In rotating machinery, aerodynamic forces and oil film forces often lead to cross-coupling stiffness. This paper is aimed at studying the stick-slip whirling vibrations induced by the piecewise smooth rotor/stator frictions in a modified flexible rotor subjected to cross-coupling stiffness. Governing equations determining the sliding region and boundaries of piecewise discontinuous friction are defined. This analytical study was conducted to discuss the complex vibrations and bifurcations. Various types of sliding motions (continuous pure rolling, continuous crossing, and grazing–sliding) were observed in this research. Further, as for discussing the impacts of the parameters on nonlinear sliding vibrations, a parametric study was conducted. The obtained results reveal that with an increase in the cross-coupling stiffness coefficient, continuous pure rolling occurs earlier, and the disk vibration time around the contact regime becomes shorter. For studying the self-excited backward whirling vibration of stick–slip nonlinear motions, analytical formulations are established. Detailed vibration amplitude and frequency studies of friction-induced backward whirling vibrations were carried out. Numerical simulations were performed to compare them with the analytical solutions and to validate the results as well. The proposed theory and results provide fresh perspectives for predicting friction-induced whirlings and creating proper designs for turbo machinery.

Brushless motors are characterized by extreme power density, thermal management and steady dynamic performance which are commonly utilized in aerospace, high-end robotics and precision medical equipment. The nonlinear solutions and the corresponding stabilities in the brushless DC motor system reveal the inherent current-speed properties. In this study, the semi-analytical solutions and the corresponding stabilities in the brushless DC motor system are obtained via a discretized mapping method. The governing equations are discretized into nonlinear polynomials through an implicit mid-point scheme. The semi-analytical solution trees from period-1 to period-2 and period-1 to period-4 are obtained. Some independent periodic solutions are observed. The stability and bifurcations are obtained quantitatively where the period-doubling bifurcations trigger the bifurcation trees and saddle-node bifurcations bound the independent solutions. Interestingly, unstable bifurcation trees are also observed. For verification, numerical simulation is conducted. The stable and unstable properties of current–velocity coupling dynamics are discussed finally.