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Research Papers

Dynamics of Cubic and Vibro-Impact Nonlinear Energy Sink: Analytical, Numerical, and Experimental Analysis

[+] Author and Article Information
Tao Li

ICA (Institut Clément Ader), CNRS,
INSA, ISAE, UPS, Mines Albi,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: tli@insa-toulouse.fr

Sébastien Seguy

ICA (Institut Clément Ader),
CNRS, INSA, ISAE, UPS, Mines Albi,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: sebastien.seguy@insa-toulouse.fr

Alain Berlioz

ICA (Institut Clément Ader), CNRS,
INSA, ISAE, UPS, Mines Albi,
Université de Toulouse,
3 rue Caroline Aigle,
Toulouse F-31400, France
e-mail: alain.berlioz@univ-tlse3.fr

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 1, 2015; final manuscript received January 18, 2016; published online April 12, 2016. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 138(3), 031010 (Apr 12, 2016) (9 pages) Paper No: VIB-15-1415; doi: 10.1115/1.4032725 History: Received October 01, 2015; Revised January 18, 2016

This paper is devoted to study and compare dynamics of primary linear oscillator (LO) coupled to cubic and vibro-impact (VI) nonlinear energy sink (NES) under transient and periodic forcing. The classic analytical procedure combining the approach of invariant manifold and multiple scales is extended from the analysis of steady-state resonance to other regimes, especially strongly modulated response (SMR). A general equation governing the variation of motion along the slow invariant manifold (SIM) is obtained. Numerical results show its convenience to explain the transition from steady-state response to SMR and the characteristics of SMR for periodic forcing. Targeted energy transfer (TET) under transient forcing can also be well understood. Experimental results from LO coupled to VI NES under periodic forcing confirm the existence of SMR and its properties (e.g., chaotic). They also verify the feasibility of the general equation to explain complicated case like SMR in experiments.

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Figures

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Fig. 1

Representation of the LO coupled to a cubic NES

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Fig. 2

SIM of cubic NES: two stable branches in thin line and one unstable branch in thick line

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Fig. 3

Representation of the LO coupled to a VI NES

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Fig. 4

Representation of the nonsmooth functions Π(z) and M(z)

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Fig. 5

SIM of VI NES: one stable branch in bold line and two unstable branches in fine line

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Fig. 6

Cubic NES under transient forcing with parameters K = 800, G = 0 and initial conditions x0=0.02, x˙0=0, y0=0, and y˙0=0. (a) Displacement of LO with cubic NES. (b) Displacement of cubic NES. (c) Relative displacement between LO and cubic NES. (d) SIM and trace between LO and cubic NES: black curve represents projected motion.

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Fig. 7

Cubic NES under periodic forcing with parameters K = 4500, G = 0.02 and initial conditions x0=0, x˙0=0, y0=0, and ∈y˙0=0. (a) Displacement of the center of gravity. (b) Displacement of cubic NES. (c) Relative displacement between LO and cubic NES. (d) SIM and trace between LO and cubic NES: black curve represents projected motion.

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Fig. 8

Cubic NES under periodic forcing with parameters K = 4500, G = 0.02 and initial conditions x0=0, x˙0=0, y0=0, and y˙0=0. (a) Envelope of v and y. (b) Phase difference between v and y. (c) Instantaneous frequency of v (HT). (d) Instantaneous frequency of y (HT). (e) WT spectrum of v. (f) WT spectrum of y.

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Fig. 9

VI NES under transient forcing with parameters B = 0.04, G = 0 and initial conditions x0=0.02, x˙0=0, y0=0.06, and y˙0=0. (a) Displacement of LO. (b) Displacement of VI NES. (c) Relative displacement between LO and VI NES. (d) SIM and trace between LO and VI NES.

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Fig. 13

Case of chaotic SMR: (a) acceleration of LO, (b) acceleration of LO in line with crosses, displacement (mm) of LO in line with the largest amplitude, and displacement (mm) of forcing in line with the smallest amplitude, and (c) enlarged view

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Fig. 12

Picture of the experimental setup: (a) global view of the system and (b) detailed view of the VI NES

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Fig. 11

VI NES under periodic forcing with parameters B = 0.04, G = 0, and σ = 0 and initial conditions x0=0.02, x˙0=0, y0=0.06, and y˙0=0. (a) Envelope of displacement of v and y. (b) Phase difference between v and y. (c) Instantaneous frequency of v (HT). (d) Instantaneous frequency of y (HT). (e) WT spectrum of v. (f) WT spectrum of y.

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Fig. 10

VI NES under periodic forcing with parameters B = 0.04, G = 0 and initial conditions x0=0.02, x˙0=0, y0=0.06, and y˙0=0. (a) Displacement of the center of gravity. (b) Displacement of VI NES. (c) Relative displacement between LO and VI NES. (d) SIM and trace between LO and VI NES.

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