Features of the dynamics of a rotating shaft with nonlinear models of internal damping and elasticity

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