关键词: 非线性动力学, 分岔, 混沌, 流体动压轴承, 稳定性, 有限元法

Abstract: Based on the variational constraint approach, the variational form of Reynolds equation in hydrodynamic lubrication is revised continuously to satisfy certain constraint conditions in the cavitation zone of oil film field. In accordance with oil film physical character, an isoparametric finite element with eight nodal points method is used to convert the revised variational form of Reynolds equation to a discrete form of finite dimensional algebraic variational equation. By means of this approach, a perturbed equation can be obtained directly on the finite element equation. Consequently, nonlinear oil film forces and their Jacobian matrices are calculated simultaneously, and compatible accuracy is obtained without increase of computational costs. A method, combining a predictor-corrector mechanism to Newton-Raphson method, is presented to calculate equilibrium position and critical speed corresponding to Hopf bifurcation point of bearing-rotor system, as by-product dynamic coefficients of bearing are obtained. The time scale I.e. the unknown whirling period of Hopf bifurcation solution of bearing-rotor system is drawn into the iterative process using PNF method. Stability of the Hopf bifurcation solution can be determined when Hopf bifurcation solution and its period are calculated. The nonlinear unbalanced T periodic responses of the system are obtained by using PNF method and path-following technique. The local stability and bifurcation behaviors of T periodic motions are analyzed by the Floquet theory. Chaotic motions are analyzed by Lyapunov exponents. Periodic, quasi-periodic, chaos, jumped, co-existed multi-solution of rich and complex nonlinear behavior of the system are revealed in the numerical results.

Key words: Nonlinear dynamics, Bifurcation Chaos, Finite element method, Hydrodynamic bearing, Stability