Multiple attractors and critical parameters and how to find them numerically: the right,the wrong and the gambling way

In recent years, several authors have proposed ‘easier numerical methods’ to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a linearisation is allowed and the curious fact that the hunting motion is more robust than the ideal stationary-state motion on the track. Concepts such as ‘multiple attractors’, ‘subcritical and supercritical bifurcations’, ‘permitted linearisation’, ‘the danger of running at supercritical speeds’ and ‘chaotic motion’ are addressed.     

Bifurcation analysis of an automobile model negotiating a curve

The paper deals with the bifurcation analysis of a rather simple model describing an automobile negotiating a curve. The mechanical model has two degrees of freedom and the related equations of motion contain the nonlinear tyre characteristics. Bifurcation analysis is adopted as the proper procedure for analysing steady-state cornering. Two independent parameters referring to running conditions, namely steering angle and speed, are varied. Ten different combinations of front and rear tyre characteristics (featuring understeer or oversteer automobiles) are considered for the bifurcation analysis. Many different dynamical behaviours of the model are obtained by slightly varying the parameters describing the tyre characteristics. Both simple and extremely complex bifurcations may occur. Homoclinic bifurcations, stable and unstable limit cycles (of considerable amplitude) are found, giving a sound and ultimate interpretation to some actual (rare but very dangerous) dynamic behaviours of automobiles, as reported by professional drivers. The presented results are cross-validated by exploiting handling diagram theory. The knowledge of the derived set of bifurcations is dramatically important to fully understand the actual vehicle yaw motions occurring while running on an even surface. Such a knowledge is a pre-requisite for robustly designing the chassis and for enhancing the active safety of vehicles.     

汽车系统的分岔研究

以1／2汽车系统为力学模型，考虑了悬架系统和轮胎刚度的非线性效应，以悬架系统的非线性系数为参变量，分析研究了在多频周期激励作用下系统的分岔特性。研究结果表明系统存在着Hopf分岔的倍周期分岔。     

Flange Force Effects on the Motion of a Train Wheelset

The motion of a train wheelset is investigated using Hopf bifurcation theory. The method takes full account of the nonlinear effects of the flange-rail contact forces which are incorporated in the model; The numerical solution is obtained over a wide range of forward speeds by transforming the bifurcation problem into a regular nonlinear boundary value problem, which is solved by standard methods. This solution is shown to be orbitally, asymptotically stable. The algorithm supplies complete information on the lateral and yaw motions and on the period of oscillation, even for very high forward speeds.     

Nonlinear dynamic behaviour of a railway wheelset

Nonlinear dynamic systems are well known to contain certain characteristic additional phenomena compared with linear systems. One example is the fact that nonlinear systems can have multiple stable solutions for one set of parameters. In that case, which one of the multiple stable solutions will be realised will depend on the initial conditions. From the domains of initial conditions which are domains of attraction, probabilities of occurrence of the stable solutions can be calculated. The described dynamical behaviour is studied in the present paper using two examples. To introduce the basic phenomena, the well-known academic Duffing oscillator with harmonic excitation is considered. Domains of attraction are shown for the two stable solutions and the probability of occurrence of the two solutions in the case of equally distributed initial conditions is calculated. The main example to be considered in this paper is the railway wheelset, which is known to show (depending on the nonlinearities in the model) a subcritical Hopf bifurcation depending on the vehicle speed. In that case a range of speeds exists, where a stable trivial solution and a stable limit cycle solution coexist in addition to an unstable limit cycle solution. Again domains of attraction of the two stable solutions are calculated and their dependence on the vehicle speed is shown. The probabilities of occurrence of the two stable solutions are calculated, where special interest is given to the probability of occurrence of the undesired limit cycle solution.     

极限工况下周期转向汽车侧向动力稳定性及分岔分析

将汽车在极限工况下的转向操作建模为周期转向激励的非线性振动系统的响应,结合打靶法和Floquet理论分析了周期转向的稳定性和分岔行为.应用Poincare映射分析了不同的质心位置和不同的车速对极限工况下汽车转向频率特性的影响.结果表明,汽车周期转向时可能发生鞍结分岔和倍周期分岔,发生鞍结分岔还常伴有跳跃现象;极限工况下汽车的转向频率特性与常规工况有很大的不同,转向频率对汽车的稳定性有显著影响,必须采用非线性分析方法.     

A Nonlinear Analysis of the Generic Types of Loss of Stability of the Steady State Motion of a Tractor-Semitrailer*

All different types of loss of stability which occur generically for a tractor semitrailer vehicle are studied when varying two parameters namely the speed V of the vehicle and the position d of the center of mass of the trailer. For a fixed value of d and varying V it turns out that only two cases either a divergence or a Hopf bifurcation can occur typically. By means of a nonlinear analysis the post-bifurcation behavior for both cases is treated showing that it is critical in both cases. This latter result means that the system is an imperfection sensitive one for which the calculation only of the linear stability limit, does not have very much practical meaning, because small perturbations of the system (changes of parameters) can lead to a drastic reduction of the critical speed. Our paper furthermore indicates how a nonlinear investigation of stability problems in vehicle dynamics with no restriction to the number of degrees of freedom of the system can be done in a straight forward manner.     

Analysis of bifurcation and stability for a tractor semi-trailer in planar motion

This paper is intended for bifurcation analysis of a nonlinear tractor semi-trailer vehicle model in planar motion and for investigating its stability under constant running conditions. Bifurcation analysis shows that bifurcation diagrams of a tractor semi-trailer are quite different from those of a single-unit vehicle. Some instability phenomena of the vehicle system such as jackknifing, sideslip, and spinning are explained by correlating them with the behaviour in the neighbourhood of unstable fixed points based on analysis of eigenvectors, phase trajectories, and status of lateral tyre force saturation. It is also found that yaw planar instability of a tractor semi-trailer is caused by lateral tyre force saturation of the tractor's rear axles and/or the trailer's axles. Moreover, the stability region in the state space is demarcated, and a stability index for evaluating size of the stability region in a feasible domain is proposed. Yaw stability under constant driving conditions is analysed by using the proposed stability index.     

A Nonlinear Analysis of the Generic Types of Loss of Stability of the Steady State Motion of a Tractor-Semitrailer∗

SUMMARY

All different types of loss of stability which occur generically for a tractor semitrailer vehicle are studied when varying two parameters namely the speed V of the vehicle and the position d of the center of mass of the trailer. For a fixed value of d and varying V it turns out that only two cases either a divergence or a Hopf bifurcation can occur typically. By means of a nonlinear analysis the post-bifurcation behavior for both cases is treated showing that it is critical in both cases. This latter result means that the system is an imperfection sensitive one for which the calculation only of the linear stability limit, does not have very much practical meaning, because small perturbations of the system (changes of parameters) can lead to a drastic reduction of the critical speed. Our paper furthermore indicates how a nonlinear investigation of stability problems in vehicle dynamics with no restriction to the number of degrees of freedom of the system can be done in a straight forward manner.     

考虑万向节间隙的汽车摆振系统动力学建模

在经典三自由度前轮摆振模型的基础上,基于非线性动力学理论,建立了考虑万向节运动副间隙的三自由度摆振模型。利用Poincare截面法绘制了车轮摆角幅值随车速变化的分岔特性曲线,分析了车速变化引起的某非独立悬架汽车前轮自激摆振的Hopf分岔,验证了在特定车速范围内发生自激励摆振的运动特性。     

非线性闭环汽车系统直线行驶稳定性分析

详细研究了驾驶员－汽车闭环系统直线行驶稳定性问题。在线性范围内分析了驾驶员预瞄时间和轮胎刚度对临界车速的影响；在非线性领域内运用Hopf定理等非线性理论研究了当系统失去直线行驶稳定性后的特殖运动形式，并以某一国产汽车为例，验证了该车失去直线行驶稳定性后，将出现稳定的蛇行运动。     

某大客车转向轮摆振及影响因素的分析

考虑侧偏角和侧偏力之间的非线性关系,建立4自由度某大客车转向轮摆振模型。运用非线性动力学理论,确定转向轮发生自激摆振的Hopf分岔点：通过MATLAB＆Simulink软件建立大客车转向轮摆振仿真模型。通过仿真结果检验理论分析,并得到车速、转向系参数、横拉杆参数、质心位置、主销后倾角与转向轮自激摆振幅值或频率的关系曲线．找出影响自激摆振的敏感参数及敏感参数的适取范围,能有效减小或消除客车的自激摆振。     

Stabilization of High-Speed Railway Vehicles

The problems of critical speeds of railway vehicles with dry friction units determination are discussed. A new approach is used which extends the field of application of dynamic response linear analysis methods to vital nonlinear multibody systems. The special features concerning the influence of dry friction forces in the body supports on the trucks and parameters of horizontal constraints of wheelsets and truck frames on critical speed are indicated. It is shown that a significant rising of railway vehicle critical speeds can be reached by changing the structure of constraints between the body and the trucks.     

Stabilization of High-Speed Railway Vehicles

SUMMARY

The problems of critical speeds of railway vehicles with dry friction units determination are discussed. A new approach is used which extends the field of application of dynamic response linear analysis methods to vital nonlinear multibody systems. The special features concerning the influence of dry friction forces in the body supports on the trucks and parameters of horizontal constraints of wheelsets and truck frames on critical speed are indicated. It is shown that a significant rising of railway vehicle critical speeds can be reached by changing the structure of constraints between the body and the trucks.     

柴油超临界喷雾大涡数值模拟研究

随着现代内燃机燃烧室内温度和压力的不断提升,液体燃料往往处于超临界状态,与亚临界状态相比,其喷雾射流行为特征发生了很大的变化.本研究采用大涡数值模拟方法对超临界环境下的单组分和多组分柴油表征燃料喷雾射流进行数值计算,对比了不同气体状态方程预测柴油表征燃料超临界射流行为的差异性,发现PR气体状态方程对正庚烷质量分数分布的...     

The effect of wheel set gyroscopic action on the hunting stability of high-speed trains

This study explores the effects of wheel set gyroscopic action on hunting stability by calculating linear and nonlinear critical speeds. First, a dynamic model for a high-speed vehicle with 23 degrees of freedom is developed by considering wheel set gyroscopic action. The linear and nonlinear critical speeds are calculated by eigenvalue analysis and drawing a bifurcation map, respectively. Two computer programs for linear and nonlinear stability analysis are developed. Second, based on an actual high-speed vehicle in China, the effects of wheel set gyroscopic action on hunting stability are quantitatively investigated using computer simulation. Furthermore, the difference between the effects of gyroscopic moments about the x-axis and z-axis is discussed. The results show that the moment about the x-axis is harmful to hunting stability, but the moment about the z-axis is beneficial to hunting stability. However, the integrated effect of these two moments can enhance the critical speeds and suppress the hunting motion.     

Fault detection and isolation for a nonlinear railway vehicle suspension with a Hybrid Extended Kalman filter

Fault detection is considered to be one way to improve system reliability and dependability for railway vehicles. The secondary lateral and anti-yaw dampers are the most critical parts in railway suspension systems. So far, the dampers have been modelled as linear components in the fault detection and isolation observer design. In this work, a Hybrid Extended Kalman filter is used to capture the nonlinear characteristics of the dampers. In order to detect and isolate faults, a nonlinear residual generator is developed, which can distinguish clearly between different types of faults. A lateral half train model serves as an example for the proposed technique. The results show that failures in the nonlinear suspension system can be detected and isolated accurately.