Introduction

Defects such as fatigue microcracks, hardened layer spalling, corrugation wear and other defects formed on the top surface of the rail profile have the potential to affect the wheel-rail contact geometry and matching relationship, increase the noise and wheel-rail wear rate during vehicle operation, and reduce the smoothness and safety of vehicle operation1,2. The most effective method for improving the wheel-rail contact condition, reducing the wheel-rail wear rate and extending the service life of the rail is the application of on-line rail profile grinding or milling to remove the damaged layer on the profile surface in a timely manner and reshape the rail profile3,4,5.

Online profiling is the process of forming the rail profile by the envelope formed by multiple grinding heads on the surface of the rail profile. This allows the grinding amount of the rail profile surface to be regulated by modifying the grinding pressure and speed6,7. The process is characterized by the rapid formation of the rail profile (the fastest grinding speed can reach 80 km/h) and the resulting minimal grinding volume (the maximum grinding thickness is only 0.12 mm)8. This makes it an appropriate method for the regular maintenance of the rail profile and it is commonly used for both preventive and restorative transformations of the rail profile9,10. As illustrated in Fig. 1, rail profile milling employs a rotary cutting combination of a right-angle cutter and a circular arc cutter mounted on the cutter disc, which is used to approximate the shape of the rail profile. The cutter disc of the cutter set moves in parallel along the upper surface of the rail, thereby achieving the objective of rail profile milling11,12,13. The maximum milling thickness of rail profile milling is 1.0 mm, which is sufficient to remove damaged and defective layers on the surface of the rail profile, thus improving the surface condition of the rail profile and extending the replacement cycle of the rail14,15. One disadvantage of this method is that the surface of the rail profile after milling will remain wavy. This phenomenon can be attributed to the waviness formed by the rotating milling trajectory of the milling cutters on the milling disc. The milling cutters are oriented radially and axially on the milling disc and mill the top surface of the rail. The radial clearance between the milling cutter grains results in uneven and corrugated surface on the rotationally milled surface as it moves16.

Figure 1
figure 1

Schematic diagram of rail profile milling principle.

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The majority of previous research on rail profile milling for reshaping has concentrated on three key areas: the maximum milling thickness, the accuracy of reshaping, and the ability to eliminate rail profile damage17,18,19,20. The relationship between parameters such as milling speed and the characteristic parameters of residual waviness on the contoured surface of the rail after milling, as well as the effect of residual waviness on the contact stresses and low fatigue cycles applied to the rail surface under wheel load, have not yet to be elucidated21,22,23,24,25. This paper will investigate the relationship between milling speed and residual waviness, as well as the contact stress between wheels and the contoured rail surfaces with residual waviness, and low cyclic fatigue. The results will provide a foundation for the analysis of the generation and expansion of damage and defects on the rail surface after milling.

Consequently, the objective of this research is to make novel contributions to the numerical modelling of the residual waviness after milling for rail profile reshaping, as well as to the analysis of the influence of the residual waviness on the surface of the rail profile in terms of contact stresses and low fatigue cycles under wheel-rail contact loading. Furthermore, the research will propose optimization parameters for rail profile reshaping. The structure of this paper is as follows.

·Firstly, the relationship between parameters such as milling speed and the characteristic parameters of residual waviness on the surface of the rail profile after milling was established. Subsequently, a numerical model of residual waviness was constructed and validated through experimentation. The experiments entailed the reshaping of the rail profile through milling.

·Secondly, a three-dimensional wheel-rail contact finite element model with residual waviness surfaces was constructed following the reshaping of the rail profile at typical milling speeds. This was based on the residual waviness observed on the numerical model. Subsequently, the impacts of residual waviness on contact stresses and low fatigue cycles on the rail surface under the wheel-rail contact loading were investigated.

·In conclusion, the following findings are presented.

Numerical modelling analysis and validation of residual waviness

Numerical modelling analysis

As illustrated in Fig. 1d, the rail profile surface is a complex surface comprising five segments of circular arcs with radii of 13, 80, 300, 80, and 13 mm, respectively. The milling grains are arranged in a vertical configuration, aligned with the straight edge of the milling cutter and the normal of the arc surface, as illustrated in Fig. 2a. A coordinate system is established with the central point of the inner circle of the R13 rail profile designated as the origin (denoted by point O). In this coordinate system, the x-axis is horizontal, the y-axis is vertical, and the z-axis is aligned with the rail. The positive milling direction is defined with respect to this coordinate system. The rail profile is divided into eight segments in the xyz coordinate system, which are milled by the circular cutters No. 2 and 3, and the right-angle cutters No. 4, 5, 6, 7, 8 and 9, respectively.

Figure 2
figure 2

Numerical modelling and validation plots for residual waviness analysis. Note: In Fig. 2, the wavelength is indicated by λL, which is the wavelength of a single visible residual waviness. Milling speed is the speed at which the rotating milling disc moves along the top surface of the rail profile.

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During the milling of the rail profile, the rotating milling unit moves in a direction parallel to the rail. The formation of milling residual waviness is illustrated in Fig. 2b. The term “HM” represents the maximum cutting thickness of the highest position on the top surface of the milled rail. Subsequently, the wavelength and wave height of the residual waviness, designated as Hλ and λL, respectively, are measured after the highest position on the top surface of the milled rail. The horizontal distance between the two adjacent milling cutters at the highest entry point on the top surface of the rail is designated as λM. The angle θ is defined as the included angle of the lowest point of the milling path and the adjacent wave crest point of the residual waviness on the outer circle of the milling disc. Finally, l is the length of the material that has been milled.

A theoretical discrepancy exists between the rail profile generated by profiling milling and the standard profile. The theoretical profile deviation value is determined by the length of the corresponding arc segment at the milling position of the right-angle milling cutter. The theoretical deviation value of δM after profiling the rail profile by profiling milling is shown in Eq. (1), which is derived from the mathematical geometrical relationship.

$$\delta_{{\text{M}}} = r_{{\text{P}}} - \sqrt {r_{{\text{P}}}^{2} - \left( {\frac{{l_{{\text{M}}} }}{2}} \right)^{2} }$$
(1)

where rP is the radius of the curve of the rail profile, lM is the length of the straight segment milled on the rail profile by the straight edge milling cutter.

In Fig. 2a, the theoretical deviation of the rail profile milling is generated in five segments, each milled by a different right-angle milling cutter. These are numbered 4, 5, 6, 7 and 8, respectively. In accordance with Eq. (1), the maximum theoretical contour deviations corresponding to the milling cutters No. 2 to No. 8 were determined to be 0, 0, ± 0.096, 0.032, ± 0.012, ± 0.082 and ± 0.025 mm, respectively.

The wavelength and wave height of the milling residual waviness are closely related to the diameter of the milling disc, the angle between the milling cutter grains, the milling speed and the milling disc speed, and are also affected by the milling thickness. If the relative sliding between the cutter and the rail profile is not taken into account, the travel of the cutter is equal to the length of the milled rail profile. The following relationship can be established:

$$N_{{\text{M}}} \lambda_{{\text{M}}} = v_{{{\text{MM}}}} t$$
(2)

where vMMt = l, and vMM is the milling speed, NM is the number of rows of milling cutter cut on the rail top surface along the horizontal direction of the rail when the milling disc is rotating and moving milling, t is the milling time.

The rail profile is milled by the milling disc at a constant rotational speed and forward speed. The relationship between NM and the rotational speed of the milling disc nM is given by Eq. (3).

$$N_{{\text{M}}} = \frac{360}{{\theta {}_{0}}}n_{{\text{M}}} t$$
(3)

where θ0 is the radial angle between two rows of milling cutters.

The combination of Eqs. (2) and (3) allows for the derivation of Eq. (4).

$$\lambda_{{\text{M}}} = \frac{{\theta {}_{0}}}{360}\frac{{v_{{{\text{MM}}}} }}{{n_{{\text{M}}} }}$$
(4)

Two milling cutter grains, situated radially adjacent to one another on the milling disc, are fixed in the same diameter position and rotate at the same speed. Therefore, the distance between the two entry points and the milling residual waviness, as well as the distance between the two peaks (i.e. the wavelength), is equal along the rail direction, adjacent to the two milling cutter particles in the rail profile surface. This can be expressed as λM = λL. Consequently,

$$\sin \theta = \frac{{0.5\lambda_{{\text{M}}} }}{{0.5d_{{\text{M}}} }}$$
(5)

In Fig. 2b, the residual waviness wave height, Hλ, following the milling of a rail profile is illustrated in Eq. (6), in accordance with the geometric relationship, as defined by Eqs. (4) and (5).

$$H_{\uplambda } = \frac{1}{2}d_{{\text{M}}} - \frac{1}{2}d_{{\text{M}}} \cos \left( {\arcsin \frac{{\lambda_{{\text{M}}} }}{{d_{{\text{M}}} }}} \right)$$
(6)

Model validation of residual waviness

To facilitate a meaningful comparison and analysis of the calculated and experimental values of the residual waviness characteristics following rail profile milling, it is essential to ensure that the respective rail profile milling speeds are identical for both the calculated and experimental values. Considering the speed limitations of the rail profile milling test machine, this was achieved by determining the milling speeds for the lowest, middle, and highest speeds of the experimental rail profile milling unit, which were found to be 300, 600, and 1000 m/h, respectively. From Eqs. (2), (5), and (7), the wavelength and wave height of the residual waviness are calculated to be 2.60 mm and 0.0021 mm, 5.21 mm and 0.0085 mm, and 8.48 mm and 0.0236 mm, respectively. The calculated values are presented in Table 1.

Table 1 Morphology of residual waviness generated by the rail profile, and calculated and experimental values of their wavelength and wave height.
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As illustrated in Fig. 1a, the precision of the previously developed numerical model of the residual waviness generated by milling the rail profile was validated through on-site milling experiments conducted on the rail profile itself. These experiments were carried out utilizing a milling apparatus mounted on a mobile rail milling vehicle. As illustrated in Fig. 1b, the milling apparatus is equipped with a milling disc with a diameter of Φ800 mm. This diameter is in alignment with the highest point of the upper surface of the milled rail profile. The electric motor, via the synchronous belt and gear transmission, drives the dynamic rotation of the milling disc, which is installed with 32 groups of milling cutter holders. As illustrated in Fig. 1c, a single set of milling cutter holders comprises two circular milling cutters with a cutting-edge radius of 13 mm and eight right-angle milling cutters (one of which is an auxiliary cutter) with a length and width of 14 mm. The experimental railway was paved with No. 60 rails, which are the most used rail designation for both metro and freight railways in China. The configuration and dimensions of the rail profile are illustrated in Fig. 1d.

The milling disc rotates at a constant speed of 60 rpm when the profile milling operation is used to reshape the rail profile. The maximum milling thickness is inversely proportional to the milling speed. Based on the performance of the profiling machine, it was determined that the milling thickness was constant at 0.3 mm. The three most used milling speeds of 300, 600, and 1000 m/h were selected for the experiment. Figure 2c depicts the morphology of the rail profile after reshaping by profile milling.

Subsequently, the reshaped rail profile is assessed using the RS2015-2W portable rail profile measuring instrument, which exhibits a measuring accuracy of ± 0.02 mm. Additionally, a laser measuring instrument with a measuring accuracy of ± 0.01 mm is employed to assess the wavelength and wave height parameters of the residual waviness of the rail profile following milling.

As shown in Fig. 2c, the initial surface roughness of the rail profile prior to milling was determined to be Ra 8.0 μm. The surface of the rail profile exhibited a relatively smooth finish following milling at three distinct speeds, with corresponding roughness values of Ra 1.1 μm, Ra 1.4 μm, and Ra 1.7 μm, respectively. It proved challenging to discern the residual waviness boundaries due to the impact reflections. To reduce the potential for error in the measurement of individual waviness, the wavelength and height values of three, five, and ten waviness were measured and averaged, respectively. The results are presented in Table 1.

Table 1 illustrates that the calculated and experimental values of the wavelength of the residual waviness after milling of the rail profile exhibit a certain degree of variation. This is attributed to the effect of the misalignment of the milling cutter particles mounted in the same row, with the calculated value of the wavelength falling within the range of variation of the experimental value. The experimental value of the wave height of the residual waviness is slightly larger than the calculated value, and the deviation between the two values is less than the error of the measuring instrument, which is ± 0.01 mm.

The actual deviation of the milled rail profile was measured, and the maximum deviation occurring at the milling position corresponding to cutter No. 5 in Fig. 2a. The maximum deviation of the rail profile was found to be ± 0.16 mm at a milling speed of 1000 m/h, ± 0.12 mm at 600 m/h and ± 0.08 mm at 300 m/h.

A comparison of the calculated rail profile deviation values derived from Eq. (1) with the experimental test values revealed that the maximum deviation values of both were within the permissible rail profile deviation values.

The preceding analysis shows that the numerical calculations of residual waviness following milling of the rail profile are in alignment with the experimental measurement outcomes. The observed discrepancy between the two is attributed to the influence of experimental conditions and measurement inaccuracies.

Effect of residual waviness on rail contact stresses and low cyclic fatigue

3D FE modelling and analysis

This paper employs the finite element method to analyze the effect of residual waviness on the wheel-rail stresses applied to the profile surface and its low fatigue cycles. Firstly, a three-dimensional finite element simulation and analysis model of wheel-rail contact must be established to analyze the effect of residual waviness on wheel-rail contact stresses and low-cycle fatigue cycles, with the latter being achieved by changing the residual waviness on the surface of the rail profile. The rail in question is of the No. 60 variety. The slope of the rail bottom is 1/40, and the wheels are LM worn tread with a wheel diameter of 820 mm. The dimensions of the milling device and the milling parameters are employed to calculate the parameters of the numerical model of the residual waviness via numerical methods, as outlined in Eqs. (4) and (6). The surface material of the rail profile is removed using the simulated machining method, thereby ensuring that the surface assumes the same shape as that of the numerical model of the residual waviness. A vertical downward load equal to the wheel weight of the vehicle is applied to the central axis of the wheel axle.

The simulation analysis does not consider the potential impact of sleepers. Once the contact has been established, the rail is fixed, and the wheel-rail contact is frictional. The coefficient of friction falls within the range of 0.05 to 0.5, and the impact of this coefficient of friction on the wheel-rail contact area, the maximum contact stress and the maximum Mises stress is relatively minor26. The friction coefficient of friction is 0.227. The calculation and solution are performed using the generalized Lagrangian method28,29,30. To reduce the computational effort and increase the mesh density, the mesh was divided, with consideration given to only 1/10 of the entire wheel. A global grid is employed, with an element size of 2.0 mm.

The rail material is U71Mn heat-treated steel, with the following physical properties: tensile strength of 1180 MPa, Young’s modulus of 2.06 × 104 MPa, Poisson’s ratio of 0.29, surface hardness of HB 407–476, and yield strength of 550 MPa. The wheel material is ER8 steel, with the following mechanical properties: tensile strength of 882 MPa, Young’s modulus of 2.04 × 104 MPa, Poisson’s ratio of 0.30, surface hardness of HB 245–276, and a yield strength of 537 MPa.

A three-dimensional finite element model was constructed for the analysis of wheel-rail contact stresses and low fatigue cycles following rail profile milling, in accordance with the configuration depicted in Fig. 3.

Figure 3
figure 3

Schematic of 3D FE modeling and analysis of residual waviness affecting wheel/rail contact stress and low fatigue cycles.

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Figure 3 demonstrates the application of a three-dimensional finite element model to investigate the influence of residual waviness following rail profile milling on wheel-rail contact stresses and low fatigue cycles. In Fig. 3a, the wheel is subjected to a vertical downward force, equivalent to the weight of the vehicle, applied at the center of the wheel axle. A further illustration of the condition of the wheel-rail contact section is provided by a partial enlargement of the contact area between the wheel tread and the rail profile. Figure 3b depicts four distinct states of the rail profile, namely the residual waviness on the surface of the rail profile without milling and after milling at three different speeds: 300, 600, and 1000 m/h, respectively. One of these states may be selected for the simulation of the wheel-rail contact stresses and low fatigue cycling analyses. Figure 3c illustrates the distribution of contact stresses on the rail profile surface across four distinct states. Figure 3d illustrates the distribution of low fatigue cycles at the wheel-rail contact point, as depicted in Fig. 3c.

Profile surface contact stress analysis

The von Mises yield criterion postulates that, under specific deformation conditions, when the equivalent stress at a given point within a stressed object reaches a certain threshold, the point will begin to transition into the plastic state. In the general state of stress, the equivalent force, σ, is

$$\overline{\sigma } = \frac{1}{\sqrt 2 }\sqrt {(\sigma_{x} - \sigma_{y} )^{2} + (\sigma_{y} - \sigma_{z} )^{2} + (\sigma_{z} - \sigma_{x} )^{2} + 6(\tau_{xy}^{2} + \tau_{yz}^{2} + \tau_{zx}^{2} )^{2} }$$
(7)

where σx is the principal stress in the x-direction; σy is the principal stress in the y-direction; σz is the principal stress in the z-direction; τxy is the shear stress in the xy-plane; τyz is the shear stress in the yz-plane; τxz is the shear stress in the xz-plane.

If the wheel weight load is 11.5 tons, the rail contour surface exhibits no residual waviness, and residual waviness of No. 1, No. 2, and No. 3, respectively, a three-dimensional finite element method is employed to simulate the wheel-rail contact stress. The contact stress on the profile surface of the rail is as illustrated in Fig. 4. The location of the maximum stress on the rail surface of Fig. 4 is taken as a base point. A cross-section of the wheel-rail contact area is then taken in a direction perpendicular to the rail to obtain Fig. 5. Figure 5 shows the stress distribution of the maximum stress section in the wheel-rail contact area.

Figure 4
figure 4

Simulation results of the equivalent stress on the rail surface under wheel-rail contact for a wheel load of 11.5 tons.

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Figure 5
figure 5

Stress distribution in the wheel-rail contact section at the point of maximum stress.

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In Fig. 4, when the surface of the rail profile exhibits no residual waviness, residual waviness No. 1, No. 2, and No. 3, the maximum value of the equivalent contact stress experienced by the contact area of the rail surface and the vertical distance from the point of occurrence to the inside of the rail profile (shown in Fig. 2a) are 936 MPa and 25 mm, 905 MPa and 27 mm, 799 MPa and 27 mm, 795 MPa and 27 mm, respectively. Following the extraction of the stress distribution map of the region exhibiting relatively high contact stress and its rotation in a clockwise direction by 90 degrees, it can be observed that the contour of the high stress distribution region exhibits a distinctive and unique shape. As illustrated by the red lines in the Fig. 4, the profiles of the high stress distribution areas appear to exhibit a crescent, snowman, flying saucer, and felt-hat shape, respectively.

In Fig. 5, when the rail profile surface exhibits no residual waviness, as well as residual waviness No. 1, No. 2, and No. 3, the maximum stress point in the wheel-rail contact area is observed at a depth of approximately 2 mm below the wheel-rail contact surface. The maximum equivalent stress values are 1240, 1120, 1170, and 1110 MPa, respectively, with the perpendicular distances to the inside of the wheel being 66 mm, 68 mm, 68 mm, and 68 mm. A comparative analysis was conducted between the two cases: one with no residual waviness on the rail profile surface and the other with residual waviness formed by milling. It was determined that the residual waviness introduced by milling resulted in a shift of the maximum equivalent stress location in the wheel-rail contact area by 2 mm outward and a reduction in the maximum equivalent stress value.

Low cyclic fatigue analysis

The wheel load on the rail surface gives rise to a high stress fatigue cycle state in the contact area of the rail surface. The low-cycle fatigue analysis data for heat-treated U71Mn rails are derived from the existing literature31,32,33.

The results of the wheel-rail contact stress simulation calculations, as illustrated in Figs. 4 and 5, were employed to analyze the low cyclic fatigue of the wheel-rail contact area under stress–strain, with a wheel weight load of 11.5 tons. The S–N data for heat-treated U71Mn was employed to obtain the distribution of low fatigue cycles on the rail profile surface of the rail, as shown in Fig. 6. The position of the lowest low fatigue cycle on the rail profile surface depicted in Fig. 6 was selected as a reference point. Subsequently, a cross-section of the wheel-rail contact area was obtained in a direction perpendicular to the rail, resulting in the data presented in Fig. 7. The distribution of low fatigue cycles of the stress–strain at this location is illustrated in Fig. 7.

Figure 6
figure 6

Simulation results of low cyclic fatigue on the rail surface under wheel-rail contact for a wheel load of 11.5 tons.

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Figure 7
figure 7

Distribution of low fatigue cycles in the wheel-rail contact area at the location of the minimum fatigue cycle.

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In Fig. 6a–d, when the surface of the rail profile exhibited no residual corrugation, No. 1, No. 2 and No. 3 residual waviness, under the load of 11.5 tons wheel weight, the minimum low fatigue cycles in the contact area of the rail surface under stress–strain were 336, 406, 752, and 786 cycles, respectively. Furthermore, the distances from the minimum low fatigue cycle occurrence location to the inside of the rail profile were 25 mm, 27 mm, 27 mm and 27 mm, respectively.

In Fig. 7a–d, when the rail profile surface has no residual corrugation, residual waviness of No. 1, No. 2 and No. 3, respectively, the minimum values of low fatigue cycles in the wheel-rail contact area appear at the location where the depth under the wheel-rail contact surface is approximately 2 mm. The minimum values of low fatigue cycles are 139, 178, 159, and 182 cycles, and the vertical distances from them to the inside of the wheels are 66 mm, 68 mm, 68 mm, and 68 mm, respectively. The results of the simulation analysis indicate that the residual waviness generated by milling causes the location of the minimum values of low fatigue cycles in the wheel-rail contact area to shift outwards by 2 mm. In addition, the maximum low fatigue cycles of the wheel-rail contact are observed when the surface of the rail profile exhibits the No. 3 residual waviness.

Conclusions

This paper characterizes the wavelength and wave height of the residual waviness after milling the rail profile surface. This is based on the mathematical relationship between the milling parameters and the characteristics of the residual waviness. A numerical analysis model of the residual waviness after milling at three commonly used milling speeds of the rail profile surface has been developed. These are 300, 600, and 1000 m/h, respectively. The residual waviness model was verified based on the experimental data of on-site milling. The three-dimensional finite element simulation method was used to analyze the contact stresses on the rail profile surfaces under a loading condition of 11.5 tons of wheel weight, with and without residual waviness, and with three types of residual waviness. Additionally, the stress–strain low fatigue cycles in the contact area were subjected to analysis. The principal findings can be summarized as follows:

  1. (1)

    The wavelength and wave height of the residual waviness produced by profile milling of rail contours depend on the diameter of the milling disc, the rotational speed, the radial angle of two adjacent milling cutters and the milling speed. It can be demonstrated that the wavelength and wave height of the residual waviness produced by profile milling of rail contours are directly proportional to the milling speed.

  2. (2)

    The residual waviness resulting from the milling of the rail profile surface has the effect of modifying the wheel-rail contact position and contact area morphology, reducing the maximum contact stress on the rail profile surface and increasing the stress–strain fatigue cycles in the contact area. When the rail profile surface was subjected to a wheel weight load of 11.5 tons, four cases of rail profile surfaces were considered: no residual waviness, residual milling waviness on the rail profile surfaces after milling speeds at 300, 600, and 1000 m/h, the morphology of the wheel-rail contact high stress areas on the profile surface are crescent, snowman, flying saucer, and felt-cap shapes, with maximum equivalent stress values of 936, 905, 799, and 795 MPa, respectively. The maximum equivalent stress values were 936, 905, 799, and 795 MPa, while the minimum low fatigue cycles were 336, 406, 752 and 786 cycles, respectively. The maximum equivalent stress values of the wheel-rail contact area sections were 1240, 1120, 1170, and 1110 MPa, while the minimum low fatigue cycles were 139, 178, 159, and 182 cycles, respectively.

  3. (3)

    In order to achieve the desired outcome of reshaping the surface profile, it is essential to prioritize a higher milling speed. This will result in a rail profile surface with a relatively low contact stress and high low fatigue cycle, which is beneficial in that it impedes the formation of surface defects on the surface of the rail profile and the expansion of the surface defects. This, in turn, prolongs the period of rail profile milling for reshaping the profile and eliminating surface defects.