Design of a Dynamic Vibration Absorber for Minimization of Maximum Amplitude Magnification Factor. Derivation of Algebraic Exact Solution.

Abstract

Fixed points design methods for dynamic vibration absorbers are very well-known in the field of vibration control and commonly applied for practical absorber design. However, they must be understood to be approximation methods from the point of view of their design criterion that the resonance amplitude magnification factor is minimized. In this study, the exact algebraic expressions of optimum tuning and damping parameters for the minimization of the largest resonance amplitude magnification factor of linear dynamic vibration absorbers have been derived for undamped primary systems. The existence of two equal height resonance points is reduced to a multiple root condition of an algebraic equation. The determinant is handled as an equation with respect to the resonance amplitude. It has became very clear that the fixed points theory design by Brock is highly accurate. In particular, it exhibits very small error in the practical mass ratio range, e.g., below unity. Algebraic solutions also exist for the resonance frequencies and the anti-resonance frequency. A numerical extension of the method is introduced in order to investigate the optimization problem for damped primary systems.