Continuation‐Enhanced Harmonic Balance Method for Nonlinear Dynamics in Rotating Machinery

Abstract

Nonlinear response conception in rotating machinery, particularly in systems with squeeze film dampers (SFDs), pose significant challenges for established time-domain numerical methods due to bifurcations and critical points along the response trajectories. Although the continuation-embedded harmonic balance method (HBM) has been widely applied to address these issues, convergence reliability near folds, bifurcations, and other strongly nonlinear regions remains a practical constraint. Furthermore, owing to limited availability of literature, the systematic evaluation of predictor–corrector continuation algorithms on solution robustness and computational efficiency also remains underexplored. Therefore, this study tends to implement a novel HB-AFT framework incorporating an adaptive step-size control strategy for the continuation parameter to maintain convergence across regions of strong nonlinearity. The solver autonomously adjusts the step size to achieve numerically stable forward and backward sweeps of the continuation parameter. Four predictor–corrector continuation methods, namely, the arc length, orthogonal, local, and normal are applied to study frequency response of two nonlinear case studies: the duffing oscillator and the SFD-supported Jeffcott rotor system. The system dynamics in each case are first expressed in time domain and then transformed into frequency domain via a truncated Fourier’s expansion followed by tracking of unstable solution branches through Floquet stability analysis. Extensive numerical simulations on the two nonlinear case studies reveal distinct performance trade-offs among the continuation schemes. Arc-length continuation provides superior robustness near bifurcations, orthogonal continuation achieves favorable computational efficiency, while local and normal parametrizations offer balanced but less reliable convergence. By quantifying computational effort alongside solution accuracy, the present study tends to deliver a practical guideline for selecting continuation strategies in nonlinear rotordynamic simulations. This robust framework is thus well suited for advanced rotordynamic analysis and has potential applications in aeroengine design, vibration control, and structural health monitoring.