基于加权轮轨缝的最佳车轮外形设计【中文5900字】
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中文5900字
基于加权轮轨缝的最佳车轮外形设计【中文5900字】
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正文：外文资料译文文章出处：WEAR期刊271（2011）218-226基于加权轮/轨缝的最佳车轮外形设计崔达斌，李丽，金雪松，李霞文摘 提出了基于加权的正常间隙，车轮和轨道之间的接触点在了火车车轮轮廓的直接优化方法。以轮轨对应，车轮LMA和中国铁路轨道chn60为例，新的优化方法用于提高车轮LMA的轮廓。车辆-轨道耦合动力学理论也被用来研究车辆的动态行为改进的分布影响，同时用滚动接触理论分析轮轨接触状态下优化轮廓线的影响。数据结果表明，LMA改进轮具有优良的弯曲行为。实验发现，与之前相同优化情况的LMA相比，用这种方法提高剖面的LMA在轨道chn60有良好的共形接触，车轮和钢轨的接触点车轨和轨道上的分布是比较均匀的和广泛的。对轮的纵断面优化后，当车辆行驶在直线轨道时，车辆对轮轨的最大正压力大大降低，以及不牺牲轮轨动态特性时的轮轨磨耗指数大大降低。1. 介绍 轮/轨（W / R）的相互作用在铁路车辆与轨道的动态行中发挥着非常重要的作用，例如，铁路车辆狩猎的临界转速，平稳舒适运行，曲线通过能力，轮轨接触应力水平，滚动接触疲劳和磨损。其中，轮廓在轮轨接触区已引起众多研究者的关注。 到目前为止，不同的轮廓设计方法，车轮和钢轨之间都已经得到良好的匹配。早期的车轮轮廓的设计方法主要是根据铁路运营商的经验。在过去的几十年中，越来越多的人更乐意使用数学模型和数值技术对轮廓进行优化，进而提高铁路车辆的动态性能。海勒和劳尔便是通过优化车轮轮廓来改善车辆的动态性能。吴提出了一种车轮轮廓的设计理念，系统地评估了车轮和钢轨在车辆的特性和操作条件方面的兼容性。张等人利用一个基于部分钢轨伸缩的改进方法（它最初是由吴开发），修改了中国的LMA 轮配置资料。修正轮可以与60kg/m 钢轨理想整合接触（chn60）。这种介于W和R之间的共形接触的形成可以有效地减少它们之间的接触应力。佩尔森、iwnicki 和后来的诺瓦莱斯等人，根据遗传算法提出了一种可以直接优化程序设计铁路车辆车轮剖面的方法。沈等人，利用所谓的“反向设计”开发了一种面向对象的方法，用于设计铁路车轮踏面和钢轨。舍夫斯托夫等人，提出了一种基于滚动圆半径差的数值优化技术（RRD）对车轮轮廓进行设计，这种该方法采用基于响应面拟合来设计一个最佳的车轮轮廓与目标点的RRD相契合。后来，舍夫斯托夫等使用相同的理念设计车轮轮廓的轮轨滚动接触疲劳和磨损。Hamid Jahed等人提出类似的方法中，RRD也可用于火车车轮轮廓的设计。 回顾近几年对车轮踏面的优化，其研究方向主要集中在逆的方法上了，在目标曲线给定的情况下，这种方法是非常有效的。然而，根据设计师经验，获得目标函数是一个需要花费很多时间的工作。在本文中，与上述的反演方法相比，根据W/R在接触点之间正常间隙的常规，提出了改善轮轨系统的动态接触行的直接解法。以轮罩使用本方法得到的改进的轮廓的剖面为例，证明了此方法的优点。 图1 车轮踏面优化区域 图2 车轮与轨道之间的正常间隙2. 优化设计方法 正常间隙（或间隙）在接触区域的W / R是评价W / R型材兼容的一个重要因素。小间隙可以在提高W / R整合接触的情况下，增加接触面积（接触片）和减少接触应力水平。轮轨滚动接触疲劳（RCF）在一定的负荷情况下，与其接触面的大小有关。W/R的产生和裂纹增长取决于轮轨接触应力/应变在接触面上的分布。 本文的研究目旨在提出一个LMA型车轮的直接数值优化设计方法，优化后的车轮与轨道的LMA chn60滚动接触有尽可能小的间隙。这种优化可以降低轮对动态行为无丧失影响情况下W/R之间的接触应力水平。2.1.数学建模 如图1所示，车轮踏面LMA从A到B之间的区域作为优化区域。在如图所示的英国-奈特系统中，起点A设置在轮缘的最大接触角所在点处，结束点B在直线上，其横坐标为30mm。曲线上的点A和B，分别为 （2） （3） 在胎面上移动的点（节点）（hi，vi），可以将胎面从A到B设置为n + 1段（i1，2，N）。Hi、vi分别为移动点的纵向坐标和横向坐标。最终，可以通过拟合这些点的三次函数得到胎面。（2）和（3）作为这样一个拟合的边界条件。 为了简化建模，每个移动节点在横坐标hi（i = 1,2，n）为一常数，移动节点的纵坐标V1，V2，则是不同的，V1，V2，Vn）为优化设计的变量。车轮踏面外形表示为f（V1，V2，Vn）。2.1.1 目标函数 W/R的正常间隙定义为平均间隙值在接触点CJ的一个特定的区域，如图2所示。轮对中心横向位移为YJ，差距DJ功能定义为 （4）其中，dji是第i个点的正常间隙，M是接触点CJ区域离散点数目。区域的边界是由C1和C2在图2中确定的。接触点的坐标CJ是由给定的W / R尺寸的轮对横向位移YJ确定。DJI是W/R给定的车轮轮廓函数f（V1，V2，VN）定义的.因此，从式（4）可知，在接触点Cj处该间隙函数Dj可以表示为Dj=Dj（yj，v1，v2,，vn)。 结合chn60轨道接触LMA剖面为例，轮对的横向位移在整个地区的间隙函数计算结果，如图3所示。应当指出的是，曲线的较大的值对应于较大的接触电脑间隙，这种情况意味着接触斑面积越小，轴载荷条件下接触应力水平越高。为了提高车轮和chn60共形接触状态，间隙值曲线应该尽可能小。利用求曲线下梯形面积的方法，可归结为 （5），K是W/R曲线上点的数量，如图3所示。从公式（5）可以得出，K是点的W / R间隙曲线数如图3所示。从公式（5）可以明显的看出，不同接触点处的Dj的差异对S有不同的影响。S越小，与W / R形接触程度越高，相应的降低轮轨接触应力水平。因此，我们希望W /R曲线中的Dj或S尽可能的小。图3 间隙函数W/R与轮对的横向位移 值得注意的是，不同的车辆/轨道运行条件，例如轨底坡，轨距，曲线超高及列车速度，导致轮轨之间不同的接触情况，如接触点的位置，接触点的分布与接触区宽度。接触点的分布表明，接触点位于车轮踏面的横向方向上。不同的加权系数被应用于控制 Di对S的影响，从而获得尽可能小的S值。根据经验，车轮的直线轨道运行上的加权因子应大于曲线轨道的。考虑到Dide 加权，方程（5）变为 （6） 在wj是轮对yj侧向位移的加权因子，其对应于接触点J加权因子可以这样确定：当车辆沿着直线或曲线轨道运行时计算轮对的横向位移，模拟车辆-轨道耦合动力学模型，找到与实际的侧向位移相近似的约束，使用较大的加权因子来确定横向位移yj的范围。 由于参数yj是在计算过程中得到，函数公式（6）也可以表示为 （7）公式（7）作为目标函数来找到最佳的车轮轮廓。2.1.2 设计约束 在优化过程中，最大法兰角应满足车辆运行安全、轮缘厚度、踏面宽度和高度等尺寸的设计要求。值得注意的是，真正的车轮踏面具有单调的斜坡，但基于三次样条函数的方法不能保证所设计的轮廓具有单调性，它可能产生波纹面。为了避免这个问题出现，在优化设计中使用用约束方程 （8）同时，下部和上部的设计边界变量vi,由式（8）给出 i=1,2,.n （9）在式（8），Gi是在第i个设计变量节点位置约束方程。在式（9），ai和bi分别为下边界和上边界。ai和bi的值选择为接近初始值，以确保尽可能高的计算速度和尽可能快的方案解决收敛速度。2.2 优化算法在这一部分中，一种改进的优化算法采用与改进的SQP（序列二次规划）相结合的方法，拟牛顿法和BFGS方法。该算法的基本思想是利用SQP方法和拟牛顿法更新迭代找到最优的搜索方向和步长的信息，从而提高计算效率。根据公式（7）（9），优化问题可以描述为 (10)方程（10）可以转换为基于拉格朗日函数的二次近似二次规划问题。二次规划问题的子问题的函数写为 （11）是拉格朗日系数，V=（v1,v2,.,vn）是设计的载体，是可变的。二次规划子问题可以通过线性化得到非线性约束。一种新的循环公式，利用子问题的解决方案构成 （12）这里是步骤参数；是在第k步循环问题的解，是第k步循环的设计变量的求解。在循环过程中，该方法用于计算拟牛顿近似矩阵，这个矩阵作为拉格朗日函数的Hessian矩阵。在第k步迭代，Hessian矩阵可以通过下式计算 （13）Hk是nxn维的Hessian矩阵，Qk=V（K+ 1）V（K）,q为n1维向量,写成如下 （14）是拉格朗日因数，S 和Gi分别为变化的S和Gi，当k = 1， H1= S1ij = 2S/vivj 是一个NN维矩阵的二阶偏微分，图4显示的优化算法流程图。本文利用Matlab软件和FORTRAN语言开发了一个计算机代码表示车轮踏面优化程序描述图。3.结果与讨论 在这一部分中，以影响铁路车辆的动态行为的优化分布的影响为例，表明了本文提出的方法的优点。3.1 案例研究：优化LMA车轮轮廓 在优化LMA车轮轮廓和分析优化配置的动态行为中，轨道倾角是1:40，轨道轨距为1435mm，车轮的名义滚动半径为457.5mm。中国的铁路客运车辆的参数用于分析 31 。在计算中所选的轨迹由60m直线轨道，一个建的弧形轨道和200m长直轨道组成。弯曲的轨道包括两个180m缓和曲线和一个半径为3000米、长为250米右转园曲线。汽车的车速为180km/h。移动节点的数目选择如下。得到可用的加权因子，当车辆在选定的轨道上运行时，采用车辆-轨道耦合动力学模型对轮对的横向位移进行计算。轨道不平顺是指在直线上发生的动态变化。前轮的横向位移，即为图5所示LMA直线。从图5中可以观察到，前轮的横向位移范围是从4mm到 4mm，所以在这样一个地区应当考虑最大的数值。前轮横向位移轨道曲线如图6所示，这一数字表明，侧向位移在8mm以内，因此，对于车轮踏面相应的接触区使用的加权系数应当小于用于4mm到4mm的加权系数。图4 车轮型面设计程序流程图 图5 在直线轨道前轮对横向位移 图6 对曲线轨道前轮对横向位移 图7 OPT和LMa 图8 车轮与轨道之间的间隙为简单起见，因子W1是用来测量侧向位移从04mm范围变化时，对应的接触点间隙Dj的变化，W2用于侧向位移变化在48mm范围时。随着车轮轮廓的其他方面的差距，在优化时不考虑间隙Dj。所以，车轮的踏面区域的优化分为两个部分，优化结果主要取决于W1和W2之间的比率，而不依赖于选择的W1和W2值的大小。使用不同的比例得到不同的优化结果。根据作者的经验，选择W1和W2首先考虑实现在车轮踏面较大的区域优化设计的目标。最合适的W1和W2的比例是通过试验论证得到的，因此，加权因子是用来控制在不同的车轮踏面优化区域的接触应力水平。W1和W2的具体值应足够大，以避免计算中出现累积误差，并对照其他相关的实际条件选择一定的比例。在这项研究中，根据先前的经验，W1和W2分别为100和50。 通过2.2节中提出的优化算法，优化配置，通过选择显示，得到如图7所示。与初始轮廓LMA相比，图中显示的优化配置明显不同于LMA型。优化前后的W / R的间隙如图8所示，从这个图可以明显看出制作轮廓曲线低于LMA侧向位移从6mm 到2mm时的曲线。小的差距意味着相应的接触应力小，间隙值范围大于LMA。3.2 轮轨接触几何 LMA的RRD的计算选择如图9所示，从这个图我们可以清楚地看到,OPT的RRD的选择大于横向位移在0-7mm区域是的LMA的RRD。这意味着在相同情况下，优化的LMA轮廓比LMA轮对有更大的等效锥度，因此车辆蛇行时的临界速度低于理想直线运行时的临界速度。当侧向位移从12mm 到12mm增大时，右轮轨接触点对的分布和LMa型接触点的分布分别如图10（a）和（b）所示。图（a）表明，侧向位移在-8mm0mm范围时，LMA轮廓接触点主要集中在同一位置，这种情况会加速车轮与钢轨的磨损。图10（b）所示的接触点分布比LMA更均匀，有利于降低轨道磨损和滚动接触疲劳。3.3 蛇行临界速度 这里计算时，该车配备了两个不同的轮廓蛇行临界速度，LMA车辆临界速度为421kmh，汇编程序的临界速度是400公里/小时，较高的等效锥度导致了较低的临界速度。但由于目前车辆运行的最大运行速度低于300km，优化仍能满足当前配置的需求。最后一个需要保持稳定性的是第三轮。图11显示，当车速是400公里/小时，第三轮对横向位移与车辆的行驶距离的关系。 图9 滚动半径差与侧向位移 图11 第三轮对横向位移与行驶距离图10 接触点与轮对横向位移分布3.4 弯曲性能具有两不同轮辐的车辆的弯曲性能用于3.1节相同的轨道来模拟。不同行驶距离的前轮横向位移如图12所示，数据表明，整体弯曲轨道时OPT轮对的偏差低于LMA轮对，振荡的幅度低于弯曲后的LMa轮对，阻尼震荡比LMA轮对更快。OPT轮对的弯曲性能是优于LMA轮对的，这是因为横向位移在-77mm范围时，OPT轮对有更高的等效锥度。 利用车辆及轨道耦合动力学的理论，对磨损指数也进行了研究，如图13所示。车轮的横向位移在8mm时，轮缘和轮轨就不接触了，因此磨损指数处于一个较低的水平。显然，当车辆通过圆形轨道时，OPT型的磨损指数比LMA型低很多，这是OPT型车轮的蠕滑率低于LMA型车轮，然而，在本文中未作过多研究，但是在圆形轨道时，LMA型和OPT型车轮的接触压力非常接近，祥见3.5节。车辆在直线轨道上行驶时，由于其蠕滑率接近零，OPT型和LMA型车轮的磨损指数均几乎接近于零。对两种不同轮廓剖面的脱轨系数和乘坐舒适性计算和比较，结果差异不大，为简介起见，相关结果不在本文表述。 图12 前轮的侧向位移与行驶距离 图13 左前轮磨损指数与行驶距离3.5 轮轨接触应力 通常接前轮轮轨的接触应力高于其他车轮。因此，应当计算车轮和钢轨最高接触应力。Kalker的非赫兹滚动接触三维弹性理论用于分析接触斑的形状和接触点处的应力分布。粘/滑区的左侧轮轨接触斑如图14所示，当车辆在直线轨道运行时，OPT型车轮接触面面积明显大于LMA型；在相同的轴荷下，OPT型车轮接触应力减少。当车辆在曲线轨道上运行时，两者有相近大小的接触面积。图14 粘/滑区当车辆在直线轨道和曲线轨道上运行图15 车辆在直线轨道（上）和曲线轨道（下）上运行是，正常压力分布图15显示了常压下左侧轮轨接触斑分布的计算结果。我们可以看到，车辆在直线轨道运行时，OPT型车轮最大正压力分布低于LMA型，其主要原因是直线轨道有较大的接触面积；当车辆行驶在弯曲的轨道时，这两种类型车轮的最大正压力差别不大。利用非赫兹接触理论计算米塞斯等效应力。车辆行驶在同一轨道时，前轮的最大轮轨接触应力有两种不同的计算方法。如图16，车辆行驶在直线轨道上时，PPT型车轮的接触应选择比LMA型低得多；如果希望车辆通过圆曲线轨道时接触应力进一步减小，LMA型胎面优化区域需要扩展到轮缘根部。目前为止，这样的研究仍在进行中。由于中国的高速铁路轨道大部分是直线或曲线，优化LMA型优化可以更有效的减少运行时磨损和滚动接触疲劳。4 结论本文提出了一种新的直接数值方法来优化车轮，它根据测量W / R之间接触点附近的正常间隙值，该方法用于优化中国LMA型车轮。利用车辆/轨道耦合动力学模型、滚动接触的力学模型、轮轨系统和轮轨接触几何模型研究了chn6与钢轨滚动接触的力学性能和优化情况，它发现与优化前相比，提高了弯曲性能，降低了水平直线轨道运行时的接触应力；同时优化轮对与chn60良好的共形接触，提高接触点的分布，降低接触应力，减少磨损和滚动接触疲劳。致谢 目前的研究已由中国国家自然科学基金（50821063，50875221），中国国家重点基础研究发展计划（2007cb714702），铁道部基础研究计划（z2006-0492008j001-a），和博士点基金（20090184110023）资助。作者非常感谢西南交通大学牵引动力国家重点实验室的严女士在英语方面给予的帮助。附件：外文资料原文Optimal design of wheel profiles based on weighed wheel/rail gapabstractA direct optimization method for railway wheel profiles is put forward based on the weighed normalgap between wheel and rail at the contact point. Taking the wheel/rail counterpart, wheel LMa and railCHN60 of China railway, as an example, the new optimization method is used to improve the profileof wheel LMa. The coupling dynamics theory of the vehicle and track is also used to investigate theeffect of the improved profile on the dynamical behavior of the vehicle, and the rolling contact theoryis hired to analysis the influence of the optimized wheel profile under the wheel/rail contact status. Thenumerical results illustrate that the improved wheelset of LMa has superior curving behavior. It is foundthat the improved profile of LMa with this method is in good conformal contact with rail CHN60, and thedistribution of contact points of the wheel and rail is relatively uniform and extensive on the wheel treadand rail top, compared to the LMa in the same case before its optimization. After the profile optimizationof the wheelset, the maximum normal pressure of the wheel/rail is greatly lowered when the vehicleruns on the tangent track, and the wear index of the wheel/rail is largely reduced without sacrificing thedynamic performance of the wheelset.1. IntroductionWheel/rail (W/R) interaction plays an important part in thedynamic behavior of railway vehicle and track, such as, the critical speed of railway vehicle hunting, running stability and comfort, the ability of curve negotiating, wheel/rail contact stress level, rollingcontact fatigue and wear. Among them, the wheel profile in W/Rcontact region has drawn attention of many researchers 15.So far, different design approaches for wheel profiles have beendeveloped to obtain the satisfactory matching of wheel and rail.Earlier methods to design wheel profiles were mainly based onthe experience of railway operators 6,7. During the last decades,there has been much greater interest in employing mathematicalmodels and numerical technology to optimize the wheel profileto improve railway vehicle dynamic behavior. Heller and Law 8optimized the wheel profile to improve the dynamic performanceof the rolling stock. Wu 9 put forward a concept of wheel pro-file design to systemically evaluate the compatibility of the wheeland rail profile based on the vehicle characteristics and the operat-ing condition. Zhang et al. 10 utilized an improved method basedon the partial rail profile expansion, which was originally devel-oped by Wu 9, to modify the whAeel profile of LMa in China. Themodified wheel has a desirable conformity contact with Chineserail of 60kg/m (CHN60). This conformal contact forming betweenW and R can effectively reduce the contact stress level betweenthem. Persson and Iwnicki 11 and later Novales et al. presented a direct optimization procedure based on genetical gorithmtodesigna wheel profile for railway vehicles 12. Shen et al. developed atarget-oriented method with so called inverse methodology forthe design of railway wheel profile involving contact angle andrail profile information 13. Shevtsov et al. proposed a numeri-cal optimization technique based on rolling circle radius difference(RRD) of wheelset to design the wheel profile 14,15. This method employed a multipoint approximation based on responsive surface fitting to design an optimum wheel profile that matches a target RRD. Later, Shevtsovetal. Used the same idea to design a wheelprofile considering wheel/rail rolling contact fatigue and wear 16. Asimilar approach was proposed by Hamid Jahed et al., wherein theRRD function was also used for the design of railway wheel profiles.As reviewed in detail in 18, the recent researches on wheelprofile optimization have mainly focus on the inverse method ology. This method ology is very efficient when a target curve is given. However, to obtain a target curve function generally by designers experience would be a trouble somework which costs much time.Inthis paper, as contrasted to the above mentioned inverse methods,adirect solution method based on the normal gap between the profiles of W/R around their contact point is put forward to improvethe dynamic and contact behavior of W/R system. The improvedprofile of wheel LMa obtained by using the present methodis given as an example to demonstrate the advantages of themethod.2. Optimal design methodThe normal gap (or normal clearance) of W/R in the contactregion is an important factor to evaluate the compatibility of W/Rprofiles 1921. The small clearance can improve the conformitycontact situation for W/R, increase the contact area (contact patch)and reduce the contact stress level. W/R rolling contact fatigue(RCF) is related to its contact patch size under the condition of the prescribed load 22. The initiation and growth of the cracks on W/R depend on the wheel/rail contact stress/strain level in the contactpatch 23.The objective of study in this paperis to propose a direct numerical optimization method to design the profile of wheel LMa. Theoptimized wheel LMa in rolling contact with rail of CHN60 has anormal clearance as small as possible. The optimization decreasesthe contact stress level between the W/R without loss of dynamicbehavior ability of the wheelset.2.1. Mathematical modelingAs shown in Fig. 1, the wheel tread of LMa from its flange root Ato its field side B is chosen as an optimization region. In the coordi-nate system as shown, the start point A is set at the point with themaximum contact angle of wheel flange. The end point B is on thestraight line and its abscissa is 30mm. The slopes of points A and Bare, respectively （2） （3）The moving points (nodes) (hi,vi), (i=1, 2, ., n) on the treadcan be set by dividing the tread from A to B into the segments ofn+1. hiand viare, respectively, the vertical and lateral coordinatesof the moving points. End for end, the tread can be generated byfitting these points with cubic spline function 17, and Eqs. (2) and(3) serve as the boundary conditions for such a fitting.To simplify modeling, the abscissa of each moving node hi(i=1,2, ., n), is selected as a constant, and the vertical coordinatesof the moving nodes, v1,v2,.,vn are considered to be varied.v1,v2,.,vn are chosen as the design variables in the optimization.The wheel tread profile is now expressed as f(v1,v2,.,vn).2.1.1. Objective functionThe normal gap of W/R is defined as the average clearance valuein a specific region around the contact point Cj, as shown in Fig. 2.When the lateral displacement of the wheelset center is yj, thefunction of the gap Djis defined as (4)in which, djiis the normal clearance at the ith point and m isthe number of discrete points in the region around the contactpoint Cj. The boundary of the region is determined by c1 andc2 in Fig. 2. The coordinates of the contact point Cjare deter-mined by the given lateral displacement yjof the wheelset forthe given W/R sizes. The value of djiis determined by the wheelprofile function f(v1,v2,.,vn) for the given profiles of the W/R.Therefore, from Eq. (4), the gap function Djcan be represented byDj= Dj(yj,v1,v2,.,vn) at the contact point Cj. Considering LMa profile in contact with CHN60 rail as anexample, their gap function in the whole region of the lateral dis-placement of wheelset is calculated, as shown in Fig. 3. It should benoted that the larger value of the curve corresponds to the largerclearance around the contact point, and this situation means thatthe area of contact patch is smaller and the level of the contactstress is higher under the condition of the same axle load.In order to improve the conformal contact status of the wheeland CHN60, the gap curve should have the values as small as pos-sible. Using the trapezoidal method of summing the area under thecurve, the area S can be formulated as （5）where K is the number of the points in the gap curve of the W/R as shown in Fig. 3. From formulae (5), it is obvious that the gap Dj at different contact points contribute different values to S. The smaller S is, the higher the W/R conformal contact degree is, and correspondingly the lower W/R contact stress level is. Therefore, it is hoped that S or Djis as small as possible in the matching of the W/R profiles.It should be noticed that different vehicle/track operation con-ditions, e.g., rail cant, track gauge, curve super elevation and trainspeed, lead to the different contact situations between W/R, suchas the contact point location, the distribution of the contact pointsand the contact area width. The distribution of the contact pointsindicates that the contact points are situated on the wheel treadin the lateral direction. Different weighting factors are applied to control the contribution to S of variable Di values a iming to obtain S value as small as possible. According to experience, weighting fac-tors for the wheel running on tangent tracks should be bigger thanthose for curved tracks.Considering the weighting Diin S, Eq. (5) becomes as （6）where wjis defined as the weighting factor of the lateral displace-ment of wheelset yj, which corresponds to contact point j. Theweighting factors can be determined in this way: calculate the lat-eral displacement of the wheelsets when the vehicle moves along tangent or curved tracks using the vehicle-track coupling dynamics model 24, find the approximate bound of the practical lateral dis-placement,uselargerweightingfactorsforthelateraldisplacementyjwithin the bound.Since the parameter yjis given in the calculation process, func-tion S in Eq. (6) can also be expressed as （7）Eq.(7) is used a sthe objective function to find the optimal wheelprofile.2.1.2. Design constraintsDuring the optimization, such size design requirements as thesafety of wheel operation, the wheel flange thickness and theheight, the tread width and the maximum flange angle should besatisfied.It is noted that the real wheel tread has the monotonic slope,but the method, based on cubic spline function, cannot ensure themonotonicity of the designed wheel profile, and may generate thecorrugated tead. To avoid this problem arising in the optimizationdesign, a constraint equation is used and reads （8）At the same time, the lower and upper boundaries of the design variables viin Eq. (8) are given as i=1,2,.n （9）In Eq. (8), Giis the constraint equation at the position of the ithdesign variable node. In Eq. (9), aiis the lower boundary and biisthe upper. The value of aiand biare selected to be as close to theinitialvalueaspossibletoensurethehighcomputationalspeedandthe fast convergence of solution as well.2.2. Optimization algorithmIn this section, a modified optimization algorithm is developed by applying the improved SQP(sequentialquadraticprogramming) 25,26 method combined with quasi-Newton method and BFGSmethod 2730. The basic idea of this algorithm is to find the opti-mal search direction and the information of the step size by usingthe SQP method and to renew the iteration by using the quasi-Newton method, thereby improving calculation efficiency.According to Eqs. (7)(9), the optimization problem can be described by (10)Eq. (10) can be converted to quadratic approximation quadraticprogramming sub-problem based on the Lagrange function. Thefunction of the quadratic programming sub-problem is written as （11）where is the Lagrange multiplier, and v = (v1,v2,.,vn)Tis the design vector that is variable.The quadratic programming sub-problem can be obtainedthrough linearization of the nonlinear constrained ones. A newiterative formula is constituted by using the solution of the sub-problem as （12）Here is the step parameter; tkis the solution of the sub-problemin the kth step iteration. v(k)iis the solution of the design variablesin the kth step iteration.In process of the iteration, the BFGS method is used to calculatethe approximate matrix of quasi-Newton, and this matrix is usedas the Hessian matrix of the Lagrangian function. In the kth stepiteration, the Hessian matrix could be calculated by using （13）where Hkis the Hessian matrix of nn dimension, Qk=v(k+1)v(k), q is the n1 dimensional vector written as follows. （14）where i(i=1, 2, ., n) is the value of the Lagrange multiplier, S() and Gi() are the gradients of the functions S and Gi, respectivelyWhen k=1, H1= S1ij = 2S/vivj is a nn dimensionalmatrix of second order partial differential.Fig. 4 shows the flow chart of the optimization algorithm. Thepresent paper develops a computer code for the wheel profile optimization procedured escribed in Fig.4 by using Matlab package and Fortran language.3. Results and discussionIn this section, the effect of the optimized profile on the dynam-ical behavior of the railway vehicle, taken as an example, isinvestigated to demonstrate the merits of the method proposedin the paper.3.1. Case study: optimizing LMa wheel profileIn optimizing LMa wheel profile and analyzing the dynamicbehavior of the optimized profile, the rail inclination is 1:40, thegauge of track is 1435mm and the nominal rolling radius of thewheel is 457.5mm. The parameters of the railway passenger vehi-cle of China are used in the analysis 31. The selected track inthe calculation consists of a 60m tangent track, a 610m curvedtrack and a 200m long straight track. The curved track includestwo 180m transition curves and 250m long right turn circle curvewith 3000m radius. The vehicle speed is 180km/h.The number of the moving nodes is selected as 17. To getthe available weighting factors, the lateral displacements of thewheelsets are calculated using the vehicle-track coupling dynam-ics model 24 while the vehicle running on the selected track. Thetrack irregularity is considered occurring on the straight line in thedynamic analysis. The lateral displacement of the front wheelset,i.e. the LMa wheelset, on the tangent track is shown in Fig. 5. Itcan be observed from Fig. 5 that the lateral displacement of thefront wheelset is almost in the range from 4mm to 4mm, so insuch a region the greatest weighting factors should be considered.The lateral displacement of the front wheelset on the curved trackis shown in Fig. 6. This figure shows that the lateral displacementis within 8mm, therefore, for the corresponding contact region ofthewheeltread,theweightingfactorsusedhereshouldbelessthanthose used in the range of 4mm to 4mm.Here, for simplicity, the factor w1is used to weight the gapDjin the contact point positions when the lateral displacementsranging from 0mm to 4mm, and w2is used when the lateral dis-placement takes place in the range of 48mm. With respect ofother parts of the wheel profile, the gap Djis not considered inthe optimization. So, the region of the wheel optimization treadis divided into two parts. The optimization result depends mainlyon the ratio between w1and w2, and do not rely on the selectedvalues of w1and w2much. Using the different ratio gets the different optimization result. According to the experience of the authors,the selection of w1and w2first considers the objective realizationof the profile optimization in the larger region of the wheel treadin the profile design. The best ratio of w1and w2for the design isdetermined by trial and error. Therefore, the weighting factors areused to control the contact stress level in different optimizationregions of the wheel tread. The specific values of w1and w2shouldbe large enough to avoid accumulative errors in the computation, and are selected with certain proportion against each other according to the condition of the practical track. In this study, w1and w2are, respectively, assigned with 100 and 50 based on the previousexperience.Through applying the optimization algorithm presented in Section 2.2, the optimized profile, indicated by the OPT, is obtained asshown in Fig. 7, compared with the initial profile LMa. It is showninFig.7thattheoptimizedprofileissignificantlydifferentfromtheLMa profile. The gaps of W/R before and after the optimization areshown in Fig. 8. It is clearly seen from this figure that the gap curveof the OPT profile is below that of LMa when the lateral displace-ment occurs ranging from 6mm to 2mm. The small gap impliesthat the corresponding contact stress is small, and out of the rangethe gap value is larger than that of LMa.3.2. Wheel/rail contact geometryThe RRD of LMa and OPT are calculated as shown in Fig. 9. Fromthis figure we can clearly see the RRD of OPT is larger than that ofLMa within a region between 0mm and 7mm of the lateral dis-placement. This means that the wheelset of the optimized profilehas a larger equivalent conicity than LMa wheelset does in thisrange, and therefore the critical hunting speed while the vehiclerunning on ideal tangent tracks will be decreased.The distribution of pairs of contact points of the right wheel/railfor OPT and LMa are shown in Fig. 10(a) and (b), respectively,when the lateral displacement increases from 12mm to 12mm.Fig.10(a)indicatesthatthecontactpointsofLMaprofilearemainlyconcentrated at the same position when the lateral displacementis in the range of 8mm to 0mm. This situation will accelerate thewear and rolling contact fatigue of the wheel and rail. Fig. 10(b)denotes that the distribution of the contact points on OPT is moreuniform than that of LMa, beneficial to reduce the rail wear androlling contact fatigue.3.3. Critical hunting speedThecriticalhuntingspeedsofthevehicleequippedwithtwodif-ferent wheel profiles are calculated here. The critical speed of thevehicle with LMa is 421km/h and with OPT is 400km/h. The lowercritical speed is due to the higher equivalent conicity. But becausethe maximum running speed required in the vehicle service opera-tion is under 300km/h so far, the optimized profile could still meetthe current requirement.The third wheelset is the last one to restore its stability. Fig. 11shows the lateral displacement of the third wheelset with OPT pro-file versus the running distance of the vehicle when the velocity is400km/h.3.4. Curving behaviorThe curving behavior of the vehicle with two different profilewheelsets is simulated on the same curved track as used in Section3.1. The lateral displacements of the front wheelset with two dif-ferentprofilesversusrunningdistancearepresentedinFig.12.ThisfigureshowsthatthemisalignmentofOPTwheelsetislessthanthatof LMa wheelset on the whole curved track. The oscillating ampli-tudeofOPTwheelsetissmallerthanthatofLMaaftertheircurving,and oscillating damps of OPT wheelset faster than that of LMawheelset. The curving behavior of OPT wheelset be superior to thatof LMa wheelset. This is because OPT wheelset has higher equivalent conicity than LMa wheelset does when the lateral displacement of wheelset center occurs in the range from 7mm to 7mm.Fig. 10. Distribution of contact points vs. lateral displacement of wheelset.By using the coupling dynamics theory of the vehicle and track,the wear index is also investigated, as shown in Fig. 13. The lat-eral displacement of the wheelset is within 8mm and the wheelflange does not contact the rail, thus the wear index is at a lowlevel. Clearly, the wear index of OPT profile is much lower thanthat of LMa profile when vehicle passes over the circle curve. Thisis because the creepages of the OPT wheelset are less than thoseof the LMa wheelset, however, their creepages are not shown inthis paper, but the contact pressures of OPT and LMa are very closeat the circle curved track, see Section 3.5. When the vehicle runson the tangent track, the wear indexes of OPT and LMa are almostclose to zero due to their creepages near to zero.The derailment ratio and ride comfort index for the two wheelprofiles are also calculated and compared. The results show little difference exist between the two wheel profiles. The relevantresults are not shown in this paper for the sake of paper brevity.3.5. Wheel/rail contact stressUsuallythecontactstresslevelonthehighrailwheelofthefrontwheelsetishigherthanthatonothers.Sothewheelandrailcontactperformances of the high rail is calculated. Kalkers theory of three-dimensional elastic bodies in rolling contact with non-Hertzian3236 is utilized to analyze the shape of the contact patch andthe distribution of the stress in the contact patch.The stick/slip areas of the left side of wheel/rail contact patches are shown in Fig.14. When the vehicle running on tangent track, the area of the contact patch of OPT profile is bigger obviously than that of LMa. The contact stress of OPT profile would be reduced underthe same axle load. When the vehicle running on curved track, theOPT and LMa profile have similar areas of the contact patches.The distributions of normal pressure on the contact patches on the left side of wheel/rail are calculated, as shown in Fig.15. Wecansee that the maximum normal pressure of OPT profile is lower thanthat of LMa profile when vehicle running on tangent track, due to alarger area of contact patch on tangent track. But when the vehicleruns on curved track, the maximum normal pressures of the twotypes of profiles have little difference.The Von-Mises equivalent stress is calculated using the non-Hertzian theory. The maximum wheel/rail contact stresses of thefront wheelset with two different profiles are calculated when thevehicleiscurvingonthesametrack.AsshowninFig.16,thecontactstress of OPT is much lower than that of LMa when the vehicleruns on the straight track; the two profiles produce similar contactstress when the vehicle passes through the circle curved track. Ifthe contact stress level of the OPT wheel is hoped to be furtherdecreased while it rolling on the circle curved track, the optimizedtread region of LMa needs to be extended to the flange root of thewheel. Such kind of researches is still under way. Yet, because themajority of high speed tracks are tangent tracks or curved trackswith large radii in China, the profile optimization of LMa at presentcould decrease the wear and the rolling contact fatigue efficientlyduring its service operation.4. ConclusionsThe paper proposes a new direct numerical method to opti-mize the railway wheel profile based on the weighed normal gapbetween W/R around their contact point. This method is used tooptimize the wheel profile of LMa of China. The dynamical behav-ior and contact situation of the optimized profile wheel in rollingcontact with CHN60 rail are detailedly investigated by using thecoupling dynamic model of the vehicle/track, the rolling contactmechanics model of wheel/rail system and the W/R contact geom-etry model. It is found that the wheelset with the optimized profilehas the superiority in the curving ability and reduces the con-tact stress level when it runs on tangent tracks, compared withthe wheelset before being optimized, namely, LMa wheelset. Alsothe optimized profile wheelset is in good conformal contact withCHN60 rail, that can improve the distribution of the contact points,reduce the contact stress level, and decrease the wear and rollingcontact fatigue.AcknowledgementsThe present work has been supported by the National Nat-ural Science Foundation of China (50821063, 50875221), theNational Basic Research Program of China (2007CB714702), theBasicResearchProgramofRailwayMinistry(Z2006-049,2008J001-A), and the Doctoral Discipline Foundation (20090184110023).The authors are grateful to thank Ms. Yan Zhang at the State KeyLaboratory of Traction Power of Southwest Jiaotong University forher kind help in improving the English text of the paper.References1 H. Fujimoto, Influence of arc and conic profile on vehicle dynamics, JSME Inter-national Journal C (1998) 15201527.2Z.M.Dong,Z.L.Wang,H.B.Jiang,Influenceonthelocomotivecurvepassingper-formancewithwheeltreadshape,ElectricLocomotives&MassTransitVehicles2 (29) (2006) 1315.3 A.F.D. Souza, Influence of the wheel and rail treads profile on the hunting of thevehicle, Transact of the ASME 107 (1985) 167174.4T.Jendel,Predictionofwheelprofilewear-comparisonwithfieldmeasurement,Wear 253 (2002) 8999.5 S. Zakharov, I. Zharov, Simulation of mutual wheel/rail wear, Wear 253 (2002)100106.6N.K. Cooperider, Analytical and experimental determination of nonlinearwheel/rail geometric constraints, U.S. Department of Transportation, FederalRailload Administration, FRA-OR&D, Report No. 76-244, 1976.7 K. Yamada, T. Hashi, M. Nakata, M. Isa, New profiled tread design method andits applied tread named “CS tread”, in: Proceedings of Contact Mechanics andWearofRail/wheelSystem,The5thInternationalConference,2528July,2000,Tokyo, Japan, 2000.8R.Heller,E.H.Law,Optimizingthewheelprofiletoimproverailvehicledynamicperformance, in: Proceedings of the 6th IAVSD-Symposium Technical Univer-sity Berlin, 1979.9 H.M. Wu, Investigations of wheel/rail interaction on wheel flange climb derail-ment and wheel/rail profile compatibility, Ph.D. thesis, The Graduate Collegeof the Illinois Institute of Technology, 2000.10 J. Zhang, Z.F. Wen, L.P. Sun, X.S. Jin, Wheel profile design based on rail profileexpansion method, Chinese Journal of Mechanical Engineering 3 (44) (2008)4449.11 I. Persson, S.D. Iwnicki, Optimisation of railway profiles using a genetic algo-rithm, Vehicle System Dynamics (2004) 517527.12M.Novales,A.Orro,M.R.Bugarin,Anewapproachforthedesignofwheelprofilegeometries, in: Proceedings of 7th World Congress on Railway Research, June48, 2006, Montreal, Canada (CD Proceedings), 2006.13 G. Shen, J.B. Ayasse, H. Chollet, I. Pratt, A u
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