A harmonic motion of a cylinder acts as a forcing source to the near wake. Depending on the amplitude and frequency of this motion, the lift exerted on the cylinder may be synchronized or not synchronized with the frequency of the motion. Here, we numerically investigate the behavior of the wake, particularly the induced forces on the cylinder, at a low Reynolds number when the forcing frequency is varied from half to twice the shedding frequency of the wake. Within the synchronization region, an abrupt change in the wake, reflected in a discontinuity in the lift, leads to two different flow modes. In the first mode, increasing the motion frequency causes the lift to increase; whereas in the second mode, this relation is reversed. We analyzed the behavior outside the synchronization region, where the lift is not synchronized with the motion. Poincar sections, phase portraits, and spectral analysis were used to characterize different behaviors (e.g., period-n, quasi-periodic, and chaos) of the lift and drag. Also, we performed nonstationary analysis in which the cylinder frequency was varied with different rates and observed typical nonstationary responses in which the jumps were eliminated.

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