Nonlinear vibrations of circular cylindrical shells with different boundary conditions and geometric imperfections

Geometrically nonlinear forced vibrations of circular cylindrical shells with different boundary conditions are investigated. The Sanders-Koiter and Donnell nonlinear shell theories are used to calculate the elastic strain energy. The shell displacements (longitudinal, circumferential and radial) are expanded by means of a double mixed series: harmonic functions for the circumferential variable and two different formulations for the longitudinal variable; these two different formulations are: (a) Chebyshev orthogonal polynomials and (b) trigonometric functions. The same formulation is applied to study different boundary conditions; results are presented for simply supported, clamped and cantilever shells. The analysis is performed in two steps: first a liner analysis is performed to identify natural modes, which are then used in the nonlinear analysis as generalized coordinates. The Lagrangian approach is applied to obtain a system of nonlinear ordinary differential equations. Different expansions involving from 14 to 52 generalized coordinates, associated with natural modes of simply supported, clamped-clamped and cantilever shells are used to study the convergence of the solution. The nonlinear equations of motion are studied by using arclength continuation method and bifurcation analysis. Numerical responses obtained in the spectral neighborhood of the lowest natural frequency are compared with results available in literature. Influence of geometrical imperfections studied as well.