Stochastic Analysis of Self-Induced Vibrations

Vortex-induced vibrations of a structural element are modelled as a non-linear stochastic single-degree-of-freedom system. The deterministic part of the governing equation represents laminar flow conditions with a stationary non-zero solution corresponding to lock-in. Across-wind turbulence is included as an additive excitation and along-wind turbulence is introduced as a parametric excitation term, both assumed to be white noise processes. An approximate closed-form solution to the corresponding Fokker–Planck equation in terms of the stationary probability density of the energy is obtained. The auto spectral density of the position at a particular energy-level is approximated by the spectral density of a linear system with energy dependent damping. The spectral density is then obtained by integration of the energy conditional spectral density over all energies weighted by the probability density. The approximate theoretical expressions for the probability density of the energy and the auto spectral density of the position compare favourably with results obtained by numerical simulation.