Minimization of the Amplitude of the Entire Span of Beams Using Mass Dampers Available to Purchase

The aim of this work is the reduction of the steady state amplitude of harmonically forced simply supported beams using mass dampers. The considered system consists of a mass damper attached to a simply supported beam with a harmonic force applied at a given point along its span. Traditionally, passive vibration devices such as mass spring dampers or mass dampers were attached to beams and carefully designed to minimize the maximum amplitude at a given point along their span. Since minimizing the amplitude at one point of the beam might increase it at another point, in this work the maximum amplitude along the entire beam span is minimized. The problem is solved first using an approximate method. For a given mass ratio, the optimal location of the mass damper is determined first, and then the optimal damping constant is calculated. Fixed-lines of the amplitude of the entire span of the beam which are independent of the damping constant are determined. The optimal placement of the mass damper is chosen such that the maximum of these lines is minimized. Then, the optimal damping constant is obtained analytically from an average of two damping ratios corresponding each to one of the peaks of the amplitude of the entire beam span to coincide with one of the two equally leveled maxima of the fixed-lines. The optimal placement and damping constant are calculated for all possible positions of the point force on the beam. These results are compared to those obtained from an exact numerical optimization procedure. The results are written in dimensionless form and can be applied to a system with any material and geometric properties.