**1. Introduction**

The study of wear in wheel-track systems is the subject of many works [1–11]. In these works, the task of wear is addressed by presenting different models of wheel–rail interaction. In most works, a constant value of normal force was assumed. In some works, simulations of rail vehicle movement on a straight track without a turnout were performed. Motion without a turnout and motion through a turnout were considered with track susceptibility (elasticity and viscous damping) as constant. According to the research conducted in [12], there are significant differences in the appearing vertical forces and normal forces when passing through a turnout with different values of the beam–subtrack system susceptibility. These forces affect the process of phenomena in wheel–rail contact and have an impact on the wear in the wheel–rail pair. The wear phenomenon was studied based on the works [13,14]. Simulation of rail vehicle movement on a straight track without a turnout and a track with a turnout was also shown.

The infrastructure of a rail transportation system consists of railroad tracks, curves, intersections, and turnouts. Turnouts are a complex structure of railroads. They connect neighbouring tracks and enable railway vehicles to change direction of travel. The basic type is the ordinary turnout consisting of switch blades (2), closure rails (2, 3), a crossing frog (4), turnout sleepers, and setting devices. The crucial element of each turnout is the

**Citation:** Kisilowski, J.; Kowalik, R. Mechanical Wear Contact between the Wheel and Rail on a Turnout with Variable Stiffness. *Energies* **2021**, *14*, 7520. https://doi.org/10.3390/ en14227520

Academic Editor: Tek Tjing Lie

Received: 3 October 2021 Accepted: 8 November 2021 Published: 11 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

frog (4) that enables the wheels of railway vehicles to roll over the place of rail crossing. Due to difficult operating conditions characterised by high-dynamic loads generated by the wheels of rail vehicles, the crossbeams are particularly exposed to the destructive character of impact loads [15–20]. Crossings can have a fixed or movable bow. The subject of the analysis is a right-hand ordinary turnout with a radius of R = 1200 m, a fixed crossbeam of 1:9, and three setting closures with a holding force of 7.5 kN (each). The individual elements of this turnout are shown in Figure 1.

**Figure 1.** Normal right turnout 1—Stock rail, 2—Switch blades, 3—Closure rails, 4—Frog, 5—Guardrail.

The mechanical destruction process of the surface layer leads to undesirable changes in the dimensions and shape of the contacting rolling surfaces of the turnout and railroad wheel elements. Degradation of turnouts, especially crossbucks, contributes to the increase in dynamic interactions, which has an adverse effect on the cooperation of the wheel–rail system [21–26].

Railroad turnouts are particularly exposed to abrasive and fatigue wear, which causes shape changes that result from the impact of high-dynamic loads of cyclic nature that occur during the passage of rail vehicles [27,28].

Railway turnouts are important elements of railway infrastructure that ensure traffic runs smoothly between different branch tracks [29]. A turnout is a structure that allows railway vehicles to pass from one track to another while maintaining a certain speed [30]. One of the most common railway turnouts is the regular turnout (Rz) [31]. It consists of three basic units, such as the switch train assembly, the connecting rail assembly, and the crossover assembly. The switch assembly is a movable turnout unit that moves the switch blades by means of a drive. A smooth and safe track change depends on the correct execution of the initial part of the switch blade, which must have the appropriate shape in order to adequately adhere to the supporting rail in the switch. In turnouts, there are often two wheel–rail contact surfaces, as well as disturbances in the nominal wheel–rail contact conditions due to wheel movement from the main rail to the switch rail [32]. The dynamic interaction between a rail vehicle and a railway turnout is more complex than that of normal or curved tracks. Severe shock loads may occur during the passage through the turnout, generating severe damage to the surfaces of the turnout components [33]. Traffic of rail vehicles in regular operation may be considered as a source of influence of high-dynamic loads of cyclic nature, which translates into damages in the form of abrasive and fatigue wear and tear, as well as changes in the shape and dimensions of the outer layer [34–38]. Calculations of dynamic loads and resulting contact and internal stresses enable rational design of railway turnouts and correct selection of material to construct their elements [39]. The results described in [40] show that profile wear disturbs the distribution of wheel–rail contact points, changes the position of wheel–rail contact points along the longitudinal direction, and affects the dynamic interaction between the vehicle

and the turnout. Profile wear disturbs the normal contact situations between the wheel and switch rail and worsens the stress condition of the switch rail [41]. This model allowed the rational design of railway turnouts and the correct selection of material from which their components are made [42].

The development of turnout constructions also results from technological progress in the production of new steel grades for railway turnouts, the development of new material testing methods, and a better understanding of the phenomena occurring in wheel–rail interaction [43–48].

#### **2. Mathematical Model of the Rail Vehicle–Track System**

To determine the durability of individual elements of the railroad switch, it is necessary to calculate the characteristics of the load in the function of time and distance, originating from the wheelsets of the railroad vehicles acting along the switch. The method used to determine the distribution of forces along the switch is the simulation of mathematical models showing the dynamics of the rail vehicle–track system using computer software. When modelling the dynamics of the wheel–rail system, the most used programs are MATLAB and Universal Mechanism. The Universal Mechanism program provides greater capabilities in modelling dynamic phenomena.

In the modelling process, a high-speed train was used, the parameters of which were taken from the work [19].

The mathematical model of the rail vehicle was built based on linear and angular coordinate systems shown in Figure 2.

**Figure 2.** Linear and angular coordinate system.

This system is used as an inertial system associated with rigid solids, of which three groups can be listed in a vehicle (typical) as shown in Figure 3.

**Figure 3.** Coordinate systems in rigid bodies of a rail vehicle.

The vehicle has a body, two bogies, and four sets of wheels. The coordinate systems originate at the centre of mass of the individual solids, and the axes lie on the axes of symmetry. There is an identical system associated with the track that is called a non-inertial system. The matrix of directional cosines between the inertial and non-inertial systems was assumed to be zero–one [14,15].

In addition, the equations of the ties were assumed according to the coordinate system and Figure 3.

The ties for analysing the system can be written as follows (Figure 4):

$$\Phi = \frac{z\_p - z\_l}{2b}, z = \frac{z\_p - z\_l}{2},$$

$$z\_{tl} = z\_l - z\_{wl} - \left(y - \frac{y\_{wl}}{2}\right)\sigma, \quad z\_{tp} = z\_p - z\_{wp} + \left(y - \frac{y\_{wp}}{2}\right)\sigma,\tag{1}$$

$$\dot{\chi} = -\frac{1}{r}\dot{\chi}$$

where 2*b* is the distance between the contact points (wheel and rail) in the middle position of the wheelset; *r* is the radius of the wheel included in the wheelset, measured in the middle position; *σ* is the coefficient linking the angular and transverse displacement of a wheelset; and *ztp*, *zp*, *zwp*, *ztl*, *zl*, and *zwl* are auxiliary nodes, used in the mathematical description of the movement of the railway vehicle.

**Figure 4.** Geometry model of the wheelset–track system (wheelset in middle position) where 1—Outer rail (left), 2—Inner rail (right) [16].

The following assumptions were made in developing the equations of motion:

The vertical loads occurring on the rail will be a variable value and will be determined from the previous step of mathematical calculations performed to determine the train parameters (wheelset and bogie spacing).

The railroad track was modelled as a Euler–Bernoulli beam on which a wheel with velocity v is rolling (motion on the straight track and motion on the turnout return track were considered).

The contact between the rolling surfaces of the wheels and the rail heads is defined based on Kalker's linear theory (defining ellipses with semi axes *a* and *b*).

In the wheel–rail contact area, the Coulomb kinetic sliding friction with a constant coefficient of friction is considered.

Such phenomena as adhesion, micro-slip, and material wear of the wheel and the rail were also considered in the dynamics of vehicle movement on the track.

In the model under consideration, the possibility of two contact ellipses occurring because of two-point rolling of the wheel on the rail within the turnout has been taken into account.

Suspension elements of the first and second stage were assumed to be linear for all assumed coordinates.

For the track without a turnout, the susceptibility was assumed according to the Voigt model (linear stiffness and linear damping). These quantities were determined by measurements on the actual object and consist of the stiffness coefficient, 0.2 × 10<sup>8</sup> N/m, and damping coefficient, 3.2 × 10<sup>3</sup> Ns/m [16].

At the turnout, the stiffness coefficient was calculated according to the parameters shown in Figure 5.

**Figure 5.** The course of variation of the vertical stiffness coefficient of the rails in the switch with different values of the bedding coefficient (real measurements on CMK Idzikowice): 1—Inner track (with cross-brace), 2—Outer track [18].

Considering geometric and structural constraints, the rail vehicle has 27 degrees of freedom. The equation can be found in the work [19].
