The non-linear coupling dynamic behaviour of a hydrodynamic bearing-flexible rotor system is analysed. A local iteration method consisting of the improved Wilson-θ method, the predictor-corrector mechanism, and the Newton-Raphson method is proposed to calculate the non-linear dynamic response. Using the proposed method, the iterations are executed only on non-linear degrees of freedom. The iteration process shows improved convergence by taking the prediction value as the initial value. The stability and bifurcation type of periodic responses are determined by the Floquet theory. According to the physical characteristics of the oil film, a variational constraint approach is proposed to revise continuously the variational form of the Reynolds equation at each step of the iterative process. Non-linear oil film forces and their Jacobians are calculated simultaneously without an increase in computational costs, and compatible accuracy is obtained. Numerical results reveal periodic, quasi-periodic, coexisting, and jump solutions of rich and complex non-linear behaviours of the system and show that the proposed methods not only save computational costs but also have high precision.
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