, 1999;Ganapathi et al, 2003Ganapathi et al, , 2004Tomabene and Fantuzzi, 2013;Tomabene et al 2014b;Khalifa, 2010Khalifa, , 2011bKhalifa, ,c, 2014Khalifa, , 2015 ).…”

“…clamped circular cylindrical shell byZhang et al (2001) and LuDai et al (2013) who have obtained their results using different shell theories and solution methods. Whereas the author has obtained the transfer matrix approach solution based on the Flügge's shell theory using the Romberg integration method for the same  ) that gives an isotropic circular cylindrical shell without thermal gradient effect.…”

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

“…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.…”