Markus Schmid Tire Modeling for Multibody Dynamics Applications Simulation‐Based Engineering Laboratory University of Wisconsin‐Madison 2011 i Abstract In vehicle dynamics, tires are one of the most important factors that govern the behavior of a moving vehicle. They are the only link between the vehicle chassis and the road and have to transmit vertical, longitudinal and lateral forces. In order to be able to describe a vehicle’s movement properly, it is necessary to understand the tires’ characteristics and their effect on certain driving situations. While empirical data gained from on‐the‐road driving experiments with real tires is obviously the most accurate (given that conditions are similar to the desired simulation), there are reasons to choose dynamical simulations instead. Since product cycles and development times are continuously reduced due to competition and progress of technology, building physical prototypes in an early stage of development is time‐consuming and very expensive. Furthermore, if the design is changed, the prototype has to be updated again and again. Clearly, with larger machinery like construction vehicles, this becomes time and cost prohibitive. Thus tending towards simulations in early development stages can be highly beneficial to the final design. Tires are very complex products and not trivial to describe. Their characteristics are influenced by several factors like design of the tire itself (e.g. material, tread pattern, carcass stiffness), air pressure inside the tire and texture of the road surface, to name a few. Several modeling approaches have been developed to describe the physics of tires so that they can be analyzed and evaluated mathematically. In the first part of this work, a combination of the steady‐state Magic Formula approach (described in [1], [2] and [3]) and the Single Contact Point transient tire model [3] is used to implement a tire model that can handle tranisent driving situations into the multibody dynamics engine Chrono::Engine [4]. For validation and experimental purposes, the tire model is then integrated with a vehicle model and different driving situations are analyzed. The second part of this research introduces a beam element type physical model that permits interaction with flexible terrain for off‐road applications. ii Acknowledgements First of all, I would like to greatly thank my advisor at the Department of Mechanical Engineering, Professor Dan Negrut, for all the support, the valuable advice and never running out of M&M’s to keep up the motivation. Thank you Dan for making me part of the team! Also, thank you very much to all the people at the Simulation‐Based Engineering Laboratory (SBEL) for the great atmosphere in the lab, helping out whenever possible and making this a great year! Finally, I would like to thank the German Academic Exchange Service (DAAD), the Institute for Machine Tools (IfW) at the University of Stuttgart and the Department of Mechanical Engineering at the University of Wisconsin‐Madison for making my studies in the United States and therefore this work possible. iii iv Table of Contents Abstract ....................................................................................................................................................................... i Acknowledgements ............................................................................................................................................... ii Table of Contents .................................................................................................................................................. iv List of Figures ......................................................................................................................................................... vi 1 Introduction .................................................................................................................................................... 1 1.1 Tire Dynamics ......................................................................................................................................................... 1 1.1.1 Tire Forces and Torques ........................................................................................................................... 1 1.1.2 Longitudinal and Lateral Slip .................................................................................................................. 2 1.1.3 Turn Spin ......................................................................................................................................................... 3 1.2 Tire Modeling .......................................................................................................................................................... 5 1.2.1 The Magic Formula Tire Model .............................................................................................................. 5 1.2.2 The Single Contact Point Transient Tire Model ............................................................................... 8 2 Implementation of the Magic Formula Tire Model ......................................................................... 14 2.1 MATLAB implementation ................................................................................................................................ 14 2.1.1 Tire property file (*.tire) ........................................................................................................................ 15 2.1.2 Transient tire model ................................................................................................................................ 16 2.1.3 Magic Formula steady‐state model .................................................................................................... 16 2.1.4 Model verification ..................................................................................................................................... 18 2.2 Chrono::Engine implementation .................................................................................................................. 24 2.2.1 The multibody dynamics simulation engine Chrono::Engine ................................................. 24 v 2.2.2 Translation of the MATLAB implementation to C++ .................................................................. 24 2.2.3 Vehicle and tire model used in Chrono::Engine ........................................................................... 25 2.2.4 The Standard Tyre Interface (STI) ..................................................................................................... 26 2.2.5 Organization of the Magic Formula tire model in Chrono::Engine ....................................... 28 2.2.6 Obtaining real‐time vehicle data ......................................................................................................... 29 2.2.7 Application of the forces and moments to the vehicle model ................................................. 31 2.2.8 Results and conclusions ......................................................................................................................... 32 3 Beam tire model for interaction with deformable terrain ........................................................... 35 3.1 Tire tread model .................................................................................................................................................. 35 3.2 Forces in the tire tread and contact patch ................................................................................................ 37 3.3 Model verification ............................................................................................................................................... 41 3.3.1 Vertical motion of the wheel ................................................................................................................ 42 3.3.2 Longitudinal motion of the wheel ...................................................................................................... 44 3.4 Results and future work ................................................................................................................................... 50 4 Summary ........................................................................................................................................................ 51 5 References ..................................................................................................................................................... 52 A Appendix ........................................................................................................................................................ 54 A.1 Magic Formula equations and factors [3] ................................................................................................. 54 A.2 .tire property file used in the Magic Formula implementations ..................................................... 62 vi List of Figures Fig. 1.1: Sign convention and tire reference frame [3] .............................................................................................. 1 Fig. 1.2: Rotational slip resulting from path curvature and wheel camber [3] .............................................. 4 Fig. 1.3: The tire as a nonlinear function with multiple inputs and outputs (steady‐state) [8] ............... 5 Fig. 1.4: Magic Formula factors [9] ...................................................................................................................................