Thickness Expansions for Higher-Order Effects in Vibrating Cylindrical Shells

In the spirit of Mindlin and others who have used series expansions to express transverse dependences in thin bodies, the present work uses Ritz expansions in a variational formulation for cylindrical shell vibrations. By expanding displacements in spatial coordinates, integral expressions for strain and kinetic energy are converted to quadratic sums involving time-dependent generalized coordinates. Hamilton’s principle provides ordinary differential equations for these coordinates. This view-point yields physical insight into the mechanisms of energy storage and avoids the geometrically thin assumption inherent to many formulations. A set of Legendre polynomials multiplied by a radial factor represent the radial dependences of displacement components, while circumferential variations are represented by sinusoidal functions. Excellent agreement in natural frequencies is found between this approach and analytical solutions over the entire range of shell thicknesses, including the limiting case of a solid cylinder. Comparisons to several thin shell theories are given, leading to conclusions about the range of validity of these theories.