{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T14:24:42Z","timestamp":1771079082964,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,23]],"date-time":"2021-08-23T00:00:00Z","timestamp":1629676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This article presents the results of tests conducted on the average and instantaneous friction coefficients of railway vehicle disc brakes. The tests were carried out independently of various states of wear on the friction linings and the brake disc. The requirements of the International Union of Railways (UIC) regarding the approval of brake linings for use were taken into account. Based on many years of research using a brake bench to test railway disc brakes, the authors developed multiple regression models for the average friction coefficient and fluctuations (tolerances) in the instantaneous friction coefficient and achieved 870 results. The models proposed three types of variables: the input braking parameters (speed, pressure, and mass to be braked), operational parameters (the wear on the friction linings and the brake disc), and design parameters (perforations in the form of holes on the disc surface). The above two models were validated on the basis of 384 brakes, and in subsequent stages a further evaluation was performed. The coefficients were determined to be, respectively, 0.99 for the model of the average friction coefficient and 0.71 for the model of tolerance (fluctuations) of the instantaneous friction coefficient.<\/jats:p>","DOI":"10.3390\/ma14164766","type":"journal-article","created":{"date-parts":[[2021,8,23]],"date-time":"2021-08-23T10:24:17Z","timestamp":1629714257000},"page":"4766","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Modeling the Average and Instantaneous Friction Coefficient of a Disc Brake on the Basis of Bench Tests"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5049-6644","authenticated-orcid":false,"given":"Wojciech","family":"Sawczuk","sequence":"first","affiliation":[{"name":"Faculty of Transport and Civil Engineering, Institute of Transport, 60-965 Poznan, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Armando Miguel Rilo","family":"Ca\u00f1\u00e1s","sequence":"additional","affiliation":[{"name":"DB Systemtechnik GmbH, 14774 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4154-1167","authenticated-orcid":false,"given":"Dariusz","family":"Ulbrich","sequence":"additional","affiliation":[{"name":"Faculty of Transport and Civil Engineering, Institute of Machines and Motor Vehicles, 60-965 Poznan, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jakub","family":"Kowalczyk","sequence":"additional","affiliation":[{"name":"Faculty of Transport and Civil Engineering, Institute of Machines and Motor Vehicles, 60-965 Poznan, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,23]]},"reference":[{"key":"ref_1","first-page":"421","article-title":"An investigation on braking systems used in railway vehicles","volume":"23","author":"Korkmaz","year":"2020","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"104360","DOI":"10.1016\/j.jweia.2020.104360","article-title":"Comparative study on the effect of aerodynamic braking plates mounted at the inter-carriage region of a high-speed train with pantograph and air-conditioning unit for enhanced braking","volume":"206","author":"Niu","year":"2020","journal-title":"J. Wind. Eng. Ind. Aerodyn."},{"key":"ref_3","first-page":"24","article-title":"Improving the performance of disc brakes on high-speed rail vehicles with a novel types of brake pad: Isobar","volume":"1","author":"Wirth","year":"1998","journal-title":"RTR"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1016\/j.wear.2006.12.022","article-title":"A cellular automaton model to describe the three dimensional friction and wear mechanism of brake systems","volume":"263","author":"Ostermeyer","year":"2007","journal-title":"Wear"},{"key":"ref_5","first-page":"176","article-title":"Electrodynamic braking in high-speed rail transport","volume":"22","author":"Lingaitis","year":"2007","journal-title":"Transport"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"214","DOI":"10.2219\/rtriqr.40.214","article-title":"Running Test Result of Electric Brake to Zero Speed","volume":"40","author":"Ogasa","year":"1999","journal-title":"Q. Rep. RTRI"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105319","DOI":"10.1016\/j.engfailanal.2021.105319","article-title":"The failure prediction of a brake disc due to nonthermal or mechanical stresses","volume":"124","author":"Chen","year":"2021","journal-title":"Eng. Fail. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.engfailanal.2014.05.022","article-title":"The effect of braking energy on the fatigue crack propagation in railway brake discs","volume":"44","author":"Li","year":"2014","journal-title":"Eng. Fail. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.wear.2019.01.124","article-title":"Crack propagation and microstructural transformation on the friction surface of a high-speed railway brake disc","volume":"428\u2013429","author":"Wang","year":"2019","journal-title":"Wear"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.surfcoat.2018.11.011","article-title":"Microstructure evolution and crack propagation feature in thermal fatigue oflaser-deposited Stellite 6 coating for brake discs","volume":"358","author":"Wu","year":"2019","journal-title":"Surf. Coat. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.ijfatigue.2016.02.024","article-title":"Thermal crack growth-based fatigue life prediction due to braking for a high-speed railway brake disc","volume":"87","author":"Wu","year":"2016","journal-title":"Int. J. Fatigue"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.promfg.2019.05.025","article-title":"FEA on different disc brake rotors","volume":"35","author":"Kallon","year":"2019","journal-title":"Procedia Manuf."},{"key":"ref_13","first-page":"79","article-title":"Numerical analyzes and a comparative study of an automotive standard brake disc with a disc drilled along the Archimedes spiral","volume":"79","author":"Sawczuk","year":"2018","journal-title":"Arch. Automtive Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kikuuwe, R., Takesue, N., Sano, A., Mochiyama, H., and Fujimoto, H. (2005, January 2\u20136). Fixed-step friction somulation: From classical Coulomb model to modern continuous models. Proceedings of the IEEE\/RSJ International Conference on Intelligent Robots and Systems, Edmonton, AB, Canada.","DOI":"10.1109\/IROS.2005.1545579"},{"key":"ref_15","first-page":"475","article-title":"A novel friction modeling and its impact on differential equations computation and Lyapunov exponents estimation","volume":"10","author":"Awrejcewicz","year":"2008","journal-title":"Vibromechanika J. Vibroeng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1109\/9.376053","article-title":"A new model for control of systems with friction","volume":"40","author":"Olson","year":"1995","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1098\/rsta.2007.2125","article-title":"Hysteretic effects in dry friction: Modelling and experimental studies","volume":"366","author":"Wojewoda","year":"2008","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.mechmachtheory.2012.06.002","article-title":"An extended bristle friction force model with experimental validation","volume":"56","author":"Liang","year":"2012","journal-title":"Mech. Mach. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Padthe, A.K., Oh, J., and Bernstein, D.S. (2006, January 14\u201316). On the LuGre model and friction-induced hysteresis. Proceedings of the 2006 American Control Conference, Minneapolis, MN, USA.","DOI":"10.1109\/ACC.2006.1657218"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1109\/9.995050","article-title":"Modyfication of the Leuven integrated friction model structure","volume":"47","author":"Lampaert","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1109\/TAC.2005.858676","article-title":"The generalized Maxwell-Slip Model: A novel model for friction simulation and compensation","volume":"50","author":"Lampaert","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","first-page":"337","article-title":"Brake system studies using numerical methods","volume":"15","author":"Baranowski","year":"2013","journal-title":"Eksploat. Niezawodn. Maint. Reliab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"355","DOI":"10.17531\/ein.2015.3.5","article-title":"Determination of the functional and service characteristics of the pneumatic system of an agricultural tractor with mechanical brakes using simulation methods","volume":"17","author":"Kulikowski","year":"2015","journal-title":"Eksploat. Niezawodn. Maint. Reliab."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.wear.2013.11.040","article-title":"Third body layer-expermental results and a model describing its influence on the traction coefficient","volume":"314","author":"Meierhofer","year":"2014","journal-title":"Wear"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.ijthermalsci.2012.05.006","article-title":"Thermomechanical modelling of dry contacts in automotive disc brake","volume":"60","author":"Belhocine","year":"2012","journal-title":"Int. J. Therm. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.icheatmasstransfer.2016.03.017","article-title":"The numerical-experimental scheme for the analysis of temperature field in a pad-disc braking system of a railway vehicle at single braking","volume":"75","author":"Grzes","year":"2016","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.engfailanal.2014.03.021","article-title":"Elevated temperature low cycle fatigue of grey cast iron used for automotive brake discs","volume":"42","author":"Peveca","year":"2014","journal-title":"Eng. Fail. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.wear.2010.11.007","article-title":"Thermal levels and subsurface damage induced by the occurrence of hot spots during high-energy braking","volume":"270","author":"Kasem","year":"2011","journal-title":"Wear"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.wear.2010.11.009","article-title":"Influence of various metallic fillers in friction materials on hot-spot appearance during stop braking","volume":"270","author":"Kumar","year":"2011","journal-title":"Wear"},{"key":"ref_30","first-page":"139","article-title":"Auslegung und Leistungsgrenzen von Scheibenbremsen","volume":"112","author":"Saumweber","year":"1988","journal-title":"ZEV Glasses Ann."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.engfailanal.2015.07.002","article-title":"Analyzing the mechanisms of thermal fatigue and phase change of steel used in brake discs","volume":"57","author":"Li","year":"2015","journal-title":"Eng. Fail. Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.engfailanal.2013.07.004","article-title":"Analyzing the mechanisms of fatigue crack initiationand propagation in CRH EMU brake discs","volume":"34","author":"Yang","year":"2013","journal-title":"Eng. Fail. Anal."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"105403","DOI":"10.1016\/j.engfailanal.2021.105403","article-title":"Influence of service fluids on carbon-carbon aircrafts brake disc oxidation","volume":"125","author":"Tartakovsky","year":"2021","journal-title":"Eng. Fail. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"10442","DOI":"10.1016\/j.ceramint.2020.12.206","article-title":"Research and prospect of ceramics for automotive disc-brake","volume":"47","author":"Li","year":"2021","journal-title":"Ceram. Int."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"106686","DOI":"10.1016\/j.triboint.2020.106686","article-title":"Comparison of braking behaviors between iron- and copper-based powder metallurgy brake pads that used for C\/C-SiC disc","volume":"154","author":"Ma","year":"2021","journal-title":"Tribol. Int."},{"key":"ref_36","first-page":"1789","article-title":"Microstructure evaluation oft he damage and wear characteristics of a failed disc brake of a provincial bus","volume":"16","author":"Reyes","year":"2019","journal-title":"Mater. Today: Proc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"203237","DOI":"10.1016\/j.wear.2020.203237","article-title":"The braking behaviors of Cu-Based powder metallurgy brake pads mated with C\/C-Sic disc for high-speed train","volume":"448\u2013449","author":"Zhao","year":"2020","journal-title":"Wear"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.1016\/j.jmps.2011.08.006","article-title":"The frequency response of dynamic friction: Model comparisons","volume":"59","author":"Woodhouse","year":"2011","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"115212","DOI":"10.1016\/j.jsv.2020.115212","article-title":"Identifying short-term variation of dynamic friction by means of its frequency response function","volume":"472","author":"Cabboi","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_40","unstructured":"(2019). Railway Applications\u2014Brake Disc for Railway Rolling Stock\u2014Part 1: Brake Disc Pressed or Shrunk onto the Axle or Drive Shaft, Dimensions and Quality Requirements."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/j.applthermaleng.2015.08.042","article-title":"Role of cross-drilled holes in enhanced cooling of ventilated brake discs","volume":"91","author":"Yan","year":"2015","journal-title":"Appl. Therm. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.fsi.2019.09.061","article-title":"Plant protein diet suppressed immune function by inhibiting spiral valveintestinal mucosal barrier integrity, anti-oxidation, apoptosis, autophagyand proliferation responses in amur sturgeon","volume":"94","author":"Wei","year":"2019","journal-title":"Fish Shellfish Immunol."},{"key":"ref_43","unstructured":"Sawczuk, W. (2016). Brake disk. (Patent P.418361)."},{"key":"ref_44","unstructured":"(2010). Brakes\u2014Disc Brakes and Their Application\u2014General Conditions for the Approval of Brake Pads, International Union of Railways. [7th ed.]. Appendix to UIC Code 541-3."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.ijfatigue.2017.05.006","article-title":"A physically short fatigue crack growth approach based on low cycle fatigue properties","volume":"103","author":"Wu","year":"2017","journal-title":"Int. J. Fatigue"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1016\/S0043-1648(03)00459-9","article-title":"An experimental investigation of hot spots in railway disc brakes","volume":"256","author":"Panier","year":"2004","journal-title":"Wear"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.jsv.2015.01.028","article-title":"Transient friction-induced vibrations in a 2-DOF model of brakes","volume":"344","author":"Pilipchuk","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.jsv.2014.11.016","article-title":"The influence of joints on friction induced vibration in brake squeal","volume":"340","author":"Kruse","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_49","first-page":"14","article-title":"Trybologiczne badanie ok\u0142adzin ciernych hamulca tarczowego pojazd\u00f3w szynowych","volume":"1","author":"Kaluba","year":"1999","journal-title":"Pojazdy Szynowe"},{"key":"ref_50","unstructured":"Rail Consult Gesellschaft f\u00fcr Verkehrsberatung mbH Wagon Osobowy Z1 02\u2014Uk\u0142ad Jezdny, Rail Consult Gesellschaft f\u00fcr Verkehrsberatung mbH. Dokumentacja Techniczno-Ruchowa."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.wear.2011.04.013","article-title":"A study of airborne wear particles generated from organic railway brake pads and brake disc","volume":"273","author":"Abbasi","year":"2011","journal-title":"Wear"},{"key":"ref_52","unstructured":"Tokaj, P. (2017). Zu\u017cycie par Ciernych Hamulc\u00f3w w Wybranych Typach Pojazd\u00f3w Szynowych, Prace Instytutu Kolejnictwa."},{"key":"ref_53","unstructured":"Gajek, L., and Ka\u0142uszka, M. (2000). Wnioskowanie Statystyczne\u2014Modele i Metody, Wydawnictwo Naukowo-Techniczne WNT."},{"key":"ref_54","unstructured":"Rabiej, M. (2018). Analizy Statystyczne z Programami Statistica i Excel, Wydawnictwo Heliot."},{"key":"ref_55","unstructured":"Krysicki, W., and W\u0142odarski, L. (2007). Analiza Matematyczna w Zadaniach, Wydawnictwo PWN."},{"key":"ref_56","first-page":"57","article-title":"Analytical model coefficient of friction (COF) of rail disc brake on the basis of multi-phase stationary tests","volume":"20","author":"Sawczuk","year":"2018","journal-title":"Maint. Reliab."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/16\/4766\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:49:49Z","timestamp":1760165389000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/16\/4766"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,23]]},"references-count":56,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["ma14164766"],"URL":"https:\/\/doi.org\/10.3390\/ma14164766","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,23]]}}}