Advances in Tribology

Volume 2018, Article ID 7298236, 11 pages

https://doi.org/10.1155/2018/7298236

## A Novel Analytical Method to Calculate Wheel-Rail Tangential Forces and Validation on a Scaled Roller-Rig

Politecnico di Torino, Department of Mechanical and Aerospace Engineering, C.so Duca degli Abruzzi 24, 10129 Turin, Italy

Correspondence should be addressed to Nicola Bosso; ti.otilop@ossob.alocin

Received 30 January 2018; Revised 25 June 2018; Accepted 12 July 2018; Published 9 August 2018

Academic Editor: Huseyin Çimenoǧlu

Copyright © 2018 Nicola Bosso and Nicolò Zampieri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The study of railway dynamic strongly depends on the estimation of the tangential forces acting between wheel and rail. Simulation of the dynamical behaviour of railway vehicles is often performed using multibody codes, and the calculation of the contact forces must be efficient and accurate, even if the contact problem is strongly nonlinear. Therefore, the contact problem is still of great interest for researchers. This work proposes an analytical and efficient algorithm to calculate wheel-rail tangential forces. The proposed method is compared with the most commonly used algorithms under different conditions. In addition, experimental tests are performed on a scaled prototype on roller-rig to demonstrate that the method can be easily adjusted using experimental results. The benefit of the proposed method is to provide an analytical and fast solution, able to obtain accurate results and to allow corrections based on empirical evidence.

#### 1. Introduction

The calculation of the tangential forces between wheel and rail is a fundamental aspect when studying railway vehicle dynamic. It plays an important role both in the case of steady state running of the vehicle and during traction or braking operations. The determination of the friction forces is a complex problem, as it involves several subproblems that lead to a strong nonlinearity on the behaviour of tangential forces, which are also affected by external conditions (contact contamination).

The problem can be solved accurately using precise but complex algorithms, which require huge computational time [1–3]. Since the problem governs vehicle dynamic, the simulation of the dynamic behaviour of railway vehicles, usually performed by means of multibody codes [4, 5], requires simpler and more efficient algorithms. However, the accuracy of the results should be preserved as much as possible. For this reason, during the last years researchers have developed several numerical methods able to solve the problem efficiently [6–12]. At first, linear models were used to study tangential forces as a function of the relative velocity between wheel and rail, which was briefly described using the kinematical creepages [2]. Later, simple nonlinear laws were proposed in order to take into account the adhesion limit at the contact, such as the hyperbolic tangent method or other heuristic nonlinear approaches based on experimental observations [8]. Kalker, after developing his complete theory, described by complex algorithms, such as CONTACT and DUVROL [2], realized that a simpler method was fundamental for vehicle simulation purposes and studied a simpler but faster method known as FASTSIM [6].

Later, Polach studied an alternative method [7], able to be faster than FASTSIM and provide similar results, that was also compared with experimental tests performed on real vehicles. Polach, in addition, developed an extension of his method in order to investigate adhesion in nonsteady condition related to traction, also considering different environmental conditions, by modifying his original method to include these aspects [13]. Also the algorithms developed by Kalker have been recently improved by Vollebregt, by increasing the accuracy of FASTSIM in the FASTSIM2 version [14] and implementing in the more complex CONTACT code the effect of “falling friction” versus the creepage [15, 16], which was already introduced by Polach in his extended method [13, 17]. Nowadays FASTSIM and Polach’s methods are commonly used in multibody codes, to investigate railway vehicle dynamic. The authors of this paper had previously proposed a simplified nonlinear method to estimate wheel-rail and wheel-roller tangential forces. That method did not consider accurately the effect of the spin creepage. The present work proposes an improvement of the previously proposed method, which correctly takes into account the spin creepage and allows considering the variation of the friction coefficient with the creepage (“falling friction”), allowing its use in traction/braking simulations. The benefit of the proposed algorithm is that it consists of two analytical equations, for the lateral and longitudinal force, written as a direct function of the three kinematical creepages. The algorithm can be easily adjusted using a set of coefficients to meet the evidence of experimental tests.

#### 2. Methods

This work proposes a new algorithm to calculate the wheel-rail tangential forces, whose peculiarity is to allow faster calculations and the possibility of improving the results, by modifying the coefficients of the method on the basis of the experimental results. This is possible because the algorithm is formulated in order to be able to modify the shape of the force-creepage behaviour acting on few coefficients.

In order to validate the algorithm, the numerical results have been compared with the most used methods (FASTSIM and Polach) considering at first simple cases and then the simulation of a vehicle on real track. Finally the method is also compared with experimental results obtained on a scaled prototype on roller-rig.

##### 2.1. Tangential Force Calculation

The proposed method for calculating the tangential forces has been improved with respect to a previous method proposed by the authors [10], where the spin creepage was not adequately considered. The new method has corrected the influence of the spin creepage in order to be comparable with the results achieved by the most common algorithms [6, 7]. This method allows calculating with two analytical equations the longitudinal and lateral force in the contact area according to (1) and (2), as a function of the longitudinal (), lateral (), and spin () creepages. The creepages are defined in general as the ratio of the relative velocity between wheel and rail in the considered direction and the vehicle velocity [1, 2]. For the specific case of a wheelset on a roller-rig, adopted in this work for the experimental tests, the expressions of the creepages are slightly different with respect to the case of the wheelset on the rail. Those expressions will be described later ((13)–(17)).The coefficients can be calculated depending on the dimension of the contact area, considered as elliptical, according to the linear theory of Kalker [2]:where* a*,* b* are the semiaxes of the contact ellipse and are the so called Kalker’s coefficients, which depend on the* a/b* ratio and on the Poisson’s modulus of the material.* G* is the shear modulus of elasticity.

The coefficient is usually set equal to 1 and* n* = ; therefore, when two of the creepages vanish, the formula leads to the Kalker’s linear law for small values of the remaining creepage, while tending to for high values of the creepage, according to (6) and (7).

The case with pure spin creepage is an exception to the described behaviour, as the lateral force tends to zero for high values of the creepage, according to (8). For low values of the spin creepage, the Kalker’s linear law is respected also in this case.All the parameters, as the normal force (*N*), the friction coefficient (), and the longitudinal, lateral, and spin creepages can be calculated according to the dynamic of the wheelset. The and exponents are the most relevant and can be used to modify the shape of the curve and to fit the experimental data. The benefits of the method are that a single analytical equation can be used to calculate the tangential forces, with a very low computational time and that the relevant coefficients can be corrected on the basis of the experimental results.

To define the effect of the spin creepage, in the denominator of the equations, two additional terms must be calculated. The first term , according to (9), depends on the ratio and on the exponent .The second term* k* is calculated according to (10), depending on the tangential modulus of elasticity, the product of the ellipse semiaxis, the normal load , and the friction coefficient , considering three different exponents, , , and .The exponents adopted on the method for the simulation of Sections 3.1 and 3.2 are shown in Table 1. These exponents are obtained using a trial and error approach in order to achieve, with the proposed algorithm, results similar to the FASTSIM [6] and Polach’s [7] algorithms.