INVESTIGATION OF STABILITY AND BIFURCATION CHARACTERISTICS OF WHEELSET NONLINEAR DYNAMIC MODEL
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摘要: 为了探究轮对系统的横向失稳问题, 考虑了陀螺效应和一系悬挂阻尼的影响作用, 建立非线性轮轨接触关系的轮对动力学模型, 研究轮对系统的蛇行稳定性、Hopf分岔特性及迁移转化机理. 通过稳定性判据获得了轮对系统失稳临界速度. 采用中心流形定理和规范型方法对轮对动力学模型进行化简, 得到与轮对系统分岔特性相同的一维复变量方程, 理论推导求得轮对系统的第一Lyapunov系数的表达式, 根据其符号即可判断轮对系统的Hopf分岔类型. 讨论了不同参数对轮对系统Hopf分岔临界速度的影响, 探究了轮对系统的超临界、亚临界Hopf分岔域在二维参数空间的分布规律. 利用数值模拟得到轮对系统的3种典型Hopf分岔图, 验证了轮对系统超临界、亚临界Hopf分岔域分布规律的正确性. 结果表明, 轮对系统的临界速度随着等效锥度的增大而减小, 随着一系悬挂的纵向刚度和纵向阻尼的增大而增大, 随着纵向蠕滑系数的增大呈先增大后减小. 系统参数变化会引起轮对系统Hopf分岔类型发生改变, 即亚临界与超临界Hopf分岔相互迁移转化. 轮对系统Hopf分岔域在二维参数空间的分布规律对于轮对系统参数匹配和优化设计具有一定的指导意义.Abstract: To explore the lateral instability of the wheelset system, the gyroscopic effect and the influence of the primary suspension damping are considered, a dynamic model of the wheelset system with a nonlinear wheel-rail contact relationship is established, and the hunting stability, Hopf bifurcation characteristics, and migration transformation mechanism are investigated. The hunting instability critical speed of the wheelset system is obtained through the stability criterion. The central manifold theorem is used to reduce the dimensions of the wheelset system. Then the reduced wheelset system is simplified using the normal form method to obtain a one-dimensional complex variable equation with the same bifurcation characteristics as the wheelset system. The expression of the first Lyapunov coefficient of the wheelset system is derived theoretically, and the Hopf bifurcation type of the wheelset system can be judged according to its sign. The influence of different parameters on the Hopf bifurcation critical speed of the wheelset system is discussed, and the distribution law of supercritical and subcritical Hopf bifurcation regions of the wheelset system in two-dimensional parameter space is explored. Three typical Hopf bifurcation diagrams of the wheelset system are obtained by numerical simulation, which verifies the correctness of the distribution law of the supercritical and subcritical Hopf bifurcation regions of the wheelset system. The results reveal that the critical speed of the wheelset system decreases with the increase of the equivalent taper, increases with the increase of the longitudinal stiffness and longitudinal damping of the primary suspension, and first increases and then decreases with the increase of the longitudinal creep coefficient. The change of system parameters will change the type of Hopf bifurcation of the wheelset system, that is, the subcritical and supercritical Hopf bifurcations migrate and transform each other. The distribution law of the Hopf bifurcation domain of the wheelset system in two-dimensional parameter space has a certain guiding significance for wheelset parameter matching and optimization design.
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Key words:
- gyroscopic effect /
- hunting stability /
- center manifold theorem /
- limit cycle /
- Hopf bifurcation
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图 2 轮对系统特征值最大实部的变化曲线
Figure 2. Variation curve of the maximum real part of the eigenvalue of the wheelset system
图 3 轮对系统的临界速度与等效锥度的关系
Figure 3. The relationship between critical speed and equivalent conicity
图 4 轮对系统的临界速度与一系纵向刚度的关系
Figure 4. The relationship between critical speed and primary longitudinal stiffness
图 5 轮对系统的临界速度与一系纵向阻尼的关系
Figure 5. The relationship between critical speed and primary longitudinal damping
图 6 轮对系统的临界速度与纵向蠕滑系数的关系
Figure 6. The relationship between critical speed and longitudinal creep coefficient
图 7 分岔类型临界线在空间
${k_x}{{ - }}{k_y}$ 内的变化规律Figure 7. Variation rule of bifurcation type critical line in parameter space
${k_x}{{ - }}{k_y}$ 
图 8 分岔类型临界线空间
${k_x}{{ - }}{c_x}$ 内的变化规律Figure 8. Variation rule of bifurcation type critical line in parameter space
${k_x{ - }}{c_x}$ 
图 9 分岔类型临界线在空间
${c_x}{{ - }}{c_y}$ 内的变化规律Figure 9. Variation rule of bifurcation type critical line in parameter space
${c_x}{{ - }}{c_y}$ 
图 10 分岔类型临界线在空间
${k_x}{{ - }}{\lambda _e}$ 内的变化规律Figure 10. Variation rule of bifurcation type critical line in parameter space
${k_x}{{ - }}{\lambda _e}$ 
图 11 点
$ {P_1} $ 位置对应的轮对系统Hopf分岔图Figure 11. Hopf bifurcation diagram of wheelset system corresponding to point
$ {P_1} $ 
图 12 点
$ {P_2} $ 位置对应的轮对系统Hopf分岔图Figure 12. Hopf bifurcation diagram of wheelset system corresponding to point
$ {P_2} $ 
图 13 点P位置对应的轮对系统Hopf分岔图
Figure 13. Hopf bifurcation diagram of wheelset system corresponding to point
$ P $ 表 A1 轮对系统参数
Table A1. The parameters of the wheelset system
Parameter Interpretation Value Unit $ m $ mass of the wheelset 1627 kg $ {I_z} $ yaw moment of inertia of wheelset 830 kg·m2 $ {I_y} $ spin moment of inertia of wheelset 100 kg·m2 $ W $ axle load of wheelset 11.22 t $ {r_0} $ centered wheel rolling radius 0.46 m $ l $ half of the swing arm 1.02 m $ a $ half of the lateral distance of rolling circle 0.7465 m $ {f_{11}} $ longitudinal creep coefficient 1.5 × 106 N $ {f_{22}} $ lateral creep coefficient 1.5 × 106 N $ {f_{23}} $ lateral spin creep force coefficient 1000 N $ {f_{33}} $ spin creep force coefficient 1000 N $ {k_y} $ primary lateral stiffness 1.0 MN·m−1 $ {k_x} $ primary longitudinal stiffness 1.0 MN·m−1 $ {c_y} $ primary lateral damping 1000 N·m·s−1 $ {c_x} $ primary longitudinal damping 1000 N·m·s−1 $ {\lambda _e} $ equivalent conicity of the wheelset 0.16 − $ {\delta _1} $ nonlinear wheel-rail contact control parameter −1.6 × 1011 − $ {\delta _2} $ nonlinear wheel-rail contact control parameter 1.6 × 1015 − 
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