Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction

Recent research advances in the dynamic behavior of shells: 1989-2000, Part 1: Laminated composite shells

The geometrically nonlinear vibration response of truncated thin conical shells is studied for the first time considering the one-to-one internal resonance, a phenomenon typically observed in symmetric structures such as conical shells. The Novozhilov nonlinear shell theory, retaining all nonlinear terms in the in-plane strain–displacement relationships of the three mid-surface displacements, is applied to study nonlinear vibrations of truncated conical shells. In-plane inertia is also taken into account, and a relatively large number of generalized coordinates, associated with the global discretization of the shell, is considered. This gives very accurate numerical solutions for simply supported, truncated thin conical shells. The effect of an exact one-to-one internal resonance, due to the axial symmetry of conical shells, is fully considered and the results are presented for different excitation levels. The numerical results show that also an almost exact one-to-one internal resonance with a mode presenting a different number of circumferential waves can also arise, which further complicates the nonlinear vibrations and leads to 1:1:1:1 internal resonance. The numerical model was augmented with additional generalized coordinates to capture this phenomenon. Pitchfork, Neimark–Sacker and period-doubling bifurcations of the forced vibration responses arising from internal resonances are detected, followed and presented, showing complex nonlinear dynamics.

Nonlinear vibrations of truncated conical shells considering multiple internal resonances

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Vibrations of highly inhomogeneous shells of revolution under static loading

The problem of axisymmetric nonlinear vibration for shallow thin spherical and conical shells when temperature and strain fields are coupled is studied. Based on the large deflection theories of von Karman and the theory of thermoelasticity, the whole governing equations and their simplified type are derived. The time-spatial variables are separated by Galerkin’s technique, thus reducing the governing equations to a system of time-dependent nonlinear ordinary differential equation. By means of regular perturbation method and multiple-scales method, the first-order approximate analytical solution for characteristic relation of frequency vs amplitude parameters along with the decay rate of amplitude are obtained, and the effects of different geometric parameters and coupling factors as well as boundary conditions on thermoelastically coupled nonlinear vibration behaviors are discussed.

Thermoelastically coupled axisymmetric nonlinear vibration of shallow spherical and conical shells

In this paper, the axisymmetric nonlinear free vibration problems of cylindrically orthotropic shallow thin spherical and conical shells under uniformly distributed static loads are studied by using MWR and Lindstedt-Poincare perturbation method, from which, the characteristic relation between frequency ratio and amplitude is obtained. The effects of static loads, geometric and material parameters on vibrational behavior of shells are also discussed.

Song Huifang

Papers

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads