Non-stationary flexural vibration behaviour of a cylindrical shell

Abstract

An analytical/numerical method involving a small number of generalized coordinates is presented for the analysis of the non-linear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. The small amplitude vibration mode is characterized by m axial half-waves and ℓ circumferential full waves. The radial excitation is assumed to have the same spatial distribution as the small amplitude vibration mode. For large excitations, a coupled mode response can occur, consisting of the directly excited ‘driven mode’ and the so-called ‘companion mode’. The latter has the same spatial form as the driven mode, but is circumferentially 90° out-of-phase. Donnell-type governing equations are used and classical laminate theory is employed. The assumed modes approximately satisfy ‘simply supported’ boundary conditions. A formulation is used which can account for a possible skewedness of the asymmetric modes. Certain axisymmetric modes which are essential for a satisfactory description of the non-linear behaviour are also included in the assumed deflection function. The effect of axial and torsional inertia, including the inertia effect of a ring or disk at the loaded end of the shell, can be taken into account approximately. Viscous modal damping is included in the analysis. The shell is statically loaded by axial compression, radial pressure, and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic state equations, Hamilton's principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time integration of the set of non-linear ordinary differential equations. The coupled mode non-linear vibration behaviour of an axially loaded isotropic shell is simulated using this approach. Non-stationary vibrations, where the response drifts between single mode and different types of coupled mode solutions, are observed in a small frequency region near resonance.