New support roller profile design for railway wheel re-profiling process by under-floor lathes with a single cutting tool

The wheel re-profiling is an important part of railway wheelset maintenance. Researchers and railway operators have been very concerned about how to minimize the loss of time during wheel re-profiling without decreasing safety. Avoiding wheelset disassembly means considerable time savings, while reducing wheel damage during operation. Underfloor wheel lathes are the most appropriate tool to achieve this double objective, and therefore the most used nowadays. Multi-cut tool lathes have the disadvantage of being extremely expensive. On the other hand, with single tool lathes, re-profiling is not smooth or safe enough when current convex profile support rollers are used. It is well known by the companies that during reprofiling the wheel suffers impacts/damaged. In this article, a methodology to optimize the profile of the support rollers used in underfloor single tool lathes for railway wheel re-profiling is proposed. This novel profile design will minimize damage and increase the safety of such lathes, since it proposes a greater smoothness in the process. Simulations of re-profiling process have been carried out by the finite element method showing that the designed roller profile reduces drastically the impact/damage during the operation. The impact generated between the re-profiling wheel and the rollers is avoided. Profile-optimized support rollers have been used in a real underfloor wheel lathe, showing good results.


Abbreviations β
Taper half-angle of the wheel γ Slope of the roller profile at the contact with the machining edge (variable as the re-profiling progresses) φ Angle between the roller-wheel contact force and its radial component, so that tan ϕ = F A F R a Roller axial length of its useful area. C 1 = (X 1 ,Y 1 ) Coordinates for the centre of curvature of the convex arc of the roller profile. C 2 = (X 2 ,Y 2 ) Coordinates for the centre of curvature of the concave arc of the roller profile. e Machining depth. Radius of arc for the convex (concave) roller profile section. s Safety section length measured along the profile (the last section of the useful width) V a Approach velocity, or velocity with which the re-profiled surface approaches the roller V m Machining velocity. X Axial coordinate X t Axial coordinate corresponding to the point of tangency between convex and concave arches. Y Radial coordinate Improving the safety and security of transport has been a major objective in the last decades. Especially, the railway industry requires careful consideration for the problems related to the contact between wheel and rail.
Scientific Reports | (2022) 12:104 | https://doi.org/10.1038/s41598-021-04190-y www.nature.com/scientificreports/ This paper proposes a method to find the optimal shape of the support rollers of a single tool underfloor wheel lathe, to avoid sudden load transfers during the process. The new shape of the roller profile has been analyzed through the dynamic simulation analysis of the wheel re-profiling process by means of a finite element method.

Description of the problem associated with a single tool under-floor lathe for railway wheel re-profiling
In this section, the problem associated with a single tool underfloor lathe equipped with support rollers (two rollers for each wheel, at least one of them motorized) is described.
The cutting tool works in a vertical plane between both rollers (see Fig. 2). Nowadays in this kind of lathes, support rollers are used whose profile is an arc of circumference, so they are totally convex. To illustrate the problems associated with these roller profiles, in Fig. 3 several phases of the re-profiling process of a wheel (in blue), of conicity semi-angle − β, supported on a generic profile roller (in red) are presented. The slope of the roller profile will be negative in the area of the roller that we will call "useful" (AC section), of width a.
At the beginning, the wheel rests on the roller at point A, where the wheel slope and the roller slope are both equal to β (Fig. 3a). This point will be the support point until it is reached by the machining edge (Fig. 3b). From then on, the support will take place on the machining edge (point P), which runs along the profile of the roller (Fig. 3c). Meanwhile, the already re-profiled surface will approach the point of the first roller support, point A. When the machining edge reaches point B, a support transfer occurs (Fig. 3d), at the end of which, the wheel will be supported again on point A, on its already re-profiled area (Fig. 3e).
Throughout the re-profiling process, the contact force between the roller and the wheel changes in its direction and in its distribution over the contact area(s). In fact, a load transfer from contact on point B to contact on point A occurs in phase (d) (Fig. 3). This load transfer supposes an abrupt decrease in its axial component, and usually causes undesirable impact and stability problems, so it is worth studying it in depth. If we consider the roller and wheel as rigid solids, the load transfer between B and A would be instantaneous; but if they are considered as deformable bodies, the contact surface at B will decrease while the contact surface in A will increase, resulting in a less abrupt load transfer.
The new design of the roller profile is first optimized considering the roller and the wheel as rigid solids. Once the optimal profile is proposed, the process will be studied considering both as deformable solids. For this purpose, the finite element technique will be used.

Roller profile design
In the intermediate phase of the re-profiling process, when the roller-wheel contact occurs on the machining edge, point P (Fig. 3), the already re-profiled surface of the wheel approaches towards point A of the roller, where it will eventually be supported again.
In Fig. 4 two moments of the re-profiling process separated by a small-time interval, Δt (in blue and purple, respectively) have been represented. V a and V m are the approach and machining speeds respectively, s is the distance between contact points at t and t + Δt. It can be seen that the greater the difference between the inclination of the wheel (taper half-angle), β and the slope of the roller profile in the contact, γ, the greater the speed with which the re-profiled surface approaches to point A.
In fact, from Fig. 4 we have: In rollers whose useful profile is totally convex, as those used at present, the slope of the roller is a monotonous diminishing function, reason why it goes away of the value of the inclination of the wheel as the machining edge www.nature.com/scientificreports/ moves away from point A. Therefore, the approach speed increases as the re-profiling process progresses, and can become considerably high at the time just before the double support (phase d in Fig. 3), producing a more abrupt load transfer than desired. Note that for rollers with a totally convex profile, the load transfer between points A and B is accompanied by a discontinuity in the axial component of the roller-wheel contact force: in B it is much larger than in A (see Fig. 3).
For the load transfer to be smooth enough, the slope of the roller profile at point B should be only slightly higher than the wheel inclination (which coincides with the slope of the roller at point A, the flank of the wheel considered linear). Thus, the curve proposed as the roller useful profile has an inflection point, where it becomes from convex to concave. The designed profile must also ensure that the second support is made actually at point A, before:  These two undesirable circumstances, represented in Fig. 5, would both imply impulsive support at point A.
In particular, the profile of the roller will be described by its radius r(x), as a function of the axial distance x. The slope of this function should be β at the ends of the useful width. The parameters to consider in the profile design are: the roller radius R at point A, the useful width a, the machining depth e, and the slope of the wheel to be re-profiled β. As a design requirement, a safety section of length s, must also be contemplated to guarantee a desirable support before circumstance (i). This length should not be too large, so that the slope of the roller at the point B (machining edge position at the double support phase), is only slightly greater than that of the wheel. Therefore, the following constraints must be met for r(x) (see Fig. 6): The load transfer occurs when the machining edge is located at point B, at the safety section s, from the last point of the useful area, C.
Limitations (i), (ii) and constraint (C2) impose an essential geometrical feature to the profile of the roller: it must be composed of a convex section followed by a concave section; therefore, it should have an inflection point in the useful area. The simplest and easiest-to-manufacture roller profile is one composed of two tangent circumference arcs. Below is the calculation process to obtain the optimal profile of this type of roller.
Optimal roller profile calculation involves the resolution of a nonlinear system of equations. For that reason, the proposed profile has been calculated in two steps. In the first one, no safety section is considered, which leads to a system of equations that can be solved exactly; the corresponding results are utilized in the second step, as the initial guess for the Newton-Raphson's iterative method.
First step: profile calculation without considering safety section. The Fig. 7 shows the roller profile composed of two circumference sections, one convex and one concave, having an inflection point between them. The position of the wheel profile is also shown at the moment of re-profiling process in which double support occurs. As can be seen, at that moment the machining edge reaches exactly the right useful limit of the roller profile, no safety section being considered.
In this first approach, the design parameters are: the wheel slope, β; the useful area length of the roller, a; the machining depth, e; and the roller radius, R (corresponding to the first and last roller-wheel contact points). Whereas the design variables are the radius of arc sections, R 1 and R 2 and the position coordinates for the corresponding centres (X 1 ,Y 1 ) and (X 2 ,Y 2 ). These variables must fulfil the following system of equations (see Fig. 7):7  www.nature.com/scientificreports/ There are 5 equations for 6 variables, so just one variable is independent. The radius of one of the arc sections (R 1 , for example) can be taken as the independent variable, the rest of variables being expressed as function of R 1 . The system is not linear but it can be solved exactly. Actually, replacing the first 4 equations in the last one and operating, we get: The value for the rest of the variables is then obtained immediately. If the profile is required to have one convex section followed by a concave one (as in Fig. 7), the value chosen for R 1 cannot be arbitrarily large, since the sum of radii R 1 + R 2 has a fixed value for a set of parameters β, a, e. This implies the following limit for R 1 : Otherwise, the value of R 2 would be negative and the useful profile area would no longer have a concave section. Of course, a negative value of R 1 implies that the entire useful profile is concave.
In theory, it is desirable a large value of the concave arc section, so that the double support occurs more smoothly. However, the convex arc section should be sufficiently large to ensure a sturdy and stable support, since the wheel will rest on that area during most of the re-profiling process. Furthermore, the smaller the radius of a section, the greater the variation of the axial component of the roller-wheel force during the passage of the machining edge through that section.
Second step: profile calculation considering safety section. When the safety section length s, measured along the concave section (magenta arc in Fig. 8), is considered as a design parameter, the system of equations becomes: www.nature.com/scientificreports/ Now, for a given value of R 1 , the first two equations (which are identical to those corresponding to the system of Eq. 2) directly provide the position of the centre of the convex arc (X 1 , Y 1 ). To obtain the radius of the concave arc, R 2 , and the position of its centre, a non-linear system of three equations (the last three Eq. 5) must be solved. The unknown vector q, and the constraint vector ϕ(q) for this system are respectively: The system will be solved numerically, applying the Newton-Raphson iterative method: using as the initial guess, the vector obtained with no safety section (Eqs. 2 and 3): Matrix φ q in Eq. (8) is the jacobian for the constraint Eq. (7): Calculation of the axial force/radial force ratio. In the re-profiling process, the contact (normal) force between each of the rollers and the wheel can be decomposed into a radial component, and an axial component (see Fig. 9). The contact force direction varies as the machining edge runs through the roller profile, so does the axial component to radial component ratio, F A F R , that can be expressed as: To express it in terms of the axial position X, note the relationship between X and φ (that depends on which arc section is in contact with the machining edge. See Fig. 10): www.nature.com/scientificreports/  A MATLAB® code has been developed to perform the described iterative calculation and the calculations necessary to obtain the output results: X 1 , Y 1 , R 2 , X 2 , Y 2 and F A /F R from the input parameters: β, a, R, e, s, R 1 . The program also provides the graph of the roller profile together with the wheel profile in the double support position, and that of the roller-wheel contact force axial to radial components ratio. As an example, in Fig. 11 these graphics are shown for the following parameters: β = 2.86º, a = 32 mm, R = 75 mm, e = 5 mm, s = 5 mm, R 1 = 30 mm. The output results are also shown.
Note that no practical axial load transfer occurs, as the final F A /F R value (support at point B) practically coincides with the initial one (support at point A). On the contrary, with conventional convex rollers, the F A / F R curve is monotonous increasing up to the point of load transfer, causing an abrupt decrease in the axial load component (see Fig. 12).
In this discussion, the cutting force exerted by the cutting tool on the wheel to be re-profiled, has been ignored. In fact, the components of the tool-to-wheel cutting force are mainly tangential and axial. Therefore, a slight reduction in the axial component of the roller-to-wheel contact force should be expected. Anyway, the axial load transfer will still occur abruptly at the "second support" when current rollers are utilized, instead of those proposed in the article, that produce no axial transfer at that moment.

Finite element simulation of the load transfer in the double support phase
In this section, finite element results for the quasi-static load transfer process (phase d in Fig. 3) are shown. The objective is to provide a criterion for choosing the convex arc radius, R 1 .
To this end, 3 rollers of parameters β = 2.86°, a = 32 mm, e = 5 mm, R = 75 mm, s = 4 mm, were simulated for 3 different values of the radius of curvature of the convex arc, R 1 = {10 mm, 50 mm, 80 mm}. The corresponding profiles are shown in Fig. 13, together with the calculated values of the second radius of curvature (concave arc), and the positions of the two centres of curvature, in Cartesian coordinates (x, y).
In Fig. 14 the beginning and the end of the double contact phase are shown for the first roller (R 1 = 10 mm, R 2 = 95.1 mm), as obtained with a finite element software. Several static simulations have been performed between these two positions for each of the three profiles mentioned above.
The most relevant results of the simulations are condensed in the graphs of Fig. 15. The green curves correspond to the roller with R 1 = 10 mm; the curves in orange, to the roller with R 1 = 50 mm; and the curves in red, Figure 11. Top: optimal roller profile for the indicated design parameters. Bottom: ratio between the axial and radial components of the contact force, throughout the re-profiling process. the axial component grows to a maximum just at the inflection point of the profile. From that point on, it decreases to the same value as in the first support. As a consequence, there is no axial load transfer. In view of these graphs, the conclusion is that the roller which concave section has greater radius is the one that produces a smoother, radial and axial load transfer. In general, the lower slope of the roller at point B, the smoother the load transfer will occur from support at point B to support at point A. This is achieved for small values of R 1 , but this radius cannot be arbitrarily small, since the contact pressure increases as it decreases. With a value of R 1 = 10 mm, a Von Mises stress less than 4500 MPa has been obtained, which we consider a fairly acceptable value.

Conclusions
In this article, a problem associated with wheel re-profiling process by under-floor lathes with a single cutting tool is first discussed. With these lathes an abrupt axial load transfer occurs when re-profiling, which is related with the profile geometry of the current support rollers used in the process.
Then, a new kind of support roller profile has been presented as a solution of the problem, being the concave-convex geometry its main feature. The methodology to design the parametrized optimum roller profile is developed and verified.
The essential difference between the process with the novel convex-concave-profile roller proposed in the article, and the conventional convex-profile roller, is shown in the lower part of Figs. 11 and 12, respectively. In fact, when the novel-profile roller is used, no axial load transfer occurs (Fig. 11), whereas currently used rollers lead to abrupt axial load transfer (Fig. 12).
Once the improvement of the convex-concave-profile rollers over ordinary convex-rollers has been revealed, an additional optimization was carried out on the former, using finite element static analysis along the process. Three profiles with different values of their concave-convex curvature radii were analysed. The results are presented in Sect. 4, where it is shown that the greater the radius of curvature of the concave section (or less that of the convex, the sum of both being fixed), the lower the load transfer in the double support. However, the radius of curvature of the convex section cannot be arbitrarily small, as it would lead to arbitrarily large values of the contact pressure, right where the roller supports most of the re-profiling process. For a roller 75 mm radius, its convex section radius of curvature of 10 mm is large enough to support a common wheel during its reprofiling process with existing lathes.
Novel-profile rollers, like those proposed in the article, have been implemented in an under-floor re-profiling lathe by a benchmark manufacturer that develop and supply advanced machine tools. This company has reported a significantly smoother behaviour of the process when using these novel rollers.