0
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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ormondroyd, J. , and Den Hartog, J. P. , 1928, “ The Theory of the Dynamic Vibration Absorber,” ASME J. Appl. Mech., 50, pp. 9–22.
Crandall, S. H. , and Mark, W. D. , 1963, Random Vibration in Mechanical Systems, Academic Press, New York, p. 71.
Yamaguchi, H. , 1988, “ Damping of Transient Vibration by a Dynamic Absorber,” Trans. JSME, Ser. C, 54(499), pp. 561–568, (in Japanese). [CrossRef]
Nishihara, O. , and Matsuhisa, H. , 1997, “ Design of a Dynamic Vibration Absorber for Minimization of Maximum Amplitude Magnification Factor (Derivation of Algebraic Exact Solution),” Trans. JSME, Ser. C, 63(614), pp. 3438–3445, (in Japanese). [CrossRef]
Nishihara, O. , and Asami, T. , 2002, “ Closed-Form Solutions to the Exact Optimizations of Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors),” ASME J. Vib. Acoust., 124(4), pp. 576–582. [CrossRef]
Asami, T. , and Nishihara, O. , 2003, “ Closed-Form Solution to H Optimization of Dynamic Vibration Absorbers (Application to Different Transfer Functions and Damping Systems),” ASME J. Vib. Acoust., 125(3), pp. 398–405. [CrossRef]
Warburton, G. B. , 1982, “ Optimum Absorber Parameters for Vibration Combinations of Response and Excitation Parameters,” Earthquake Eng. Struct. Dyn., 10(3), pp. 381–401. [CrossRef]
Nishihara, O. , and Matsuhisa, H. , 1997, “ Design and Tuning of Vibration Control Devices Via Stability Criterion,” Dynamics and Design Conference’97 (in Japanese), No. 97-10-1, pp. 165–168.
Febbo, M. , and Vera, S. A. , 2008, “ Optimization of a Two Degree of Freedom System Acting as a Dynamic Vibration Absorber,” ASME J. Vib. Acoust., 130(1), p. 011013. [CrossRef]
Chun, S. , Baek, H. , and Kim, T. , 2014, “ Discussion: Optimization of a Two Degree of Freedom System Acting as a Dynamic Vibration Absorber,” ASME J. Vib. Acoust., 136(2), p. 025502. [CrossRef]
Sinha, A. , 2015, “ Optimal Damped Vibration Absorber: Including Multiple Modes and Excitation Due to Rotating Unbalance,” ASME J. Vib. Acoust., 137(6), p. 064501. [CrossRef]
Tursun, M. , and Eskinat, E. , 2014, “ H2 Optimization of Damped-Vibration Absorbers for Suppressing Vibrations in Beams With Constrained Minimization,” ASME J. Vib. Acoust., 136(2), p. 021012. [CrossRef]
Iwanami, K. , and Seto, K. , 1984, “ An Optimum Design Method for the Dual Dynamic Damper and Its Effectiveness,” Bull. JSME, 27(231), pp. 1965–1973. [CrossRef]
Kamiya, K. , Kamagata, K. , Matsumoto, S. , and Seto, K. , 1996, “ Optimal Design Method for Multi Dynamic Absorber,” Trans. JSME, Ser. C, 62(601), pp. 3400–3405, (in Japanese). [CrossRef]
Yasuda, M. , and Pan, G. , 2003, “ Optimization of Two-Series-Mass Dynamic Vibration Absorber and Its Vibration Control Performance,” Trans. JSME, Ser. C, 69(688), pp. 3175–3182, (in Japanese). [CrossRef]
Pan, G. , and Yasuda, M. , 2005, “ Robust Design Method of Multi Dynamic Vibration Absorber,” Trans. JSME, Ser C, 71(712), pp. 3430–3436, (in Japanese). [CrossRef]
Zuo, L. , 2009, “ Effective and Robust Vibration Control Using Series Multiple Tuned-Mass Dampers,” ASME J. Vib. Acoust., 131(3), p. 031003. [CrossRef]
Zuo, L. , and Cui, W. , 2013, “ Dual-Functional Energy Harvesting and Vibration Control: Electromagnetic Resonant Shunt Series Tuned Mass Dampers,” ASME J. Vib. Acoust., 135(5), p. 051018. [CrossRef]
Den Hartog, J. P. , 1956, Mechanical Vibrations, 4th ed., McGraw-Hill, New York, pp. 96–100.
Issa, J. S. , 2013, “ Optimal Design of a Damped Single Degree of Freedom Platform for Vibration Suppression in Harmonically Forced Undamped Systems,” ASME J. Vib. Acoust., 135(5), p. 051003. [CrossRef]
Tong, X. , Liu, Y. , Cui, W. , and Zuo, L. , 2015, “ Analytical Solutions to H2 and H Optimizations of Resonant Shunted Electromagnetic Tuned Mass Damper and Vibration Energy Harvester,” ASME J. Vib. Acoust., 138(1), p. 011018. [CrossRef]
Argentini, T. , Belloli, M. , and Borghesani, P. , 2015, “ A Closed-Form Optimal Tuning of Mass Dampers for One Degree-of-Freedom Systems Under Rotating Unbalance Forcing,” ASME J. Vib. Acoust., 137(3), p. 034501. [CrossRef]
Tamura, S. , and Shiotani, T. , 2012, “ Design of Dynamic Absorber for Two DOF System by Fixed Points Theory (1st Report, Case of Two DOF System With Identical Mass and Stiffness),” Trans. JSME, Ser. C, 78(786), pp. 372–381, (in Japanese). [CrossRef]
Tomimuro, T. , and Tamura, S. , 2015, “ Design of Dynamic Absorber for Two DOF System by Fixed Points Theory (2nd Report, Case of the System with Different Mass and Stiffness, and Excited Secondary Mass),” Trans. JSME, 81(825), p. 14-00622, (in Japanese).
Weisstei , and Eric, W. , 2016, “ Parseval's Theorem,” from MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/ParsevalsTheorem.html.
Hahnkamm, E. , 1932, “ Die Dämpfung von Fundamentschwingungen bei veränderlicher Erregergrequenz,” Ing. Arch., 4, pp. 192–201, (in German). [CrossRef]
Brock, J. E. , 1946, “ A Note on the Damped Vibration Absorber,” ASME J. Appl. Mech., 13(4), p. A-284.
Asami, T. , Nishihara, O. , and Baz Amr, M. , 2002, “ Analytical Solutions to H and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems,” ASME J. Vib. Acoust., 124(2), pp. 284–295. [CrossRef]
Kreyszig, E. , 1999, Advanced Engineering Mathematics, 8th ed., Wiley, New York, p. 784.
Binmore, K. , and Davies, J. , 2007, Calculus Concepts and Methods, Cambridge University Press, Cambridge, p. 190.

Figures

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
Fig. 1

Analytical models of DVAs attached to an undamped primary system

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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)

Tables

Errata

Discussions

Related

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In