Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft

The paper considers the free vibration of a Jeffcott rotor whose shaft has a strong non-linear elastic property. The mathematical model of the Jeffcott rotor is a second-order non-linear differential equation with a complex deflection function. An analytic procedure based on the Krylov-Bogolubov method is developed to solve this differential equation. Two different types of initial conditions are considered. The obtained solution describes the oscillatory motion of the rotor center. The influence of the damping, hydrodynamic and gyroscopic force and the variation of the mass of the rotor on the vibrations of the rotor is then analyzed.