Abstract

The dynamics of spinning shafts with non-constant rotating speed is described by a nonlinear system that under certain conditions might exhibit also chaotic behavior. In this article chaotic dynamics of the spinning shaft is examined. Initially, the trajectories in phase space around the equilibrium manifolds are determined. Then by choosing a set of initial conditions, nearby to an equilibrium, corresponding to eigenvalues of the Jacobian with a nonzero real part, identification of chaos is examined. Approximations of the trajectory, with the linearization curves around the equilibria, are defined and they are good in a region very close to the associated equilibrium point. It is shown that the eigenvalues, as Lyapunov exponents indicators, are not parameter dependent but state dependent. The eigenvalues of the linearized system within an orbit are varying from positive to zero, therefore the Lyapunov exponent is not defined through this limit as an explicit number but variant. The existence of eigenvalues with positive real parts in certain parts of the orbit is an indication of chaos since it shows a divergence of nearby orbits. One orbit starting from an initial condition which corresponds to eigenvalues with positive real part is crossing the threshold and pass to points that the eigenvalues with zero real parts, therefore this ‘threshold’ is not discriminating chaotic with regular regions as expected. The variant positive Lyapunov exponents have been examined also with numerical investigations and it is an indication of chaos. The Poincare section indicates irregular motion and the approximated Information Entropy is relatively high, and both are indicating chaos. It should be highlighted that this is a mechanical system with variant real parts of eigenvalues as Lyapunov exponents within one orbit and the threshold is insufficient to distinguish chaotic from regular regions. Further work is needed to determine the chaotic regions of the spinning shaft. Further developments in the mathematics of nonlinear dynamical systems associated with the equilibrium manifolds are needed to examine the significance of variant Lyapunov exponents for this kind of systems. Also, the necessity to reexamine the validity of existing algorithms and the development of new ones for the determination of variant Lyapunov exponents, become evident.