We study cross-flow vortex-induced vibration (VIV) of a linearly sprung circular cylinder equipped with a dissipative oscillator with cubic stiffness nonlinearity, restrained to move in the direction of travel of the cylinder. The dissipative, essentially nonlinear coupling between the cylinder and the oscillator allows for targeted energy transfer (TET) from the former to the latter, whereby the oscillator acts as a nonlinear energy sink (NES) capable of passively suppressing cylinder oscillations. For fixed values of the Reynolds number (Re = 48, slightly above the fixed-cylinder Hopf bifurcation), cylinder-to-fluid density ratio, and dimensionless cylinder spring constant, spectral-element simulations of the Navier–Stokes equations coupled to the rigid-body motion show that different combinations of NES parameters lead to different long-time attractors of the dynamics. We identify four such attractors which do not coexist at any given point in the parameter space, three of which lead to at least partial VIV suppression. We construct a reduced-order model (ROM) of the fluid–structure interaction (FSI) based on a wake oscillator to analytically study those four mechanisms seen in the high-fidelity simulations and determine their respective regions of existence in the parameter space. Asymptotic analysis of the ROM relies on complexification-averaging (CX-A) and slow–fast partition of the transient dynamics and predicts the existence of complete and partial VIV-suppression mechanisms, relaxation cycles, and Hopf and Shilnikov bifurcations. These outcomes are confirmed by numerical integration of the ROM and comparisons with spectral-element simulations of the full system.

References

References
1.
Williamson
,
C. H. K.
,
1996
, “
Vortex Dynamics in the Cylinder Wake
,”
Annu. Rev. Fluid Mech.
,
28
(
1
), pp.
477
539
.
2.
Williamson
,
C. H. K.
, and
Govardhan
,
R.
,
2004
, “
Vortex-Induced Vibrations
,”
Annu. Rev. Fluid Mech.
,
36
(
1
), pp.
413
455
.
3.
Gabbai
,
R. D.
, and
Benaroya
,
H.
,
2005
, “
An Overview of Modeling and Experiments of Vortex-Induced Vibration of Circular Cylinders
,”
J. Sound Vib.
,
282
(
3
), pp.
575
616
.
4.
Bearman
,
P. W.
,
2011
, “
Circular Cylinder Wakes and Vortex-Induced Vibrations
,”
J. Fluids Struct.
,
27
(
5
), pp.
648
658
.
5.
Tumkur
,
R. K. R.
,
Domany
,
E.
,
Gendelman
,
O. V.
,
Masud
,
A.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2013
, “
Reduced-Order Model for Laminar Vortex-Induced Vibration of a Rigid Circular Cylinder With an Internal Nonlinear Absorber
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
7
), pp.
1916
1930
.
6.
Tumkur
,
R. K. R.
,
Calderer
,
R.
,
Masud
,
A.
,
Pearlstein
,
A. J.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2013
, “
Computational Study of Vortex-Induced Vibration of a Sprung Rigid Circular Cylinder With a Strongly Nonlinear Internal Attachment
,”
J. Fluids Struct.
,
40
, pp.
214
232
.
7.
Tumkur
,
R. K. R.
,
2014
, “
Modal Interactions and Targeted Energy Transfers in Laminar Vortex-Induced Vibrations of a Rigid Cylinder With Strongly Nonlinear Internal Attachments
,”
Ph.D. thesis
, University of Illinois at Urbana-Champaign, Champaign, IL.
8.
Tumkur
,
R. K. R.
,
Pearlstein
,
A. J.
,
Masud
,
A.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2017
, “
Effect of an Internal Nonlinear Rotational Dissipative Element on Vortex Shedding and Vortex-Induced Vibration of a Sprung Circular Cylinder
,”
J. Fluid Mech.
(submitted).
9.
Vakakis
,
A. F.
,
Gendelman
,
O. V.
,
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
.
10.
Sigalov
,
G.
,
Gendelman
,
O. V.
,
Al-Shudeifat
,
M. A.
,
Manevitch
,
L. I.
,
Vakakis
,
A. F.
, and
Bergman
,
L.
,
2012
, “
Resonance Captures and Targeted Energy Transfers in an Inertially-Coupled Rotational Nonlinear Energy Sink
,”
Nonlinear Dyn.
,
69
(
4
), pp.
1693
1704
.
11.
Gendelman
,
O. V.
,
Sigalov
,
G.
,
Manevitch
,
L. I.
,
Mane
,
M.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2012
, “
Dynamics of an Eccentric Rotational Nonlinear Energy Sink
,”
ASME J. Appl. Mech.
,
79
(
1
), p.
011012
.
12.
Sigalov
,
G.
,
Gendelman
,
O. V.
,
Al-Shudeifat
,
M. A.
,
Manevitch
,
L. I.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2012
, “
Alternation of Regular and Chaotic Dynamics in a Simple Two-Degree-of-Freedom System With Nonlinear Inertial Coupling
,”
Chaos
,
22
(
1
), p.
013118
.
13.
Mehmood
,
A.
,
Nayfeh
,
A. H.
, and
Hajj
,
M. R.
,
2014
, “
Effects of a Non-Linear Energy Sink (NES) on Vortex-Induced Vibrations of a Circular Cylinder
,”
Nonlinear Dyn.
,
77
(
3
), pp.
667
680
.
14.
Dai
,
H.
,
Abdelkefi
,
A.
, and
Wang
,
L.
,
2016
, “
Vortex-Induced Vibrations Mitigation Through a Nonlinear Energy Sink
,”
Commun. Nonlinear Sci. Numer. Simul.
,
42
, pp.
22
36
.
15.
Gendelman
,
O. V.
,
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
.
16.
Gendelman
,
O.
, and
Bar
,
T.
,
2010
, “
Bifurcations of Self-Excitation Regimes in a Van der Pol Oscillator With a Nonlinear Energy Sink
,”
Physica D
,
239
(
3
), pp.
220
229
.
17.
Domany
,
E.
, and
Gendelman
,
O. V.
,
2013
, “
Dynamic Responses and Mitigation of Limit Cycle Oscillations in Van der Pol–Duffing Oscillator With Nonlinear Energy Sink
,”
J. Sound Vib.
,
332
(
21
), pp.
5489
5507
.
18.
Benarous
,
N.
, and
Gendelman
,
O. V.
,
2016
, “
Nonlinear Energy Sink With Combined Nonlinearities: Enhanced Mitigation of Vibrations and Amplitude Locking Phenomenon
,”
Proc. Inst. Mech. Eng.
,
230
(
1
), pp.
21
33
.
19.
Blanchard
,
A. B.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2016
, “
Capture Into Slow-Invariant-Manifold in the Fluid–Structure Dynamics of a Sprung Cylinder With a Nonlinear Rotator
,”
J. Fluids Struct.
,
63
, pp.
155
173
.
20.
Hartlen
,
R. T.
, and
Currie
,
I. G.
,
1970
, “
Lift-Oscillator Model of Vortex-Induced Vibration
,”
J. Eng. Mech. Div.
,
96
(
5
), pp.
577
591
.
21.
Iwan
,
W.
, and
Blevins
,
R.
,
1974
, “
A Model for Vortex Induced Oscillation of Structures
,”
ASME J. Appl. Mech.
,
41
(
3
), pp.
581
586
.
22.
Nayfeh
,
A. H.
,
Owis
,
F.
, and
Hajj
,
M. R.
,
2003
, “
A Model for the Coupled Lift and Drag on a Circular Cylinder
,”
ASME
Paper No. DETC2003/VIB-48455.
23.
Facchinetti
,
M. L.
,
De Langre
,
E.
, and
Biolley
,
F.
,
2004
, “
Coupling of Structure and Wake Oscillators in Vortex-Induced Vibrations
,”
J. Fluids Struct.
,
19
(
2
), pp.
123
140
.
24.
Fischer
,
P. F.
,
Lottes
,
J. W.
, and
Kerkemeier
,
S. G.
,
2008
, “
Nek5000
,” Argonne National Laboratory, Lemont, IL, accessed June 7, 2017, http://nek5000.mcs.anl.gov
25.
Blanchard
,
A. B.
,
Bergman
,
L. A.
,
Vakakis
,
A. F.
, and
Pearlstein
,
A. J.
,
2016
, “
Multiple Long-Time Solutions for Intermediate Reynolds Number Flow Past a Circular Cylinder With a Nonlinear Inertial and Dissipative Attachment
,”
69th Annual Meeting of the APS Division of Fluid Dynamics, Portland, OR, Nov. 20–22
.
26.
Blanchard
,
A.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2017
, “
Targeted Energy Transfer in Laminar Vortex-Induced Vibration of a Sprung Cylinder With a Nonlinear Dissipative Rotator
,”
Physica D
,
350
, pp. 26–44.
27.
Mittal
,
S.
, and
Singh
,
S.
,
2005
, “
Vortex-Induced Vibrations at Subcritical Re
,”
J. Fluid Mech.
,
534
, pp.
185
194
.
28.
Giannetti
,
F.
, and
Luchini
,
P.
,
2007
, “
Structural Sensitivity of the First Instability of the Cylinder Wake
,”
J. Fluid Mech.
,
581
, pp.
167
197
.
29.
Sipp
,
D.
, and
Lebedev
,
A.
,
2007
, “
Global Stability of Base and Mean Flows: A General Approach and Its Applications to Cylinder and Open Cavity Flows
,”
J. Fluid Mech.
,
593
, pp.
333
358
.
30.
Zebib
,
A.
,
1987
, “
Stability of Viscous Flow Past a Circular Cylinder
,”
J. Eng. Math.
,
21
(
2
), pp.
155
165
.
31.
Noack
,
B. R.
, and
Eckelmann
,
H.
,
1994
, “
A Global Stability Analysis of the Steady and Periodic Cylinder Wake
,”
J. Fluid Mech.
,
270
, pp.
297
330
.
32.
Dušek
,
J.
,
Le Gal
,
P.
, and
Fraunié
,
P.
,
1994
, “
A Numerical and Theoretical Study of the First Hopf Bifurcation in a Cylinder Wake
,”
J. Fluid Mech.
,
264
, pp.
59
80
.
33.
Joseph
,
D. D.
,
1967
, “
Parameter and Domain Dependence of Eigenvalues of Elliptic Partial Differential Equations
,”
Arch. Ration. Mech. Anal.
,
24
(
5
), pp.
325
351
.
34.
Chen
,
K. K.
,
Tu
,
J. H.
, and
Rowley
,
C. W.
,
2012
, “
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
,”
J. Nonlinear Sci.
,
22
(
6
), pp.
887
915
.
35.
Noack
,
B. R.
,
Afanasiev
,
K.
,
Morzynski
,
M.
,
Tadmor
,
G.
, and
Thiele
,
F.
,
2003
, “
A Hierarchy of Low-Dimensional Models for the Transient and Post-Transient Cylinder Wake
,”
J. Fluid Mech.
,
497
, pp.
335
363
.
36.
Deane
,
A. E.
,
Kevrekidis
,
I. G.
,
Karniadakis
,
G. E.
, and
Orszag
,
S. A.
,
1991
, “
Low-Dimensional Models for Complex Geometry Flows: Application to Grooved Channels and Circular Cylinders
,”
Phys. Fluids A
,
3
(
10
), pp.
2337
2354
.
37.
Ma
,
X.
, and
Karniadakis
,
G. E.
,
2002
, “
A Low-Dimensional Model for Simulating Three-Dimensional Cylinder Flow
,”
J. Fluid Mech.
,
458
(1), pp.
181
190
.
38.
Noack
,
B. R.
, and
Eckelmann
,
H.
,
1994
, “
A Low-Dimensional Galerkin Method for the Three-Dimensional Flow Around a Circular Cylinder
,”
Phys. Fluids
,
6
(
1
), pp.
124
143
.
39.
Schmid
,
P. J.
,
2010
, “
Dynamic Mode Decomposition of Numerical and Experimental Data
,”
J. Fluid Mech.
,
656
, pp.
5
28
.
40.
Tumkur
,
R. K. R.
,
Fischer
,
P. F.
,
Bergman
,
L. A.
,
Vakakis
,
A. F.
, and
Pearlstein
,
A. J.
,
2017
, “
Stability of the Steady, Two-Dimensional Flow Past a Linearly-Sprung Circular Cylinder
,”
J. Fluid Mech.
(submitted).
41.
Dowell
,
E.
,
Crawley
,
E.
,
Curtiss
,
H.
, Jr.
,
Peters
,
D.
,
Scanlan
,
R.
, and
Sisto
,
F.
,
1995
,
A Modern Course in Aeroelasticity
,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
.
42.
Iooss
,
G.
, and
Adelmeyer
,
M.
,
1992
,
Topics in Bifurcation Theory and Applications
,
World Scientific
,
London
.
43.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1983
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer
,
Berlin
.
44.
Kuznetsov
,
Y. A.
,
1995
,
Elements of Applied Bifurcation Theory
,
Springer Verlag
,
New York
.
45.
Habib
,
G.
, and
Kerschen
,
G.
,
2015
, “
Suppression of Limit Cycle Oscillations Using the Nonlinear Tuned Vibration Absorber
,”
Proc. R. Soc. A
,
471
(
2176
), p.
20140976
.
46.
Gai
,
G.
, and
Timme
,
S.
,
2016
, “
Nonlinear Reduced-Order Modelling for Limit-Cycle Oscillation Analysis
,”
Nonlinear Dyn.
,
84
(
2
), pp.
991
1009
.
47.
Malher
,
A.
,
Touzé
,
C.
,
Doaré
,
O.
,
Habib
,
G.
, and
Kerschen
,
G.
,
2016
, “
Passive Control of Airfoil Flutter Using a Nonlinear Tuned Vibration Absorber
,”
11th International Conference on Flow-Induced Vibrations
(
FIV
), The Hague, The Netherlands, July 4–6.
48.
Manevitch
,
L. I.
,
2001
, “
The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables
,”
Nonlinear Dyn.
,
25
(
1–3
), pp.
95
109
.
49.
Gendelman
,
O.
, and
Starosvetsky
,
Y.
,
2007
, “
Quasi-Periodic Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink Under Periodic Forcing
,”
ASME J. Appl. Mech.
,
74
(
2
), pp.
325
331
.
50.
Starosvetsky
,
Y.
, and
Gendelman
,
O.
,
2008
, “
Strongly Modulated Response in Forced 2DOF Oscillatory System With Essential Mass and Potential Asymmetry
,”
Physica D
,
237
(
13
), pp.
1719
1733
.
51.
Guckenheimer
,
J.
,
Hoffman
,
K.
, and
Weckesser
,
W.
,
2005
, “
Bifurcations of Relaxation Oscillations Near Folded Saddles
,”
Int. J. Bifurcation Chaos
,
15
(
11
), pp.
3411
3421
.
52.
Guckenheimer
,
J.
,
Wechselberger
,
M.
, and
Young
,
L.-S.
,
2006
, “
Chaotic Attractors of Relaxation Oscillators
,”
Nonlinearity
,
19
(
3
), pp.
701
720
.
53.
Benoit
,
E.
,
Callot
,
J. L.
,
Diener
,
F.
, and
Diener
,
M.
,
1981
, “
Chasse au canard (première partie)
,”
Collect. Math.
,
32
(
1
), pp.
37
76
.
54.
Shilnikov
,
L.
,
1965
, “
A Case of the Existence of a Countable Number of Periodic Motions (Point Mapping Proof of Existence Theorem Showing Neighborhood of Trajectory Which Departs From and Returns to Saddle-Point Focus Contains Denumerable Set of Periodic Motions)
,”
Sov. Math.
,
6
, pp.
163
166
.
55.
Shilnikov
,
L.
,
1967
, “
The Existence of a Denumerable Set of Periodic Motions in Four-Dimensional Space in an Extended Neighborhood of a Saddle-Focus
,”
Sov. Math. Dokl.
,
8
, pp.
54
58
.
56.
Wiggins
,
S.
,
1990
,
Introduction to Applied Nonlinear Dynamical Systems and Chaos
,
Springer-Verlag
,
Berlin
.
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