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Research Papers

Optimal Design of Double-Mass Dynamic Vibration Absorbers Arranged in Series or in Parallel

[+] 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 July 16, 2016; final manuscript received September 10, 2016; published online November 23, 2016. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 139(1), 011015 (Nov 23, 2016) (16 pages) Paper No: VIB-16-1352; doi: 10.1115/1.4034776 History: Received July 16, 2016; Revised September 10, 2016

This paper considers the optimal design of double-mass dynamic vibration absorbers (DVAs) attached to an undamped single degree-of-freedom system. Three different optimization criteria, the H optimization, H2 optimization, and stability maximization criteria, were considered for the design of the DVAs, and a performance index was defined for each of these criteria. First, the analytical models of vibratory systems with double-mass DVAs were considered, and seven dimensionless parameters were defined. Five of these parameters must be optimized to minimize or maximize the performance indices. Assuming that all dimensionless parameters are non-negative, the optimal value of one parameter for a double-mass DVA arranged in series (series-type DVA) was proven to be zero. The optimal adjustment conditions of the other four parameters were derived as simple algebraic formulae for the H2 and stability criteria and numerically determined for the H criterion. For a double-mass DVA arranged in parallel (parallel-type DVA), all five parameters were found to have nonzero optimal values, and these values were obtained numerically by solving simultaneous algebraic equations. Second, the performance of these DVAs was compared with a single-mass DVA. The result revealed that for all optimization criteria, the performance of the series-type DVA is the best among the three DVAs and that of the single-mass DVA is the worst. Finally, a procedure for deriving the algebraic or numerical solutions for H2 optimization is described. The derivation procedure of other optimal solutions will be introduced in the future paper.

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References

Figures

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Fig. 1

Analytical models of DVAs attached to an undamped primary system

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Fig. 2

Example of the characteristic roots of a three degrees-of-freedom system in the complex plane

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Fig. 3

Single degree-of-freedom system subjected to simple harmonic excitation

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Fig. 4

Frequency response curves of a single degree-of-freedom system at various damping ratios

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Fig. 5

Comparison of exact and approximate solutions for H optimization: (a) optimal tuning condition νopt, (b) optimal damping ratio ζ2opt, and (c) enlarged view of the frequency response curve near the resonance frequencies

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Fig. 6

Steady-state and transient responses of primary system with optimal single-mass DVA: (a) steady-state response to sinusoidal excitation and (b) transient response to the initial velocity υ0

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Fig. 7

Optimized parameters and minimized or maximized performance indices for a single-mass DVA: (a) H optimization (algebraic solution), (b) H2 optimization (algebraic solution), and (c) stability maximization (algebraic solution)

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Fig. 8

Optimized frequency response curves of a primary system with an attached single-mass DVA: (a) H optimization, (b) H2 optimization, and (c) stability maximization

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Fig. 9

Optimized parameters and frequency response curves for negative absorber damping at μ = 0.1: (a) optimal values of four dimensionless parameters and minimized performance index Iamin and (b) frequency response curves of the primary system for several negative damping values

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Fig. 10

Optimized parameters and minimized or maximized performance indices for a series-type double-mass DVA: (a) H optimization (numerical solution), (b) H2 optimization (algebraic solution), and (c) stability maximization (algebraic solution)

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Fig. 11

Optimized frequency response curves of a primary system with an attached series-type double-mass DVA: (a) H optimization, (b) H2 optimization, and (c) stability maximization

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Fig. 12

Optimized parameters and minimized or maximized performance indices for a parallel-type double-mass DVA: (a) H optimization (numerical solution), (b) H2 optimization (numerical solution), and (c) stability maximization (numerical solution)

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Fig. 13

Optimized frequency response curves of the primary system with an attached parallel-type double-mass DVA: (a) H optimization, (b) H2 optimization, and (c) stability maximization

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Fig. 14

Minimized or maximized performance indices of systems containing a single- or double-mass DVA: (a) resonance amplitude ratio hmin, (b) squared area Iamin under the frequency response curve, and (c) stability index Λmax

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Fig. 15

Frequency responses of the primary mass and DVAs optimized using the H criterion: (a) response curves of a system with a single-mass DVA, (b) response curves of a system with a series-type DVA, and (c) response curves of a system with a parallel-type DVA

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Fig. 16

Graphical representation of the eigenvalues λi of Hessian matrices: (a) eigenvalues of the Hessian matrix given by Eq. (A19) and (b) eigenvalues of the Hessian matrix given by Eq. (B6)

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