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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.

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References

Figures

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Fig. 1

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

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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.

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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

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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.

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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

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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.

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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

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Fig. 8

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

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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

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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

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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

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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.

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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.

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Fig. 14

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

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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.

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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

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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.

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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.

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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

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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

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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

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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

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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)

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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)

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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)

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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)

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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.

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