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.
Highlights
When designing vibration systems such as vibration-based energy harvesters and vibration absorbers, the magnitudes of resonant peaks are considered more important because the bandwidth is affected by the magnitudes of peaks
The synthesis method of a spatial vibration system with a realizable stiffness matrix is developed through geometrical investigation into vibration modes
It is shown that six vibration axes and six lines of action obtained by transforming the vibration modes by the mass matrix form two orthocentric tetrahedra that share orthocenter coincident with mass center
Summary
When designing vibration systems such as vibration-based energy harvesters and vibration absorbers, the magnitudes of resonant peaks are considered more important because the bandwidth is affected by the magnitudes of peaks. To derive the geometric constraints on vibration axes from orthogonality with respect to the mass matrix M, we first investigate the geometric properties of M as a linear one-toone transformation, which maps a screw expressed in axis coordinates into one in ray co-ordinates [15]. The above statement of Constraint 1 can be proved as follows: Referring to Fig. 4(a), x3 lies on the plane P since the mass matrix M transforms the lines passing through the point A into the lines lying on the plane P, and vice versa (see APPENDIX in [15]) It is shown in (20) and (21) that x3 meets both X1 and X2. Constraint 2: The tetrahedra share orthocenter, which coincides with the center of mass To prove this statement, we consider three planes (P1, P2, and P3) respectively containing the axes of Xi(i = 1, 2, 3) and perpendicular to the opposite edges xi(i = 1, 2, 3) (Fig. 5). The aforementioned geometric properties of the 6-DOF spatial vibration system can be considered an extension of the planar system
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