Nonlinear vibration of axially moving simply-supported circular cylindrical shell

Abstract This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.