This report describes the impulsive dynamics of a system of two coupled oscillators with essential (nonlinearizable) stiffness nonlinearity. The system considered consists of a grounded weakly damped linear oscillator coupled to a lightweight weakly damped oscillating attachment with essential cubic stiffness nonlinearity arising purely from geometry and kinematics. It has been found that under specific impulse excitations the transient damped dynamics of this system tracks a high-frequency impulsive orbit manifold (IOM) in the frequency-energy plane. The IOM extends over finite frequency and energy ranges, consisting of a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system and being initially at rest and subjected to an impulsive force on the linear oscillator. The damped nonresonant dynamics tracking the IOM then resembles continuous resonance scattering; in effect, quickly transitioning between multiple resonance captures over finite frequency and energy ranges. Dynamic instability arises at bifurcation points along this damped transition, causing bursts in the response of the nonlinear light oscillator, which resemble self-excited resonances. It is shown that for an appropriate parameter design the system remains in a state of sustained high-frequency dynamic instability under the action of repeated impulses. In turn, this sustained instability results in strong energy transfers from the directly excited oscillator to the lightweight nonlinear attachment; a feature that can be employed in energy harvesting applications. The theoretical predictions are confirmed by experimental results.

References

References
1.
Gendelman
,
O.
,
Vakakis
,
A. F.
,
Bergman
,
L. A.
, and
McFarland
,
D. M.
,
2010
, “
Asymptotic Analysis of Passive Nonlinear Suppression of Aeroelastic Instabilities of a Rigid Wing in Subsonic Flow
,”
SIAM J. Appl. Math.
,
70
(
5
), pp.
1655
1677
.10.1137/090754819
2.
Lee
,
Y. S.
,
Kerschen
,
G.
,
Vakakis
,
A. F.
,
Panagopoulos
,
P.
,
Bergman
,
L. A.
, and
McFarland
,
D. M.
,
2005
, “
Complicated Dynamics of a Linear Oscillator With a Light, Essentially Nonlinear Attachment
,”
Physica D
,
204
, pp.
41
69
.10.1016/j.physd.2005.03.014
3.
Andersen
,
D.
,
Vakakis
,
A. F.
,
Starosvetsky
,
Y.
, and
Bergman
,
L.
,
2012
, “
Dynamic Instabilities in Coupled Oscillators Induced by Geometrically Nonlinear Damping
,”
Nonlinear Dyn.
,
67
, pp.
807
827
.10.1007/s11071-011-0028-0
4.
Den Hartog
,
J. P.
,
1956
,
Mechanical Vibrations
,
McGraw-Hill
,
New York
.
5.
Vakakis
,
A. F.
,
2001
, “
Inducing Passive Nonlinear Energy Sinks in Linear Vibrating Systems
,”
ASME J. Vibr. Acoust.
,
123
(
3
), pp.
324
332
.10.1115/1.1368883
6.
Vakakis
,
A. F.
,
Manevitch
,
L. I.
,
Gendelman
,
O.
, and
Bergman
,
L. A.
,
2003
, “
Dynamics of Linear Discrete Systems Connected to Local Essentially Nonlinear Attachments
,”
J. Sound Vib.
,
264
, pp.
559
577
.10.1016/S0022-460X(02)01207-5
7.
Kerschen
,
G.
,
Lee
,
Y.
,
Vakakis
,
A. F.
,
McFarland
,
D. M.
, and
Bergman
,
L. A.
,
2006
, “
Irreversible Passive Energy Transfer in Coupled Oscillators With Essential Nonlinearity
,”
SIAM J. Applied Math.
,
66
(
2
), pp.
648
679
.10.1137/040613706
8.
Gendelman
,
O.
,
Gorlov
,
D.
,
Manevitch
,
L. I.
, and
Musienko
,
A.
,
2005
, “
Dynamics of Coupled Linear and Essentially Nonlinear Oscillators With Substantially Different Masses
,”
J. Sound Vibr.
,
286
, pp.
1
19
.10.1016/j.jsv.2004.09.021
9.
Lee
,
Y. S.
,
Nucera
,
F.
,
Vakakis
,
A. F.
,
McFarland
,
D. M.
, and
Bergman
,
L. A.
,
2009
, “
Periodic Orbits, Damped Transitions and Targeted Energy Transfers in Oscillators With Vibro-Impact Attachments
,”
Physica D
,
238
, pp.
1868
1896
.10.1016/j.physd.2009.06.013
10.
Gendelman
,
O. V.
,
2001
, “
Transition of Energy to Nonlinear Localized Mode in Highly Asymmetric System of Nonlinear Oscillators
,”
Nonlinear Dyn.
,
25
, pp.
237
253
.10.1023/A:1012967003477
11.
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
, pp.
522
551
.10.1016/j.jsv.2006.06.074
12.
Vakakis
,
A. F.
,
Gendelman
,
O.
,
Bergman
,
L. A.
,
McFarland
,
D. M.
,
Kerschen
,
G.
, and
Lee
,
Y. S.
,
2008
,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems
,
Springer-Verlag
,
New York
.
13.
Anderson
,
D.
,
Starosvetsky
,
Y.
,
Mane
,
M.
,
Hubbard
,
S.
,
Remick
,
K.
,
Wang
,
X.
,
Vakakis
,
A.
, and
Bergman
,
L.
,
2012
, “
Non-Resonant Damped Transitions Resembling Continuous Resonance Scattering in Coupled Oscillators With Essential Nonlinearities
,”
Physica D
,
241
, pp.
964
974
.10.1016/j.physd.2012.02.009
14.
Itin
,
A. P.
,
Neishtadt
,
A. I.
, and
Vasiliev
,
A. A.
,
2000
, “
Captures Into Resonance and Scattering on Resonance in Dynamics of a Charged Relativistic Particle in Magnetic Field and Electrostatic Wave
,”
Physica D
,
141
, pp.
281
296
.10.1016/S0167-2789(00)00039-7
15.
Neishtadt
,
A. I.
,
2006
, “
Scattering by Resonances
,”
Celest. Mech. Dyn. Astron.
,
65
(
1-2
), pp.
1
20
.10.1007/BF00048435
16.
Vainchtein
,
D. L.
,
Neishtadt
,
A. I.
, and
Mezic
,
I.
,
2006
, “
On Passage Through Resonances in Volume-Preserving Systems
,”
Chaos
,
16
, p.
043123
.10.1063/1.2404585
17.
Kerschen
,
G.
,
Gendelman
,
O.
,
Vakakis
,
A. F.
,
Bergman
,
L. M.
, and
McFarland
,
D. M.
,
2008
, “
Impulsive Periodic and Quasi-Periodic Orbits of Coupled Oscillators With Essential Stiffness Nonlinearity
,”
Commun. Nonlinear Sci. Numer. Simul.
,
13
, pp.
959
978
.10.1016/j.cnsns.2006.08.001
18.
Masri
,
S. F.
and
Caughey
,
T. K.
,
1979
, “
A Nonparametric Identification Technique for Nonlinear Dynamic Problems
,”
ASME J. Appl. Mech.
,
46
, pp.
433
447
.10.1115/1.3424568
19.
Worden
,
K.
,
1990
, “
Data Processing and Experiment Design for the Restoring Force Surface Method, Part I: Integration and Differentiation of Measured Time Data
,”
Mech. Syst. Signal Process.
,
4
(
4
), pp.
295
319
.10.1016/0888-3270(90)90010-I
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