Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell's nonlinear shell theory and Amabili-Reddy's third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency-response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode (m, n), apart from the regular modes (i x m, j x n) (i = 1, 3, 5,., and j = 0, 1, 2, 3, ...) that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency-responses. Results show that present iteration procedure is efficient and successful to get the frequency-responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.