IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 5, OCTOBER 2006 513 Recent Advances in Control-Oriented Modeling of Automotive Power Train Dynamics Joˇ sko Deur, Member, IEEE, Joˇ sko Petri´ c, Jahan Asgari, and Davor Hrovat, Senior Member, IEEE Abstract—This paper presents a survey of the recent research results of the authors in the field of modeling of automotive power train systems and components. The goal of the research is to pro- pose simple and accurate power train models for controller design and to propose computationally efficient simulations. The mod- eling includes typical power train components such as electronic throttle, SI engine, torque converter, planetary gear set, wet clutch, differential, half shaft, and tire. Experimental model validation re- sults are presented. Index Terms—Control, dynamics, modeling, power train, road vehicles. NOMENCLATURE a Inner clutch diameter. A 0 Cross-sectional flow area of the torque con- verter. b Outer clutch diameter. b a Equivalent half-shaft damping. b ii ,b it ,b ti ,b tt Damping coefficients of the second-order torque converter model (calculated from the torque converter static curves). b eb Equivalent damping of the engine block mounts in the pitch direction. f 0 Steady-state functions of the torque con- verter. F Tire force. F app Applied clutch force. F z Normal force to tire. g Speed ratio of the second planetary gear. g(v r ) Sliding tire friction function (friction poten- tial function). h Manifold heat-transfer coefficient; clutch fluid film thickness; speed ratio of the first planetary gear. i Differential speed ratio. I Engine inertia. I ii ,I it ,I ti ,I tt Equivalent cross-coupling inertia of the second-order torque converter model (calcu- lated from I i, t and b ii, it, ti, tt ). I i, t, s Impeller, turbine, and stator inertia. Manuscript received October 29, 2005; revised June 16, 2006. Recommended by Guest Editor K. Jezernik. This work was supported in part by the Ford Motor Company, and in part by the Ministry of Science, Education, and Sports of the Republic of Croatia. J. Deur and J. Petri´ c are with the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb HR-10002, Croatia (e-mail: josko.deur@fsb.hr; josko.petric@fsb.hr). J. Asgari and D. Hrovat are with the Ford Research and Advanced Engineer- ing, Dearborn, MI 48121 USA (e-mail: jasgari@ford.com; dhrovat@ford.com). Digital Object Identifier 10.1109/TMECH.2006.882980 k a Equivalent half-shaft stiffness. k eb Equivalent stiffness of the engine block mounts in the pitch direction. L Tire–road contact patch length. L f Equivalent fluid inertia length of the torque converter. M Engine torque. M b Engine load torque. p Manifold air pressure. p app ,p Applied clutch pressure(s). r Effective tire radius. R Gas constant. s Operator of the Laplace transform. S i, t, s Characteristic torque converter area con- stants. t Time. T a Ambient temperature; dc motor armature time constant. T Manifold air temperature; clutch torque. T d Engine combustion delay. T p Nondominant time constant of the second- order torque converter model. T r Engine runner air temperature. T v Time constant due to the torque converter fluid inertia effect. v Wheel center speed. v r Tire–road relative speed (slip speed). V Manifold volume. V f Torque converter fluid velocity. W i Throttle air-mass flow. W o Engine port air-mass flow. ˜ z Average horizontal tire tread deflection. z Horizontal tire tread deflection. α b Half of the transmission backlash angle. ζ Bristle position inside tire–road contact patch. θ Throttle angle. θ LH Limp-home throttle position. θ R Reference throttle angle. κ Ratio of the specific thermal capacities; char- acteristic coefficient of lumped tire friction model. ν Empiric scaling factor of the clutch model squeeze-speed equation. ρ Fluid density. σ 0, 1 Tire tread stiffness and damping coefficients. σ 2 Tire–road viscous friction coefficient. τ dif = 2(1 − i −1 )τ hs . Reactive differential torque. 1083-4435/$20.00 © 2006 IEEE