Design Method of Rail Grinding Profile Based on Wheel Rail Contact Relationship

Abstract

When the railway track is damaged, it is essential to grind it in in a timely manner to enhance the wheel-rail contact performance. This paper relies primarily on the rolling radius difference function as the main design basis, taking the expected distribution between the wheel and the rail as the boundary condition. In cases where the abscissa of the rail is known, the Euler method is used to solve the differential equation to obtain the optimized ordinate of the rail. The profile of the rail is optimized, and the feasibility of the design method is verified through programmed algorithm. Based on the application results of the rail grinding profile design on the Wuhan-Guangzhou line, the rail light strip position and contact point distribution are more reasonable after grinding, the dynamic performance of the vehicle passing is improved, and the wheel-rail wear rate is reduced.

1 Introduction

In the process of vehicle operation, the rail will gradually develop concave wear, fatigue, wave wear and other adverse conditions that require maintenance. As an important method of railway maintenance, rail grinding technology can eliminate and mitigate surface damage on rails, thereby extending the rail’s service life [1, 2].

In railway operation, the dynamic interaction between wheels and rails, rail wear and train operation safety are closely linked to the geometric shape matching between wheels and rails. By reasonably improving the wheel rail contact state can ensure that train operation safety, dynamic performance and wheel and rail wear are in a more reasonable state, so as to effectively extend the service life of rails [3, 4]. Chen [5] studied the rail profile in the turnout area of passenger lines and heavy haul lines, and proposed design methods under different application scenarios. Shevtsov et al. [6] proposed the optimal design of wheel tread based on the function of rolling circle radius difference (RRD, the same below). The numerical analysis method was used to optimize the design of wheel tread, improving vehicle running performance and reducing wheel rail wear. Shen et al. [7] took the RRD function as the design goal to push back the wheel tread of railway vehicles, reducing the wheel rail contact stress and making the wear more uniform, and developed the corresponding calculation program. Jiang [8] used NURBS curve fitting method to parametrically reduce the grinding of the rail profile before grinding, and proposed a grinding mode optimization decision-making method for the quality of rail grinding profile. This method can effectively select the optimal grinding scheme from the rail grinding mode library. Ha [9] aimed at the phenomenon of serious rail wear in small radius curve sections, genetic optimization algorithm was used to optimize the asymmetric profile design, and the verification showed that the optimized profile shows good dynamic performance and wear reduction effect.

Most of the existing references carry out wheel rail optimization design for a single optimization objective, with relatively simplistic considerations. The traditional design of rail profile based on wheel diameter difference function deduces the position of rail contact point based on the position of contact point on the wheel as known condition. In this way, one-to-one correspondence between design objectives and known conditions is established, which will lead to large errors in design results. In order to make up for this defect, this paper proposes directly deducing the rail profile based on the rail contact point position (target contact light band) to obtain a better rail grinding profile, and comparisons are made between the performance of vehicle system dynamics and wheel-rail contact mechanical characteristics before and after optimization.

2 Mathematical Modeling

First, given the wheel profile, the rail profile to be optimized, and the basic wheel rail contact parameters, calculate the RRD curve. Then, optimize the RRD curve based on the requirements of vehicle operation. Take the optimized RRD curve as the main objective function of the design, and deduce the design rail profile based on the expected distribution of wheel rail contact. Subsequently, obtain the whole optimized rail shape by splicing the original rail profile. After that, the designed rail profile is tested for wheel rail geometric contact, contact stress, dynamics performance and other indicators. If it does not meet the requirements, return to modify the parameters and redesign the rail profile until it meets the requirements. The design flow chart is shown in Fig. 1.

Fig. 1

Flow chart of grinding rail profile design process

The optimized RRD function must satisfy three conditions: First when the wheelset has a slight lateral displacement, the RRD should be minimal, to ensure that the vehicle can travel steadily on a straight track. Second, when the wheelset has a large lateral displacement, the RRD should be increased in order to enhance the curve passing performance and reduce the wheel-rail wear; third, the RRD function curve should be as smooth as possible.

The definitions of wheel-rail coordinate systems are established, according to the right-hand rule, as shown in Fig. 2. Black solid lines represent the wheelset, the blue solid line represents the rail.

Fig. 2

Wheel-rail contact coordinates

The wheel-rail contact points must coincide at the same point in space [10]. Therefore,

$$ \left( {\begin{array}{*{20}l} {y_{rl} } \\ {z_{rl} } \\ \end{array} } \right) = \left[ {\begin{array}{*{20}l} {\cos \phi_{w} } \\ {\sin \phi_{w} } \\ \end{array} \begin{array}{*{20}l} { - \sin \phi_{w} } \\ {\cos \phi_{w} } \\ \end{array} } \right]\left( {\begin{array}{*{20}l} {y_{wl} } \\ {z_{wl} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}l} {y_{w0} } \\ {z_{w0} } \\ \end{array} } \right) $$

(1)

$$ \left( {\begin{array}{*{20}l} {y_{rr} } \\ {z_{rr} } \\ \end{array} } \right) = \left[ {\begin{array}{*{20}l} {\cos \phi_{w} } \\ {\sin \phi_{w} } \\ \end{array} \begin{array}{*{20}l} { - \sin \phi_{w} } \\ {\cos \phi_{w} } \\ \end{array} } \right]\left( {\begin{array}{*{20}l} {y_{wr} } \\ {z_{wr} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}l} {y_{w0} } \\ {z_{w0} } \\ \end{array} } \right) $$

(2)

There must be a common section and a common normal at the contact point due to the overlap of contact points in space, such that

$$ \frac{{dz_{rl} }}{{dy_{rl} }} = \tan \left( {a\tan \frac{{dz_{wl} }}{{dy_{wl} }} + \phi_{w} } \right) $$

(3)

$$ \frac{{dz_{rr} }}{{dy_{rr} }} = \tan \left( {a\tan \frac{{dz_{wr} }}{{dy_{wr} }} + \phi_{w} } \right) $$

(4)

The geometric relationship between the roll angle and other contact parameters can be written as follows:

$$ \tan \phi_{w} = \frac{{z_{wl} - z_{wr} - \frac{{z_{rl} - z_{rr} }}{{\cos \phi_{w} }}}}{{\left| {y_{wr} \left| + \right|y_{wl} } \right|}} $$

(5)

The RRD curve is defined as:

$$ \Delta R = z_{wl} - z_{wr} $$

(6)

These equations hold true in situations where there is a solitary contact point between the wheel and the rail in space:

$$ z_{wl} \left( {y_{w} } \right) - z_{wr} \left( {y_{w} } \right) = \min \left\{ {z_{wl} \left( {y_{w} } \right) - z_{wr} \left( {y_{w} } \right)|_{y = yw} } \right\} $$

(7)

$$ sign\left( {\frac{{dz_{wl} }}{{dy_{wl} }}} \right) = sign\left( {\frac{{dz_{rl} }}{{dy_{rl} }}} \right) \equiv 1;\,sign\left( {\frac{{dz_{wr} }}{{dy_{wr} }}} \right) = sign\left( {\frac{{dz_{rr} }}{{dy_{rr} }}} \right) \equiv - 1 $$

(8)

Based on assumption (2), the rail is a convex curve such that:

$$ sign\left( {\frac{{d^{2} z_{rl} }}{{dy_{rl}^{2} }}} \right) = sign\left( {\frac{{d^{2} z_{rr} }}{{dy_{rr}^{2} }}} \right) \equiv 1 $$

(9)

To address the problem using numerical integration, it is necessary to transform the differential–algebraic equations (DAE) given in Eqs. (1–9) into ordinary differential equations (ODE). Simultaneously, derivatives with respect to y_w must be taken at both ends of Eqs. (1–9). There are:

$$ \left\{ {\begin{array}{*{20}l} \begin{gathered} \frac{{dz_{{rl}} \left( {y_{w} } \right)}}{{dy_{w} }} = \frac{{dy_{{wl}} \left( {y_{w} } \right)}}{{dy_{w} }}\sin \phi _{w} \left( {y_{w} } \right) + y_{{wl}} \left( {y_{w} } \right)\cos \phi _{w} \left( {y_{w} } \right)\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} + \frac{{dz_{{wl}} \left( {y_{w} } \right)}}{{dy_{w} }}\cos \phi _{w} \left( {y_{w} } \right) \hfill \\ \quad \quad \quad \quad - z_{{wl}} \left( {y_{w} } \right)\sin \phi _{w} \left( {y_{w} } \right)\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{dz_{{rr}} \left( {y_{w} } \right)}}{{dy_{w} }} = \frac{{dy_{{wr}} \left( {y_{w} } \right)}}{{dy_{w} }}\sin \phi _{w} \left( {y_{w} } \right) + y_{{wr}} \left( {y_{w} } \right)\cos \phi _{w} \left( {y_{w} } \right)\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} + \frac{{dz_{{wr}} \left( {y_{w} } \right)}}{{dy_{w} }}\cos \phi _{w} \left( {y_{w} } \right) \hfill \\ \quad \quad \quad \quad - z_{{wr}} \left( {y_{w} } \right)\sin \phi _{w} \left( {y_{w} } \right)\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{\frac{{dz_{{rl}} \left( {y_{w} } \right)}}{{dy_{w} }}}}{{\frac{{dy_{{rl}} \left( {y_{w} } \right)}}{{dy_{w} }}}} = \tan \left( {a\tan \frac{{dz_{{wl}} \left( {y_{w} } \right)}}{{dy_{{wl}} \left( {y_{w} } \right)}} + \phi _{w} \left( {y_{w} } \right)} \right) \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{\frac{{dz_{{rr}} \left( {y_{w} } \right)}}{{dy_{w} }}}}{{\frac{{dy_{{rr}} \left( {y_{w} } \right)}}{{dy_{w} }}}} = \tan \left( {a\tan \frac{{dz_{{wr}} \left( {y_{w} } \right)}}{{dy_{{wr}} \left( {y_{w} } \right)}} + \phi _{w} \left( {y_{w} } \right)} \right) \hfill \\ \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{\left( {y_{{wr}} \left( {y_{w} } \right) - y_{{wl}} \left( {y_{w} } \right)} \right)}}{{\cos ^{2} \phi _{w} \left( {y_{w} } \right)}}\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} + \left( {\frac{{dy_{{wr}} \left( {y_{w} } \right)}}{{dy_{w} }} - \frac{{dy_{{wl}} \left( {y_{w} } \right)}}{{dy_{w} }}} \right)\tan \phi _{w} \left( {y_{w} } \right) = \hfill \\ \quad \quad \frac{{d\Delta R\left( {y_{w} } \right)}}{{dy_{w} }} - \frac{{\frac{{dz_{{rl}} \left( {y_{w} } \right) - dz_{{rr}} \left( {y_{w} } \right)}}{{dy_{w} }}}}{{\cos ^{2} \phi _{w} \left( {y_{w} } \right)}}\frac{{d\phi _{w} \left( {y_{w} } \right)}}{{dy_{w} }} \hfill \\ \hfill \\ \end{gathered} \hfill \\ {\frac{{d\Delta R\left( {y_{w} } \right)}}{{dy_{w} }} = \frac{{dz_{{wl}} \left( {y_{w} } \right)}}{{dy_{w} }} - \frac{{dz_{{wr}} \left( {y_{w} } \right)}}{{dy_{w} }}} \hfill \\ \end{array} } \right. $$

(10)

The column vectors of all unknowns are separated into matrices in the form of:

$$ A \times u = b $$

(11)

To solve the system of obtained six independent differential algebraic equations with six unknowns, the above equations are converted into ordinary differential equations, which are then solved by applying the Euler method:

$$ \left\{ {\begin{array}{*{20}l} {y_{k + 1} = y_{k} + h \times f\left( {x_{k} ,y_{k} } \right)} \\ {y_{0} = y\left( {x_{0} } \right)} \\ \end{array} } \right. $$

(12)

Indeed, the design challenge related to rail profiles in turnout areas is a multi-objective optimization problem. To ensure the dynamic performance of the vehicle, as discussed in this paper, the objective function can be formulated as follows:

$$ Obj:f_{1} = \min \left\{ {\frac{{\mathop \int \nolimits_{{ - y_{w\max } }}^{{y_{w\max } }} \left( {\left| {\Delta R_{real} \left( {y_{w} } \right) - \Delta R_{opt} \left( {y_{w} } \right)} \right|} \right)dy_{w} }}{{\mathop \int \nolimits_{{ - y_{w\max } }}^{{y_{w\max } }} \Delta R_{opt} \left( {y_{w} } \right)dy_{w} }}100\% } \right\} $$

(13)

where $\Delta {{\text{R}}}_{{\text{real}}}$ is the computed RRD after the optimization, while $\Delta {{\text{R}}}_{{\text{opt}}}$ is the targeted optimization of RRD.

3 Application Examples

In order to verify the reliability of this paper, a section of rail on the Wuhan-Guangzhou line railway was used as an example to optimize the grinding profile. The original rail profile is paved with 60 kg/m standard profile (Type: CN60, gauge 1435 mm, cant: 1/40), and the LMA type tread with the most passes was selected as the wheel reference profile.

The measured rail surface state before grinding is shown in Fig. 3. From the figure, there are double light bands on the rail, and the rail contact light band is too wide. According to the feedback from the railway section, the vehicle shaking alarm often occurs when running on this section.

Fig. 3

Rail surface state

Ignoring the influence of the wheelset yaw angle on the geometric contact characteristics, the assembly set is selected as the given wheel profile and average worn rail profile. The wheel rail geometric contact characteristics are shown in Fig. 4. Figure 4 shows that there is an obvious jum** phenomenon at the contact points during the lateral movement of the wheelset, which will cause multi-point contact during the movement of the vehicle, resulting in double light bands, uneven wear and large impact, thus shortening the service life of the rail.

Fig. 4

Original contact distribution of LMA and worn CN60 rail

The RRD curve and equivalent conicity of wheel rail original contact are shown in Fig. 5. Figure 5 shows that the RRD and equivalent conicity, which are the main indicators of the vehicle system dynamics, exhibit obvious linearity in the section where the wheelset lateral displacement is small ([−3, 3] mm), and the equivalent conicity slope is low (the equivalent conicity is about 0.01). This is not favourable for the stability of the wheelset when running on a straight line, and it may cause the vehicle to shake due to excessive wheelset lateral displacement. Figure 4 shows that the wheel rail contact points are obviously concentrated within this range, which is not conducive to uniform wear. When the lateral displacement of the wheelset is large ([±8, ±10] mm), there is an obvious step phenomenon in the wheel diameter difference curve, which may cause a large impact when the vehicle passes through the curved section, posing potential safety risks. Combined with Figs. 3 and 4, when the wheel rail contact point transitions from the rail head area to the gauge angle side contact area, the wheel rail contact point has a large jump phenomenon, which leads to multi-point contact when the wheelset displacement is large, resulting in uneven wear, thus reducing the service life of the wheel rail.

Fig. 5

RRD function (a) and equivalent conicity (b) of LMA and worn CN60 rail

According to the above analysis, a reasonable RRD curve should have the following characteristics:

• (1) In a small range of wheelset lateral displacement, maintain a low slope (the equivalent conicity is about 0.05–0.08) to ensure the straight-line operation stability of the wheelset.

• (2) The distribution of wheel rail contact points shall be as uniform as possible to ensure that the transition zone of the wheelset from small to large displacement is smooth as soon as possible, to reduce the dynamic impact of the wheel rail and the uniform wear between the wheel rail.

• (3) A large RRD curve shall be ensured in the region of large lateral displacement to ensure sufficient anti-derailment safety and reduce wheel rail wear and stress concentration.

The optimized RRD curve is shown in Fig. 6.

Fig. 6

Optimized RRD function of LMA and worn CN60 rail

According to the optimized RRD curve, the rail grinding profile is obtained, as shown in Fig. 7.

Fig. 7

Comparison of left and right rail profile before and after griding

The wheel rail contact calculation and analysis of the optimized profile are performed again, as shown in Fig. 8. From the figure, the distribution of wheel rail contact points after the shape of the rail is matched with the LMA tread is relatively uniform and continuous, which overcomes the defect that the wheel rail contact points jump with the lateral displacement, and avoids the occurrence of excessive local stress and local wear.

Fig. 8

Optimized contact distribution of LMA and designed CN60 rail

Figure 9a shows the verification results of RRD curve. The overall difference between the target RRD and the calculated RRD is less than 4%, and the area with the largest difference has a small probability of wheel rail contact when the wheel set traverse is ±12 mm. In general, the design effect meets the design requirements. From the perspective of wear, the optimized profiles tend to have longer service life due to the increase of contact pieces and better control of local stress concentration. The optimized equivalent conicity is shown in Fig. 9b. When the transverse displacement of the wheel set is within ±3 mm, the equivalent conicity is about 0.05–0.06, indicating that the stability on the straight line is good and can suppress the shaking phenomenon. In a large range of lateral displacement, different equivalent conicity calculation methods maintain a high value, which is conducive to the passage of the curve.

Fig. 9

Error statistics of the calculated RRD (a) and equivalent conicity (d) of LMA and designed CN60 rail

Figure 10 shows the comparison of the normal contact forces. The optimized non-Hertz contact stress is greatly reduced in the rail top contact area, and the combined contact points are more evenly distributed in this area, indicating that the wheel rail wear will be more uniform, and the service life of the rail will be extended.

Fig. 10

Comparisons of normal stress of LMA and designed CN60 rail

4 Dynamic Verification

A vehicle turnout coupling dynamic model was established based on MATLAB and Simulink platform to verify the dynamic response of the rail profile in the turnout area before and after optimization. Using the LMA wheel tread, a rail vehicle passage of a CN60 single straight turnout was simulated, as shown in Fig. 11. From the dynamic simulation results, the optimized vehicle has better performance when going through the turnout, and the wear index between the wheel and rail is lower.

Fig. 11

The comparison of dynamic interaction. a Lateral displacement of wheelset, b lateral acceleration, c lateral force, d lateral force Wear index

5 Conclusions

This paper introduced a rail grinding profile design method for the heavy-haul train. It is based on the wheel-rail contact relationship and aims at improving vehicle dynamic performance and wheel rail wear rate by controlling wheel rail contact distribution. It utilizes the idea of inverse design method and multigoal optimization. The results of computer software program simulation and the application results of rail grinding profile design on the Wu-Guang line show that the grinding profile design using this algorithm can optimize the dynamic performance of the wheel and rail while ensuring a good match with the target wheel diameter difference curve, and provide a feasible and efficient profile customization method for rail grinding. With the help of the proposed method, it can be expected that the rail could have longer service life under proper grinding operation.