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Research Papers

Response of Pounding Dynamic Vibration Neutralizer Under Harmonic and Random Excitation

[+] Author and Article Information
Sami F. Masri

Viterbi School of Engineering,
University of Southern California,
Los Angeles, CA 90089-2531
e-mail: masri@usc.edu

John P. Caffrey

Viterbi School of Engineering,
University of Southern California,
Los Angeles, CA 90089-2531

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 27, 2018; final manuscript received November 1, 2018; published online November 22, 2018. Assoc. Editor: Ahmet S. Yigit.

J. Appl. Mech 86(2), 021003 (Nov 22, 2018) (15 pages) Paper No: JAM-18-1496; doi: 10.1115/1.4041910 History: Received August 27, 2018; Revised November 01, 2018

Exact steady-state solutions are obtained for the motion of an single-degree-of-freedom (SDOF) system that is provided with a highly nonlinear auxiliary mass damper (AMD), which resembles a conventional dynamic vibration neutralizer (DVN), whose relative motion with respect to the primary system is constrained to remain within a specified gap, thus operating as a “pounding DVN.” This configuration of a conventional DVN with motion-limiting stops could be quite useful when a primary structure with a linear DVN is subjected to transient loads (e.g., earthquakes) that may cause excessive relative motion between the auxiliary and primary systems. Under the assumption that the motion of the nonlinear system under harmonic excitation is undergoing steady-state motion with two impacts per period of the excitation, an exact, closed-form solution is obtained for the system motion. This solution is subsequently used to develop an approximate analytical solution for the stationary response of the pounding DVN when subjected to random excitation with white spectral density and Gaussian probability distribution. Comparison between the analytically estimated rms response of the primary system and its corresponding response obtained via numerical simulation shows that the analytical estimates are quite accurate when the coupling (tuning parameters) between the primary system and the damper are weak, but only moderately accurate when the linear components of the tuning parameters are optimized. It is also shown that under nonstationary, the pounding DVN provides slightly degraded performance compared to the linear one but simultaneously limits the damper-free motion to specified design constraints.

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Figures

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Fig. 1

Model of 2DOF system with bumpers

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Fig. 2

Model of 2DOF system with rigid stops

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Fig. 3

Simulated transient motion segment of a system with μ = 0.05, ζ1 = 0.01, d = 40, e = 0.25 under harmonic excitation of the form f1cost), and f2 = 0, with Ω/ω1 = 1.0, over a period of about three normalized time periods t/T1, after steady-state conditions are achieved. (a) The three curves shown correspond to the normalized displacement of the primary system q1(t)/(f1/k1), the displacement of the auxiliary mass q2(t)/(f1/k1), and the relative displacement between the two masses (q2(t) − q1(t))/(f1/k1); (b) The four curves shown correspond to the normalized displacement of the primary system q1(t)/(f1/k1), the normalized velocity of the primary system q˙1(t)/ω1, the normalized velocity of the auxiliary mass q˙2(t)/ω1, and the normalized relative velocity between the two masses (q˙2(t)−q˙1(t))/ω1, over the same time segment shown in (a). Note the nature of the displayed records corresponding to two symmetric impacts per excitation period.

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Fig. 4

Analytical steady-state solution for a system with μ = 0.05, ζ1 = 0.01, r = 1.0, d = 40, e = 0.25 under harmonic excitation of the form f1cos(Ωt), and f2 = 0, with Ω/ω1 = 1.0, over a period of about three normalized time periods t/T1. (a) The three curves shown correspond to the normalized displacement of the primary system q1(t)/(f1/k1), the displacement of the auxiliary mass q2(t)/(f1/k1), and the relative displacement between the two masses (q2(t) − q1(t))/(f1/k1); (b) the three curves shown correspond to the normalized velocity of the primary system q˙1(t)/ω1, the normalized velocity of the auxiliary mass q˙2(t)/ω1, and the normalized relative velocity between the two masses (q˙2(t)−q˙1(t))/ω1, over the same time segment shown in (a). Note the nature of the displayed records corresponding to two symmetric impacts per excitation period. The displayed segment is constructed by “stitching” four steady-state solutions, each covering half-a-period of the excitation.

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Fig. 5

Simulated motion segment of a system with μ = 0.10, ζ1 = 0.01, r = 1.0, d = 10, e = 0.75 under harmonic excitation of the form f1cos(Ωt), and f2 = 0, with Ω/ω1 = 1.0, over a period of about three normalized time periods t/T1, after steady-state conditions are achieved. (a) The three curves shown correspond the normalized displacement of the primary system q1(t)/(f1/k1), the displacement of the auxiliary mass q2(t)/(f1/k1), and the relative displacement between the two masses (q2(t) − q1(t))/(f1/k1); (b) the four curves shown correspond the normalized displacement of the primary system q1(t)/(f1/k1), the normalized velocity of the primary system q˙1(t)/ω1, the normalized velocity of the auxiliary mass q˙2(t)/ω1, and the normalized relative velocity between the two masses (q˙2(t)−q˙1(t))/ω1, over the same time segment shown in (a). Note the nature of the displayed records corresponding to two symmetric impacts per excitation period.

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Fig. 6

Analytical steady-state solution for system with μ = 0.05, ζ1 = 0.01, r = 1.0, d = 10, e = 0.75 under harmonic excitation of the form f1cos(Ωt), and f2 = 0, with Ω/ω1 = 1.0, over a period of about three normalized time periods t/T1. (a) The three curves shown correspond the normalized displacement of the primary system q1(t)/(f1/k1), the displacement of the auxiliary mass q2(t)/(f1/k1), and the relative displacement between the two masses (q2(t)−q1(t))/(f1/k1); (b) the three curves shown correspond to the normalized velocity of the primary system q˙1(t)/ω1, the normalized velocity of theauxiliary mass q˙2(t)/ω1, and the normalized relative velocity between the two masses (q˙2(t)−q˙1(t))/ω1, over the same time segment shown in (a). Note the nature of the displayed records corresponding to two symmetric impacts per excitation period. The displayed segment is constructed by stitching four steady-state solutions each covering half a period of the excitation.

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Fig. 7

Analytical steady-state frequency response curves for multiple solutions, for system with μ = 0.10, ζ1 = 0.01, d = 40, and e = 0.25 under harmonic excitation of the form f1cos(Ωt), and f2 = 0, with r = Ω/ω1, over a normalized frequency range from 0.96 to 1.04. The three curves shown correspond to the response of a linear SDOF system without a DVN, and the response of the same SDOF when provided with the pounding DVN. The two displayed steady-state solutions correspond to different choices of the phase angle ψ as determined from Eq. (31).

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Fig. 8

Analytical steady-state peak response curves for systems with different combinations of mass ratios μ, coefficients of restitution e, and gap sizes d, when operating under harmonic excitation of the form f1cos(Ωt), and f2 = 0, with r = Ω/ω1 = 1, and ζ1 = 0.01. (a) The three curves shown correspond to a system with μ = 0.05 and three different values of e of 0.25, 0.50, and 0.75. (b) The three curves shown correspond to a system with μ = 0.10 and three different values of e of 0.25, 0.50, and 0.75. For ease of comparison, identical abscissa and ordinate scales are used in both panels of Fig. 8.

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Fig. 9

Excitation and response of reference SDOF system without any vibration control devices; ζ = 0.01

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Fig. 10

Comparison of PSD of excitation and response of reference SDOF system under stationary random excitation. ζ = 0.01.

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Fig. 11

Time history of the reference SDOF system when provided with a pounding DVN in which the coupling elements between m1 and m2 are weak (i.e., ω2/ω1 ≪ 1), and with the primary system damping ζ1 = 0.01, having a mass ratio μ = 0.10 with e = 0.25, under stationary random excitation. (a) Normalized time history of the primary system displacement q1(t); (b) normalized time history of the relative motion (q2(t) − q1(t)).

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Fig. 12

Comparison of numerical simulation results for a nonlinear pounding DVN with a primary system ratio of critical damping ζ1 = 0.01 and a coefficient of restitution e = 0.25. In one case there is soft coupling between m1 and m2 (i.e., ω2/ω1 ≪ 1); in the second case, the coupling parameters (ω2/ω)opt and ζ2opt are optimized, based on a linear 2DOF system with a DVN having the same mass ratio μ: (a) μ = 0.05 and (b) μ = 0.10.

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Fig. 13

Comparison of the PSD of the stationary response of the reference SDOF system with ζ1 = 0.01, μ = 0.10, and e = 0.25 PSD under four conditions: (a) coupling parameters between m2 and m1 are small (i.e., ω2/ω1 ≪ 1 and ζ2 ≪ 1), and d/σ0 = 2; (b) same system as in (a) but with d/σ0 = 4; (c) coupling parameters between m2 and m1 are optimized (i.e., ω2/ω1 = (ω2/ω1)opt and ζ2=ζ2opt), and d/σ0 = 2; and (d) same system as in (c) but with d/σ0 = 4

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Fig. 14

Comparison of estimated and simulated rms response of a pounding DVN under stationary random excitation, in which the coupling elements between m1 and m2 are weak (i.e., ω2/ω1 ≪ 1) and with the primary system damping ζ1 = 0.01. Variation of the normalized rms response with the normalized gap ratio for two different mass ratios μ: (a) μ = 0.05 and (b) μ = 0.10.

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Fig. 15

Nonstationary random excitation and response of an example SDOF system with damping ζ1 = 0.01 when provided with an optimized DVN having a mass ratio μ = 0.10. The broad-band noise modulating envelope has the form g(t)=a1  exp(−a2t)+a3 exp(−a4t). (a) Normalized nonstationary excitation f1(t)/k1 and (b) comparison of motion of primary system m1 without and with an optimized linear DVN (i.e., without gap constraint).

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Fig. 16

Comparison of the nonstationary response of the SDOF system under the earthquake-like excitation shown in Fig. 15 when provided with an optimized DVN having a mass ratio μ = 0.10, and with different gap constraints. (a) Comparison of the motion of the primary system m1 for two different gap ratios and (b) comparison of the relative motion between the primary system m1 and the auxiliary mass m2 for two different gap ratios: d/(f1/k1) ≫ 1 and d/(f1/k1) = 1.5.

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