An Application of Chebyshev’s Min-max Criterion to the Optimal Design of a Damped Dynamic Vibration Absorber

Watts [1] in 1883 and Frahm [2] in 1909 reported on the first use of a dynamic vibration absorber. The range of industrial applications of such device is wide and includes the minimization of the rolling of ships, the damping of torsional vibration of internal combustion engines and the smoothing of vibrations in a barber cutter. The minimization of oscillations of tall buildings by means of dynamic vibrations absorbers is considered in reference [3]. In particular, (see Figure 1) a secondary mass (the absorber) and a spring–dashpot system can be effectively designed to reduce the vibrations of the primary mass. The inertial force exherted on the secondary mass counteracts the disturbing force on the primary mass. Thus the vibratory motion of such mass is reduced. The main advantage of adopting a dynamic vibration absorber is the significant reduction of a primary mass vibration amplitude. The drawbacks are: the sensitivity to forcing frequency change; the high stresses in the elastic element connecting the secondary to the primary mass; the introduction of two other resonant frequencies in the neighborhood of the suppressed frequency. The first mathematical theory on the passive dynamic vibration absorber is described in a paper by Ormondroyd and Den Hartog [4]. Moreover, in the book authored by Den Hartog [5] a closed form optimal solution for the system with viscous damping only between the primary and secondary mass (i.e.; b1 =0) can be found. In the following treatment such a model