椭圆轴承-转子系统非线性运动及稳定性分析

运行中的轴承—转子系统,由于油膜出现气穴,存在破裂区域,此时Reynolds方程的变分形式已不能满足。基于变分约束原理,按照油膜的物理特性,在动力积分、迭代过程中实时形成修正的Reynolds方程变分形式的有限元方程及其扰动方程,在不增加计算量的情况下,同时求得了非线性油膜力及其Jacobian矩阵,并且使其具有相互协调一致的精度。将预估—校正机理和Newton-Raphson方法相结合给出了一种轴承—转子系统Hopf分岔点所对应线性失稳转速及轴承动力学系数的计算方法。将时间尺度引入PNF(Poincare-Newton-Floquetl方法求得了系统Hopf分岔极限环解及其涡动周期,判断了该解的稳定性。基于PNF法及将延续算法和PNF法相结合的轨迹预测追踪算法研究了系统非线性不平衡周期响应,结合Floquet理论分析了非线性轴承—转子系统T周期运动的局部稳定性和分岔行为。运用Lyapunov指数分析了系统响应的混沌现象。数值结果展现了系统响应具有丰富复杂的周期、拟周期、多解共存、跳跃和混沌等非线性现象。

STABILITY AND NONLINEAR RESPONSE OF ROTOR SYSTEM WITH ELLIPTICAL SLIDING BEARING SUPPORT

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.