Design for Prespecified Ratios Between Six Energy Peaks Using a Planar Symmetric Dual-Body Vibration System

When the design of a vibration system requires a broad bandwidth, the numbers and the ratios of energy peaks become major design factors. In particular, within a specific range of frequency, an increase in the number of peaks can widen the valid working bandwidth by decreasing the distances between peaks. In this paper, a planar symmetric dual-body vibration system is used to implement desired work ratios at six target frequencies. The geometrical relation between vibration modes and energy peaks is investigated to develop the design method and introduces geometrical representation of vibration modes of a symmetric dual-body system together. Six vibration modes of a symmetric dual-body system are divided into two groups with three vibration modes that represent the centers of vibration. It is shown that the orthocenters of two modal triangles of each of two rigid bodies coincide with its center of mass. The frequency responses to both direct and base excitations are derived in terms of vibration centers and target frequencies, and thus work ratios are obtained. Finally, the derived equation of work ratios is used to determine the modal matrix composed of the vibration modes when the desired mass, specific work ratios, and target resonant frequencies are given. Consequently, the corresponding stiffness matrix is found and realized. Numerical examples of four cases with different work ratios are presented to illustrate the proposed design method.