0
Research Papers

Passive Suppression Mechanisms in Laminar Vortex-Induced Vibration of a Sprung Cylinder With a Strongly Nonlinear, Dissipative Oscillator

[+] Author and Article Information
Antoine Blanchard

Department of Aerospace Engineering,
University of Illinois,
Urbana, IL 61801
e-mail: ablancha@illinois.edu

Lawrence A. Bergman

Department of Aerospace Engineering,
University of Illinois,
Urbana, IL 61801

Alexander F. Vakakis

Department of Mechanical Science
and Engineering,
University of Illinois,
Urbana, IL 61801

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 1, 2017; final manuscript received May 30, 2017; published online June 14, 2017. Assoc. Editor: Walter Lacarbonara.

J. Appl. Mech 84(8), 081003 (Jun 14, 2017) (19 pages) Paper No: JAM-17-1120; doi: 10.1115/1.4036942 History: Received March 01, 2017; Revised May 30, 2017

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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Williamson, C. H. K. , 1996, “ Vortex Dynamics in the Cylinder Wake,” Annu. Rev. Fluid Mech., 28(1), pp. 477–539. [CrossRef]
Williamson, C. H. K. , and Govardhan, R. , 2004, “ Vortex-Induced Vibrations,” Annu. Rev. Fluid Mech., 36(1), pp. 413–455. [CrossRef]
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. [CrossRef]
Bearman, P. W. , 2011, “ Circular Cylinder Wakes and Vortex-Induced Vibrations,” J. Fluids Struct., 27(5), pp. 648–658. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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).
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.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
Hartlen, R. T. , and Currie, I. G. , 1970, “ Lift-Oscillator Model of Vortex-Induced Vibration,” J. Eng. Mech. Div., 96(5), pp. 577–591.
Iwan, W. , and Blevins, R. , 1974, “ A Model for Vortex Induced Oscillation of Structures,” ASME J. Appl. Mech., 41(3), pp. 581–586. [CrossRef]
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.
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. [CrossRef]
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
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.
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.
Mittal, S. , and Singh, S. , 2005, “ Vortex-Induced Vibrations at Subcritical Re,” J. Fluid Mech., 534, pp. 185–194. [CrossRef]
Giannetti, F. , and Luchini, P. , 2007, “ Structural Sensitivity of the First Instability of the Cylinder Wake,” J. Fluid Mech., 581, pp. 167–197. [CrossRef]
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. [CrossRef]
Zebib, A. , 1987, “ Stability of Viscous Flow Past a Circular Cylinder,” J. Eng. Math., 21(2), pp. 155–165. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Joseph, D. D. , 1967, “ Parameter and Domain Dependence of Eigenvalues of Elliptic Partial Differential Equations,” Arch. Ration. Mech. Anal., 24(5), pp. 325–351. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Schmid, P. J. , 2010, “ Dynamic Mode Decomposition of Numerical and Experimental Data,” J. Fluid Mech., 656, pp. 5–28. [CrossRef]
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).
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.
Iooss, G. , and Adelmeyer, M. , 1992, Topics in Bifurcation Theory and Applications, World Scientific, London.
Guckenheimer, J. , and Holmes, P. , 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin.
Kuznetsov, Y. A. , 1995, Elements of Applied Bifurcation Theory, Springer Verlag, New York.
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. [CrossRef]
Gai, G. , and Timme, S. , 2016, “ Nonlinear Reduced-Order Modelling for Limit-Cycle Oscillation Analysis,” Nonlinear Dyn., 84(2), pp. 991–1009. [CrossRef]
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.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Guckenheimer, J. , Hoffman, K. , and Weckesser, W. , 2005, “ Bifurcations of Relaxation Oscillations Near Folded Saddles,” Int. J. Bifurcation Chaos, 15(11), pp. 3411–3421. [CrossRef]
Guckenheimer, J. , Wechselberger, M. , and Young, L.-S. , 2006, “ Chaotic Attractors of Relaxation Oscillators,” Nonlinearity, 19(3), pp. 701–720. [CrossRef]
Benoit, E. , Callot, J. L. , Diener, F. , and Diener, M. , 1981, “ Chasse au canard (première partie),” Collect. Math., 32(1), pp. 37–76.
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.
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.
Wiggins, S. , 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, Berlin.

Figures

Grahic Jump Location
Fig. 1

Linearly sprung circular cylinder in cross-flow equipped with a translational nonlinear energy sink

Grahic Jump Location
Fig. 2

Time series of y, z, w, and CL for a type-I solution with ε = 0.03 and λ = 0.18. The gray-dashed lines indicate LCO amplitudes for the NES-less standard-VIV case.

Grahic Jump Location
Fig. 3

Wavelet spectra of y, z, w, and CL for a type-I solution with ε = 0.03 and λ = 0.18: (a) wavelet spectrum of y, (b) wavelet spectrum of z, (c) wavelet spectrum of w, and (d) wavelet spectrum of CL

Grahic Jump Location
Fig. 4

Time series of y, z, w, and CL for a type-II solution with ε = 0.01 and λ = 0.26. The gray-dashed lines indicate LCO amplitudes for the NES-less standard-VIV case.

Grahic Jump Location
Fig. 5

Wavelet spectra of y, z, w, and CL for a type-II solution with ε = 0.01 and λ = 0.26: (a) wavelet spectrum of y, (b) wavelet spectrum of z, (c) wavelet spectrum of w, and (d) wavelet spectrum of CL

Grahic Jump Location
Fig. 6

Time series of y, z, w, and CL for a type-III solution with ε = 0.04 and λ = 0.42. The gray-dashed lines indicate LCO amplitudes for the NES-less standard-VIV case.

Grahic Jump Location
Fig. 7

Wavelet spectra of y, z, w, and CL for a type-III solution with ε = 0.04 and λ = 0.42: (a) wavelet spectrum of y, (b) wavelet spectrum of z, (c) wavelet spectrum of w, and (d) wavelet spectrum of CL

Grahic Jump Location
Fig. 8

Time series of y, z, w, and CL for a type-IV solution with ε = 0.12 and λ = 0.3

Grahic Jump Location
Fig. 9

Wavelet spectra of y, z, w and CL for a type-IV solution with ε = 0.12 and λ = 0.3: (a) wavelet spectrum of y, (b) wavelet spectrum of z, (c) wavelet spectrum of w, and (d) wavelet spectrum of CL

Grahic Jump Location
Fig. 10

Snapshots of the spanwise vorticity at τ = 3600 for type-I solution in (c), and in the limit cycle at an instant when the cylinder crosses the midplane with positive velocity otherwise: (a) fixed cylinder, (b) NES-less linearly sprung cylinder, (c) type-I solution with ε = 0.01 and λ = 0.26, (d) type-II solution with ε = 0.03 and λ = 0.18, (e) type-III solution with ε = 0.04 and λ = 0.42, and (f) type-IV solution with ε = 0.12 and λ = 0.3

Grahic Jump Location
Fig. 11

Strouhal number (top) and lift coefficient (bottom) for an NES-less linearly sprung cylinder at Re = 48 with m *= 10 and fn*=0.167 obtained by spectral-element computation

Grahic Jump Location
Fig. 12

Slow invariant manifold (SIM) for λ = 0.32 and ϖ=ℑ[λ1]=0.9333 (here, λ/ϖ<1/3). The unstable branch is shown with a dashed line, and the stable branches with solid lines.

Grahic Jump Location
Fig. 13

Evolution of the super-slow flow on the slow invariant manifold for (a) complete LCO suppression for ε = 0.12 and λ = 0.3, (b) partial LCO suppression for ε = 0.07 and λ = 0.2, (c) no LCO suppression for ε = 0.02 and λ = 0.12, and (d) relaxation oscillations for ε = 0.03 and λ = 0.15. Arrows represent the path of the trajectory along the SIM.

Grahic Jump Location
Fig. 14

Theoretical predictions for regions of existence of LCO suppression mechanisms. The dashed line indicates Shilnikov bifurcations.

Grahic Jump Location
Fig. 15

Comparison between integration of the 3DOF ROM (7) and theoretical predictions for regions of existence of the LCO suppression mechanisms. Upright squares correspond to SMRs, diamonds to no LCO suppression, triangles to partial LCO suppression, and dots to complete LCO suppression.

Grahic Jump Location
Fig. 16

Projection of the trajectory computed by integration of the ROM (7) onto the slow invariant manifold for (a) an SMR with ε = 0.02 and λ = 0.3, (b) a no-suppression mechanism with ε = 0.01 and λ = 0.2, and (c) a partial-suppression mechanism with ε = 0.05 and λ = 0.26

Grahic Jump Location
Fig. 17

Comparison between spectral-element computations and theoretical predictions for regions of existence of the LCO suppression mechanisms. Upright squares correspond to type-I solutions, diamonds to type-II solutions, triangles to type-III solutions, and dots to type-IV solutions.

Grahic Jump Location
Fig. 18

Projection of the trajectory from spectral-element computation onto the slow invariant manifold for (a) a type-I solution with ε = 0.01 and λ = 0.26, (b) a type-II solution with ε = 0.04 and λ = 0.42, and (c) a type-III solution with ε = 0.12 and λ = 0.3. The closed curve in (b) denotes the long-time periodic orbit.

Grahic Jump Location
Fig. 19

Illustration of the bifurcation between regions B and C for λ = 0.3 and (a) ε = 0.08 and (b) ε = 0.07

Grahic Jump Location
Fig. 20

Illustration of the bifurcation between regions C and D for λ = 0.12 and (a) ε = 0.085 and (b) ε = 0.055

Grahic Jump Location
Fig. 21

Illustration of the bifurcation between regions A and C for λ = 0.03 and (a) ε = 0.19 and (b) ε = 0.14

Grahic Jump Location
Fig. 22

Illustration of the bifurcation between regions A and D for λ = 0.1 and (a) ε = 0.04 and (b) ε = 0.03

Grahic Jump Location
Fig. 23

Time series of y, z, w, and CL for an SMR with ε = 0.02 and λ = 0.3 obtained by numerical integration of the 3DOF ROM(7)

Grahic Jump Location
Fig. 24

Time series of y, z, w, and CL for a no-suppression mechanism with ε = 0.01 and λ = 0.2 obtained by numerical integration of the 3DOF ROM (7)

Grahic Jump Location
Fig. 25

Time series of y, z, w, and CL for a partial-suppression mechanism with ε = 0.05 and λ = 0.26 obtained by numerical integration of the 3DOF ROM (7)

Grahic Jump Location
Fig. 26

Time series of y, z, w, and CL for a complete-suppression mechanism with ε = 0.12 and λ = 0.3 obtained by numerical integration of the 3DOF ROM (7)

Grahic Jump Location
Fig. 27

Time series of y, z, w, and CL for a complete-suppression mechanism with ε = 0.12 and λ = 0.3 obtained by numerical integration of the 3DOF ROM (7). Integration is performed with zero initial conditions, except for C˙L(0)=10−12.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In