GD1031輕型貨車(chē)設(shè)計(jì)—總體設(shè)計(jì),gd1031,輕型,貨車(chē),設(shè)計(jì),總體,整體 Optimal design of wheel profiles based on weighed wheel/rail gap abstract A 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. Introduction Wheel/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 [1–5]. 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 [8]optimized 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 designer’s 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 method The normal gap (or normal clearance) of W/R in the contactregion is an important factor to evaluate the compatibility of W/Rprofiles [19–21]. 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 modeling As 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 function The 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 constraints During 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 algorithm In this section, a modified optimization algorithm is developed by applying the improved SQP(sequentialquadraticprogramming) [25,26] method combined with quasi-Newton method and BFGSmethod [27–30]. 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 n×n dimension, Qk=v(k+1)?v(k), q is the n×1 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, respectively When k=1, H1= [S1ij] = [?2S/?vi?vj] is a n×n 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 discussion In 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 profile In 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 4–8mm. 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 geometry The 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 speed Thecriticalhuntingspeedsofthevehicleequippedwithtwodif-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 behavior The 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 ar