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.

References

References
1.
Gendelman
,
O. V.
,
Manevitch
,
L. I.
,
Vakakis
,
A. F.
, and
M’Closkey
,
R.
,
2001
, “
Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
68
(
1
), pp.
34
41
.10.1115/1.1345524
2.
Vakakis
,
A. F.
, and
Gendelman
,
O. V.
,
2001
, “
Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture
,”
ASME J. Appl. Mech.
,
68
(
1
), pp.
42
48
.10.1115/1.1345525
3.
Gendelman
,
O. V.
,
2004
, “
Bifurcations of Nonlinear Normal Modes of Linear Oscillator With Strongly Nonlinear Damped Attachment
,”
Nonlinear Dyn.
,
37
, pp.
115
128
.10.1023/B:NODY.0000042911.49430.25
4.
Quinn
,
D. D.
,
Gendelman
,
O. V.
,
Kerschen
,
G.
,
Sapsis
,
T. P.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2008
, “
Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures: Part I
,”
J. Sound Vib.
,
311
(
3–5
), pp.
1228
1248
.10.1016/j.jsv.2007.10.026
5.
Sapsis
,
T. P.
,
Vakakis
,
A. F.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
,
Kerschen
,
G.
, and
Quinn
,
D. D.
,
2009
, “
Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures: Part II, Analytical Study
,”
J. Sound Vib.
,
325
(
1–2
), pp.
297
320
.10.1016/j.jsv.2009.03.004
6.
Vakakis
,
A. F.
,
McFarland
,
D. M.
,
Bergman
,
L. A.
,
Manevitch
,
L. I.
, and
Gendelman
,
O. V.
,
2004
, “
Isolated Resonance Captures and Resonance Capture Cascades Leading to Single-or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators
,”
ASME J. Vib. Acoust.
,
126
(
2
), pp.
235
244
.10.1115/1.1687397
7.
Gourdon
,
E.
,
Alexander
,
N. A.
,
Taylor
,
C. A.
,
Lamarque
,
C. H.
, and
Pernot
,
S.
,
2007
, “
Nonlinear Energy Pumping Under Transient Forcing With Strongly Nonlinear Coupling: Theoretical and Experimental Results
,”
J. Sound Vib.
,
300
(
3–5
), pp.
522
551
.10.1016/j.jsv.2006.06.074
8.
McFarland
,
D. M.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2005
, “
Experimental Study of Non-Linear Energy Pumping Occurring at a Single Fast Frequency
,”
Int. J. Nonlinear Mech.
,
40
(
6
), pp.
891
899
.10.1016/j.ijnonlinmec.2004.11.001
9.
Kerschen
,
G.
,
McFarland
,
D. M.
,
Kowtko
,
J. J.
,
Lee
,
Y. S.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2007
, “
Experimental Demonstration of Transient Resonance Capture in a System of Two Coupled Oscillators With Essential Stiffness Nonlinearity
,”
J. Sound Vib.
,
299
(
4–5
), pp.
822
838
.10.1016/j.jsv.2006.07.029
10.
Sapsis
,
T. P.
,
Quinn
,
D. D.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2012
, “
Effective Stiffening and Damping Enhancement of Structures With Strongly Nonlinear Local Attachments
,”
ASME J. Vib. Acoust.
,
134
(
1
), p.
011016
.10.1115/1.4005005
11.
Jiang
,
X.
,
McFarland
,
D. M.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2003
, “
Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results
,”
Nonlinear Dyn.
,
33
, pp.
87
102
.10.1023/A:1025599211712
12.
Malatkar
,
P.
, and
Nayfeh
,
A. H.
,
2007
, “
Steady-State Dynamics of a Linear Structure Weakly Coupled to an Essentially Nonlinear Oscillator
,”
Nonlinear Dyn.
,
47
, pp.
167
179
.10.1007/s11071-006-9066-4
13.
Bellet
,
R.
,
Cochelin
,
B.
,
Herzog
,
P.
, and
Mattei
,
P. O.
,
2010
, “
Experimental Study of Targeted Energy Transfer From an Acoustic System to a Nonlinear Membrane Absorber
,”
J. Sound Vib.
,
329
(
14
), pp.
2768
2791
.10.1016/j.jsv.2010.01.029
14.
Mariani
,
R.
,
Bellizzi
,
S.
,
Cochelin
,
B.
,
Herzog
,
P.
, and
Mattei
,
P. O.
,
2011
, “
Toward an Adjustable Nonlinear Low Frequency Acoustic Absorber
,”
J. Sound Vib.
,
330
(
22
), pp.
5245
5258
.10.1016/j.jsv.2011.03.034
15.
Lamarque
,
C. H.
,
Gendelman
,
O. V.
,
Ture Savadkoohi,
A.
, and
Etcheverria
,
E.
,
2011
, “
Targeted Energy Transfer in Mechanical Systems by Means of Non-Smooth Nonlinear Energy Sink
,”
Acta Mech.
,
221
(
1
), pp.
175
200
.10.1007/s00707-011-0492-0
16.
Manevitch
,
L. I.
,
2001
, “
The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables
,”
Nonlinear Dyn.
,
25
, pp.
95
109
.10.1023/A:1012994430793
17.
Gourc
,
E.
,
Michon
,
G.
,
Seguy
,
S.
, and
Berlioz
,
A.
,
2011
, “
Experimental Investigation and Theoretical Analysis of a Nonlinear Energy Sink Under Harmonic Forcing
,”
ASME
Paper No. DETC2011-48090. 10.1115/DETC2011-48090
18.
Ture Savadkoohi
,
A.
,
Lamarque
,
C. H.
, and
Dimitrijevic
,
Z.
,
2012
, “
Vibratory Energy Exchange Between a Linear and a Nonsmooth System in the Presence of the Gravity
,”
Nonlinear Dyn.
,
70
(2), pp.
1473
1483
.10.1007/s11071-012-0548-2
19.
Starosvetsky
,
Y.
, and
Gendelman
,
O. V.
,
2008
, “
Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink With Harmonic Forcing and Frequency Detuning
,”
J. Sound Vib.
,
315
(
3
), pp.
746
765
.10.1016/j.jsv.2007.12.023
20.
Starosvetsky
,
Y.
, and
Gendelman
,
O. V.
,
2008
, “
Strongly Modulated Response in Forced 2DOF Oscillatory System With Essential Mass and Potential Asymmetry
,”
Physica D
,
237
(
13
), pp.
1719
1733
.10.1016/j.physd.2008.01.019
21.
Starosvetsky
,
Y.
, and
Gendelman
,
O. V.
,
2008
, “
Attractors of Harmonically Forced Linear Oscillator With Attached Nonlinear Energy Sink. II: Optimization of a Nonlinear Vibration Absorber
,”
Nonlinear Dyn.
,
51
, pp.
47
57
.10.1007/s11071-006-9168-z
22.
Starosvetsky
,
Y.
, and
Gendelman
,
O. V.
,
2010
, “
Bifurcations of Attractors in Forced System With Nonlinear Energy Sink: The Effect of Mass Asymmetry
,”
Nonlinear Dyn.
,
59
(
4
), pp.
711
731
.10.1007/s11071-009-9572-2
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