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Research Papers

Exact Algebraic Solution of an Optimal Double-Mass Dynamic Vibration Absorber Attached to a Damped Primary System

[+] Author and Article Information
Toshihiko Asami

Professor
Mem. ASME
Department of Mechanical Engineering,
University of Hyogo,
2167 Shosha, Himeji,
Hyogo 671-2280, Japan
e-mail: asami@eng.u-hyogo.ac.jp

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received November 5, 2018; final manuscript received May 13, 2019; published online June 14, 2019. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 141(5), 051013 (Jun 14, 2019) (13 pages) Paper No: VIB-18-1480; doi: 10.1115/1.4043815 History: Received November 05, 2018; Accepted May 13, 2019

This article presents exact algebraic solutions to optimization problems of a double-mass dynamic vibration absorber (DVA) attached to a viscous damped primary system. The series-type double-mass DVA was optimized using three optimization criteria (the H optimization, H2 optimization, and stability maximization criteria), and exact algebraic solutions were successfully obtained for all of them. It is extremely difficult to optimize DVAs when there is damping in the primary system. Even in the optimization of the simpler single-mass DVA, exact solutions have been obtained only for the H2 optimization and stability maximization criteria. For H optimization, only numerical solutions and an approximate perturbation solution have been obtained. Regarding double-mass DVAs, an exact algebraic solution could not be obtained in this study in the case where a parallel-type DVA is attached to the damped primary system. For the series-type double-mass DVA, which was the focus of the present study, an exact algebraic solution was obtained for the force excitation system, in which the disturbance force acts directly on the primary mass; however, an algebraic solution was not obtained for the motion excitation system, in which the foundation of the system is subjected to a periodic displacement. Because all actual vibration systems involve damping, the results obtained in this study are expected to be useful in the design of actual DVAs. Furthermore, it is a great surprise that an exact algebraic solution exists even for such complex optimization problems of a linear vibration system.

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Figures

Grahic Jump Location
Fig. 1

Analytical model of a series-type dynamic vibration absorber (DVA) attached to a damped primary system subjected to (a) force and (b) motion excitation

Grahic Jump Location
Fig. 2

Optimal mass ratios μBopt obtained using three different optimization criteria ((c) solid line, numerical solution; dashed line, perturbation solution): (a) H-optimal solution for the force excitation system, (b) H2-optimal solution for the force excitation system, (c) H2-optimal solution for the motion excitation system, and (d) stability maximization optimal solution

Grahic Jump Location
Fig. 3

Optimal tuning ratios νopt obtained using three different optimization criteria ((c) solid line, numerical solution; dashed line, perturbation solution): (a) H-optimal solution for the force excitation system, (b) H2-optimal solution for the force excitation system, (c) H2-optimal solution for the motion excitation system, and (d) stability maximization optimal solution

Grahic Jump Location
Fig. 4

Optimal tuning ratios νBopt obtained using three different optimization criteria ((c) solid line, numerical solution; dashed line, perturbation solution): (a) H-optimal solution for the force excitation system, (b) H2-optimal solution for the force excitation system, (c) H2-optimal solution for the motion excitation system, and (d) stability maximization optimal solution

Grahic Jump Location
Fig. 5

Optimal damping ratios ζ3opt obtained using three different optimization criteria ((c) solid line, numerical solution; dashed line, perturbation solution): (a) H-optimal solution for the force excitation system, (b) H2-optimal solution for the force excitation system, (c) H2-optimal solution for the motion excitation system, and (d) stability maximization optimal solution

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Fig. 6

Minimized or maximized performance indices obtained using the three optimization criteria: (a) resonance amplitude minimized by the H criterion, (b) Ia minimized by the H2 criterion for the force excitation system, (c) Ia minimized by the H2 criterion for the motion excitation system, and (d) stability index maximized by the stability criterion

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Fig. 7

Optimized responses of the primary system with an attached DVA with μ = 0.05 based on the three different criteria: (a) and (d) show mobility transfer functions, and (b) and (c) show compliance transfer functions. (a) Response optimized by the H criterion, (b) response optimized by the H2 criterion for force excitation, (c) response optimized by the H2 criterion for motion excitation, and (d) response optimized by the stability criterion.

Grahic Jump Location
Fig. 8

Optimized responses of the primary system with an attached DVA with μ = 0.1 based on the three different criteria: (a) and (d) show mobility transfer functions and (b) and (c) show compliance transfer functions. (a) Response optimized by the H criterion, (b) response optimized by the H2 criterion for force excitation, (c) response optimized by the H2 criterion for motion excitation, and (d) response optimized by the stability criterion.

Grahic Jump Location
Fig. 9

Analytical model of a damped single degree-of-freedom system subjected to (a) force and (b) motion excitation

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Fig. 10

Three transfer functions of the force excitation system shown in Fig. 9(a): (a) compliance transfer function, (b) mobility transfer function, and (c) accelerance transfer function

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Fig. 11

Three transfer functions of the motion excitation system shown in Fig. 9(b): (a) compliance transfer function, (b) mobility transfer function, and (c) accelerance transfer function

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