Features of Interaction of the New and Worn Wheel and Rail Profiles

Abstract

An increase in the friction path and sliding velocity contributes to an increase in thermal loading and wear of heavily loaded rubbing surfaces of wheels and rails. One of the reasons contributing to the development of these phenomena is the change in the shape of the rubbing profiles of wheels and rails due to wear. Conditions of power and thermal loading and increased wear rate of profiles of the wheels and rails are considered, and friction paths and sliding velocities are calculated at movement of new and worn wheels of the wheelset in the straight and curved sections of a rail track. It was revealed that wear of the profiles significantly increases friction path, sliding (friction) velocity, and frequency of lateral oscillations (zigzag movement) of the wheelset in the straight sections of the path along the rolling surface and in curved sections—the friction path of the inner wheel. Therefore, change of the wheel profile due to wear negatively affects the frequency of the wheelset lateral oscillations, friction path, sliding velocity, and wear rate that should be taken into account when determining wear norms of the wheels and rails.

1. Introduction

An increase in the speed and axial load of wheelsets leads to an increase in the power and thermal loads of the interacting surfaces of the wheels and rails, resistance to rotation of the wheels, and their damage. When the wheelset moves in a straight segment, it makes periodic lateral movements (lateral vibrations), which is due to the conical shape of the wheel rolling surface. This leads to increased wear of the wheel rolling surface and its angle of inclination, an increase in the frequencies of transverse vibrations, friction path, sliding velocity, and wear intensity. In addition, changes in the profiles of the interacting surfaces of wheels and rails due to wear affect the dynamic characteristics of railway transport both, on straight and curved sections of the track, on the stability of the movement of the rolling stock and the comfort of passengers. It can also contribute to wheel derailment, increased wear rate, etc. Therefore, the identification of parameters and the determination of their influence on tribological properties of interacting wheels and rails is a fundamental task of the railway industry [1, 2, 3]. However, insufficient knowledge of the processes occurring in the contact zone of wheels and rails, in particular thermal, adhesive, fatigue, and other phenomena, makes it difficult to prevent the noted undesirable phenomena.

Wheel and rail wear rates are seen as one of the fundamental problems of rail transport. Different parts of interacting surfaces of wheels and rails have different functions, different creep, and therefore they should have different properties. The friction coefficient of the steering surfaces of the wheels and rails should be as low as possible—not more than 0.1. Excessively high friction on rolling surfaces lead to their increased shear stresses, plastic deformations, adhesion phenomena, and fatigue damage; low friction can lead to poor traction and braking, as well as heavy wear during traction and braking.

The wheel slipping causes an increase in thermal and power loads on contact surface layers, the vibrations, characteristic noise, and the most dangerous type of wear—scuffing [4]. Figure 1 shows the standard geometric parameters of interacting surfaces of the wheels and rails (a), and a schematic view of various radii inside of the contact zone (b) [5], which causes an increase in the relative sliding velocity with an increase of the ratio R₂/R₁.

Figure 1.

Standard geometric parameters of interacting surfaces of the wheels and rails (a), and a schematic view of various radii inside the contact zone of the wheel flange (b).

Friction path and sliding velocity depend on difference of radii of contacting circles, angle of inclination of the tangent to the flange transitional curve, and coordinates of the wheel and rail contact points.

The nature of movement of a wheelset along a rail track, traffic safety, energy consumption for traction, and environmental pollution are largely determined by both the tribological properties of the interacting surfaces of wheels and rails and the design and condition of the wheelsets and rails. At the same time, in straight and curved sections, the nature of movement of wheelsets along the rail track, their power and thermal load, and damageability differ significantly.

Attempts to increase the wear resistance of wheels and rails by choosing the initial roughness of their materials, by increasing hardness (about 400 HB) and other measures are known [6, 7, 8].

For a given axial load, at other conditions being equal, the wear of the wheel profile mainly depends on the sliding velocity and friction path. These parameters can change in straight sections, during lateral movements of the wheelset, and in curves, when the deflected from the radial position axis of the wheelset returns back to the radial position, contributing to the rail corrugation.

The research results show that to date there are no reasonable recommendations for improving tribological properties of the wheels and rails. The most effective method of improving tribological properties of the wheels and rails is considered to be lubrication of steering surfaces and modification of rolling surfaces. However, the specific conditions for applying lubricants and friction modifiers to rubbing surfaces, the direct environmental impact, as well as the constant change in the profiles of wheels and rails negatively affect the effectiveness of these measures. Apparently, a necessary condition for improving the tribological properties of the wheel and rail profiles is to increase the stability of the third body with due properties and reduce the thermal and power loads of the contact zone.

The destruction of the third body leads to direct contact of rubbing surfaces and a sharp deterioration in their tribological properties. With a given axial load and other equal conditions, the sliding velocity and friction path can have the greatest impact on the loading of the contact zone. The work considers the effect of the wheel profile wear on the sliding velocity and friction path, which increase the loading and damage to the rubbing surfaces of the working profiles of wheels and rails.

In straight and curved sections, the operating conditions of the wheel and rail differ significantly. In the straight sections, the contact zone is preferably located in the middle part of the rolling surface of the wheel, and the movement of the wheelset in the track occurs with the periodic contact of the wheel flange and the rail head.

In the curves, the contact surface of the first wheelset is displaced to the flange of the outer wheel and hence to the lateral surface of the rail head. In this process of movement, the contact surfaces are under the influence of a changing set of both, relative sliding and vertical and horizontal loads. Due to wear and movement of the contact point during displacements of the wheelset in the track, the initial conical shape of the rolling surface and the angle of inclination of the wheel flange are short-term. Constant changes of the wheel and rail profiles lead to an increase of power and thermal loads of the contact zone, seizure in places of destruction of the third body, increase of the friction coefficient, its instability and intensity of wear, decrease of the angle of inclination of the flange and lateral surface of the rail, and increase of probability of the wheel rolling on the rail.

In search of reducing the wheel damage, a large number of studies have been performed, and there are many recommendations: clarification of the initial profile of the wheel and rail; increased hardness of interacting surfaces; selection of materials; application of “conformal profiles” of rolling surfaces, etc. However, they do not always have reliable theoretical or experimental justifications and are not effective enough.

2. Features of movement of the wheelset on straight sections of the track

At movement of a wheelset in the straight segments, because of conical form of the wheel tread surface, it performs periodic lateral displacements (or “hunting” oscillations) that affect friction path, friction velocity, and wear rate (Figure 2).

Figure 2.

Zigzag movement of a wheelset on a straight line.

The cyclic interaction of wheels and rails in the conditions of rolling with sliding, as well as the noted types of their damage, which have a different nature and are established visually without clear signs of their manifestation, are typical, limiting the efficiency of the wheels and rails. It should also be noted that the processes preceding the rise of adhesive wear and fatigue phenomena, in particular the processes of formation and destruction of the third body, have not been sufficiently studied, and some of the parameters that contribute to increasing the intensity of various types of damage have not been revealed.

The standard geometric parameters of the interacting surfaces of wheels and rails, in particular the radius of the base of the flange and the rounding of the corner of the rail, are different, which in some conditions excludes direct contact of the corner of the rail and the base of the wheel flange. However, the geometric parameters of the wheel and rail profiles are short-lived, wear out during operation, and lead to conformal contact. Worn wheels and rails under certain conditions are characterized by conformal contact, increased power and thermal loads of the contact zone and friction work, and, accordingly, increased wear intensity and instability of friction forces.

A typical feature of interaction of the wheel flange surface and lateral surface of the rail head is a comparatively small magnitude of the coefficient of friction (no more then 0.1). But this depends on the operating conditions of the wheel flange with the lateral surface of the rail head, and in severe operating conditions, the friction coefficient becomes unpredictable and can increase sharply, leading to catastrophic wear and rolling of the wheel on the rail.

The dependences of the radius of the wheel R on the coordinate of the contact point x on the transition curve, the angle of inclination β of the tangent passing through this point (Figure 1a), and the sliding path L per one revolution of the wheel, respectively, have the form:

$$\beta =\mathit{\text{arctg}}\frac{x}{\sqrt{r^2-x^2}};R=R_1+k=R_1+r-{\sqrt{r^2-x}}^2;L=2Rarccos\frac{R_1}{R};h=\mathit{IL},$$E1

where r is the radius of curvature of the flange base, x-coordinate of the point of contact of the flange base along the x-axis, R₁—radius of the wheel at the beginning of the transitional curve; k—wheel radius increment: I—wear rate; L—friction path; h—wear value.

As can be seen from Figure 3, the thermal load of the rolling surfaces is relatively low. At operation of the wheel in traction and braking modes, at rotation around vertical axis, and at slipping, the values of sliding velocity and friction path increase. The flange root and the rail corner in the contact zone have sufficiently high tangential stresses, temperature, and creep level. They serve the role of both rolling surfaces and steering surfaces, and the value of the friction coefficient 0.1 for both cases is unacceptable. The flange root and the rail corner are involved in traction, braking and turning the wheel, which requires mutually exclusive properties: a relatively high value of the friction coefficient for traction and braking and a low value of the friction coefficient for steering.

Figure 3.

Power and thermal loads of the wheel and rail contact zones.

Figure 4 schematically shows the conditions of loading and damage to the rubbing surfaces of wheels and rails when moving in straight and curved sections and the consequences of changing the initial profiles of wheels and rails, in particular: increased frequencies of lateral vibrations, friction paths and intensity of damage to rubbing surfaces, as well as environmental pollution with vibrations, noise, and wear products.

Figure 4.

Loading and damage conditions of rubbing surfaces of wheels and rails when moving in straight and curved sections.

3. Power and thermal loads and damageability of wheels and rails

Figure 3 shows power and thermal loads of the wheel and rail contact zones (of tread and steering surfaces).

As can be seen from the figure, the root of the wheel flange and rail corner are the most loaded parts of their profiles where the greatest friction work is done. Table 1 shows the contact stresses, temperatures, and friction coefficients in the contact zone of the tread surface as well as the surfaces of the wheel flange root and rail corner.

Parameter	Contact stress GPa	Required coefficient of friction	Temperature
Tread surface	1.5	0.2 -0.4	400 °C
Surfaces of the wheel flange root and rail corner	3	≤ 0.1	800°C

Table 1.

Contact stresses, temperatures, and friction coefficients in the contact zone of the tread surface.

This causes a difficult-to-predict, destructive effect of surface layers, and many studies are devoted to its prevention [9, 10].

The actual contact area is much smaller than the nominal contact area (Figure 5) [8]. In the process of friction, with direct contact of surfaces, on actual contact zones, significant stresses arise, corresponding to deformation and heat release, leading to destruction of the third body and to seizure of directly interacting irregularities.

Figure 5.

Zones of actual contact and corresponding surface deformations.

In the process of friction, separation of the seized places causes a sharp increase in tangential stresses and deformations.

Considering the heat distribution in irregularities as a boundary value problem describing heat transfer in the cylinder, the results of solving the problem for various surrounding materials—for water and lubricating oil—will have the form shown in Figure 6.

Figure 6.

Dependence of the temperature on the distance from the micro-asperity base for various ambient:

water

oil lubricant.

As it is seen from the figure, intensity of the temperature reduction toward the base of the micro-asperity is higher for water than for oil lubricant. The raised heat capacity and thermal conductivity of water contributes to better cooling conditions of the micro-asperity, and at increased distance from the heat source, the temperature for water is lower than for oil lubricant.

On the wheels and rails, in addition to the steering and tread surfaces, there are also surfaces of the flange root and the rail corner, which simultaneously play the roles of the steering and tread surfaces, though the requirements for these surfaces are different. Existing profiles of wheels and rails can be divided into rolling surfaces (participating in “free” rolling, traction, and braking) and steering surfaces (wheel flange and lateral surface of the rail, taking part in steering mainly in curves and preventing the wheelset from derailment). The root of the flange can roll along the corner of the rail, participate in traction, braking, and steering. But traction (braking) and steering require mutually exclusive properties: a relatively high value of the coefficient of friction during traction and braking and, as low as possible coefficient of friction during steering. And the “ideal” value of the coefficient of friction (μ ≥ 0.1) in the contact area of the wheel flange and rail corner is not acceptable for both cases. By gradually shifting the points of interaction of the wheel and rail from the rolling surface to the root of the flange and the rail corner and then to the flange and the lateral surface of the rail, the relative sliding velocity, friction path, the likelihood of fatigue damage, scuffing, vibration, noise, and wear intensity of the rubbing surfaces increase. In addition, the close proximity of the rolling and steering surfaces facilitates mixing of lubricants and friction modifiers. Therefore, the rolling and steering surfaces must be separated and modified with friction modifiers having corresponding properties.

Processes accompanying the interaction of profiles, especially with an increase in creep (with an increase in traction or when the interacting places move toward the flange of the wheel and the lateral surface of the rail), increase the likelihood of destruction of the third body and interacting surfaces. However, an increase in the relative sliding of the wheels causes an increase in thermal and power loads in the contact of surface layers, generating vibrations, characteristic noise, and the most dangerous type of wear—scuffing. Therefore, maintaining a third body between interacting surfaces is critical. The above problems are relevant for any railways, and their solution requires special experimental and theoretical studies. It is known that the working conditions of wheelsets in curved segments are very difficult, especially of the first wheelset. This issue has become especially acute in recent years with an increase in train speed, and a lot of work has appeared on increasing the resistance of wheel flanges to operational impacts.

Figure 7 shows wheels and rails interacting surfaces (tread and steering surfaces) and rail corner and root of the wheel flange (a) and friction work values (b) [11].

As seen in Figure 7, the greatest friction work falls on the root of the wheel flange and the rail corner, which contributes to the greatest resistance to wheel rotation, energy consumption to overcome this resistance, and damage to the wheels and rails.

Figure 7.

Tread and steering surfaces, rail corner, and root of the wheel flange (a) and friction work values (b).

Figure 8 shows damages of interacting surfaces of wheels and rails.

Figure 8.

Damages of interacting surfaces of wheels and rails.

Figure 8a shows traces of fatigue damage to the rolling surface and adhesive wear of the steering surface, Figure 8b—grooved rolling surface, Figure 8c—traces of the wheel rolling surface melting, and Figure 8d and e shows traces of fatigue damage to the rolling surface and delamination of the rail steering surface. These types of damage are formed simultaneously, and the dominant type depends on the working conditions.

4. Evaluation of the wheels and rails tribological parameters

Currently, there are no reliable and recognized methods for predicting the tribological parameters of rubbing surfaces of machines, including wheels and rails. To do this, they usually use methods used in mechanical engineering. The Amonton–Coulomb formula for determining the coefficient of friction ƒ = FT/FN is currently the most popular; to estimate the volume of worn material, Archard’s formula (1952) W = KPS/H is used. It is assumed that the friction power under limiting conditions is constant and can be determined by the formulas ƒPV = const., Pm Vn = const., and the contact stress is determined by the Hertz formula (1892) σH = 0.418(q. Ered/ρred).where ƒ is the sliding friction coefficient; FN and FT—normal and tangential forces; W—volume of wear products; K—empirical coefficient, usually called the wear coefficient; P—normal contact load; S—friction path; H—hardness of surface layers of the material of rubbing parts; V—sliding velocity; m, n—exponents; σH—normal contact stress; q = linear load; Ered—reduced modulus of elasticity; ρred—reduced radius of curvature. However, these formulas are characterized by limited parameters and do not always give a satisfactory result.

As can be seen from the above formulas, tribological parameters largely depend on the sliding velocity, friction path, and contact load.

5. Wheelset movement in a straight segment

It is known that when the wheelset moves in a straight segment, due to the conical shape of the rolling surface of the wheel, it performs periodic zigzag movements (lateral oscillations), which affects the friction path, sliding velocity, and wear intensity. Therefore, we determine the friction path of the wheels and the average sliding velocity.

Consider new standard wheelset with the diameter of the rolling circumference D = 957 mm (radius R = D/2 = 478.5 mm) and a slope of the profile $\mathrm{n}=tan\alpha =\frac{1}{20}=0.05$ and the same wheelset worn-out by 7 mm with the profile slope n = 0.34. For solution of the problem, we will use the graphs of periodic lateral displacements and the axle yaw of the free wheelset (without bogie) with the data: amplitude—for new wheelset $y_0=0.295!!=7.5\mathrm{mm}$, for worn-out one $y_0=0.885!!=22.5\mathrm{mm}$; wave length—for new wheelset λ = 53I = 1615.44 cm, for worn-out one λ = 20I = 609.6 cm (Figure 9) [12].

Figure 9.

Lateral oscillatory movement of new and worn wheels.

As seen from Figure 9, the amplitude and frequency of lateral oscillations and, accordingly, the friction path of a new and worn wheels are different. Let us determine numerical values of the friction paths for new and worn wheels.

Maximum value of the new wheelset axle yaw may be found by the formula [12]:

$$Q_{\mathit{max}}^0={sin}^{-1}\left(\frac{2y_0}{R+n{y}_0}\right)={sin}^{-1}\left(\frac{2\times 7.5}{478.5+0.05\times 7.5}\right)=1.7950;$$E2

At rotation of the wheelset axle by this angle, the contact points of its each wheel will slide on the rail by the distance

$$\mathrm{T}1/4=\frac{d}{2}{Q}_{\mathit{max}}=\frac{d}{2}{Q}_{\mathit{max}}^0\frac{\pi }{{180}^0}=\frac{10}{2}\times 1.7950\frac{3.14}{{180}^0}=0.1566\mathrm{mm}.$$E3

where d = 10 mm is the diameter of the contact spot.

Consequently, friction path of the wheels during one cycle

$$\mathrm{Tc}=\mathrm{T}1/4\times 4=0.1566\times 4=0.6264\mathrm{mm}$$E4

and for distance of 1 km will be

$$\mathrm{T}1\mathrm{km}=\mathrm{Tc}\frac{1\mathit{km}}{\lambda }=0.6264\mathit{mm}\frac{1000m}{16.1544m}=38.77\mathrm{mm}.$$E5

For the worn-out by 7 mm wheelset, with radius of the rolling circumference R = 478.5-7 = 471.5 mm and parameters $\mathrm{n}=0.34;y_0=0.885!!=22.5$ mm, using successively formulas (1), (2), (3)), and (4), we obtain T1km = 308.4 mm.

The frequency f and period P of the zigzag oscillations, as well as friction velocity U depend on the train speed V. Numerical values of these parameters at V = 100 km/h = 27.77 m/s are calculated by the formulas $\mathrm{f}=\frac{V}{\lambda };\mathrm{P}=\frac{1}{f}\text{and}\mathrm{U}=\frac{\mathit{Tc}}{P}.$

Table 2 shows the results of calculation of the friction path per 1 km, the frequency of zigzag oscillations, and the sliding velocity at the speed V = 100 km/h.

Parameter	Designation	The wheel condition	Value
Friction path per 1 km	T1km	New; Worn	38.74 mm; 308.4 mm
Frequency of the zigzag motion at V = 100 km/h	f	New; Worn	1.72 Hz; 4.55 Hz
Friction (sliding) velocity at V = 100 km/h	U	New; Worn	1.077 mm/s; 8.553 mm/s

Table 2.

Results of calculation of the friction path per 1 km, the frequency of zigzag movement, and the sliding velocity at a speed of 100 km/h.

As can be seen from Table 2, the friction path, frequency of the zigzag movement and sliding velocity and, accordingly, the wear rate, vibration, and noise of worn wheels are much greater than that of new ones.

Let us consider the sliding distance of the wheel flange during rolling, rolling with sliding, and sliding of the wheel tread. When the tread surface slides without rolling, the sliding path of the flange is maximum and equal to the arc cd (Figure 10a). At rolling of the wheel tread with sliding (Figure 10b), length of the arc of contact cd is decreased, and at rolling of the wheel without sliding of the wheel tread (Figure 10f) length of the arc of contact is minimal.

Figure 10.

Arc of contact of the wheel flange and lateral surface of the rail head at various ratios of the sliding velocity and rolling velocity: (a) sliding without rolling; (b) rolling with sliding; (f) rolling without sliding.

Therefore, sliding distance and hence sliding velocity depend on the ratio of the sliding velocity and rolling velocity of the wheel tread surface.

6. Peculiarities of a wheelset movement in curves

At movement of the wheelset along a curved section of the track, in order to maintain the radial position of the wheelset, the wheels must travel different distances due to the difference in the radii of curvature of the outer and inner rails. But the lateral displacement of the wheelset with the initial tilt of the rolling surface of the wheels 1:20 (which is constantly changing due to wear) does not always compensate for the difference in the paths traveled by the outer and inner wheels and does not provide the radial position of the wheelset axle. The root of the flange or the flange of the outer wheel pressing against the lateral surface of the rail prevents the wheel from rolling onto the rail, leading to additional resistance to wheel rotation. In this case, the axle of the wheelset is twisted, bended, and deviated from the radial position, creating a two-point or conformal contact, increasing the angle of attack, and creating a danger of the wheel going off the rail. In the extreme case, when the axle of the wheelset is extremely deviated from the radial position, in order to continue movement, one of the wheels must slide along the rail (outer wheel in the direction of movement or inner wheel in the opposite direction) in order to return in the radial position, or the wheelset will roll onto the rail head and come off it. Figure 11 shows: (a) a curved section of track; (b, c) movement of a wheelset in curves.

Figure 11.

(a) Curved section of a track; (b) movement of a wheelset in curves; (c) slippage of the inner wheel [13].

Intermittent slippage of one of the wheelset wheels (Figure 11c) is accompanied by torsional vibrations of the wheelset shaft and longitudinal vibrations of the carriage (which are identified as flange noises [14, 15, 16]) and the corresponding wear of the rails—the so-called corrugation that occurs mainly on the lower rail in the curves [17, 18, 19, 20]. Despite numerous attempts, there is no reliable solution to the problem of rail corrugation, except for expensive grinding technology.

Figure 12 illustrates typical rail profile changes that arise in curves [21, 22]: (a, b) severe lateral wear and (c) severe rail corrugation. Such wear causes the great change of rail profile and, therefore, strongly affects the running behavior of railway vehicles, such as motion stability, riding comfort, and derailment safety.

Figure 12.

Wear magnitudes of different parts of the wheel working surface: W1—vertical wear; W2—horizontal wear; W3—450 wear (a, b) severe lateral wear; (c) severe rail corrugation.

Determine numerical values of the friction parameters of a new (unworn) wheels of a wheelset moving in a curve with radius ρ (Figure 11a).

The outer wheel displaced laterally by y = 10 mm, rolling along the arc l1 = 1 km = 1000 m and its flange sliding on the lateral surface of the rail, will make n1 = l1/2 π R1 = l1/2π(R + ny) =1000/2 × 3.14(478.5 + 0.05 × 10) = 332.265 revolutions.

The flange points at one revolution describe an extended cycloid with a full loop below the rolling line (Figure 13), whose length presents the friction path. The maximum friction path will have the points K located at the greatest distance Rk from the wheel center (Figure 14).

Figure 13.

Extended cycloid.

Figure 14.

Wheel rolling on the rail.

The value of this radius will be Rk = R1 + hk, where R1 = 478.5 + 0.05 × 10 = 479 mm, and hk = 30-13-ny = 30-13-0.05 × 10 = 16.5 mm, or.

$$\mathrm{Rk}=479+16.5=495.5\mathrm{mm}$$

Here 30 mm is the height of the flange and 13 mm—the height at which the peripheral part of the flange is rounded.

The length of the loop below the rolling line can be replaced by 2hk with sufficient accuracy. Then the friction path for the outer wheel per revolution will be.

$$\text{F1}=\text{2hk}=2\times 16.5=33\mathrm{mm},$$

and at rolling along the arc l1 = 1 km, the friction path will be.

$$\mathrm{F}=\mathrm{F}1\text{n1}=33\times 332.265=\mathrm{10,965}\mathrm{mm}=10.965\mathrm{м}.$$

The wheel friction velocity depends on the train speed. To determine it, we divide the friction path of the wheel for its one revolution F1, by the time t required for rotation of the wheel by the angle 2φ (Figure 14), where

$$\mathrm{\phi }={cos}^{-1}\frac{R_1}{R_k}={cos}^{-1}\frac{479}{495.5}=14.8270.$$E6

Consequently, 2φ = 2 × 14.8270 = 29.650, which is 29.650/3600 = 0.08236 revolutions.

For the train speed v = 50 km/h = 50 · n1 = 50 · 332.265 = 16613.25 rph = 4.615 rps, the time t will be

$$\mathrm{t}=\text{2φ}/\mathrm{v}=0.08236/4.615=0.0178\mathrm{с}$$E7

and sliding velocity

$$\text{VslF}1/\mathrm{t}33/001781854\mathrm{mm}/\mathrm{s}1854\mathrm{m}/\mathrm{s}.$$E8

Consider now the inner wheel. Its effective radius of the wheel rolling circle.

$$\text{R2}=\text{R-ny}=478.5-0.05\times 10=478\mathrm{mm},$$E9

Determine the height of the wheelset cone ρw from the proportion (Figure 15):

Figure 15.

Cone of the wheelset.

$$\frac{R_1}{R_2}=\frac{{\rho }_w+S}{{\rho }_w-S}=\frac{R+\mathit{ny}}{R-\mathit{ny}},\text{from where}{\rho }_w=\frac{\mathit{SR}}{\mathit{ny}}=\frac{790\times 478.5}{0.05\times 10}=756030\mathrm{mm}=756.03\mathrm{m}.$$E10

The inner wheel will not slide on the rail if the radius of curvature of the rail is ρ = ρw. If ρ < ρw, then the wheel will have a negative slip in the opposite to movement direction, and if ρ > ρw, then a positive slip in the direction of movement. Consider the last two cases.

1. (ρ < ρw); Suppose ρ = 420 m. In this case, the arc length l2 (Figure 11a) will be

   l2 = (ρ -S) β, where β = l1/ (ρ + S) = 1000/(420 + 0.79) = 2.376 rad. and substituting we get.

   l2 = (420-0.79) · 2.376 = 996.244 m.

   The inner wheel will roll over the distance

   $$l_2^1=\text{2πR}2\mathrm{n}1=2\times 3.14\times 0.478\times 332.265=997.912\mathrm{m},$$E11

   which is greater than l2. The total friction path will be

   $$\mathrm{SF}=\text{l2}-l_2^1=996.244-997.912=-1.668\mathrm{m},$$E12

   that is, sliding occurs in the direction opposite to the movement.

2. (ρ > ρw); Suppose ρ = 1020 m. Determine the angle β = l1/(ρ + S) = 1000/(1020 + 0.79) = =0.9796 rad. The arc length l2 = (ρ -S)β = (1020-0.79) · 0.9796 = 998.448 m. The total friction path in this case will be

$$\mathrm{SF}=\text{l2}-l_2^I=998.448-997.912=0.537\mathrm{м},$$E13

that is, the wheel will slide in the direction of movement.

Determine the sliding velocity of the inner wheel. When rolling, the inner wheel periodically slides along the rail for the time t required to return the twisted axle of the wheelset to its original position. This time is equal to the quarter of the free oscillation period of the single-mass torsional oscillatory system “wheelset axle - inner wheel.”

$$\mathrm{t}=\frac{T}{4}=\frac{1}{4}\text{2π}\sqrt{\frac{I}{C}}=\frac{1}{4}2\times 3.14\sqrt{\frac{54.96}{4.29\times {10}^6}}=0.0056\mathrm{s},$$E14

where I = 54.96 kgm2—moment of inertia of the internal wheel (mass m = 398 kg) relative to the axis of rotation; С = 4.29 × 106 Nm—torsional stiffness of the wheelset axis (diameter d = 173 mm).

The maximum angle by which the axle of the wheelset will rotate each time will be

$$\text{γmax}=\frac{f\mathcal{Q}{R}_{2}2S}{I_{p}G}=\frac{0.4\times {10}^5\times 478\times 2\times 790}{87.94\times {10}^6\times 77\times {10}^3}=0.00446\mathrm{rad}=0.25560,$$E15

where f = 0.4—friction coefficient between rail and wheel rolling surface; Q = 10 t = 105 N—vertical load on the wheel; Ip = 87.94 × 104 mm4—polar moment of inertia of the wheelset axle cross section; G = 77 GPa = 77 × 103 mPa = 77 × 103 N/mm²—shear modulus of the wheel axle material.

The inner wheel twisted by the angle γmax will slide during return to the initial position along the arc

$$\mathrm{l}=\text{R2}\times \text{γmax}=478\times 0.00446=2.13\mathrm{mm},$$E16

representing elemental component of the total friction path. Consequently, sliding velocity of the inner wheel will be

$$\text{Vsl}=\text{l}/\mathrm{t}=2.13/0.0056=380.36\mathrm{mm}/\mathrm{s}=0.3804\mathrm{m}/\mathrm{s}.$$E17

The friction parameters of the worn wheels are calculated similarly considering their parameters y = 25 mm and n = 0.34.

Table 3 shows numerical values of the friction parameters of a new (unworn) and worn wheels of a wheelset moving in a curved section of the track.

Parameter	Designation	Location of the rubbing element	The wheel condition	Haight of the wheelset cone ${\rho }_w$	Radius of curvature of the track ρ	Value
Friction path per 1 km	F; SF	On the flange of the outer wheel; On the rolling surface of the inner wheel	New; Worn; New; Worn	------; ------; 756.03 m; 43.82 m	------; ------; 420 m; $\left(\mathrm{\rho }<{\rho }_w\right)$; 1020 м; $\left(\mathrm{\rho }>{\rho }_w\right)$; 1020 м; $\left(\mathrm{\rho }>{\rho }_w\right)$	10.965 m; 9.615 m; −1.668 m; 0.537 m; 33.86 m
Friction (sliding) velocity at V = 50 km/h	Vsl	On the flange of the outer wheel; On the rolling surface of the inner wheel	New; Worn; New; Worm	------; ------; ------; ------	------; ------; ------; ------	1.854 m/s; 1.729 m/s; 0.380 m/s; 0.357 m/s

Table 3.

Friction parameters of a new (unworn) and worn wheels of a wheelset moving in a curved section of the track.

The most dangerous type of damage to the wheels of a wheelset moving along a curved section of the track is wear of the wheel flanges. As can be seen from Table 3, the friction path and sliding velocity of the flange of a new and worn wheels differ little from each other. The sliding velocities of a new and worn rolling and flange surfaces also differ little from each other; however, the friction path for a worn rolling surface of the wheel is much higher than that for a new one, and, therefore, its wear at the same wear rate will also prevail.

Due to the complexity and interdisciplinary of the interaction processes between wheels and rails, further research is planned in the direction of creating conditions for the generation of stable solid third bodies with the necessary tribological properties on rolling and directional surfaces, devices for their application to friction surfaces, and the development of wheel pairs with a reduced load of the contact zone rubbing surfaces.

7. Conclusions

At a given load of wheels and rails, contact stresses, and thermal loads of the zone of their friction contact, damage to the third body and the wear rate of rubbing surfaces mainly depend on the sliding velocity and friction path. The calculations have shown that at pure rolling of a worn wheelset in straight sections, the friction path, the frequency of zigzag movement, and the sliding velocity of rolling surfaces are much higher than those for new wheels.

At pure rolling of new (unworn) and worn wheelsets in curved sections, the friction paths of the wheel flanges are close, the friction path of the rolling surface of a worn wheel exceeds that of a new one about 20 times, and the sliding velocities of worn and new wheels are almost the same.

Also shown a change in the friction path of the wheel flange from pure rolling to pure sliding of the rolling surface, and that the rail corrugation is formed by periodic slippage of the inner wheel on the rail. The amplitude and frequency of lateral movement and, accordingly, the friction path of a new and worn wheels are different. During pure rolling of new (unworn) and 7 mm worn wheelsets on curved sections, the friction paths of the wheel flanges are close, and the friction path of the tread surface of the worn wheel exceeds the friction path of the new wheel by approximately 20 times, and the sliding speeds of worn and new wheels are almost the same. It also shows the change in the friction path of the wheel flange when the movement of the rolling surface changes from pure rolling to pure sliding.

Acknowledgments

This research was financially supported by Project # 22-2204 of Shota Rustaveli National Science Foundation of Georgia.

Conflict of interest

The authors declare no conflict of interest.