Closed-form exact solution to H-infinity optimization of dynamic vibration absorber (First report: development of an algebraic approach and its application to standard problem)

The fixed-points method for the dynamic vibration absorber (DVA) is widely accepted and the results are prevalent for practical applications. However, they usually have to fall back to a heuristic approach from the point of view of its optimization criterion. A typical design problem to minimize the maximum amplitude magnification factor of the primary system, for which the fixed-points method was originally developed, is an example of such common cases. In the present paper, a new algebraic formulation is developed to this classic problem and closed-form exact solutions to both the optimum tuning ratio and the optimum damping parameters are derived, on the assumption of undamped primary system. This algebraic approach is based on an observation of trade-off between two resonance amplitude magnification factors. Thus, the problem reduces to a solution of an algebraic equation, which is derived as a discriminant of quartic algebraic equation. In undamped case, it was proven that the optimum parameters, the minimum amplitude magnification factor, the resonance and antiresonance frequencies, and sensitivities of the amplitude magnification factors are totally algebraic. A numerical extension enables efficient solutions for the damped primary system and has more direct applicability.