A Simple Analysis Method for Contact Deformation of Rolling Tire

A method is presented for the dynamic analysis of contact deformation for rolling tire on a road. The tire is modeled as an inextensible ring on elastic foundation with damping. Under known radial deflection, the ground contact region is predicted by solving simplified ring motion equation, treating ground contacting as a boundary condition. Furthermore, the effect of internal damping and rotating speed is analyzed on contact region.     

Index

A method is presented for the dynamic analysis of contact deformation for rolling tire on a road. The tire is modeled as an inextensible ring on elastic foundation with damping. Under known radial deflection, the ground contact region is predicted by solving simplified ring motion equation, treating ground contacting as a boundary condition. Furthermore, the effect of internal damping and rotating speed is analyzed on contact region.     

Survey of Wheel—Rail Rolling Contact Theory

This paper describes the theory of frictional rolling contact as far as it is significant for the wheel-rail system. It is divided into two parts.

The first part, mostly non-mathematical, contains a historical survey from the times of Carter and Fromm (1926) to the present day, in which all aspects of rolling contact theory are discussed. Included are a quantitative account of the results of Hertz theory (Section 3), and a table of the creepage and spin coefficients.

The second part gives a present day account of the simplified theory (Section 4), and of the exact linear and non-linear theory (Section 5).

The paper closes with some recommendations for future research, of which the most pressing is a thorough investigation of the accuracy of simplified theory.     

Survey of Wheel—Rail Rolling Contact Theory

SUMMARY

This paper describes the theory of frictional rolling contact as far as it is significant for the wheel-rail system. It is divided into two parts.

The first part, mostly non-mathematical, contains a historical survey from the times of Carter and Fromm (1926) to the present day, in which all aspects of rolling contact theory are discussed. Included are a quantitative account of the results of Hertz theory (Section 3), and a table of the creepage and spin coefficients.

The second part gives a present day account of the simplified theory (Section 4), and of the exact linear and non-linear theory (Section 5).

The paper closes with some recommendations for future research, of which the most pressing is a thorough investigation of the accuracy of simplified theory.     

In-plane Dynamics of Tires on the Road Based on an Experimentally Verified Rolling Ring Model

The dynamic behavior of tires rolling on the road is predicted based on a ring model that is experimentally verified to well represent a real tire. The road contact is kinematically represented by two displacement constraints, one in radial and one in tangential direction. The displacement constraints are enforced via the receptance theory. The receptance matrix developed for the rolling tire is found to be Hermitian. Numerical results show that the natural (or eigen-) frequencies of the rolling tire with road contact are, as expected, higher than those of the tire with no contact. The influence of the rotating speed to the eigen-frquencies of the tire is also shown. Analytic expressions for the mode shapes of a rolling tire with ground contact are derived.     

In-plane Dynamics of Tires on the Road Based on an Experimentally Verified Rolling Ring Model

SUMMARY

The dynamic behavior of tires rolling on the road is predicted based on a ring model that is experimentally verified to well represent a real tire. The road contact is kinematically represented by two displacement constraints, one in radial and one in tangential direction. The displacement constraints are enforced via the receptance theory. The receptance matrix developed for the rolling tire is found to be Hermitian. Numerical results show that the natural (or eigen-) frequencies of the rolling tire with road contact are, as expected, higher than those of the tire with no contact. The influence of the rotating speed to the eigen-frquencies of the tire is also shown. Analytic expressions for the mode shapes of a rolling tire with ground contact are derived.     

Kalker's algorithm Fastsim solves tangential contact problems with slip-dependent friction and friction anisotropy

This paper presents two extensions of Kalker's algorithm Fastsim of the simplified theory of rolling contact. The first extension is for solving tangential contact problems with the coefficient of friction depending on slip velocity. Two friction laws have been considered: with and without recuperation of the static friction. According to the tribological hypothesis for metallic bodies shear failure, the friction law without recuperation of static friction is more suitable for wheel and rail than the other one. Sample results present local quantities inside the contact area (division to slip and adhesion, traction) as well as global ones (creep forces as functions of creepages and rolling velocity). For the coefficient of friction diminishing with slip, the creep forces decay after reaching the maximum and they depend on the rolling velocity. The second extension is for solving tangential contact problems with friction anisotropy characterised by a convex set of the permissible tangential tractions. The effect of the anisotropy has been shown on examples of rolling without spin and in the presence of pure spin for the elliptical set. The friction anisotropy influences tangential tractions and creep forces. Sample results present local and global quantities. Both extensions have been described with the same language of formulation and they may be merged into one, joint algorithm.     

A novel method to model wheel–rail normal contact in vehicle dynamics simulation

An approximate analytical method is proposed for calculating the contact patch and pressure distribution in the wheel–rail interface. The deformation of the surfaces in contact is approximated using the separation between them. This makes it possible to estimate the contact patch analytically. The contact pressure distribution in the rolling direction is assumed to be elliptic with its maximum calculated by applying Hertz' solution locally. The results are identical to Hertz's for elliptic cases. In non-elliptic cases good agreement is achieved in comparison to the more accurate but computationally expensive Kalker's variational method (CONTACT code). Compared to simplified non-elliptic contact methods based on virtual penetration, the calculated contact patch and pressure distribution are markedly improved. The computational cost of the proposed method is significantly lower than the more detailed methods, making it worthwhile to be applied to rolling contact in rail vehicle dynamics simulation. Such fast and accurate estimation of contact patch and pressure paves the way for on-line modelling of damage phenomena in dynamics simulation packages.     

Investigation of Features of Tyre Rolling at Non-Small Velocities on the Basis of a Simple Tyre Model with Distributed Mass Periphery

SUMMARY

On the basis of the brush-type tyre model the paper considers the interaction between steady-state rolling deformable wheel and flat road surface as well as corresponding force and moment characteristics of the wheel.

At least two zones of sliding, anisotropic dry friction, sliding friction coefficient speed-dependent and instantaneous leap of the friction coefficient when transition from sliding to adhesion zone occurs, have been taken into account, as well as distributed peripheral mass of tyre, elasticity, pseudo-dry friction and damping properties in radial, tangential and lateral directions of the elements at the wheel periphery, including a visco-elastic belt. Vertical force distribution in the contact area is not supposed to be known in advance and follows from the calculation. As a result, sliding zone lengths, distributed forces in contact area, six components of generalized road reaction reduced to the wheel center, and rolling resistance moment are found as functions of vertical load, movement velocity, longitudinal and side slip, friction in contact area with road, stiffnesses, dry friction and damping in the tyre model elements and of distributed peripheral mass.

A computer program developed in Fortran and results of calculations are of particular interest for qualitative analysis including steady rolling of studded tyre and also racing car and aircraft tyres which peripheral mass shows itself in a special way because of great movement velocities.     

基于损伤扩容理论的圆形隧洞围岩松动圈位移计算方法

为解决传统弹塑性力学理论在计算时未考虑实际围岩卸载时的扩容效应导致的位移设计值预测误差较大的问题,通过考虑不同应力状态下围岩损伤扩容系数的不同,建立圆形隧洞开挖松弛位移的分层总和法理论计算方法,提出以洞壁处围岩损伤扩容状态为控制值的围岩开挖位移预警值计算方法。结合分水江引水隧洞工程实例,对不同围岩级别、不同埋深、洞壁围岩的不同扩容效应（即不同支护状态）时圆形隧洞开挖后洞壁围岩位移变化规律进行分析,提出以圆形隧洞为基础的围岩预警值理论计算经验公式。由实际应用结果表明： 考虑扩容效应的围压位移分层总和法计算值比弹塑性理论计算值大3倍以上,计算结果更符合实际情况。     

Creep force modelling for rail traction vehicles based on the Fastsim algorithm

The evaluation of creep forces is a complex task and their calculation is a time-consuming process for multibody simulation (MBS). A methodology of creep forces modelling at large traction creepages has been proposed by Polach [Creep forces in simulations of traction vehicles running on adhesion limit. Wear. 2005;258:992–1000; Influence of locomotive tractive effort on the forces between wheel and rail. Veh Syst Dyn. 2001(Suppl);35:7–22] adapting his previously published algorithm [Polach O. A fast wheel–rail forces calculation computer code. Veh Syst Dyn. 1999(Suppl);33:728–739]. The most common method for creep force modelling used by software packages for MBS of running dynamics is the Fastsim algorithm by Kalker [A fast algorithm for the simplified theory of rolling contact. Veh Syst Dyn. 1982;11:1–13]. However, the Fastsim code has some limitations which do not allow modelling the creep force – creep characteristic in agreement with measurements for locomotives and other high-power traction vehicles, mainly for large traction creep at low-adhesion conditions. This paper describes a newly developed methodology based on a variable contact flexibility increasing with the ratio of the slip area to the area of adhesion. This variable contact flexibility is introduced in a modification of Kalker's code Fastsim by replacing the constant Kalker's reduction factor, widely used in MBS, by a variable reduction factor together with a slip-velocity-dependent friction coefficient decreasing with increasing global creepage. The proposed methodology is presented in this work and compared with measurements for different locomotives. The modification allows use of the well recognised Fastsim code for simulation of creep forces at large creepages in agreement with measurements without modifying the proven modelling methodology at small creepages.     

High-speed wheel/rail contact determining method with rotating flexible wheelset and validation under wheel polygon excitation

A rotating flexible wheelset model is developed and integrated into a vehicle/track dynamic model. Flexible wheelset modes with natural frequencies less than 1000?Hz are considered in the wheelset modelling. An innovation of the paper is that wheel/rail rolling contact calculation considers the effect of the wheelset flexibility and the rotating effect. By introducing two half dummy rigid wheelsets the rolling contact between the flexible wheelset and the two rails can be transformed to that between a rigid wheelset and the rails. As an extension application, the wheel OOR (Out-Of-Round) wears with the 11th, 15th, and 17th orders are used to the vehicle system dynamic model with rigid, flexible and rotating-flexible wheelset model. The results of the three models are compared to study the influence of wheelset flexibility and rotation. The ‘online searching contact method’ developed in this paper is compared with the traditional contact method with considering the rotating flexible wheelset. And then a measured OOR is used to excite the rotating flexible wheelset, the response of which is analysed and verified.     

Shear forces in the contact patch of a braked-racing tyre

This article identifies tyre modelling features that are fundamental to the accurate simulation of the shear forces in the contact patch of a steady-rolling, slipping and cambered racing tyre. The features investigated include contact patch shape, contact pressure distribution, carcass flexibility, rolling radius (RR) variations and friction coefficient. Using a previously described physical tyre model of modular nature, validated for static conditions, the influence of each feature on the shear forces generated is examined under different running conditions, including normal loads of 1500, 3000 and 4500 N, camber angles of 0° and?3°, and longitudinal slip ratios from 0 to?20%. Special attention is paid to heavy braking, in which context the aligning moment is of great interest in terms of its connection with the limit-handling feel. The results of the simulations reveal that true representations of the contact patch shape, carcass flexibility and lateral RR variation are essential for an accurate prediction of the distribution and the magnitude of the shear forces generated at the tread–road interface of the cambered tyre. Independent of the camber angle, the contact pressure distribution primarily influences the shear force distribution and the slip characteristics around the peak longitudinal force. At low brake-slip ratios, the friction coefficient affects the shear forces in terms of their distribution, while, at medium to high-slip ratios, the force magnitude is significantly affected. On the one hand, these findings help in the creation of efficient yet accurate tyre models. On the other hand, the research results allow improved understanding of how individual tyre components affect the generation of shear forces in the contact patch of a rolling and slipping tyre.     

Practical Frequency and Time Optimal Design of Passive Linear Vibration Isolation Mounts

Summary In this paper we examine a linear one-degree of freedom vibration isolator mount. The linearity of the system allows us to analyze its frequency and time response characteristics analytically. Optimal damping and stiffness values for the isolator are obtained by minimizing certain cost functions, which are the Root Mean Square (RMS) of the absolute acceleration and the relative displacement. These RMS cost functions are used to create a design chart for the isolator parameters. This is very useful particularly in the presence of physical constraints such as a limit in relative displacement. The time response of the system for a unit step input is also considered to gain an insight into the transient characteristics of the system. We obtain an optimal value for the damping ratio of the system in order to minimize the transmitted acceleration. Combining the frequency and time response analyses leads to an optimal value for the mount natural frequency and damping ratio satisfying both time and frequency domains. The results are verified numerically using measured acceleration as input.     

Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons

Friction dampers of mechanical systems are frequently exposed to medium-frequency (M-F) dither generated in the surrounding environment. A dithered system of technical importance is the railway freight wagon with friction dampers in the primary suspension developing two-dimensional friction, where dither is generated by the rolling contact of wheel and rail. This paper presents some results of the investigation of the influence of dither on dry friction damping. This influence has been studied experimentally and theoretically, and the parameters of dither influencing dry friction damping have been indicated. An experimental set-up has been designed that allows investigating friction damping in the presence of dither. The experiments have shown that friction damping in the presence of the M-F dither behaves like viscous damping. This means that dither smoothes dry friction as far as damping is concerned. To investigate this phenomenon theoretically, a rheological model of dry friction has been proposed that is applicable to one- and two-dimensional friction. In the latter case, the model takes into account friction anisotropy. According to performed numerical simulations of freight wagon motion, with dither supplied to the model through measured vertical accelerations of axle boxes, smoothing dry friction by dither strongly influences ride dynamics of the wagon with friction dampers in the primary suspension. Smoothing dry friction by dither should be accounted for in numerical simulations of motion of vehicles with friction dampers in the primary suspension by employing a proper model of the two-dimensional friction and application of realistic dither generated by rolling contact.     

基坑土层等效m值反演分析

工程勘察报告尚不能正确提供各类土层的水平基床系数,而现有规程和指南给出的基坑土层水平反力系数的比例系数m值参考范围较大,致使设计中对m值的选取存在很大的随意性。通过假设及简化分析,提出一个综合考虑土体重度、基坑开挖深度、土层水平反力系数和抗剪强度指标的基坑土层等效m值计算公式,并利用水平位移实测数据,通过非参数回归方法反演确定系数Δ1、Δ2,以修正等效m值计算公式。以昆明市某浅埋公路隧道工程为例,对该反演方法进行分析验证。验证结果表明,该方法不仅能缩小规程或指南给定的m值取值区间,还可以根据基坑开挖深度不断地修正基坑土层等效m值,实现基坑动态控制。     

大跨悬索桥MTMD抖振控制的参数优化

针对悬索桥抖振控制问题,建立有限元模型,应用神经网络和遗传算法对多重调频质量阻尼器(MTMD)进行双参数优化。以某大跨悬索桥为例,利用神经网络改进的谐波合成模拟方法(RBF-WAWS法)对脉动风速进行模拟,并换算成抖振力作用主梁上,通过时程分析及后处理获取主跨跨中横桥向响应值。将响应值的均方差作为优化目标函数,以MTMD总质量、个数及阻尼比作为优化变量和约束条件,采用神经网络拟合目标函数并应用改进的自适应遗传算法进行寻优。结果表明,优化后的MTMD能有效控制悬索桥在脉动风作用下的抖振响应,减振率达48%。提出的理论与计算方法对悬索桥中MTMD的设置及参数选取具有实际工程意义。     

小阻力扣件有荷阻力取值分析

进行有荷载情况下的扣件阻力试验可验证和完善桥上无缝线路扣件纵向阻力取值计算理论。试验表明,有载情况下小阻力扣件的弹性位移值、纵向阻力值均与《送审稿》取值不符;在各级垂向荷载情况下,小阻力扣件的摩阻系数可偏安全取0.20,纵向阻力可采用垂向荷载值与摩阻系数的乘积而得。     

Low-frequency vibration control of floating slab tracks using dynamic vibration absorbers

This study aims to effectively and robustly suppress the low-frequency vibrations of floating slab tracks (FSTs) using dynamic vibration absorbers (DVAs). First, the optimal locations where the DVAs are attached are determined by modal analysis with a finite element model of the FST. Further, by identifying the equivalent mass of the concerned modes, the optimal stiffness and damping coefficient of each DVA are obtained to minimise the resonant vibration amplitudes based on fixed-point theory. Finally, a three-dimensional coupled dynamic model of a metro vehicle and the FST with the DVAs is developed based on the nonlinear Hertzian contact theory and the modified Kalker linear creep theory. The track irregularities are included and generated by means of a time–frequency transformation technique. The effect of the DVAs on the vibration absorption of the FST subjected to the vehicle dynamic loads is evaluated with the help of the insertion loss in one-third octave frequency bands. The sensitivities of the mass ratio of DVAs and the damping ratio of steel-springs under the floating slab are discussed as well, which provided engineers with the DVA's adjustable room for vibration mitigation. The numerical results show that the proposed DVAs could effectively suppress low-frequency vibrations of the FST when tuned correctly and attached properly. The insertion loss due to the attachment of DVAs increases as the mass ratio increases, whereas it decreases with the increase in the damping ratio of steel-springs.     

离散元法的沥青路面车-路动力学响应分析

为了分析车辆荷载作用下沥青路面结构的细观状态力学响应，建立了二自由度1/4车辆模型与多层路基路面耦合离散元模型，通过各结构层单轴压缩应力-应变试验与相同工况试验数据比较，经迭代运算得到路面离散元模型各结构层细观参数，应用试验得到的沥青路面细观参数建立多层路基路面模型，在离散元模型的上表面设定一定不平度，在一定速度作用下，1/4车辆模型在路基路面离散元模型上表面匀速移动，从而求解车辆动荷载作用下沥青路面各结构位移、应力等细观受力状态。进而改变1/4车辆模型的车体悬架刚度、悬架阻尼系数、轮胎刚度，轮胎阻尼系数，从而获得在改变车辆参数作用下沥青路面内部的应力变化规律。研究结果表明：基于离散元理论不但可以求得沥青路面在车-路相互作用下各层的应力与变形，而且还可以求得沥青路面各结构层颗粒流的变化趋势，在车辆移动荷载作用下，随着路基路面深度增加，各结构层颗粒流竖直方向动态位移与应力响应依次减少，其中上基层颗粒流动位移比上面层颗粒流动位移减少25%，下面层颗粒流竖向应力约为上面层颗粒流竖向应力的50%，水平方向上颗粒流既有压应力又有拉应力，变化比较复杂，上面层颗粒流水平方向主要承受压应力，其余结构层主要承受拉应力；增加轮胎与悬架刚度系数对模型颗粒流水平方向拉应力影响较大，增加轮胎与悬架阻尼系数对垂直方向颗粒流压应力与水平方向拉应力影响较小。