The first goal of this work was to develop models based on nonlinear ordinary-differential equations or nonlinear algebraic equations, which produce the lift and drag coefficients on a cylinder or a cylinder-like structure. We introduced an improved wake oscillator for the lift, which combines the van der Pol and Duffing equations. We proposed a two-term quadratic model that relates the drag and lift coefficients, which reproduces the phase relationship between the drag and lift and its variation with the Reynolds number. We found that a mixed-type (external and parametric) forcing is needed to represent the effects of the cylinder motion.

The second goal of this work was to develop a deeper understanding of the shedding and fluid forces on a cylinder and how they depend on its oscillatory motion within and outside the synchronization (or lock-in) band of frequencies. We performed extensive CFD (computational fluid dynamics) simulations and solved the unsteady Reynolds-averaged Navier-Stokes equations that govern the flow fields around fixed and moving (in either the cross-flow or in-line direction) cylinders. We identified various wake modes that can exist, depending on the cylinder motion (direction, amplitude, and frequency) by using modern methods of nonlinear dynamics. The possible responses can be period-one, periodic with large period, quasiperiodic, or chaotic. Moreover, we found that the route to chaos is torus breakdown. We investigated how four frequency sweeps of the cross-flow motion affect the response curves and the hysteresis phenomenon. We studied in detail the effect of the in-line motion on the wake and related this effect to the reduction in the lift and mean drag due to a synchronization type that is very different from the one due to cross-flow motion.