Features of the Dynamics of a Rotating Shaft with Nonlinear Models of Internal Damping and Elasticity

Abstract

The article analyzes the influence of nonlinear (cubic) internal damping (in the Kelvin-Voigt model) and cubic nonlinearity of elastic forces on the dynamics of a rotating flexible shaft with distributed mass. The shaft is modeled by a Bernoulli-Euler rod using the Green function; discretization and reduction of the rotating shaft dynamics problem to an integral equation are performed. It is revealed that in such a system there always exists a branch of limited periodic motions (autovibrations) at a supercritical rotation speed. In addition, with small internal damping, the periodic branch continues into the subcritical region: upon reaching the critical speed, a subcritical Poincare-Andronov-Hopf bifurcation is realized and there is an unstable branch of periodic motions below the branch of stable periodic autovibrations (the occurrence of hysteresis when the rotation speed changes). With an increase in the coefficient of internal friction, the hysteresis phenomenon disappears and at a critical rotation speed, soft excitation of autovibrations of the rotating shaft occurs via the supercritical Poincare-Andronov-Hopf bifurcation.