Dynamic response of circular plates subjected to moving massive loads

Techniques are presented for studying the dynamic response of circular disks excited by moving loads. The loading system, consisting of a mass, spring, and dashpot, travels in a circular path concentric with the disk at constant angular velocity. For cases involving elastically- supported rigid disks, the equations of motion for the disk and moving load may be written as a set of coupled Hill-Mathieu equations, typical of moving mass problems. By applying relatively simple transformations, the equations may be rewritten as a set of coupled linear differential equations with constant coefficients. The problem is then reduced to solving an ordinary eigenvalue problem. When the eigenvalues are pure imaginary numbers, they correspond to the frequency components in the motion of the moving mass, and describe the disk motion as well. In certain regions the eigenvalues have positive real parts, corresponding to motions which are unbounded in time. There are three distinct regions of instability which appear in the rigid disk problem. A stiffness instability region occurs immediately above the critical speed of the disk, and is caused by load stiffness. At higher speeds, a region of instability due to modal coupling appears. Finally, if the load speed exceeds a certain terminal velocity (determined primarily by the mass of the load), an unstable solution will always exist. The dynamic response of circular elastic disks with similar loading is investigated using the conventional eigenfunction expansion technique. The system of coupled Hill-Mathieu equations obtained by applying this method reduces to an ordinary eigenvalue problem when certain transformations are made. Thus, many modes may be included in the solution, although it is generally sufficient to consider only a few modes. Solutions to the eigenvalue problem reveal regions of instability directly analogous to those observed in the rigid disk examples