Nonlinear vibration, stability, and bifurcation analysis of axially moving and spinning cylindrical shells

The nonlinear vibration characteristics of the rotating axially moving circular cylindrical shells in subharmonic regions are investigated in the present paper. The motion equations are carried out based on the Hamilton principle in cylindrical coordinates utilizing Donnell's nonlinear shell theory. By introducing the suitable airy stress function, three equilibrium equations in the cylindrical coordinates are simplified into two nonlinear coupled nonhomogeneous PDEs, including a compatibility equation and the transverse motion equation. The compatibility equation solution is obtained employing the seven degrees of freedom for the flexural mode shape of the system. By implementation of the Galerkin method, the motion equation would be projected into seven nonlinear coupled nonhomogeneous ODEs. This set of equations is solved using a direct normal form method validated by the numerical method and available data. The effect of angular velocity and axial speed is investigated employing frequency and force response curves, bifurcation diagrams, time history, and the system's phase portraits. Always axially moving and rotation speed of the system intensifies the nonlinear behavior of the frequency responses.