Hopf Bifurcation Characteristic Analysis of Straddle-Type Monorail Vehicle Bogie Based on Spatial Perturbations
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摘要:
为探究跨座式单轨车辆转向架的稳定性问题，基于轮轨接触关系下轮胎力学特性的空间激扰，对跨座式单轨车辆转向架系统进行Hopf分岔特性分析. 首先，建立考虑空间激扰的三自由度车辆转向架非线性动力学模型；其次，采用Hurwitz稳定性判据求解转向架系统的临界速度，引入中心流形稳定性指标判断系统的Hopf分岔类型，并利用MATCONT工具包对理论分析结果进行数值验证；最后，探讨不同空间激扰条件对转向架系统稳定性的影响. 研究结果表明：在低速运动状态下转向架以2.4 Hz的侧滚运动为主，中高速运动状态下主要表现为摇头和侧滚的耦合运动为主，频率在1.3 Hz～2.4 Hz；当速度达到163.563 24 km/h时，系统发生超临界Hopf分岔，出现稳定极限环，且在速度为163.563 6 km/h时发生鞍结分岔，系统又出现不稳定极限环；空间激扰作用下，转向架的临界速度随导向轮径向刚度和稳定轮径向阻尼的增大而减小，随导向轮径向阻尼、稳定轮径向刚度以及走行轮径向刚度和阻尼的增加而提高；此外，转向架结构参数的改变可引发系统超临界与亚临界Hopf分岔之间的迁移，为避免亚临界Hopf分岔导致系统运动状态突变，应合理设计转向架的结构参数.
Abstract:To investigate the stability of the bogie in a straddle-type monorail vehicle, the Hopf bifurcation characteristics of the straddle-type monorail vehicle bogie system were analyzed based on the spatial perturbations of the mechanical properties under the wheel-rail contact relationship. Firstly, a three-degree-of-freedom nonlinear dynamic model of the vehicle bogie, incorporating spatial perturbations, was established. Secondly, the Hurwitz stability criterion was employed to solve for the critical speed of the bogie system, and the type of Hopf bifurcation was identified via the center manifold stability index. The theoretical results were further validated numerically using the MATCONT toolbox. Finally, the influence of different spatial perturbation conditions on the stability of the bogie system was discussed. The research results show that at a motion state of low speeds, the bogie primarily exhibits rolling motion with a frequency of 2.4 Hz; at a motion state of medium and high speeds, the motion is dominated by coupled yawing and rolling, with frequencies ranging from 1.3 Hz to 2.4 Hz. When the speed reaches 163.563 24 km/h, the system undergoes a supercritical Hopf bifurcation, resulting in the emergence of a stable limit cycle; at a speed of 163.563 6 km/h, a saddle-node bifurcation occurs, leading to an unstable limit cycle. Under spatial perturbations, the critical speed of the bogie decreases with increasing radial stiffness of the guiding wheel and radial damping of the stabilizing wheel but increases with greater radial damping of the guiding wheel, radial stiffness of the stabilizing wheel, as well as the radial stiffness and damping of the running wheel. Moreover, changes in the structural parameters of the bogie can induce transitions between supercritical and subcritical Hopf bifurcations. To avoid subcritical Hopf bifurcations that may cause abrupt changes in the system’s motion state, the structural parameters of the bogie should be carefully designed.
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Key words:
- straddle-type monorail vehicle /
- stability /
- Hopf bifurcation /
- spatial perturbation /
- critical speed
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图 2 走行轮轮胎侧偏力与侧偏角的关系
Figure 2. Relationship between cornering force and sideslip angle of running wheel tire
图 3 转向架系统的特征值最大实部变化曲线
Figure 3. Maximum real part change curve of eigenvalue of bogie system
图 8 导向轮力学模型与临界速度的关系
Figure 8. Relationship between mechanical model of guiding wheel and critical speed
图 9 稳定轮力学模型与临界速度的关系
Figure 9. Relationship between mechanical model of stabilizing wheel and critical speed
图 10 走行轮力学模型与临界速度的关系
Figure 10. Relationship between mechanical model of running wheel and critical speed
图 11 双参数空间中分岔类型分界线的分布规律
Figure 11. Distribution rule of dividing line of bifurcation types in two-parameter space
表 1 转向架系统模型参数
Table 1. Model parameters of bogie system
参数 定义 数值 m/kg 转向架质量 2400 Jz/(kg·m2) 转向架z向转动惯量 3898 Jx/(kg·m2) 转向架x向转动惯量 980 v/(km·h−1) 转向架运行速度 — $ K_{\alpha }^{\prime} $/(KN·m·rad−1) 走行轮的回正刚度 10.75 Kτ/(KN·rad−1) 走行轮的切向刚度 33.333 Kr/(KN·m−1) 走行轮的径向刚度 1400 Cr/(KN·s·m−1) 走行轮的径向阻尼 9.8 Kg/Ks/(KN·m−1) 导向轮/稳定轮的径向刚度 625 Cg/Cs/(KN·s·m−1) 导向轮/稳定轮的径向阻尼 5 Lg/m 导向轮纵向跨距的1/2 0.6085 b/m 两走行轮轴距的1/2 0.15 hs/m 稳定轮到转向架质心的垂向距离 0.92 hg/m 导向轮到转向架质心的垂向距离 0.14 ht/m 走行轮中心到转向架质心的高度 0.5 
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表 2 轮胎模型参数拟合结果相关性检验
Table 2. Correlation test of tire model parameter fitting results
决定系数 调整决定系数 误差平方和 均方根误差 0.9997 0.9997 1.5972 × 10589.3745
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表 3 系统的特征值
Table 3. Eigenvalues of system
特征值 临界速度 1 8.403i 2 −8.403i 3 −7.793 + 34.946i 4 −7.793 - 34.946i 5 −0.955 + 15.570i 6 −0.955 - 15.570i
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