Design of a Spatial Vibration System for Prescribed Ratio of Energy Peaks

Abstract

The work ratio is a primary factor in the applications relevant to wide bandwidth requirements because a bandwidth depends on resonant peaks. Especially, since a single-body system can have up to six resonant peaks, a six degree-of-freedom spatial system is more desirable for a broad bandwidth design. This paper presents a novel design method of a spatial vibration system for any prescribed ratio of energy peaks. Firstly, we introduce an important and concise geometric nature of a spatial vibration system with a single rigid body when it has only rotational vibration modes. The vibration modes represent vibration axes and six lines of action can be obtained by transforming the vibration modes by the mass matrix. It is shown that the six axes of vibration and six lines of action form two orthocentric tetrahedra that share the orthocenter coincident with the mass center. It is also shown that the stiffness matrix determined from two tetrahedra can always be realized by means of parallel connection of line springs. Using the orthocentric tetrahedra, we acquire analytical expressions for the energy produced by external forces at resonant frequencies, which is used to determine vibration modes that satisfy the requirements for given mass properties and six target resonant frequencies. Finally, the stiffness matrix that satisfies requirements is found and realized. To illustrate the process of the presented method, we use four numerical examples with different work ratios and demonstrate that the method is useful for a wide bandwidth.

Keywords

• Broad bandwidth
• spatial vibration system
• resonant frequency
• rotational vibration mode
• work ratio