In this paper we present a computational study of the forces acting on a circular cylinder vibrating both transversely and in-line to a uniform stream. The in-line vibration frequency is equal to twice the transverse frequency. The cylinder thus follows a figure-eight trajectory, emulating the trajectory of a free vortex-induced vibration. We consider three values of transverse oscillation frequency, in the regime of the natural frequency of the Ka´rma´n street, for a Reynolds number of 400. We find that the fluid forces are greatly influenced by the direction in which the figure-eight is traversed. We also find that the spectrum of the lift force is characterized by the strong presence of odd-numbered higher harmonics. Moreover, depending on the combination of oscillation amplitude and frequency, the lift force exhibits aperiodic time dependence.

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