  Vortex-induced vibration is a nonlinear phenomenon that can damage buildings
or produce energy. Here, the range of synchronization between fluid and
structure is an important parameter. Using a coupled van der Pol and linear
oscillator system, we derived an analytic, closed-form expression for the
synchronization range as a function of fluid shedding frequency, or lock-in as
it's also called. Our new definition is an improvement to an earlier
synchronization definition as it includes the effect of dissipation; without it
hysteresis is missing. The fluid speed range with hysteresis is found to be
non-monotonic with forcing strength while the region of absolute
synchronization increases monotonically. Our proposed synchronization
definition can be a useful tool when designing structural dampers or energy
harvesters but is a general solution for all systems that can be modeled with
similarly coupled equations.
