1. Introduction

2. Materials and Methods

2.4. Calculation Model

The calculation model is defined in this subsection, starting with the reference frames definition and following with the mathematical description of each of the equation blocks shown in Figure 3 (Section 2.3) The blocks belonging to the left main branch (those related with kinematics) are presented first, while the blocks of the right main branch (related with dynamics) are presented then.

2.4.1. Reference Frames Definition

Four reference frames have been defined for the kinematics and dynamics analyses described in the next pages. These frames are described below and shown in Figure 4 for a wheelset (for the whole bogie does not need a specific reference frame):

• Absolute reference frame $XYZ$, clockwise, fixed and whose origin set on the rolling plane, anchored to the track beginning and centered between the rails.
• Track reference frame $\tilde{x}\tilde{y}\tilde{z}$, clockwise, mobile at the vehicle speed and whose origin is set on the rolling plane and along the track middle line, holding the $\tilde{x}$ axis always tangent to that line.
• Axle reference frame $\stackrel{̿}{x}\stackrel{̿}{y}\stackrel{̿}{z}$, clockwise, mobile at the axle speed and whose origin is set at the gravity center of the wheelset.
• Contact area reference frame $x_c{y}_c{z}_c$, clockwise, mobile at the contact area speed and whose origin is set on the center of the area.

2.4.2. Kinematics Equations Blocks

References (Fissette, 2016), (Moody, 2014), (Oldknow, 2015), (Ortega, 2012), (Rovira, 2012) and (Sichani, 2016) explain how to obtain the kinematic parameters for the wheelsets through relations dependent on the railway line geometry. Not only does Reference (Pellicer & Larrodé, 2021) collect these relations, but it also extends them to all of the possible geometries that can be found in a railway line: straight section, circular curve and transition curve (clothoid, quadratic parabola or cubic parabola).

The parameters obtained at these geometries are described in Reference (Pellicer & Larrodé, 2021): uncentering and uncentering speed, average uncentering and uncentering speed, yaw angle and yaw angle variation speed / rate, average sinus of yaw angle and of yaw angle variation, average yaw angle, combination of the uncentering and yaw angle effects, angle of longitudinal displacement of the contact area and, finally, tilt and tilt speed / rate.

As explained in References (Oldknow, 2015), (Ortega, 2012), (Rovira, 2012) and (Sichani, 2016), the total uncentering of a wheelset ($y$*) can be computed by adding the original uncentering and the uncentering coming from wheelset rotation (this rotation is in reality that of the bogie pivot with respect to the tangent line to the track centerline).

The reason why uncentering must be saturated is because there exists a geometrical constraint: total uncentering cannot be greater than the addition of half the track play / slack (the so – called “flangeway clearance) and the existing gauge widening (equal or different to 0). When total uncentering reaches that value, then the flange belonging to the outer wheel touches the outer rail.

As for the creepages, they are the rigid slip velocities divided by the vehicle speed in order to turn them into non-dimensional (although the spin creepage is dimensional as the resulting units are “rad/m”). References (Fissette, 2016), (Ortega, 2012) and (Sichani, 2016) explain how to compute creepage from kinematics parameters, whereas Reference (Pellicer & Larrodé, 2021) collects this information and proves the formulae. The creepages can be longitudinal, lateral and spin, as explained below:

• Longitudinal creepage: Difference between the nominal wheel radius and the real rolling one (generating $V_x^I$), application of tractive or braking torques to the wheel ($V_x^{II}$) and variation of yaw angle ($V_x^{III}$).
• Lateral creepage: Not null yaw angle (generating $\mathsf{\Delta }{V}_y^I$), adoption of a new equilibrium position by the wheelset ($\mathsf{\Delta }{V}_y^{II}$) and not null tilt angle ($\mathsf{\Delta }{V}_y^{III}$).
• Spin creepage: Conicity (generating $\mathsf{\Delta }{\mathsf{\Phi }}^I$, alternatively known as the camber effect (Ortega, 2012)) and variation of yaw angle (generating $\mathsf{\Delta }{\mathsf{\Phi }}^{II}$).

Finally, Figure 5 shows how a wheelset is positioned on a narrow curve and its main kinematics parameters are shown. Below the figure, the main kinematics equations are presented: uncentering (1), its saturation (2 – 5), creepages definition (6 – 8), longitudinal creepage (9 – 11), transversal creepage (12 – 14) and spin (15 – 17):

$$y=\frac{r_o{b}_o}{kR}$$

(1)

$$y^{*} \left(1^{st} wheelset\right)=y+e\frac{\pi \left|\psi \right|}{360}$$

(2)

$$y^{*} \left(2^{nd} wheelset\right)=y-e\frac{\pi \left|\psi \right|}{360}$$

(3)

$$y_{lim}^{*}=\frac{\eta }{2}+\xi$$

(4)

$$y^{*}=y_{\mathrm{l}\mathrm{i}\mathrm{m}}^{*}\left(\mathrm{if}\mathrm{the}\mathrm{former}\mathrm{was}\mathrm{greater}\mathrm{before}\right)$$

(5)

$$v_x=\mathsf{\Delta }{V}_x/V$$

(6)

$$v_y=\mathsf{\Delta }{V}_y/V$$

(7)

$$\phi =\mathsf{\Phi }/V$$

(8)

$$\mathsf{\Delta }{V}_x=V_x^I+V_x^{III}+V_x^{III}$$

(9)

$$\mathsf{\Delta }{V}_x=-\mathsf{\Delta }r \omega -r_o{\omega }^{\prime }\pm b_o\dot{\psi }$$

(10)

$$v_x=\frac{-\mathsf{\Delta }r}{r_o}+\frac{-r_o{\omega }^{\prime }\pm b_o\dot{\psi } }{V}$$

(11)

$$\mathsf{\Delta }{V}_y=V_y^I+V_y^{III}+V_y^{III}$$

(12)

$$\mathsf{\Delta }{V}_y=-Vsen\psi cos{\gamma }_o+\dot{y}\cos {\gamma }_o-r_i\dot{\mathsf{\Phi }}\cos {\gamma }_o$$

(13)

$$v_y=\left(-sen\psi +\frac{\dot{y}-r_i\dot{\mathsf{\Phi }}}{V}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\gamma }_o$$

(14)

$$\mathsf{\Delta }\mathsf{\Phi }=\mathsf{\Delta }{\mathsf{\Phi }}^I+\mathsf{\Delta }{\mathsf{\Phi }}^{II}$$

(15)

$$\mathsf{\Delta }\mathsf{\Phi }=\pm \omega sen{\gamma }_o+\dot{\psi }\cos {\gamma }_o$$

(16)

$$\phi =\pm \frac{sen {\gamma }_o}{r_o}+\frac{\dot{\psi } cos{\gamma }_o}{V}$$

(17)

2.4.3. Dynamics Equation Blocks

The normal force is exerted by the rail on the wheel as a response to the opposite force (due to gravity or components of accelerations such as the centrifugal one) that the latter exerts on the former. References (ADIF, 1983 – 2021), (ADIF, 2023), (Andrews, 1986), (Fissette, 2016), (Jiménez, 2016), (Moody, 2014), (Rovira, 2012), (Santamaría et al., 2009) and (Tipler & Mosca, 2014) provide some information on how to compute the normal force on each wheel.

However, the most important Reference is (Pellicer & Larrodé, 2021), as it is the one which fills the gaps and obtains the normal force on each wheel as a function of these factors: axle load (${\lambda }_{eje}$), center of gravity of the axle load ($H_{CdG}$), gradient angle (${\beta }_{rp}$) (directly inferred from the inclination ($i$)), cant angle (${\vartheta }_r$), lateral acceleration ($a_{lat}$) and, finally, wheel contact angle (${\gamma }_o$) and longitudinal displacement angle of the contact patch ($\varsigma$). This allows obtaining the normal force on the outer and inner wheel in relation to a curve ($N_e$ and $N_i$, respectively) and decomposing it in their perpendicular and parallel components ($N_{\perp }$ and $N_{\parallel }$). It should be noted that at a straight section, $N_e$ and $N_i$ would be identical ($N$).

On the other hand, an isolated wheel transmits its own weight and its load to the rail, with which is shares an interface: the contact area. The contact area must be greater than zero in order to avoid an infinite normal stress. Both the contact area (geometry problem) and the normal stress must be determined so as to compute the wear and know where it acts (normal problem). As explained in Reference (Sichani, 2016), whenever two bodies make contact, that contact can be non-conformal or conformal. In the first type of contact, the contact area is relatively small in comparison with the characteristic size of the bodies; while in the second type of contact, the geometry of a body adapts to the geometry of the other body, resulting in a relatively big contact area (this could happen when the wheel and rail are so worn-out that their geometries coincide). Conformal contacts can be further simplified if the quasi-identity relation is fulfilled, which means that a relation between the shear modulus and the Poisson’s ratio exists; this condition is fulfilled in this case given that the materials in contact are the same (steel).

Both the geometric and the normal contact problem are solved together, and in References (Ortega, 2012), (Rovira, 2012) and (Sichani, 2016), these theories for solving them are commented upon. As stated in Reference (Pellicer & Larrodé, 2021), which collects the theories, the Herztian contact theory is the most common due to its high accuracy, low computing effort and because the hypotheses it brings are fulfilled for most of the cases:

• The bodies in contact are homogeneous, isotropic and linear elastic.
• Displacements are supposed to be infinitesimal (much smaller than the bodies’ characteristic dimensions).
• The bodies are smooth at the contact zone, that is, without any roughness.
• Each body can be modeled as an elastic half-space, which requires non-conformity.
• The bodies’ surfaces can be approximated by quadratic functions in the vicinity of the maximum interpenetration point. This implies that the curvatures (the second derivates of the functions) are constant.
• The distance between the undeformed profiles of both bodies at the maximum interpenetration point can be approximated by a paraboloid.
• The contact between the bodies is made without friction, so only normal pressure can be transmitted.

More details on Hertz’s solution (theory, parameters, equations, etc.) are given in References (Greenwood, 2018), (Hertz, 1882).

Nonetheless, the Hertzian model ignores the forces and torques due to friction: as a consequence of the relative motion between the wheel and the rail in the longitudinal and lateral directions and around the vertical axis $(z_c$), opposing forces and torques appear. These are associated with tangential stresses and deformations at the contact area, specifically at the slip region of the ellipse (split into one stick and one slip region). There are two ways to compute these variables:

• Analytical: The values are globally computed for the whole contact patch. A set of analytical equations are used, and the tangential problem can be decoupled from the geometric and normal ones because non-conformity and quasi-identity are satisfied.
• Finite-element: The values of the variables are locally computed and are added thereafter so as to obtain the global values. For that, the contact patch is meshed.

In the current work, the analytical way is chosen, inasmuch as that it allows tackling the problem with an algorithm which comes to the results at a good accuracy – computational effort ratio. For the computation of these tangential forces and also the spin torque, in References (Rovira, 2012), (Sichani, 2016) and (Ortega, 2012) some models are commented upon. Reference (Pellicer & Larrodé, 2021) collects them, concluding that Polach’s method is the most appropriate for considering the spin effect, since it brings accurate results with a low effort.

Reference (Pellicer & Larrodé, 2021) collects all of these theories and concludes that Polach’s method is the most appropriate for considering the spin effect on the variables, inasmuch that it brings accurate results with a low computational effort. References (Polach, 2000) and (Polach, 2005) provide more details on the method.

Another important part of dynamics is the flange – rail contact. This is an aggressive contact appearing at tight / narrow curves where gauge widening is not enough for a smooth curve negotiation, so the wheel flange presses laterally against the rail and the rail exerts a reaction force on the flange. It is important to remark that under the hypotheses considered, the tread – rail contact does not cease to exist.

For finding the reaction that the rail exerts on the flange, Reference (Andrews, 1986) proposes the center of friction model. This model states that every bogie, when curving, has a point at which, if a wheel were mounted there, this wheel would spin ideally, that is, with no slip. This point is called the center of friction and determining it allows computing the forces exerted by the rail on the flange – rail contact through force and torque balances. There can be one flange – rail contact (free motion) or two (restricted motion): the latter occurs at the tightest curves when the two wheels of a diagonal touch the rails.

As for the load distribution on the flange and the tread contact areas, Reference (Piotrowski & Chollet, 2005) explains the Sauvage model, a heuristic method to obtain the total indentation (${\delta }_o$) as the sum of the indentation at the tread – rail contact (${\delta }_{br}$) and that at the flange – rail contact (${\delta }_{pe}$). Reference (Pellicer & Larrodé, 2021) simplifies the Sauvage model by introducing the load distribution coefficient (${\alpha }_{fn}$), ranging from 0.5 (same normal load for both contacts) and 1 (the tread contact would become discharged). Its usual values are taken from the results of the Sauvage model: 0.7 – 0.8. At the end, this method is combined with the center of friction one.

Finally, Figure 7 shows the tangential forces and torques associated with the creepages at each wheel of a four-wheeled bogie. The main dynamics equations are introduced thereafter: those for normal force computation on each wheel (Eqs. 18 – 26), those which allow applying Hertz’s solution (Eqs. 27 – 36), those which allow applying Polach’s method (Eqs. 37 – 46) and, finally, those of the center of friction model (Eqs. 47 – 51):

$${\lambda }_{eje}=\frac{{\lambda }_u+{\lambda }_{tara}}{n_{ejes}}$$

(18)

$$H_{CdG}=\frac{{\frac{1}{n_{ejes}}(\lambda }_u{H}_u+{\lambda }_{tara}{H}_{tara}) }{\frac{1}{n_{ejes}}\left({\lambda }_u+{\lambda }_{tara}\right)}$$

(19)

$${\beta }_{rp}=arctan\left(\frac{i}{1000}\right)$$

(20)

$${\vartheta }_r=arcsen \left(\frac{h_r}{2b_o}\right)$$

(21)

$$a_{lat}=\frac{V^2}{R+y}-\frac{h_r^{\prime }}{2b_o} g cos{\beta }_r$$

(22)

$$N_e=\frac{{\lambda }_{eje}}{2}\left(1+\frac{y}{b_o}\right)g cos{\vartheta }_r cos{\beta }_{rp}+\frac{{\lambda }_{eje}}{2b_o}{a}_{lat}{H}_{CdG}$$

(23)

$$N_i=\frac{{\lambda }_{eje}}{2}\left(1-\frac{y}{b_o}\right)g cos{\vartheta }_r cos{\beta }_{rp}-\frac{{\lambda }_{eje}}{2b_o}{a}_{lat}{H}_{CdG}$$

(24)

$$N_{\perp }={ N}_{e|i}\mathit{cos}\left(\varsigma \right)\mathit{cos}\left({\gamma }_o\right)$$

(25)

$$N_{\parallel }=N_{e|i}\mathit{cos}\left(\varsigma \right)\mathit{sin}\left({\gamma }_o\right)$$

(26)

$$A=\frac{1}{2}\left(\frac{1}{{R_y}_1}+\frac{1}{{R_y}_2}\right)$$

(27)

$$B=\frac{1}{2}\left(\frac{1}{{R_x}_1}+\frac{1}{{R_x}_2}\right)$$

(28)

$${R_y}_2=\frac{r}{cos{\gamma }_o}$$

(29)

$$A_c=\pi ab$$

(30)

$$a=m_H{\left(\frac{3}{2}N\frac{1-{\nu }^2}{E}\frac{1}{A+B}\right)}^{\frac{1}{3}}$$

(31)

$$b=n_H{\left(\frac{3}{2}N\frac{1-{\nu }^2}{E}\frac{1}{A+B}\right)}^{\frac{1}{3}}$$

(32)

$$\frac{1-{\nu }^2}{E}=\frac{1}{2}\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)$$

(33)

$$cos\theta =\frac{\left|B-A\right|}{A+B}$$

(34)

$${p_z}_o=\frac{3F_z}{2\pi ab}$$

(35)

$${\delta }_o=r_H{\left({\left(\frac{3}{2}N\frac{1-{\nu }^2}{E}\right)}^2\left(A+B\right)\right)}^{\frac{1}{3}}$$

(36)

$$s_i=\frac{\mu N_{\perp }}{Gab{ C}_{jj}}{v}_i, i,j=x,1; i,j=y,2$$

(37)

$$s_{y,C}=s_y+\left(-\phi \right)a, \left|s_y+\left(-\phi \right)a\right|>\left|s_y\right|$$

(38)

$$s_{y,C}=s_y, \left|s_y+\left(-\phi \right)a\right|\le \left|s_y\right|$$

(39)

$$F=-\frac{2\mu N_{\perp }}{\pi }\left(\frac{\epsilon }{1+{\epsilon }^2}+arctan\epsilon \right)$$

(40)

$$F_i=F\frac{s_i}{s}, i=x,y$$

(41)

$$F_{y,S}=-\frac{9}{16}a \mu N_{\perp } K_M\left[1+\mathrm{6,3} \left(1-e^{-\frac{a}{b}}\right)\right]\frac{\left(-\phi \right)}{s_C}$$

(42)

$$F_{y,C}=F_y+F_{y,S}$$

(43)

$$F=-\frac{2\mu N_{\perp }}{\pi }\left(\frac{k_A\epsilon }{1+{\left(k_A\epsilon \right)}^2}+arctan{(k}_S\epsilon )\right)$$

(44)

$$\mu ={\mu }_o\left[\left(1-A_f\right)e^{-{wB}_f}+A_f\right]$$

(45)

$$w_i=s_{i}V i=x,y$$

(46)

$${\zeta }_v={\alpha }_{fn}{N}_e$$

(47)

$$N_p={\zeta }_v cos{\gamma }_o+{\zeta }_h sen{\gamma }_o$$

(48)

$$\left(-F_t|+F_f\right)=-{\sum }_{i=1}^{i=Z_w+2}{F}_{x,i}^{\prime }$$

(49)

$${\zeta }_{h,1}-{\zeta }_{h,Z_w}={\sum }_{i=1}^{i=Z_w+2}{F}_{y,i}^{\prime }$$

(50)

$${\zeta }_{h,1}{\mathcal{u}}_{fl,1}-{\zeta }_{h,Z_w}{\mathcal{u}}_{fl,4}={\sum }_{i=1}^{i=Z_w+2}\left(F_{y,i}^{\prime } {\mathcal{u}}_{f,i}\right)+{\sum }_{i=1}^{i=Z_w+2}\left(F_{x,i }^{\prime }{\mathcal{v}}_{f,i}\right)$$

(51)

Figure 6. Tangential forces and torques associated with the creepages at each wheel of a four-wheeled bogie. Source: Own elaboration.

Figure 6. Tangential forces and torques associated with the creepages at each wheel of a four-wheeled bogie. Source: Own elaboration.

2.4.4. Calculation of Wear and Prediction of RCF

Wear is the damage to the wheels which reduces their useful life drastically. This wear is due to abrasive and adhesive wear and there exist models based on a wear rate which enable obtaining the wear depth and, hence, characterizing the damage (Sichani, 2016). Abrasive wear is due to the relative movement between the wheel and rail surfaces and their roughness, which cause friction and this, in turn, the loss of wheel and rail material. In contrast, adhesive wear is due to plastic deformation and to the cohesive forces appearing between both surfaces (Van der Waals, electrostatic or chemical), which ends up producing a material transfer from one surface to the other (Rovira, 2012).

For wheel wear characterization, Reference (Rovira, 2012) listed the following hypotheses:

• The equations are parametrized for abrasive wear and not for adhesive wear as both phenomena are already included in the resulting wear law if they have been experimentally calibrated.
• The different mathematical tools study the wear on the wheel profile, where the wear estimated at every instant is cumulative.
• Wear is assumed to be regular: the variation of the transversal profile is studied, not pattern formation along the longitudinal (circumferential) direction. Thus, the wear at a certain position and instant is extrapolated to the whole circumference.
• At the contact interface there are not any pollutants. The effect of pollutants is considering by modifying the friction coefficient or introducing new wear laws.

Considering these hypotheses, the models commented upon in references (Rovira, 2012) and (Sichani, 2016) can be applied to wheel wear characterization. In reference (Pellicer & Larrodé, 2021), energy transfer models and the RAK model are collected and assessed and it was determined that the energy transfer using the USFD wear law since its wear law is continuous, so small errors do not lead to great errors in the end.

Moreover, under high axle loads, the stress distribution around the contact patch may cause fatigue cracks on the wheel surface or inside it. For only predicting if RCF is to appear or not, the fatigue index model developed by reference (Dirks, Ekberg & Berg, 2015) and presented in reference (Sichani, 2016) is useful. The fatigue index ($F{I}_{suf}$) is simply the utilized friction term (${\mu }_u$) minus the shakedown limit ($L_{RCF}$) and by comparing its value with zero, 3 situations can be observed: if $F{I}_{suf}<0$, RCF is not enough for initiating cracks, while if $F{I}_{suf}=0$, this is the limit situation and cracks are not initiated yet. However, if $F{I}_{suf}>0$, RCF initiates cracks on the surface since the tangential force is elevated.

Lastly, Figure 7 shows the wear calculation according to the USFD, which can be eliminated by reprofiling when its depth reaches a certain threshold (Alba, 2015), (Peng et al., 2019), (RENFE, 2020). The main equations of the USFD model are displayed thereafter (Eqs. 52 -56) and the main ones of the fatigue index model (Eqs. 57 – 59) are right next:

$$\frac{T\gamma }{A_c}=\frac{\left|F_x{v}_x\right|+\left|F_y{v}_y\right|+\left|M_z\phi \right|}{A_c}$$

(52)

$$W_{R, USFD}=5.3 \frac{T\gamma }{A_c}, for \frac{T\gamma }{A_c}\le 10.4$$

(53)

$$W_{R,USFD}=55.0, for 10.4<\frac{T\gamma }{A_c}\le 77.2$$

(54)

$$W_{R,USFD}=55.0+61.9\left(\frac{T\gamma }{A_c}-77.2\right), for \frac{T\gamma }{A_c}>77.2$$

(55)

$$H_{USFD}=W_{R,USFD} \frac{a L_{rr} }{\rho \pi r_{rr}}{10}^3$$

(56)

$$F{I}_{surf}={\mu }_u-L_{RCF}$$

(57)

$$F{I}_{surf}=\frac{\sqrt{F_x^2+F_y^2}}{N}-\frac{{\tau }_{lim}}{p_{z_o}}$$

(58)

$$F_{max,RCF}=\frac{2}{3}{\tau }_{lim}\pi ab$$

(59)

Figure 7. Reprofiling of a wheel which has lost material to wear. Source: Own elaboration.

Figure 7. Reprofiling of a wheel which has lost material to wear. Source: Own elaboration.

3. Results

4. Discussion

• Scenarios from (a) to (d) can compare to the life of a 920-mm wheel with $e=1.800$ m and ${\lambda }_{eje}=\mathrm{18,574}$ kg: 115,476 km, computed in reference (Pellicer & Larrodé, 2021).
• The 920-mm wheel can operate for 147,844 km in scenario (a), which implies a 28.03 % increase; for 98,501 km in scenario (b), implying a 14.70 % decrease; for 134,992 km in scenario (c), yielding a 16.90 % increase; and, finally, for 83,808 km in scenario (d), so a 27.42 % decrease.
• Scenarios from (e) to (h) can compare to the life of a 355-mm wheel with $e=1.800$ m and ${\lambda }_{eje}=\mathrm{6,996}$ kg: 35,311 km (real life end) and 129,066 km (fictional life end), calculated in reference (Pellicer & Larrodé, 2021).
• Regarding real life ends, the 355-mm wheel can operate for 60,474 km in scenario (e), which implies a 71.26 % increase; for 47,089 km in scenario (f), implying a 33,36 % increase; for 44,699 km in scenario (g), yielding a 26,59 % increase; and, finally, for 25,742 km in scenario (h), so a 27.10 % decrease.
• Regarding fictional life ends, the 355-mm wheel can run for 221,042 km in scenario (e), which implies a 71.26 % increase; for 171,116 km in scenario (f), implying a 32.58 % increase; for 163,382 km in scenario (g), yielding a 26.59 % increase; and, finally, for 94,089 km in scenario (h), so a 27.10 % decrease.
• These trends are plotted in Figure 11. The scenarios associated with axle load variation ((a), (b), (e) and (f)) are shown in Figure 11(a), while those associated with wheelbase variation (c), (d), (g) and (h)) are in Figure 11(b):
• As it can be seen, increasing axle load is worse than increasing wheelbase (which has an enormous percentual increase). This is because increases in axle load augment both wear depth and the width of the contact patch, whereas increases in wheelbase only augment the former.
• The distance difference between reprofiling (the reprofiling span) is very variable. Should the wagons perform $n$ routes Albarque – Zacarín – Albarque (75.272 km) a week, then reprofiling periodicity should be $Reprofiling span·\left(7/(75.272n)\right)$. Because the reprofiling span is not constant inside any of the scenario, the mean value must be computed for everyone.
• According to the reprofiling periodicity criterion, some scenarios are much more unfavorable than others. Scenarios (b), (d) and (h) have a mean reprofiling span below 4,000 km, while in scenario (e) more than 8,000 km are reached. The next bar plot, in Figure 12, displays this information:
• From a maintenance economy perspective, scenarios (e), (f) and (g) are preferrable over (a), (b), (c) and (d), and even scenario (h) is superior to (b) and (d). This means that using Graz Pauker 702 bogies (with 355-mm wheels) with any axle load below 7 t/axle and with a wheelbase equal or less than 1.800 m is better than using Y-25 (with 920-mm wheels) when it comes to maintenance economy.
• In spite of that, Y-25 bogies are commonly used for rail motorways due to its high load capacity. If every wagon incarnates a couple of Y-25 bogies, it will have 4 axles in total, which means that the wagon total load (tare plus payload) will reach 90 t at 22.5 t/axle. By contrast, wagons with a couple Graz Pauker 702 bogies have 8 axles in total, which means that the wagon total load will only reach 40 t at 5 t/axle.
• Comparing the scenarios symmetrically (that is, (a) to (e), (b) to (f), (c) to (g) and (d) to (h)), it can be checked that the fictional life of 355-mm wheels is always higher that the real life of 920-mm ones. The (fictional) life increases, unexpected at first, respond to the different kinematic response of reduced-diameter wheels when negotiating curves. As demonstrated by Redtenbacher’s formula, uncentering is proportional to wheel radius (to wheel diameter in turn, as radius is the half), so not only do reduced-diameter wheels uncenter less than ordinary-diameter ones, but also their flanges will push against the rails less intensely. Moreover, the bogies where reduced-diameter wheels are mounted are less loaded, which will further reduce the force exerted by the rail on the flange (coming from force and torque balances). Figure 13(a) illustrates partial uncentering (differential effect) for the three scenarios and shows how saturation ($y_{lim}$) is reached at a lower radius threshold for reduced-diameter wheels, while Figure 13(b) shows total uncentering (adding bogie rotation) in the worst case (leading wheelset, outer wheel), but even in this case, flange – rail contact is less aggressive owing to dynamics:
• RCF is predicted for every flange – rail contact (except for isolated cases where the 355-mm wheel is negotiating curves with radii close to the threshold radius) as a consequence of the high normal pressure (4 – 5 GPa) at the flange contact area with the rail. This pressure stacks up hydrostatically over a small contact area.
• The effects caused by RCF can be mitigated by setting a reduced wear depth limit (as in the current work). In real operation, the economical factor forces the operators to find the trade-off between crack growth and wear depth limit. Moreover, due to operational safety reasons, their internal regulations forbid eliminating more than 80 mm in diameter for a 920-mm wheel and more than 20 for a 355-mm one.

5. Conclusions

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