Nonlinear vortex induced vibrations of a rotating beam

Abstract

Theoretical, numerical and experimental study of nonlinear dynamic behaviour of a rotating beam subjected to mechanical excitation and vortex induced vibrations is presented in the dissertation. The theoretical model has been formulated on the basis of extended Euler-Bernoulli beam theory taking into account geometrical nonlinearities and an effect of rotation. Kinetic, potential energies and virtual work of nonconservative forces have been formulated taking into account coupled transversallongitudinal- torsional vibrations of the rotating beam, varied angular velocity and an arbitrary preset angle. On this basis, by means of Hamilton principle the full model composed of a set of partial differential equations and associated boundary conditions have been derived. Then, the model has been reduced to transversal oscillations in susceptible to bending direction combined with longitudinal motion. The partial differential equation of motion have been reduced to a one-degree-of-freedom system by Galerkin method considering the first vibration mode. The reduced model demonstrated hardening of frequency response curve for small angular velocity with a transition to softening for the increase angular velocity. Additionally, the beam has been modelled in a Finite Element Method package and modal analysis have been performed. The created Campbell diagrams demonstrated impact of angular velocity on vibration frequencies and vibration modes. The possibility of a few important modal interactions have been presented. Vortex induced vibration have been studied numerically for 2D problem considering fluid flow through a undeformable beam cross-section. The aerodynamic forces and arising vortices are studied for different preset angles and flow velocities. The simulation results indicated the bifurcation points when vortices occur and Fourier transform allowed to extract the amplitude and frequency of the vortices. Moreover, lift and drag forces have been approximated for different preset angles. The dynamics of rotating beam under aerodynamic forces has also been tested on the dedicated experimental setup. The signals measured and recorded from strain gauges and from high speed cameras have been used for the reduced model validation. The proposed van der Pol model allowed simulations of vortex induced vibrations. By varying van der Pol’s equation parameters various profiles of fluid flow and vortex shedding frequency have been obtained. It has been shown that not only amplitude coming from van der Pol oscillator but also shedding frequency and the profile of the excitation input may essentially modify the beam response leading to its complex dynamics.