*2.3.2.3 Contact patch energy (Tγ)*

Not only the lateral force is determined by a multiple-body simulation (MBS), but also the contact patch energy Tγ. Tγ represents the input parameter for Burstow's [20] evaluating function for rail surface wear in curved tracks. For determining Tγ on the


basis of MBS, again a s-curve has been used. The applied curves in the simulation belong to the radii classes R1 to R4 and their reference radius (215 m for R1, 315 m for R2, 500 m for R3 and 800 m for R4). Further, the simulation includes superelevation and transition curves. **Table 3** in Section 2.3.2.3 depicts the input parameters for the MBS of Tγ. As the simulation is not only done for powered but also unpowered wheelsets up to 8 Tγ values can be assigned to one specific vehicle due to 4 radii classes. The contact patch energy Tγ is given in Nm/m for the TDM. A detailed specification for the MBS is described in the SBB guidance for vehicle pricing [21].

### *2.3.2.4 Traction power value (TPV)*

As damage term D3 describes rail surface wear due to the impact of vehicles power, the traction power value (TPV) is calculated. TPV describes the power of a vehicle (PWheel in kW) related to the contact area between rail and wheel (Aeff in mm<sup>2</sup> ) as shown in Eq. (17). The TPV therefore only exists for powered wheelsets.

$$\text{TPV} = \frac{\text{P}\_{\text{Wheel}}}{\text{A}\_{\text{eff}}} \tag{17}$$

TPV—traction power value (kW/mm<sup>2</sup> ); PWheel—power per wheel (kW); Aeff effective contact area between rail and vehicle wheel (mm2 ).

The vehicle's power is a value that should be available from the data sheet of the vehicle, whereas the contact area between rail and wheel can be calculated. Basis for this is the methodology due to the Hertzian contact area for arbitrarily curved surfaces as it is described in [29]. However, the Hertzian contact area is downsized by a factor of 2/3 to include system-related uncertainties. Eq. (18) depicts the general approach of the downsized Hertzian contact area due to the major (a) and minor (b) radius. In Eq. (19) the major and minor radius of the ellipse are specified as well as the auxiliary angle (ϑ). Due to the formula of Hertz, the major and minor radius are functions dependent on static wheel force (P0), Poisson's ratio (ν), modulus of elasticity (E) and the wheel and rail radius (RWheel, RRail). The coefficients η and ξ are functions of ϑ. Summarizing the general Eq. (18) of Hertz with the parameters in Eq. (19) leads to a simplified Hertzian formula, as depicted in Eq. (20).

$$\mathbf{A}\_{\rm eff} = \frac{2}{3}\pi \,\mathbf{a} \,\mathbf{b} \tag{18}$$

$$\mathbf{a} = \sqrt[3]{\frac{3\,\mathrm{g}^3 \left(1 - \nu^2\right) \,\mathrm{P\_0}}{\mathrm{E}\left(\frac{1}{\mathrm{R\_{\mathrm{Wheel}}}} + \frac{1}{\mathrm{R\_{\mathrm{fail}}}}\right)}} \times 10^6 \quad \text{and} \quad \mathbf{b} = \sqrt[3]{\frac{3\,\mathrm{\eta}^3 \left(1 - \nu^2\right) \,\mathrm{P\_0}}{\mathrm{E}\left(\frac{1}{\mathrm{R\_{\mathrm{Wheel}}}} + \frac{1}{\mathrm{R\_{\mathrm{fail}}}}\right)}} \times 10^6$$

$$\boldsymbol{\theta} = \arccos\left(\frac{\mathrm{R\_{\mathrm{Wheel}}} - \mathrm{R\_{\mathrm{fail}}}}{\mathrm{R\_{\mathrm{Wheel}}} + \mathrm{R\_{\mathrm{fail}}}}\right) \tag{19}$$

$$\mathbf{A\_{\mathrm{eff}}} = \frac{2}{3}\pi \,\xi \,\eta \left(\frac{3\,\left(1 - \nu^2\right)}{\mathrm{E}}\right)^{\frac{3}{2}\times} \left(\frac{1}{\left(\frac{1}{\mathrm{R\_{\mathrm{Dual}}}}, \frac{1}{\mathrm{R\_{\mathrm{fail}}}}\right)}\right)^{\frac{3}{2}\times 40^6} \tag{20}$$

Aeff—effective contact area between rail and vehicle wheel (mm<sup>2</sup> ); a—major radius of ellipse (mm); b—minor radius of ellipse (mm); P0—vehicle static wheel

E

3

force (kN); RWheel—wheel radius (m); RRail—rail radius (m); ϑ—auxiliary angle (rad); ξ, η—coefficients; ν—Poisson's ratio; E—modulus of elasticity (kN/m<sup>2</sup> ).

For the analytic calculation of the Hertzian contact area, the vehicle is considered to be standing on a straight track. The rail head radius (RRail) is therefore expected to be 0.3 m. Further, the Poisson's ratio (ν) is uniformly set to 0.3 and the modulus of elasticity (E) to 2.1 � 108 kN/m2 . These parameters are constant values in the TDM and do not get changed. The coefficients ξ and η can be summarized (approximation) in a function for each, as shown in Eq. (21) due to [21] and [29].

$$
\xi(\ $) = \textbf{1.5281739} \times \$ ^{-0.8571601}
$$

$$
\eta(\ $) = \textbf{0.4724037} \times \$  + \textbf{0.2366389} \tag{21}
$$

ϑ—auxiliary angle (rad); ξ, η—coefficients.

Implementing the functions of the coefficients ξ and η into Eq. (20) gives the final description of the effective contact area due to the Hertzian methodology, as shown in Eq. (22). This formula is valid for the Austrian TDM due to constant values for RWheel, E and ν.

$$\mathbf{A}\_{\text{eff}} = \frac{8.3593707 \times 8 + 4.1874191}{\ $^{0.8571601}} \times \left(\frac{\mathbf{P}\_0}{\frac{1}{\mathbf{R}\_{\text{Wheel}}} + \frac{1}{0.3}}\right)^{\frac{1}{6}}$ 
$$\text{with } \$$
 = \arccos\left(\frac{\mathbf{R}\_{\text{Wheel}} - \mathbf{0}.\mathbf{3}}{\mathbf{R}\_{\text{Wheel}} + \mathbf{0}.\mathbf{3}}\right) \tag{22}$$

Aeff —effective contact area between rail and vehicle wheel (mm<sup>2</sup> ); P0—vehicle static wheel force (kN); RWheel—wheel radius (m); ϑ—auxiliary angle (rad).

The determined TPV in kW/mm<sup>2</sup> corresponds directly to the damage term D3. A multiplication by the number of driven axles is not necessary, since the vehicle's power per wheel is related to the Hertzian contact area between wheel and rail.

### **2.4 Exemplary calculations**

In the following subchapters, exemplary calculations are done for a sample vehicle. A universal locomotive is chosen as a sample vehicle that goes through a track segment of radius class R3. The maximum permissible speed of this vehicle is 230 kmph at an axle load of 22 t. In **Table 4**, all input parameters are given that are necessary to calculate the rolling stock parameters and the damage increments for the sample vehicle.

#### *2.4.1 Rolling stock parameters*

In this chapter, the rolling stock parameters of the sample vehicle are calculated due to the described formula in Section 2.3.2. The vertical wheel forces P1 and P2 and the TPV are determined for powered axles as the sample vehicle has four powered axles and no unpowered axles. Furthermore, Tγ and YR are not treated in this chapter as they are estimated by MBS. Determining the dynamic P2 force for radius class R3 at 90 kmph as shown in Eq. (23) and for 40 kmph (damage term D5) in Eq. (24):


### **Universal locomotive: axle load = 22 t | Smax,Veh = 230 kmph R3: 400 < R** ≤ **600 m | Sreference = 90 kmph | Rreference = 500 m**

#### **Table 4.**

*Input parameter of the sample vehicle.*

P2,90 ¼ 107,910 þ 25 � 0*:*02 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1,247*:*5 1,247*:*<sup>5</sup> <sup>þ</sup> <sup>245</sup> <sup>r</sup> � <sup>1</sup> � 55,400 � <sup>π</sup> <sup>4</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 62,000,000 � ð Þ 1,247*:*<sup>5</sup> <sup>þ</sup> <sup>245</sup> <sup>p</sup> ! � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 62,000,000 � 1,247*:*<sup>5</sup> <sup>p</sup> ! � <sup>10</sup>�<sup>3</sup> ¼ 216*:*8 kN (23)

$$\begin{split} \mathbf{P}\_{2,40} &= \left( 107, 910 + 11 \times 0.02 \times \sqrt{\frac{1,247.5}{1,247.5 + 245}} \times \left( 1 - \frac{55,400 \times \pi}{4 \times \sqrt{62,000,000 \times (1,247.5 + 245)}} \right) \right) \\ &\times \sqrt{62,000,000 \times 1,247.5} \right) \times 10^{-3} \\ &= \underline{156.3 \text{ kN}} \end{split} \tag{24}$$

Determining the dynamic P1 force for radius class R3 as shown in Eq. (25). to Eq. (28):

$$\mathbf{m}\_{\mathbf{e}} \approx \mathbf{0}.4 \times \left( \mathbf{60} + \frac{\mathbf{150}}{\mathbf{0.6}} \right) = \mathbf{124} \text{ kg/m} \tag{25}$$

$$\mathbf{G} = \frac{\mathbf{3.86}}{\mathbf{1.15}^{0.115}} \times \mathbf{10}^{-8} = \mathbf{3.798} \times \mathbf{10}^{-8} \text{ m/N}^{2/3} \tag{26}$$

$$\mathbf{K\_{H}} = \frac{\mathbf{P\_{1,est}}^{\*} - 107.910}{3.798 \times 10^{-8} \times \left(\mathbf{P\_{1,est}}^{\*} - 107.910^{2/3}\right)} = 2.386 \times 10^{9} \text{N/m} \tag{27}$$

$$P\_{1,90} = \left(107.910 + 25 \times 0.02 \times \sqrt{\frac{2.386 \times 10^9 \times 124}{1 + u\zeta\_{1245}}}\right) \times 10^{-3} = 367.3 \text{ kN} \quad \text{(28)}$$

*\*At the end of the iteration process P1,est is expected to be* �*367,283.5 N.* Determining the TPV as shown in Eq. (29):

$$\mathbf{A}\_{\rm eff} = \frac{8.3593707 \times 8 + 4.1874191}{9^{0.8571601}} \times \left(\frac{107.910 \times 10^{-3}}{\frac{1}{1.15} + \frac{1}{0.3}}\right)^{\frac{1}{6} = 110.4 \text{ mm}^2}$$

$$\text{with } \theta = \arccos(\frac{1.15 - 0.3}{1.15 + 0.3}) = 0.94442737\tag{29}$$

$$\text{TPV} = \frac{\mathbf{P}\_{\rm Vhelel}}{\mathbf{A}\_{\rm eff}} = \frac{800}{110.4} = 7.243 \text{ kW/mm}^2$$

**Table 5** summarizes the parameter values for the powered wheels of a universal locomotive in a track segment of radius class R3. This locomotive does not have unpowered wheelsets. In the case of vehicles with both powered and unpowered wheelsets there exist another table with vehicle parameters for unpowered wheels. In the following table, the parameters YR and Tγ that belong to MBS are added too for further calculations. P2 and YR values at 40 kmph are needed later on for damage term D5.

For every vehicle type (such as locomotives, freight wagons, passenger wagons or multiple units) that appears in a network these five physical parameters shown in **Table 5** need to be estimated. As not every parameter occurs in every radius and/or speed class in the following their appearance is summarized:



**Table 5.**

*Vehicle parameters for powered wheels of a sample vehicle (universal locomotive).*


If those input parameters are available for all vehicles in the network, the seven damage increments of the TDM can be estimated for each vehicle. This is illustrated for the sample vehicle in the following Section 2.4.2.

*2.4.2 Damage increment (Dn)*

In this chapter exemplary calculations for the seven damage increments of the TDM are given. Required data and information to do so are.


The basic formula of the vehicle-specific damage increments Dn can be seen in Eq. (30). Since the vehicle parameters are given per wheel, they must be multiplied by the number of wheelsets (powered and unpowered).

$$\mathrm{Dn\_{Veh,Si\langle R\rangle}} = \text{formula for damage termn} \left[ \mathrm{VehPn\_{Si\langle R\rangle\_p(S\_{\min})}} \right] \times \mathrm{n\_{Veh\_p}}$$

$$+ \text{formula for damage termn} \left[ \mathrm{VehPn\_{Si\langle R\rangle\_{op}(S\_{\min})}} \right] \times \mathrm{n\_{Veh\_{op}}}$$

$$\text{with : S = } \min[\mathrm{S\_{max,Veh}, S\_{T\,rack}}] \tag{30}$$

DVeh,Si∣Rj —damage increment Dn of the sample vehicle at speed (i) and radius class (j) (with i=1-8 und j=1-4); nVeh —number of powered (p) or unpowered (up) wheelsets of the sample vehicle; VehPSi∣Rj Sð Þ min —vehicle parameter of powered/ unpowered wheelsets at R/S-class, depending on relevant speed; S max,Veh —maximum permissible vehicle speed in kmph; STrack—reference speed at a certain speed and radius class in kmph; S—relevant speed for determining the vehicle parameter in kmph.

Calculating the vehicle-specific damage increments, it is important to distinguish between powered and unpowered wheelsets. On the other hand, attention must be paid to the relevant speed due to the vehicle and the radius/speed-class. The minimum of both is considered in the calculation.

Exemplary calculation for damage increment D1 is given in Eq. (31).

$$\text{Formula for D}\_1: \text{P}\_{2, \text{S}}^3$$

$$\text{S} = \min[230 \text{ kmph}, 90 \text{ kmph}] = 90 \text{ kmph}$$

$$\text{D}\_{1, \text{R}3} = 216.8^3 \times 4 + \text{O} = 40,793,916 \text{ kN}^3 \tag{31}$$

D1,R3—damage increment D1 of the sample vehicle in radius class R3 (kN3 /vehicle); nVehp—4; nVehup —0; P2,R3p —216.8 (kN/wheel); P2,R3up —0 (kN/wheel). Exemplary calculation for damage increment D2 is given in Eq. (32).

$$\text{Formula for D}\_2: \text{P}\_{2, \text{S}}^{1, 2}$$

$$\text{S} = \min[230 \text{ kmph}, 90 \text{ kmph}] = 90 \text{ kmph}$$

$$\text{D}\_{2, \text{R}} = 216.8^{12} \times 4 + 0 = 2,543.7 \text{ kN}^{1, 2} \tag{32}$$

D2,R3—damage increment D2 of the sample vehicle in radius class R3 (kN1.2/ vehicle); nVehp—4; nVehup —0; P2,R3p —216.8 (kN/wheel); P2,R3up —0 (kN/wheel). Exemplary calculation for damage increment D3 is given in Eq. (33).

$$\text{Formula for D}\_3: \text{TPV} = 7.24 \text{\AA} \text{W/mm}^2 \tag{33}$$

D3—damage increment D3 of the sample vehicle (kW/mm2 ); TPVVeh—7.243 (kW/mm<sup>2</sup> )

Exemplary calculation for damage increment D4.1 and D4.2 is given in Eq. (34).

Formula for D4*:*1ð Þ area a : D4*:*<sup>1</sup> ¼ 0 for T<sup>γ</sup> ≥ 175Nm*=*m D4*:*1,R3 ¼ 0*:*00

$$\text{Formula for D}\_{4.2}(\text{area } \mathbf{e}): \text{D}\_{4.2} = \left(\frac{\text{T}\_{\text{\gamma}} - 65}{\text{110}}\right) \quad \text{for T}\_{\text{\gamma}} \ge 65 \text{Nm/m}$$

$$\text{D}\_{4.2, \text{R}3} = 2 \ast \left(\frac{291 - 65}{\text{110}}\right) + \mathbf{0} = \mathbf{4.11} \tag{34}$$

D4*:*1R3—damage increment D4.1 of the sample vehicle in radius class R3 (-/vehicle); D4*:*2R3—damage increment D4.2 of the sample vehicle in radius class R3 (-/vehicle); n<sup>0</sup> <sup>p</sup> —2; n<sup>0</sup> up —0; Tγ,p —291 (Nm/m); Tγ,up —0 (Nm/m).

Exemplary calculation for damage increment D5 is given in Eq. (35).

$$\text{Formula for D}\_5: \sqrt{\left(\mathbf{0.5 \times P\_{2, \text{S}=40}}^2 + \mathbf{0.5 \times Y\_{\text{R}=190}}^2\right)}$$

$$\text{D}\_{5, \text{R}3} = \sqrt{\left(\mathbf{0.5 \times 156.3}^2 + \mathbf{0.5 \times 69.0}^2\right) \ast 2 + \mathbf{0}} = 241.6 \text{ kN} \tag{35}$$

D5,R3—damage increment D5 of the sample vehicle in radius class R3 (kN/vehicle); n0 <sup>p</sup> —2; n<sup>0</sup> up —0; P2,40,p —156.3 (at 40 kmph) (kN/wheel); P2,40,up —0 (at 40 kmph) (kN/wheel); YR¼190,p —69.0 (at 40 kmph and radius 190 m) (kN/wheel); YR¼190,up—0 (at 40 kmph and radius 190 m) (kN/wheel).

Exemplary calculation for damage increment D6 is given in Eq. (36).

$$\text{Formula for D}\_6: \text{P}\_{1, \text{S}}^3$$

$$\text{S} = \min[230 \text{ kmph}, 90 \text{ kmph}] = 90 \text{ kmph}$$

$$\text{D}\_{6, \text{R}3} = \text{367.3}^3 \times 4 + \text{0} = \text{198,208,728 kN}^3 \tag{36}$$

D6,R3—damage increment D6 of the sample vehicle in radius class R3 (kN<sup>3</sup> /vehicle); nVehp—4; nVehup —0; P1,R3p —367.3 (kN/wheel); P1,R3up —0 (kN/wheel). Exemplary calculation for damage increment D7 is given in Eq. (37).

Formula for D7 : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 71,R � P2,S <sup>2</sup> <sup>þ</sup> <sup>f</sup> 72,R � YR 2 q� � 3 S ¼ min 230 kmph, 90 kmph ½ � ¼ 90 kmph D7,R3 ¼ 2 ∗ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>0</sup>*:*<sup>7</sup> � <sup>216</sup>*:*8<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>3</sup> � <sup>40</sup>*:*02 q� � 3 <sup>þ</sup> <sup>0</sup> <sup>¼</sup> 12,207,944 kN<sup>3</sup> (37)

D7,R3—damage increment D7 of the sample vehicle in radius class R3 (kN<sup>3</sup> /vehicle); n0 <sup>p</sup> —2; n<sup>0</sup> up —0; P2,R3p —216.8 (kN/wheel); P2,R3up —0 (kN/wheel); YR3,p — 40.0 (kN/wheel); YR3,up —0 (kN/wheel); f 71,R —0.7; f72,R —0.3.
