Numerical simulation of the chaotic lateral vibrations of long rotating beams

The purpose of the present study is to investigate the global nonlinear dynamic behavior of long circular beams spinning about their longitudinal axes. Special attention is given to the stimulation of chaotic regimes. Contrary to the usual formulations which apply a fortiori energy methods, the dynamic equilibrium system is obtained using a Newtonian derivation procedure. The final mathematical model that governs the three dimensional nonlinear dynamics of the rotating beam is further elaborated in the time domain by employing a combination of finite difference solvers that rely on the second-order accurate Keller Box and Crank-Nicolson schemes. Several numerical tests have been performed assuming a drillstring structural model. The calculations show that the evolutions of the dynamic components that govern the kinematics of the concerned mechanical system are indeed chaotic. That feature was demonstrated in several different ways and in particular through Lyapunov exponents, phase space Poincaré maps and Power Spectral Densities of the time dependent variables.