# Modelling of Frictional Conditions in the Wheel–Rail Interface Due to Application of Top-of-Rail Products

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Introduction

_{x}, lateral creepage c

_{y}, spin creepage c

_{z}, the TOR product mass m

_{0}applied at the application site, and the number of wheel passes N. The model calculates the coefficient of adhesion T/N (ratio of the tangential contact friction force to the normal contact force) in the wheel–rail contact at the distance d from the application site as an output depending on the contact conditions.

#### 2.2. Model Discretization

_{i}that are spaced by the mean circumference of the wheels that roll over the rail (see Figure 3). This means that a specific point on the wheel surface that is in repeated contact with the rail surface is followed in the simulation. The assumption of equidistant calculation points on the rail is a simplification. In railway operation, wheels roll at varying effective diameters, which are determined by the nominal diameter, the wheel and rail profile shapes, and the lateral shift of the wheelset. In addition, longitudinal creepage occurs, which changes the circumferential length of the wheel surface that is in contact with the rail per distance rolled.

#### 2.3. Creep Force Model

#### 2.4. Modelling of Product Consumption

_{0}+ k

_{m}

_{1}· (1 − exp(−m/(m

_{0}·k

_{m}

_{2}))) · f

_{p}(p

_{m}) · f

_{c}(c

_{x},c

_{y})

_{0}·k

_{m}

_{2}and assuming f

_{p}= f

_{c}= 1, the product consumption rate dm/dN = k

_{0}+ k

_{m}

_{1}is constant. For m ≈ (m

_{0}·k

_{m}

_{2}), the consumption rate decreases in an exponential way. k

_{0}, k

_{m}

_{1}, and k

_{m}

_{2}are parameters that have been determined from experiments (see Section 3). m

_{0}represents the initially applied product mass. Figure 4 shows the product mass decrement dm/dN for varying values of k

_{m}

_{1}, k

_{m}

_{2}, and m

_{0}.

_{0}in Equation (1). f

_{p}describes the influence of the mean normal contact pressure p

_{m}on the consumption rate, while f

_{c}describes the influence of the longitudinal creepage c

_{x}and the lateral creepage c

_{y}. Equations (2) and (3) satisfactorily describe the experimental results in twin disc experiments, taking the reference condition as 1178 MPa mean normal pressure and 1% longitudinal creep. k

_{p}and k

_{c}are empirical parameters that are fitted to experimental results.

_{p}= 1 + k

_{p}· (p

_{m}/1178 MPa − 1)

_{c}= 1 + k

_{c}· (sqrt(c

_{x}

^{2}+ c

_{y}

^{2})/0.01 − 1)

#### 2.5. Modelling of Coefficient of Friction

_{C}, Stage II has an increasing coefficient of friction, and Stage III represents dry and clean surface conditions where the coefficient of friction stabilizes on a high level described by µ

_{D}.

_{C}+ (µ

_{D}− µ

_{C}) · exp(−m/k

_{m})

_{D}is the limiting coefficient of friction with no product in the contact area (dry and clean surface), while µ

_{C}is the limiting coefficient of friction with surplus product in the contact area. k

_{m}is a parameter that describes the exponential decay of the coefficient of friction with increasing product mass m. Figure 6 shows the change of the coefficient of friction µ as a function of the product mass m for varying values of k

_{m}.

_{0}. Applying little product (case m

_{0}= 0.1 g) causes a rapid increase of the coefficient of friction towards µ

_{D}. When the applied product amount is increased (cases m

_{0}= 0.3 g and m

_{0}= 0.5 g), the coefficient of friction drops initially towards µ

_{C}, followed by an increase towards µ

_{D}. If a large amount of product is applied, the coefficient of friction is low for a number of cycles before the coefficient of friction gradually increases following an S-shaped characteristic. Such curves have been observed for grease-based lubricants in lubrication starvation tests in twin disc experiments [31].

_{0}shifts the transition between µ

_{C}and µ

_{D}to higher numbers of disc revolutions and decreases the slope of the curve in the transition region. In the twin disc lubrication starvation tests with grease in [31], this effect was weak, but it was prominent in ball-on-disc experiments with oil-based TOR products in [30].

#### 2.6. Modelling of Product Carry-On

_{m}

_{1}that is transferred to the counter surface:

_{m}

_{1}= −t · (m

_{1}− m

_{2}) with m

_{1}> m

_{2}

_{1}and m

_{2}on two surfaces for different constant values of the transfer function t without product consumption in a twin disc set-up. After a certain number of disc revolutions N, the product amount on both surfaces is equal.

_{t}· (1 − exp(−m/c

_{m})) · exp(−(N − 1)/c

_{N}))

_{m}and c

_{N}are constants that are determined from experiments (see Section 3). Parameter k

_{t}is related to the product pick-up efficiency at the application site which depends on the details of the application system and the local contact conditions. This value is set to k

_{t}= 0.5 by default.

_{N}) accounts for the combined effect of the decrease of product transfer in the contact patch due to the squeezing out of product and the change of transfer properties due to mixing of third body layer constituents on the surface. This term may also implicitly account for the reduction of transfer in the case of water-based, drying products when the product changes from a “wet” state to a “dry” state. The drying time for the transition from the liquid state after application to the solid state at some distance from the application at a typical vehicle speed may be approximated by the number of wheel revolutions.

## 3. Experimental Results and Model Parameterization

_{x}at a rotational speed of 400 rpm. During the experiment, the evolution of the torque on the driving shaft of the machine was continuously recorded from which the coefficient of adhesion T/N was calculated as a function of the number of disc revolutions N. The experiments start with 1000 disc revolutions in dry conditions, followed by five product applications at intervals of 3000 revolutions using a syringe. Table 1 states the experimental conditions that were used for the model parameterization of a water-based TOR friction modifier. Experiments were carried out at room temperature.

## 4. Modelling Results and Discussion

#### 4.1. Introduction

#### 4.2. Simulation of Wayside Application

_{r}, the corresponding mass on the wheel surface m

_{w}, and the resulting coefficient of friction µ as a function of distance from the application site d for steady-state conditions after 100 wheel passes. The product masses are internal model parameters that were used to calculate the friction values. These masses may differ from the actual masses found in the field.

#### 4.3. Influence of Product Transfer Behaviour

_{m}and c

_{N}of the transfer function (see Equation (6)) on the coefficient of friction. For the reference case, the model parameters are taken from Table 2.

_{m}reduces the product transfer function t for a given mass m. This reduces the carry-on distance and the amount of product past the application site, which increases the coefficient of friction. A reduction of parameter c

_{m}should increase the carry-on distance of the product and reduce the coefficient of friction as this facilitates the transfer of product between the surfaces. However, in the current parameterization, the limiting factor for the carry-on behaviour of the product was the dependence of the transfer function on the number of wheel revolutions N. Therefore, the curve for 0.2·c

_{m}is almost identical to the reference condition in Figure 14.

_{N}influences the distance from the application site where a marked reduction of the coefficient of friction is observed (see Figure 15). A reduction of c

_{N}reduces the number of wheel revolutions during which product transfer between the wheel and rail surfaces takes place. This shifts the location of the transition between coefficient of friction 0.3 and 0.4 towards the application site.

#### 4.4. Influence of Consumption Rate

_{m}

_{1}and k

_{m}

_{2}(see Equation (1)) on the coefficient of friction is shown in Figure 16. Increasing k

_{m}

_{1}leads to a fast consumption of the product after the application site so that the coefficient of friction rises quickly. When k

_{m}

_{1}is reduced, the evolution of the coefficient of friction stabilizes at an intermediate friction level that is determined by the transfer behaviour between the wheel and rail. k

_{m}

_{2}changes the product consumption when little product is present in the contact and has therefore a larger influence on the coefficient of friction further away from the application site. Increasing k

_{m}

_{2}reduces product consumption. In the example, the coefficient of friction stabilized at an intermediate value away from the application site that was determined by the product transfer behaviour (similar to the case with the reduced value of k

_{m}

_{1}).

#### 4.5. Influence of Application Pattern

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of wayside application of TOR products. Products are spread along the rail and are consumed by passing wheels. Product heights (in grey) are exaggerated.

**Figure 4.**Change of product mass decrement dm/dN according to Equation (1) with k

_{0}= 0, f

_{p}= f

_{c}= 1: (

**a**) k

_{m}

_{2}= 0.53, variation of k

_{m}

_{1}, and (

**b**) k

_{m}

_{1}= 1.2 mg, variation of k

_{m}

_{2}or applied product mass m

_{0}.

**Figure 5.**Schematic evolution of the coefficient of friction µ with the number of disc revolutions N in a twin disc experiment from an initially low coefficient of friction µ

_{C}after the application of product to a high coefficient of friction µ

_{D}typical for dry and clean contact conditions without product.

**Figure 6.**Change of the coefficient of friction µ as a function of product mass m according to Equation (4) for µ

_{D}= 0.50 and µ

_{C}= 0.15.

**Figure 7.**Change of coefficient of friction µ with the number of disc revolutions N for different applied product masses m

_{0}according to Equations (1) and (4) with parameters according to Table 2 in Section 3. Limiting values for dry and clean condition µ

_{D}and fully conditioned surface condition µ

_{C}are plotted as thin lines.

**Figure 8.**Change of product masses m

_{1}and m

_{2}with the number of disc revolutions N due to the transfer of product between the surfaces in a twin disc setup. Three examples without product consumption for different transfer function values t and initial product masses are presented.

**Figure 9.**Cross-sectional view of the rail and wheel. Bottom: when the wheel and rail are in contact, some product is squeezed out of the contact patch laterally. Top: when the wheel and rail separate, product is carried on near the contact patch edges on the wheel surface, which is partially re-transferred to the rail if contact width and position do not change along the track.

**Figure 11.**Check of the model parameterization for different product amounts; full-scale wheel–rail test rig experiment at 80 kN wheel force and 5% longitudinal creepage; thin lines indicate the experimental data and thick lines indicate the model results.

**Figure 12.**Parameterization of the transfer function t according to Equation (6): pick-up experiments at the full-scale wheel–rail test rig at 80 kN wheel load and 5% longitudinal creepage.

**Figure 13.**Product masses (internal model variables) and associated coefficient of friction µ as a function of the distance from the application site d.

**Figure 14.**Coefficient of friction µ as a function of the distance from the application site d. Variation of parameter c

_{m}of the transfer function (see Equation (6)).

**Figure 15.**Coefficient of friction µ as a function of the distance from the application site d. Variation of parameter c

_{N}of the transfer function (see Equation (6)).

**Figure 16.**Coefficient of friction µ as a function of the distance from the application site d. Variation of parameters k

_{m}

_{1}and k

_{m}

_{2}that describe the consumption behaviour according to Equation (1).

**Figure 17.**Comparison of two product application scenarios: blue solid, wheel pass 100, 0.02 g of product application before every wheel pass.; red dashed, wheel pass 100 (before reapplication), 0.20 g of product application before every 10th wheel pass.; and red solid, wheel pass 101 (after reapplication), 0.20 g of product application before every 10th wheel pass.

Condition Number | Maximum Normal Pressure p _{0} | Longitudinal Creepage c _{x} | Applied Product Mass m _{0} | Comment |
---|---|---|---|---|

1 | 1500 MPa | 1.0% | 0.05 g | Reference condition |

2 | 900 MPa | 1.0% | 0.05 g | Decrease of maximum normal pressure |

3 | 1500 MPa | 0.5% | 0.05 g | Decrease of longitudinal creepage |

4 | 1500 MPa | 1.0% | 0.10 g | Increase of applied product mass |

Parameter | Value | Equation |
---|---|---|

k_{0} | 1.0449 × 10^{−9} kg | (1) |

k_{m}_{1} | 1.1617 × 10^{−6} kg | (1) |

k_{m}_{2} | 0.53 | (1) |

k_{p} | 1.84 | (2) |

k_{c} | 0.57 | (3) |

µ_{D} | 0.58 | (4) |

µ_{C} | 0.15 | (4) |

k_{m} | 1.6290 × 10^{−4} kg | (4) |

k_{t} | 0.5 | (6) |

c_{m} | 0.035 g | (6) |

c_{N} | 12.1 | (6) |

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**MDPI and ACS Style**

Trummer, G.; Lee, Z.S.; Lewis, R.; Six, K.
Modelling of Frictional Conditions in the Wheel–Rail Interface Due to Application of Top-of-Rail Products. *Lubricants* **2021**, *9*, 100.
https://doi.org/10.3390/lubricants9100100

**AMA Style**

Trummer G, Lee ZS, Lewis R, Six K.
Modelling of Frictional Conditions in the Wheel–Rail Interface Due to Application of Top-of-Rail Products. *Lubricants*. 2021; 9(10):100.
https://doi.org/10.3390/lubricants9100100

**Chicago/Turabian Style**

Trummer, Gerald, Zing Siang Lee, Roger Lewis, and Klaus Six.
2021. "Modelling of Frictional Conditions in the Wheel–Rail Interface Due to Application of Top-of-Rail Products" *Lubricants* 9, no. 10: 100.
https://doi.org/10.3390/lubricants9100100