1 Introduction Sliding electrical contacts, such as current collectors, slip rings or grounding contacts are widely used in industrial applications. They form a special class of tribological systems in which both classical mechanical and electrical stresses occur. In the last century it has been shown that technical bodies - due to their rough topography - only touch over a relatively small region, the so-called real contact area [1]. On one hand this has a substantial effect on the overall characteristics of the contact system, such as its tribological and electrical performance. On the other hand, both of former influence the highly topical question of thermal management, since not only frictional energy, but also Joule heat generation is to be considered. Besides that, changes in already established concepts of automotive industry, increasing importance of electric drives and that of transmitting signal and power current between moving machine parts rises new challenges in development. Electrical contact resistance and arc discharge, for example, have a significant influence on both efficiency and service life of the systems [2]. Consequently, in tribological research it became necessary to investigate the effect of electrical conduction through rough surfaces in Aus Wissenschaft und Forschung 31 Tribologie + Schmierungstechnik · 65. Jahrgang · 6/ 2018 Numerical determination of load-dependent electrical contact resistance L. Katona* Im Fall von elektrisch beaufschlagten Bauteilen in Kontakt schnürt sich der Strom im Interface auf die reale Kontaktfläche (auf einzelne Berührflächen) ein, was zu einem sogenannten elektrischen Kontaktwiderstand führt. Die Qualität eines elektrischen Kontaktes bezüglich Kontaktwiderstand und Lichtbogenentladung beeinflusst dabei sowohl das tribologische, als auch das thermische Verhalten des Gesamtsystems. Entsprechend besteht ein Bedarf in der industrieorientierten Forschung den elektrischen Kontaktwiderstand durch numerische Simulationen berechnen, bzw. vorhersagen zu können. Dieser Beitrag behandelt zwei komplementäre Methoden zur numerischen Berechnung des lastabhängigen elektrischen Kontaktwiderstandes. Die eine basiert auf der Randelementmethode und berücksichtigt gemessene Topographien, wogegen die zweite gemäß dem Modell von Greenwood und Williamson auf einer statistischen Beschreibung der Oberflächen beruht. Ergebnisse der numerischen Rechnungen werden anhand experimenteller Daten validiert, Vor- und Nachteile der einzelnen Modelle diskutiert. Schlüsselwörter Elektrischer Kontaktwiderstand, Simulation, reale Kontaktfläche, Greenwood-Williamson, Randelementmethode In case of electrically charged machine parts in contact current is constricted in the interface to the real contact surface (onto individual contact spots), which leads to the so-called electrical contact resistance. The quality of an electrical contact in terms of contact resistance and arc discharge influences both the tribological and thermal behaviour of the overall system. Accordingly, there is a demand in industry oriented research to determine / predict electrical contact resistance via numerical simulations. This work deals with two complementary methods for the numerical determination of the load-dependent contact resistance. One is based on the Boundary Element Method and takes digitalized real topographies into account, whereas the second is based on a statistical description of the surfaces according to the model of Greenwood and Williamson. Results of the numerical calculations are validated with experimental data, the advantages and disadvantages of the individual models are discussed. Keywords Electrical contact resistance, simulation, real contact area, Greenwood-Williamson, Boundary Element Method Kurzfassung Abstract * Dr. techn. László Katona AC2T research GmbH, 2700 Wiener Neustadt, Austria T+S_6_18.qxp_T+S_2018 29.10.18 17: 05 Seite 31 ter constants was subtracted from experimental data to receive the load-dependent electrical contact resistance R c (L). Furthermore, experimental uncertainties of both electrical resistance and mechanical load are coloured with blue and red, respectively. Experimental investigations on contact resistance were followed by digitalization of the nominal contact surfaces using optical microscopy. The individual topographies of the contact pairs were aligned to each other by numerical post processing, making use of the circular wear tracks. In this way the so-called combined topographies of commutator and carbon brush were determined together with topographical characteristics, namely the root mean square height S q , the density of peaks S pd and their arithmetic mean curvature S pc . As long the S q roughness parameters could be derived directly from the combined topographies, latter ones, i. e. S pd and S pc were determined with the software MountainsMap [6] and the corresponding ISO standards [7]. Selected results are given in Table 1. Former steps, i. e. to align optical data and to determine the combined topographies were prerequisites for the la- Aus Wissenschaft und Forschung 32 Tribologie + Schmierungstechnik · 65. Jahrgang · 6/ 2018 detail and to consider it both regarding wear and thermal construction. In accordance with above motivation, it is the aim of this work to review two common contact mechanical models, namely a Boundary Element Method (BEM) [3] based and that of Greenwood and Williamson (GW model) [4] - both in combination with the Holm equation [5] -, for their applicability to determine / predict the load-dependent contact resistance of rough mating surfaces. 2 Experimental investigations Experiments were performed on pairs of so-called flat commutators and carbon brushes of direct current electrical motors at a range between 0 to 4 N of mechanical loads to investigate the load-dependent electrical resistance. Both contact pairs are made of electro graphite, hence the effect of oxide layers in the interface does not had to be considered further. Moreover, the motors were operated previously for different times and at different supply voltages. Accordingly, the contact surfaces were worn together and showed - due to different arc intensity at varying source voltages - a variety regarding topographical characteristics, e. g. S q roughness parameter, or number and mean curvature of the asperities, see Table 1. For the measurements a test rig - specially designed for this purpose - was used, where the shaft of the DC electrical motor positioned the contact pairs towards each other, guaranteeing for the original contact position. Mechanical load was applied through a sprig, which was deflected by a micrometre screw and allowed a precise adjustment. Figure 1 shows a typical result of such a loaddependent electrical resistance measurement, green trend. The exponential decay of resistance over load agrees with previous reports, such as for example in Refs. [4] and [5] well. Furthermore, one should note, that due to technical reasons electrical resistance can only be measured with respect to the entire system, i. e. carbon brush - contact interface - commutator, neglecting further components like the connecting wires. Hence, the load-dependent data R tot (L) of Figure 1 can be considered as (1) with the constant terms R br and R co , which mark the resistance of brush and commutator, respectively. However, both can be calculated analytically or numerically, if geometry and electric resistivity are known. Subsequently, the sum R ∞ (see level marked with black) of lat- Figure 1: Typical result of a load-dependent electrical resistance measurement ! "! #$% & '( ) *" ) * #$% , # %&'( ! $ ) 1 '' * + ! " ) 1 ''+ sample a 8.121 39.82 239.70 sample b 8.499 18.79 242.03 sample c 4.613 49.81 239.12 sample d 4.024 29.26 410.91 Table 1: Roughness parameters of selected combined topographies of commutator and carbon brush pairs T+S_6_18.qxp_T+S_2018 29.10.18 17: 05 Seite 32 ter application of a Boundary Element Method based contact mechanical scheme. Furthermore, it was verified that it is sufficient to approximate the surface height distributions of these combined topographies with Gaussian functions, which - if centred around a zero mean - are defined by the particular S q roughness parameters. This, together with the S pd and S pc values motivated to choose the contact model of Greenwood and Williamson as an alternative approach to BEM. 3 Computational models The Boundary Element Method enables one to consider real surfaces in the calculations, just like the previously mentioned combined topographies with their large number of arbitrarily shaped asperities. This is, because only the boundary is to be mashed, which in fact requires to stay in the elastic deformation regime, however allows to perform calculations on surfaces of several tens of mm 2 with comparably high resolution. In particular, the nominal contact areas were of around 20 mm 2 . The resolution of optical data was 1.76 µm in both in plane directions and 410 nm in the vertical direction. The BEM scheme itself is based on the Boussinesq equation and minimizes the total strain energy, as described in detail in Ref. [3]. Result of such a calculation is the spatially resolved pressure distribution, i. e. domains of contact and no contact (equal to or exceeding, and below a defined pressure threshold, respectively). Subsequently, the n contact regions were approximated by n individual circles (with radii r n ), each having the same area as the corresponding patch. To these, the so-called Holm equation [5] was applied, which describes the electrical constriction resistance R s of two semi-infinite bodies touching over a circular area of radius r as: (2) Here ρ co and ρ br mark the electric resistivity of the two bodies, i. e. in the present case that of commutator and carbon brush. Finally, the load-dependent contact resistance R cBEM (L) was determined by parallel connection of the particular constriction resistances at each step of mechanical load L BEM . The alternative way, given by the Greenwood-Williamson model describes rough surfaces in a statistical way, assuming for example a Gaussian height distribution Φ(σ,z) of surface heights, recall section 2. According to Ref. [4], the governing equations for mechanical load L GW (d) - needed to compress the surface by a rigid approach d - and the corresponding contact resistance R cGW (d) are given by: (3) and (4) respectively. In Eqs. (3) and (4) z marks the coordinate vertical to the approaching surfaces, E * is the combined Young’s modulus, N 0 is the amount of all asperities (i. e. S pd roughness parameter multiplied by the nominal contact area) and r asp denotes the mean radius of curvature of latter, which is the inverse of S pc value, recall Table 1. 4 Results and discussion Resulting load-dependent electrical contact resistances from both experimental investigations (red, together with their particular uncertainties in black), and those of the numerical calculations are depicted in the left graphs of Figure 2. Labels a to d correspond to the samples as listed in Table 1. The corresponding right graphs show the ratio between experimental R c(exp) and numerical R c(num) load-dependent contact resistance to estimate the accuracy of the applied model calculations. Particular ratios, i. e. R c(exp) / R c(BEM) and R c(exp) / R c(GW) are coloured in magenta and in light blue, respectively. In terms of BEM related calculations contact resistance (green curves), notable deviation form experimental data appears in case of samples c and d. This might result from the accuracy of numerical alignment of optical surface data, i. e. from the combined topographies. Furthermore, the effect of contact resistance rising with increasing mechanical load - which can be noticed at sample d - is contradictory to physical expectations. This arises as an artefact, namely when neighbouring patches merge at a subsequent higher load, and simultaneously a number of small contacts appear, which altogether give raise to the contact resistance. Finally, the fact that in BEM topographies are considered in accordance with their resolution can also result in reduced or increased contact resistance, as for example in the range between 0 and 1 N at sample b. Regarding the results from the Greenwood-Williamson model (blue curves) all behaviours of matching, overand underestimating experimental data occur. This can be explained by the uncertainty of required input parameters, however, especially for loads above 1 N the trends resemble experimental data well. Aus Wissenschaft und Forschung 33 Tribologie + Schmierungstechnik · 65. Jahrgang · 6/ 2018 # $ %! & ' ( )* + ( ,- 4! . # )9: %; & ' 3 9: %; & ' 4< = > ? @ A! B$C D %8 E ; & F G H 5%67 8& ; 8 I J K 4 ( )* + ( ,- ? @ A! B$C D %8 E ; & L G H 5%67 8& ; 8 I J M NL 7 T+S_6_18.qxp_T+S_2018 29.10.18 17: 05 Seite 33 Aus Wissenschaft und Forschung 34 Tribologie + Schmierungstechnik · 65. Jahrgang · 6/ 2018 Figure 2: Numerically determined load-dependent electrical contact resistance, together with experimental results (left panels) and their respective ratios (right panels) a b c d T+S_6_18.qxp_T+S_2018 29.10.18 17: 05 Seite 34 5 Conclusions Altogether it can be concluded that both methods (BEM and GW model) are sufficient for the numerical prediction of contact resistance. A distinction between which approach to choose can be drawn from the available input data and computational power. Further studies on load-dependent electrical contact resistance, its application to determine real contact area in-situ and detailed information about the mentioned experimental and numerical models can be found in Ref. [8]. Acknowledgements This work was funded by the Austrian COMET-Program (Project K2 of XTribology, Grant No. 849109) and carried out at the Excellence Centre of Tribology. References [1] F. P. BOWDEN and D. TABOR: The area of contact between stationary and between moving surfaces. Proceedings of Royal Society of London, (A169): 391 - 413, 1939. [2] M. BRAUNOVIC, V. V. KONCHITS and N. K. MYSHKIN: Electrical contacts: Fundamentals, applications and technology. CRC Press, 2006. [3] S. ILINCIC: Combined finite element-boundary element method for contact mechanics of rough engineering surfaces. Doctoral dissertation, Technical University Vienna, Institute for Electrical Engineering and Information Technology, 2012. [4] J. A. GREENWOOD und J. B. P WILLIAMSON: Contact of Nominally Flat Surfaces. Proceedings of Royal Society of London, (A295): 300 - 319, 1966. [5] R. HOLM: Electrical contacts: Theory and application. Springer, reprint of 4 th edition, 1981. [6] http: / / www.digitalsurf.fr/ en/ mntspm.html (January 17, 2018) [7] DIN EN ISO 25178-2 Geometrische Produktspezifikation (GPS) - Oberflächenbeschaffenheit: Flächenhaft Teil 2: Begriffe und Oberflächen-Kenngrößen (ISO 25178- 2: 2012); German version EN ISO 25178-2: 2012, 2012. [8] L. KATONA: Determination of the real contact area via load-dependent electrical contact resistance. Doctoral dissertation, Technical University Vienna, Institute of Applied Physics, 2018. Aus Wissenschaft und Forschung 35 Tribologie + Schmierungstechnik · 65. Jahrgang · 6/ 2018 Für eine Veröffentlichung bitten wir Sie, uns die Daten als Word-Dokument und als PDF zur Verfügung zu stellen sowie zusätzlich die Original-Bilddaten. Hilfreich ist es ferner, wenn die Bilder durchnummeriert und bereits an der richtigen Stelle platziert sowie mit den zugehörigen Bildunterschriften versehen sind. 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