A Generalized Analytical Framework for Prediction of Self-Excited Backward Whirl in Geometrically Nonlinear Rotor/Stator Rubbing Systems

Traditional analytical models for dry friction backward whirl predominantly rely on linear shaft assumptions, failing to capture amplitude-dependent frequency shifts and nonlinear response dynamics. For the geometrically nonlinear rotor/stator rubbing systems characterized by isotropic, smooth odd-polynomial restoring forces with arbitrary truncation orders, a generalized analytical framework for prediction of self-excited backward whirl is presented in this paper. By expanding the restoring force via a Taylor series with an arbitrary truncation order, a response decomposition strategy is established to treat the steady-state response as a superposition of a synchronous forced forward response and a self-excited backward whirl. A harmonic balance procedure is subsequently employed to decouple these components, yielding closed-form analytical solutions for both stability thresholds and orbital topologies. With the aid of numerical simulations, the proposed framework is validated across multiple degrees of nonlinearity, demonstrating excellent agreement with low root-mean-square errors when the relative sliding duration ratio exceeds 0.7. Crucially, compared with traditional linear rotors, geometric nonlinearity introduces a strong stiffness-hardening effect, resulting in reduced, self-limiting whirl amplitudes and pronounced upward frequency shifts. Furthermore, the instability boundaries of dry friction backward whirl and backward full-annular quasi-periodic rub are revealed to be governed by a nonlinear synergistic excitation driven by the dynamic coupling of dry friction, unbalance forces and geometric nonlinearity. This study provides a computationally efficient analytical tool capable of evaluating the physical qualitative trends and quantitative boundaries of nonlinear rotors for the stability assessment and vibration control of modern nonlinear rotor systems.