Minimax design of vibration absorbers for linear damped systems

This paper addresses the issue of design of a passive vibration absorber in the presence of uncertainties in the forcing frequency. A minimax problem is formulated to determine the parameters of a vibration absorber which minimize the maximum motion of the primary mass over the domain of the forcing frequency. The limiting solutions corresponding to the forcing frequency being unrestricted and to that where the forcing frequency is known exactly, are shown to match those available in the literature. The transition of the optimal vibration absorber parameters between the extreme two cases is presented and the solutions are generalized by permitting the mass ratio of the absorber mass and the primary mass to be design parameters. For the specific case where the primary system is undamped, detailed analysis is presented to determine the transition of the optimal vibration absorber parameters between three distinct domains of solutions.