Recently, a new vibration absorber setup was proposed where the absorber is placed between the dynamic system and its moving support. The problem was solved and design guidelines were proposed using the classical absorber design technique. In this work, the unique optimal absorber parameters are determined with the aim of minimizing the maximum of the primary system amplitude. For a given stiffness ratio of the system, the optimal mass and damping ratios are obtained analytically using an optimization method based on invariant points of the objective function. Similar to the case of the classical vibration absorber setup, these points are independent of the system damping ratio. It is shown that a trade-off relationship exists between these points, therefore the optimal mass ratio is determined first by a proper placement of the invariant points. Two suboptimal damping ratios are determined by forcing one of the two peaks of the objective function to coincide with one of the invariant points. Then, the optimal damping ratio is obtained from the average of the two suboptimal damping ratios. This approximate analytical solution is validated through comparison with the exact optimal parameters which were calculated numerically using two different numerical optimization methods. The first is based on the genetic algorithm technique and the second on the downhill simplex method. The optimal parameters are plotted and several examples are considered where the objective function is plotted in its approximate and exact optimal shapes.

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