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Research Papers

Optimal Design of an Inerter-Based Dynamic Vibration Absorber Connected to Ground

[+] Author and Article Information
Shaoyi Zhou

LaMCoS, CNRS UMR5259, INSA-Lyon,
University of Lyon,
Lyon, F-69621, France
e-mail: shaoyi.zhou@insa-lyon.fr

Claire Jean-Mistral

LaMCoS, CNRS UMR5259, INSA-Lyon,
University of Lyon,
Lyon, F-69621, France
e-mail: claire.jean-mistral@insa-lyon.fr

Simon Chesne

LaMCoS, CNRS UMR5259, INSA-Lyon,
University of Lyon,
Lyon, F-69621, France
e-mail: simon.chesne@insa-lyon.fr

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received March 19, 2019; final manuscript received May 28, 2019; published online June 19, 2019. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 141(5), 051017 (Jun 19, 2019) (11 pages) Paper No: VIB-19-1125; doi: 10.1115/1.4043945 History: Received March 19, 2019; Accepted May 28, 2019

This paper addresses the optimal design of a novel nontraditional inerter-based dynamic vibration absorber (NTIDVA) installed on an undamped primary system of single degree-of-freedom under harmonic and transient excitations. Our NTIDVA is based on the traditional dynamic vibration absorber (TDVA) with the damper replaced by a grounded inerter-based mechanical network. Closed-form expressions of optimal parameters of NTIDVA are derived according to an extended version of fixed point theory developed in the literature and the stability maximization criterion. The transient response of the primary system is optimized when the coupled system becomes defective, namely having three pairs of coalesced conjugate poles, the proof of which is also spelt out in this paper. Moreover, the analogous relationship between NTIDVA and electromagnetic dynamic vibration absorber is highlighted, facilitating the practical implementation of the proposed absorber. Finally, numerical studies suggest that compared with TDVA, NTIDVA can decrease the peak vibration amplitude of the primary system and enlarge the frequency bandwidth of vibration suppression when optimized by the extended fixed point technique, while the stability maximization criterion shows an improved transient response in terms of larger modal damping ratio and accelerated attenuation rate.

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References

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Figures

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Fig. 1

A SDOF undamped primary structure under force excitation controlled by (a) TDVA and (b) NTIDVA

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Fig. 2

Normalized vibration amplitude responses of undamped primary system coupled with (a) TDVA and (b) NTIDVA. The same mechanical damping ratios are used in both cases (dotted: ξ = 0, solid: ξ = 0.1, dash-dotted: ξ = ∞). Other tuning parameters remain unchanged for three simulations: (a) μ = 0.05 and α = 0.95 and (b) μ = 0.05, ν = 0.005, α = 1, and β = 0.975.

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Fig. 3

Evolution of three reference frequencies λR12, λR22, and λR32 as a function of the mass ratio μ. Dotted line, λR12; dash-dotted line, λR22; black circle marker, exact solution of λR32; solid line, curve-fitted solution of λR32.

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Fig. 4

(a) A SDOF primary structure controlled by EMDVA and (b) equivalent electrical model of EMSD enclosed by a RLC series shunt circuit

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Fig. 5

Comparison of frequency responses of Eqs. (45) and (46) relevant to the inerter-based mechanical network and the EMSD coupled with RLC series shunt circuit

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Fig. 6

Frequency responses of the primary system controlled by TDVA (marked by dash-dotted line) and NTIDVA (marked by solid line) with tuning parameters optimized according to (a) fixed point theory and (b) stability maximization criterion. The dotted lines correspond to frequency responses without any control. With the mass ratio given as μ = 0.05, the suppression bandwidths normalized by the fundamental frequency of primary system are, respectively, SB1 = 0.164 for TDVA and SB2 = 0.221 for NTIDVA.

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Fig. 7

Frequency responses of relative displacement between the main mass and mass of TDVA (or NTIDVA) as marked by dash-dotted line (or solid line) with mass ratio being μ = 0.05 and other parameters tuned by (a) fixed point theory and (b) stability maximization criterion

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Fig. 8

Evolution of two performance indices characterizing the transient response against the mass ratio μ: (a) degree of stability Λ and (b) system damping ratio ξ*. Dash-dotted line, fixed point theory; solid line, stability maximization criterion; lines without marker, TDVA; lines with circle marker, NTIDVA.

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Fig. 9

Root loci of SDOF primary system coupled with (a) TDVA and (b) NTIDVA. The square markers correspond to eigenvalues when tuned by the fixed point theory, and the circle markers are relevant to poles when calibrated by the stability maximization criterion. With the mass ratio set as μ = 0.05, four performance indices read as follows: ΛTDVA,fpt = 0.061, ΛTDVA,smc = 0.109, ΛNTIDVA,fpt = 0.051, and ΛNTIDVA,smc = 0.210.

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Fig. 10

Normalized transient responses of the primary system controlled by a TDVA (as marked by dash-dotted line) and a NTIDVA (marked by solid line) with parameters tuned by (a) fixed point theory and (b) stability maximization criterion. The mass ratio is set as: μ = 0.10, and the initial conditions are x1(0) = z0 = 0.1, x1(0) = x2(0) = x2(0) = x3(0) = x3(0) = x4(0) = x4(0) = 0.

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Fig. 11

Normalized transient responses of relative displacement between the main mass and mass of TDVA (or NTIDVA) as marked by dash-dotted line (or solid line) with parameters tuned by (a) fixed point theory and (b) stability maximization criterion. The mass ratio is imposed as: μ = 0.10 and the initial conditions are x1(0) = z0 = 0.1 and x1(0) = x2(0) = x2(0) = x3(0) = x3(0) = x4(0) = x4(0) = 0.

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Fig. 12

Comparison of TIDVA from Ref. [33] and the proposed NTIDVA: (a) normalized frequency response of primary system when optimized by the extended fixed point technique for the mass ratio given as μ = 0.05; (b) and (c) evolution of two performance indices, Λ and ξ*, against the mass ratio μ when calibrated by the stability maximization criterion. Dotted line, without any control; dash-dotted line, TIDVA; solid line, NTIDVA.

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