Deterministic and Stochastic Oscillations of a Flexible Cylinder in Arrays of Static Tubes

This study concentrates on vortex-induced vibrations of one flexiblecylinder in an array of fixed tubes. To describe approximately thedynamics of this system we generalize a previously developed linearsemi-empirical model that includes memory effects. We choose a cubicdamping term to model adequately vortex-induced oscillations and weobtain thus a nonlinear integro-differential equation governing thedisplacement of the cylinder. We use a two-variable expansion to derivecriteria for the appearance of stable, periodic, nonlinear oscillations.This approach predicts the appearance of a limit cycle and gives acriterion for the stability of the oscillations. In a stochasticoscillation model we apply additive white noise to the otherwisedeterministic oscillator model. For small values of the noise intensitywe can approximately solve the Fokker–Planck equation. A comparison ofthis approximation with numerical simulations shows a satisfactorydegree of agreement.