Optimal design of a damped single degree of freedom platform for vibration suppression in harmonically forced undamped systems

Abstract

Vibration reduction in harmonically forced undamped systems is considered using a new vibration absorber setup. The vibration absorber is a platform that is connected to the ground by a spring and damper. The primary system is attached to the platform, and the optimal parameters of the latter are obtained with the aim of minimizing the peaks of the primary system frequency response function. The minimax problem is solved using a method based on invariant points of the objective function. For a given mass ratio of the system, the optimal tuning and damping ratios are determined separately. First, it is shown that the objective function passes through three invariant points, which are independent of the damping ratio. Two optimal tuning ratios are determined analytically such that two of the three invariant points are equally leveled. Then, the optimal damping ratio is obtained such that the peaks of the frequency response function are equally leveled. The optimal damping ratio is determined in a closed form, except for a small range of the mass ratio, where it is calculated numerically from two nonlinear equations. For a range of mass ratios, the optimal solution obtained is exact, because the two peaks coincide with the two equally leveled invariant points. For the remaining range, the optimal solution is semiexact. Unlike the case of the classical absorber setup, where the absorber performance increases with increasing mass ratios, it is shown that an optimal mass ratio exists for this setup, for which the absorber reaches its utmost performance. The objective function is shown in its optimal shape for a range of mass ratios, including its utmost shape associated with the optimal mass ratio of the setup.