Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2022.35.0001 Acta Polytechnica CTU Proceedings 35:1–7, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague KALKER’S COEFFICIENT c11 AND ITS INFLUENCE ON THE DAMPING AND THE RETUNING OF A MECHANICAL DRIVE TORSION SYSTEM OF A RAILWAY VEHICLE Vojtěch Dybala Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Automotive, Combustion Engines and Railway Engineering, Technická 4, 160 00 Prague 6, Czech Republic correspondence: vojtech.dybala@fs.cvut.cz Abstract. Within the research of electromagnetically excited torsion oscillations in the mechanical part of traction drive systems of modern railway vehicles, which has been realized at the Faculty of mechanical engineering at the CTU in Prague, there are two separate simulation models in use. The basic calculation model, which is utilized to gain basic characteristics of the torsion system as natural frequencies and natural modes of oscillations. And the complex simulation model, which simulates a drive of the vehicle. This contribution is focused on the basic calculation model, which has been built in MATLAB. This model in its first version did not apply the contact between wheels and rails. It was necessary to find out, if this simplification is relevant with respect to subsequent simulations within the complex simulation model and its results. Therefore, the contact interaction as a traction force in longitudinal direction in the wheel-rail contact was realized via the Kalker’s linear theory. This article deals with the comparison between models with and without the implementation of the wheel-rail contact and its influence on the damping within the torsion system and retuning of the torsion system. Keywords: Kalker, natural frequency, railway vehicle, torsion system, wheel-rail contact. 1. Introduction One of the typical features of modern electrical railway vehicles is the individual mechanical drive of the wheel- set, e.g. see Figure 1. In principal it means that each wheel-set has its own propelling unit. This unit mostly consists of an electrical traction motor, a gearbox and a coupling connecting both of them. For sure there are more concepts of such a mechanical drive unit. Figure 1. A partly-suspended drive of a locomotive [1]. For purposes of this research a fully-suspended drive layout has been applied. This type of an individual drive consists of an electrical traction motor, a gear- box and a hollow shaft, which transmits driving and braking power between the gearbox and the wheel-set – Figure 2. Because generally this research is focused on high-speed and high-power railway vehicles, the fully-suspended type of a drive train was chosen, as it is a typical and an appropriate layout for this type of vehicles. Figure 2. A fully-suspended drive of a locomotive [2]. 2. Basic Mathematical Model This mathematical model, built in MATLAB, is uti- lized to provide basic characteristics of a torsion sys- tem, which schematically represents the design of a mechanical drive train of a railway vehicle. The basic characteristics are: • natural frequencies of oscillations • natural modes of oscillations 1 https://doi.org/10.14311/APP.2022.35.0001 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Vojtěch Dybala Acta Polytechnica CTU Proceedings Knowledge of these characteristics is supposed to be important for evaluation of frequency analysis, which will be carried out in subsequent simulations in the above-mentioned complex simulation model, built in MATLAB Simulink. Now the complex simulation model applies some simplifications to do simulations effective in terms of simulation time, amount of data and goals of the research. The basic calculation model supposes also some simplifications as the interaction in the wheel-rail contact. Because of the frequency analysis it is appropriate to do a review if the simplifi- cation of the basic calculation model is reasonable and if there can be a significant impact on the evaluation frequency analysis. 2.1. Model with no Wheel-Rail Contact The mathematical model is based on the scheme of the torsion system (Figure 3), which respects the layout of the fully-suspended drive presented in Figure 2. Figure 3. Torsion system scheme – fully-suspended drive [3]. Equations of motion describing this system were derived via the Lagrange method. The matrix nota- tion of these equation of motion is (1). To calculate natural modes and natural frequencies of oscillations these equations are solved as a system without the vector of excitation [M ], see (2). [J ][φ̈] + [k][φ] = [M ] (1) [J ][φ̈] + [k][φ] = [0] (2) The system (2) was solved via the function eig(k, J ), which is a pre-programed function in MATLAB. This function returns: • an eigen vector representing rotation angle φi,j for a rotation mass Ji and j natural mode of oscillation • an eigen values vector [λj ] representing j natural angle frequency Ωj = √ λj The eigen values are transformed into natural fre- quencies fj via the formula (3). fj = √ λj 2π (3) 2.2. Model with a Wheel-Rail Contact The wheel-rail contact was implemented into the cal- culation model via longitudinal traction forces T1, which can be seen in the top view of the drive train – Figure 4. The wheel-rail contact is simplified so, that lateral forces and spin moment between the wheel and the rail are neglected. Practically it means that it represents rolling of a cylindrical wheel on rails, not conical profile wheels and sinus movements of the wheel-set is neglected. Also forces T1 on both sides are supposed to be same. Figure 4. Visualization of wheel-rail forces – top view. The longitudinal traction force T1 is calculated via Kalker’s linear theory for the wheel-rail contact (4). T1 = c11 ael bel G sx = C1 sx = k1 sx (4) In this article details about the Kalker’s theory will not be presented, but they can be found in [4– 6]. For purposes of this contribution the value of Kalker’s coefficient c11 = 4, 984 is taken as a fact. That coefficient itself can varies. Because that theory is linear, its validity is limited for small values of slip with respect to adhesion characteristic – phase I in Figure 5. For the area of higher slip (phase II) the direction of the curve k1* decreases. While the value of c11 can be supposed for good conditions in the wheel-rail contact (dry rails), for worst conditions (wet or dirty rails) it can decrease as well. This fact is presented in Figure 6 within Popovici’s adhesion characteristics. And therefore because of both effects, the calcu- lations were carried out also for variations of c11, specifically for c11/2 and c11/4. sx = rk ωk − vx v (5) Due to above mentioned simplifications also only the wheel slip sx in the longitudinal direction is sup- posed. Its deduction is presented in Figure 7 and its mathematical representation is formula (5). [J ][φ̈] + [b][φ̇] + [k][φ] = [0] (6) Within the torsion system scheme (Figure 8) and subsequently the equations of motion in the matrix representation (6) the force T1 creates a damping ele- ment in the wheel-rail contact. This damping defines 2 vol. 35/2022 Kalker’s Coefficient c11 and its Influence Figure 5. Traction characteristic and adhesion char- acteristic of wheelset [4–6]. Figure 6. Popovici’s adhesion characteristics [7]. the formula (7). rk is constant, C1 = k1 (Figure 5 and Figure 6) will varies according to c11 and the vehicle velocity v is a variable as well. The calculations for this system were carried out as an analysis of the influence of C1 and v on the damping factor. bW −R = C1 r2k v (7) The velocity as a calculation parameter was spec- ified based on the supposed traction characteristics (Figure 9) of the vehicle, which is a high-speed locomo- tive for purposes of the research. The applied values of velocity were from 0 km/h to 200 km/h. The system (6) was solved by the MATLAB func- Figure 7. Wheel slip deduction [2]. Figure 8. Torsion system scheme – fully-suspended drive. Figure 9. Vehicle traction characteristic. tion polyeig(k, b, M ), which returns eigenvectors and eigenvalues for defined matrices. Eigen vectors re- turns values of angle of rotation as described in 2.1. Eigen values are complex numbers in the form accord- ing to (8) for this system with damping. λj = −δj ± i Ωdmp,j (8) Resulting natural frequencies of oscillation are then calculated according to (9). fdmp,j = Ωdmp,j 2π (9) 3 Vojtěch Dybala Acta Polytechnica CTU Proceedings 3. Calculation Results Results of calculations in sections 3.1 and 3.2 below are respective to the theory from sections 2.1 and 2.2 and the subsequent conclusion aims to make an analysis of the velocity influence on the retuning of the torsion system via damping. 3.1. Model with no Wheel-Rail Contact The torsion system scheme, which respects the design of the fully-suspended drive in Figure 2 is a system with 7 degrees of freedom and seven natural frequen- cies of oscillation were calculated, see Table 1. Because the torsion system in this state was considered as a free system the first natural frequency is 0Hz and it matches with a free rotation of the system. Natu- ral modes of oscillation complying with the natural frequencies are presented in Figure 10 to Figure 16. Such a considered state of the torsion system can be agreed with the situation when a railway vehicle does not generate any force in the wheel-rail contact – standstill of the vehicle or drive without traction or brake force. Natural frequencies of torsion system [Hz] 1. 2. 3. 4. 5. 6. 7. 0 6 57 337 572 857 2403 Table 1. Natural frequencies overview [3]. Figure 10. First natural mode of torsion oscilla- tions [3]. Figure 11. Second natural mode of torsion oscilla- tions [3]. Table 2 provides a description of these natural modes with respect to dominant oscillation of a spe- cific rotation mass. Order Respective Dominant Less of natural oscillations significant natural frequency of a mass oscillations modes [Hz] 1. 0 Own free – rotation 2. 6 Wheel-set towards – hollow shaft 3. 57 Wheels of – wheel-set 4. 337 Wheel-set Pinion towards towards hollow shaft rotor 5. 572 Pinion towards – rotor 6. 857 Wheel-set towards Hollow hollow shaft shaft joints Gear wheel towards hollow shaft 7. 2403 Pinion towards rotor – Pinion towards gear wheel Table 2. Description of natural modes [3]. Figure 12. Third natural mode of torsion oscilla- tions [3]. Figure 13. Fourth natural mode of torsion oscilla- tions [3]. 4 vol. 35/2022 Kalker’s Coefficient c11 and its Influence Figure 14. Fifth natural mode of torsion oscilla- tions [3]. Figure 15. Sixth natural mode of torsion oscilla- tions [3]. Figure 16. Seventh natural mode of torsion oscilla- tions [3]. 3.2. Model with a Wheel-Rail Contact In this section the attention will be put on calculated eigenvalues and respective natural frequencies. The first group of calculations was done for the value of the Kalker’s coefficient c11 = C1. With respect to the damping Figure 17 shows that, the damping reaches very high values in low velocities and strongly decrease with increasing velocity. This fact can be concluded with respect to the third eigenvalue, which relates to torsion oscillations of the wheel-set, see Table 2. Regarding natural frequencies see Figure 18. A small change in the value of the second one can be observed, from 6Hz to 4Hz. A significant change can be observed regarding the third one when its value decreased from 57Hz to 0Hz. This means that the damping in the wheel-rail contact is so high, that it suppresses torsion oscillations of the wheel-set. For the second group of calculations, where C1 = c11/2 = k1∗, see Figure 5 and Figure 6, the situation changed. The damping (Figure 19) was very high in low velocities again and suppressed oscillations of the wheel-set on the frequency of 57Hz. With increasing velocity, the damping decreases. Between 100 km/h and 105 km/h retuning of the system appeared. There Figure 17. Damping as a function of velocity – c11. Figure 18. Natural frequencies as a function of ve- locity – c11. is a visible jump in damping and change in the third natural frequency (Figure 20), which started to grow. Then the third natural frequency increases with de- creasing damping and approximate to the value of 57 Hz. Also, a small change in the value of the second natural frequency from 6Hz to 4Hz was observed. Figure 19. Damping as a function of velocity – c11/2. In the third group of calculations, where C1 = c11/4 = k1∗, see Figure 5 and Figure 6, the results were equivalent to results from the second group. The difference, which Figure 21 and Figure 22 presents, is that the point of torsion system retuning occurs in lower velocity, between 50 km/h and 55 km/h. A small 5 Vojtěch Dybala Acta Polytechnica CTU Proceedings Figure 20. Natural frequencies as a function of ve- locity – c11/2. change in the value of the second natural frequency from 6Hz to 4Hz was observed again. Figure 21. Damping as a function of velocity – c11/4. Figure 22. Natural frequencies as a function of ve- locity – c11/4. The fourth, the fifth, the sixth and the seventh eigen value and respective damping parameters and natural frequencies are not mentioned, because the calculations did not show any influence of the variation of values of the velocity and the Kalker’s coefficient. 4. Conclusions Results of calculations presented in Figure 17 to Fig- ure 22 proved, that the wheel-rail contact can signifi- cantly influence a behavior of a torsion system, which means to change some natural frequencies. Specifi- cally, the third natural frequency related to the oscil- lations of the wheel-set itself in a very strong way and weakly the second natural frequency related to oscilla- tions of the wheel-set towards the hollow shaft. On the other hand, it was presented, that the variability of parameters, which characterize the wheel-rail contact, don’t influence the rest of the torsion system. With respect to a research it can practically mean, that for a research oriented on a wheel-set torsion oscillations and related problematics, the effect of the wheel-rail contact should be considered. On the other hand, within a research of torsion phenomenon regarding the rest of a torsion system, as traction motor, gears and coupling, the wheel-rail contact can be neglected. List of symbols ael main half-axis of contact ellipse [m] bel secondary half-axis of contact ellipse [m] b, bW −R damping parameter [Nms.rad−1] c11 Kalker’s coefficient [-] f natural frequency [Hz] fdmp natural frequency of damped system [Hz] J mass of rotation [kg.m2] k torsion stiffness [Nm.rad−1] M external torque [Nm] T1 Tangential force [N] rk Wheel radius [m] sx Wheel slip [%] φ angle rotation [rad] φ̇ time derivative of angle rotation [rad.s−1] φ̈ second time derivative of angle rotation [rad.s−2] λ eigenvalue [-] Ω natural angle frequency [rad.s−1] Ωdmp natural angle frequency of damped system [rad.s−1] ωk angular speed of a wheel [rad/s] δ oscillation damping [rad/s] Acknowledgements This research has been realized using the support of The Technology Agency of the Czech Republic, programme Na- tional Competence Centres, project #TN01000026 Josef Bozek National Center of Competence for Surface Trans- port Vehicles This support is gratefully acknowledged. References [1] T. Fridrichovský. Studie disertační práce. Praha, 2017. [2] V. Dybala, M. Libenský, B. Šulc, C. Oswald. Slip and Adhesion in a Railway Wheelset Simulink Model Proposed for Detection Driving Conditions Via Neural Networks. In SBORNÍK vědeckých prací Vysoké školy báňské – Technické univerzity Ostrava, Řada strojní, Ostrava, 2018. [3] V. Dybala. The Electromagnetically Excited Resonance of the Pinion in Fully-Suspended Drive of a Locomotive and its Sensitivity on the Torsion Stiffness of the Rotor Shaft. In Sborník abstraktů konference STČ, Praha, 2021. 6 vol. 35/2022 Kalker’s Coefficient c11 and its Influence [4] J. Kolář. Teoretické základy konstrukce kolejových vozidel. Praha: Česká technika – nakladatelství ČVUT, 2009, p. 276. [5] J. Kolář. Zborník prednášok II. – XX. Medzinárodná konferencia – Súčasné problémy v kolajových vozidlách. In Dynamika individuálního pohonu dojkolí s nápravovou převodovkou, Žilina, 2011. [6] J. Kolář. Problémy modelování vlivu svislých nerovností trati do dynamiky pohonu dvojkolí. In Železničná doprava a logistika XI, pp. 38–47, 2015. [7] R. Popovici. Friction in Wheel-Rail Contacts. Enschede, The Netherlands: University of Twente, 2010. 7 Acta Polytechnica CTU Proceedings 35:1–7, 2022 1 Introduction 2 Basic Mathematical Model 2.1 Model with no Wheel-Rail Contact 2.2 Model with a Wheel-Rail Contact 3 Calculation Results 3.1 Model with no Wheel-Rail Contact 3.2 Model with a Wheel-Rail Contact 4 Cocnlusions List of symbols Acknowledgements References