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Research Papers

# Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing

[+] Author and Article Information
Etienne Gourc

Université de Toulouse,
INSA, Toulouse F-31077, France
e-mail: gourc@insa-toulouse.fr

Guilhem Michon

Université de Toulouse,
ISAE, Toulouse F-31055, France
e-mail: guilhem.michon@isae.fr

Sébastien Seguy

Université de Toulouse,
INSA, Toulouse F-31077, France
e-mail: seguy@insa-toulouse.fr

Alain Berlioz

Université de Toulouse,
UPS, Toulouse F-31062, France
e-mail: alain.berlioz@univ-tlse3.fr

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 25, 2013; final manuscript received November 13, 2013; published online February 5, 2014. Assoc. Editor: Steven W Shaw.

J. Vib. Acoust 136(2), 021021 (Feb 05, 2014) (8 pages) Paper No: VIB-13-1091; doi: 10.1115/1.4026432 History: Received March 25, 2013; Revised November 13, 2013

## Abstract

In this paper, the dynamic response of a harmonically forced linear oscillator (LO) strongly coupled to a nonlinear energy sink (NES) is investigated both theoretically and experimentally. The system studied comprises an LO with an embedded, purely cubic NES. The behavior of the system is analyzed in the vicinity of $1:1$ resonance. The complexification-averaging technique is used to obtain modulation equations and the associated fixed points. These modulation equations are analyzed using asymptotic expansion to study the regimes related to relaxation oscillation of the slow flow, called strongly modulated response (SMR). The zones where SMR occurs are computed using a mapping procedure. The slow invariant manifolds (SIM) are used to derive a proper optimization procedure. It is shown that there is an optimal zone in the forcing amplitude-nonlinear stiffness parameter plane, where SMR occurs without having a high amplitude detached resonance tongue. Two experimental setups are presented. One is not optimized and has a relatively high mass ratio ($≈13%$) and the other one is optimized and exhibits strong mass asymmetry (mass ratio $≈1%$). Different frequency response curves and associated zones of SMR are obtained for various forcing amplitudes. The reported experimental results confirm the design procedure and the possible application of NES for vibration mitigation under periodic forcing.

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## Figures

Fig. 1

Schema of the 2 DOF system comprising an LO and an NES

Fig. 2

Example of SIM for K = 100, λ2 = 0.2

Fig. 3

Illustration of the mapping procedure for K = 100, σ = 1, F = 0.15, ɛ = 0.01, λ1 = 0.1, λ2 = 0.2

Fig. 4

First experimental setup (ε=12.9%)

Fig. 5

Experimental (green) and analytical (blue) frequency response curve of the LO and the NES for the first experiments and F = 2.7N (o: experimental periodic oscillation; *: experimental modulated oscillation; thin line: analytical stable fixed points; thick line: analytical unstable fixed point)

Fig. 6

Experimental measure of weak quasi-periodic response for F = 2.7N, f = 14.5 Hz, ε=12.9%

Fig. 7

Boundary of the saddle-node bifurcation for K = 100, ɛ = 0.01, λ1 = 0.1, λ2 = 0.2

Fig. 8

Critical forcing amplitude as a function of the nonlinear stiffness ɛ = 0.01, λ1 = 0.1, λ2 = 0.2

Fig. 9

Zone of SMR as a function of the forcing amplitude for K = 1000, ɛ = 0.01, λ1 = 0.1, λ2 = 0.2

Fig. 10

General view of the second experimental setup (ɛ = 1.2%)

Fig. 11

Design curve corresponding to physical parameters

Fig. 12

Force-displacement relationship of the designed NES

Fig. 13

Experimental (green) and analytical (blue) frequency response curve of the LO and the NES for the second experiments and G = 0.25 mm (vertical red line: analytically determined zone of SMR; vertical dashed line: experimentally determined zone of SMR; thin black line: theoretical FRF without NES)

Fig. 14

Experimental measurement of SMR with the second experiment with G = 0.25 mm, f = 8.5 Hz, ε=1.2% (point A in Fig. 13)

Fig. 15

Experimental (green) and analytical (blue) frequency response curve of the LO and the NES for the second experiments and G = 0.325 mm (vertical red line: analytically determined zone of SMR; vertical dashed line: experimentally determined zone of SMR)

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