Dynamic stability of periodic shells with moving loads

A moving load causes the radial displacements of an axi-symmetric shell to be several times higher than that produced by the static application of the same load. The travel velocity of the moving load affects the amplitude of the radial response and a critical velocity above which the shell response becomes unstable can be identified. A finite element model (FEM) is developed to analyze the dynamic response of axi-symmetric shells subjected to axially moving loads. The model accounts for the effect of periodically placing stiffening rings along the shell, on the dynamic response and stability characteristics of the shell. Shape functions obtained from the steady-state solution of the equation of motion for a uniform shell are utilized in the development of the FEM. The model is formulated in a reference frame moving with the load in order to enable studying the shell stability using wave propagation and attenuation criteria. Hence, the critical velocity can be identified as the minimum velocity allowing the propagation of applied perturbations. Such stability boundaries are conveniently identified through a transfer matrix formulation. The model is used to determine the critical velocities of the moving load for various arrangements and geometry of the stiffening rings. The obtained results indicate that stiffening the shell generally increases the critical velocity and generates a pattern of alternating stable and unstable regions. The presented analysis provides a viable means for designing a wide variety of stable dynamic systems operating with fast moving loads such as crane booms, robotic arms and gun barrels.