Abstract
Suppressing vortex-induced vibration (VIV) has recently attracted numerous researchers due to its practical significance in many engineering applications. Most of the previous studies have focused on a passive or active flow control. A structurally active control approach to mitigate a two-dimensional, nonlinear coupled, cross-flow/in-line VIV has not been well studied. This paper presents a reduced-order fluid-structure dynamic model and combined analytical–numerical solutions for the efficient suppression of two-dimensional VIV of a flexibly mounted circular cylinder in uniform flows. The theoretical model is based on the use of coupled Duffing–Rayleigh oscillators with three variables describing the cylinder cross-flow/in-line displacements and the strength of the fluid vortex circulation in the cylinder wake. These equations of fluid-structure motions contain geometric and hydrodynamic nonlinearities. Closed-loop linear and nonlinear velocity feedback controllers are implemented in the transverse direction governing the larger cross-flow response than the associated in-line counterpart. Approximated analytical expressions are derived by using the harmonic balance to explicitly capture the system nonlinear dynamic features and the effects of key dimensionless parameters. Parametric investigations are carried out to evaluate the linear versus nonlinear controller performance in terms of the maximum response suppression capability and the power requirement in a wide range of reduced flow velocities, mass ratios, and control gains. Over the main lock-in resonance region with coupled cross-flow/in-line responses, the linear controller is found to be more efficient in suppressing the two-dimensional VIV by also modifying the system frequencies and phase relationships. Nevertheless, based on the power comparison, the nonlinear control is superior to the linear control for very small targeted controlled amplitudes of a very low-mass cylinder. These active control strategies may be further applied to the multimode VIV suppression of a long flexible cylinder with multi degrees of freedom.
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Abbreviations
- \(A_{x}/D, A_{y}/D\) :
-
Cylinder amplitude per diameter
- \(a_{1}-a_{10}\) :
-
Analytical solution coefficients
- \(C_{D0} (C_{L0})\) :
-
Unsteady drag (lift) coefficient of stationary cylinder
- \(c_{s}\) :
-
Structural viscous damping
- \(c_{f}\) :
-
Fluid-added damping
- \(C_{a}\) :
-
Added mass coefficient
- D :
-
Cylinder external diameter
- \(D_{t}\) :
-
Mean drift effect caused by geometric nonlinear coupling
- \(F_x^*, F_y^*\) :
-
Dimensional fluid force
- \(F_{yc}\) :
-
Dimensionless control force
- \(F^{*}_{yc}\) :
-
Dimensional control force
- \(f_{o}\) :
-
Cylinder oscillation frequency
- \(f_{n}\) :
-
Cylinder natural frequency
- G :
-
Dimensional nonlinear gain
- Q :
-
Dimensional linear gain
- K :
-
Cylinder stiffness
- \(m_{s}\) :
-
Cylinder mass
- m :
-
System total mass
- \(m^{*}\) :
-
Mass ratio
- \(m_{a}\) :
-
Fluid-added mass
- p :
-
Wake variable transformation
- P :
-
Dimensionless average power
- \(q (q_{0})\) :
-
Dimensionless displacement (amplitude) of vortex circulation
- Re:
-
Reynolds number
- R :
-
Maximum amplitude reduction percentage
- St:
-
Strouhal number
- t :
-
Dimensional time
- U :
-
Dimensional flow velocity
- \(V_{r}\) :
-
Reduced flow velocity
- X(x):
-
Dimensional (dimensionless) in-line displacement
- Y(y):
-
Dimensional (dimensionless) cross-flow displacement
- \(x_{0}, y_{0}\) :
-
Dimensionless amplitude
- \(q_{M}, x_{M}, y_{M}\) :
-
Dimensionless amplitude at ideal lock-in frequency (\(\delta = \omega =1\))
- \(\alpha _{x}^* \beta _{x}^{*} \alpha _{y}^{*} \beta _{y}^{*}\) :
-
Dimensional geometrically nonlinear coefficients
- \(\alpha _{x} \beta _{x} \alpha _{y} \beta _{y}\) :
-
Dimensionless geometrically nonlinear coefficients
- \(\alpha _{xq}, \alpha _{yq}\) :
-
Hydrodynamic force coefficients
- \(\beta _{K}\) :
-
Dimensionless linear control gain
- \(\beta , \varepsilon , \lambda \) :
-
Empirical wake coefficients
- \(\gamma \) :
-
Stall parameter
- \(\gamma _{\mathrm{G}}\) :
-
Dimensionless nonlinear control gain
- \(\delta \) :
-
Cylinder-to-vortex-shedding frequency ratio
- \(\theta _{xy}\) :
-
x-y relative phase angle
- \(\theta _{qy}\) :
-
q-y relative phase angle
- \(\mu \) :
-
Dimensionless mass parameter
- \(\xi \) :
-
Structural damping ratio
- \(\rho \) :
-
Fluid density
- \(\tau \) :
-
Dimensionless time
- \(\omega \) :
-
Dimensionless oscillation frequency in analytical solution
- \(\omega _{n}\) :
-
Cylinder angular natural frequency
- \(\omega _{st}\) :
-
Vortex-shedding angular frequency
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Ma, B., Srinil, N. Two-dimensional vortex-induced vibration suppression through the cylinder transverse linear/nonlinear velocity feedback. Acta Mech 228, 4369–4389 (2017). https://doi.org/10.1007/s00707-017-1946-9
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DOI: https://doi.org/10.1007/s00707-017-1946-9