Nonlinear responses of unbalanced flexible rotating shaft passing through critical speeds

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for the present work. The shaft is modeled as a beam with a circular cross-section and the Euler Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method is used. Furthermore, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. It is concluded that the magnitude and position of mass unbalance in both longitudinal and radial directions, significantly affect the dynamic behavior of the rotor system. It is also observed that applied torque greatly influence dynamic responses leading to passing through the first 3 critical speeds. These influences are also presented graphically and discussed.