Lock-in regimes for Vortex-Induced Vibrations of a cylinder attached to a bistable spring

Abstract

We study Vortex-Induced Vibrations (VIV) of a cylinder placed in uniform flow, attached to bistable springs, with the goal of understanding the effect of the non-linearity of the spring on the size of the lock-in regime. A reduced order Wake Oscillator Model (WOM), used to numerically simulate this system, also forms the basis of our theoretical analysis. We first characterize the natural frequency of the bistable spring as a function of the distance between the potential minima of the spring potential and the maximum displacement amplitude. Using one-way coupled simulations of the wake variable, we also construct a model for oscillations of the wake variable as a function of the displacement amplitude and frequency of structure oscillations. The budget for the average kinetic energy of the structure, involving the balance of rate of energy production (via vortex shedding) with rate of energy dissipation (via fluid damping) implies a universal relationship between the oscillation amplitude and structure frequency, which we term as the ”Equilibrium Constraint” (EC). At high mass ratios, the lock-in condition, coupled with the EC, can be used to obtain a set of possible solutions for the oscillation amplitude of the cylinder, given a particular reduced velocity. We are able to then propose a theoretical model for dependence of oscillation amplitude on reduced velocity for VIV involving linear and bistable springs. For a given value of inter-well distance of the spring potential, the theory is able to predict the range of reduced velocity over which the structure undergoes double-well and single-well oscillations. The WOM simulations confirm that the regimes in parameter space corresponding to larger amplitudes can typically be attributed to the lock-in of the lift force acting on the cylinder with the natural frequency of the spring. We find that the lock-in characteristics of the linear and bistable springs are quite similar if the amplitude dependence of the natural frequency of the bistable spring is taken into account. The theory is able to accurately describe the qualitative behavior of the oscillation amplitude with respect to the spring potential and flow parameters. The theory is also able to successfully explain why the range of reduced velocities over which large amplitude VIV is observed increases for cases involving bistable springs. The results here are potentially useful towards designing systems involving VIV with non-linear springs, such as for enhancing the range of flow velocity where power extraction is achieved in VIV-based energy harvesting devices.