An improved Magic Formula/Swift tyre model that can handle inflation pressure changes

This paper describes extensions to the widely used TNO MF-Tyre 5.2 Magic Formula tyre model. The Magic Formula itself has been adapted to cope with large camber angles and inflation pressure changes. In addition the description of the rolling resistance has been improved. Modelling of the tyre dynamics has been changed to allow a seamless and consistent switch from simple first order relaxation behaviour to rigid ring dynamics. Finally the effect of inflation pressure on the loaded radius and the tyre enveloping properties is discussed and some results are given to illustrate the capabilities of the model. INTRODUCTION Since its conception over 20 years ago, the Magic Formula has fairly quickly been adopted as the industry standard tyre model for vehicle handling simulations. Over the years various developments have been made to improve the accuracy and to extend the capabilities of the model, for example the method to describe combined slip has been improved and a special Magic Formula has been developed to handle the large camber angles occurring on motorcycles. In parallel research has been done to increase the frequency range by introducing rigid ring dynamics, contact patch transients and an obstacle enveloping model, also known as the SWIFT or MF-Swift model. An overview and description of these developments can be found in [1]. The fact that from the start the tyre model equations were published in the open literature has certainly contributed to the popularity of the Magic Formula. On the other hand, over the years it has also resulted in a number of different, at times incompatible, implementations. Different versions of the Magic Formula may be used, equations are sometimes partially implemented, in-house extensions are added, different axis system or units are used, etc. Notwithstanding these issues, the TNO MF-Tyre 5.2 tyre model has reached a mature status and is widely used in the industry for vehicle handling studies. The model equations are documented in [2]. To move forward from MF-Tyre 5.2 several targets have been defined: • To improve the description of camber, i.e. to have an explicit formulation and control over the camber stiffness and to extend the capabilities of the model for handling very large camber angles. This will make the special “motorcycle” Magic Formula superfluous and improves processing of measurements according to the TIME procedure. • To include the effect of inflation pressure changes in the Magic Formula. This will eliminate the need to have separate parameters sets (tyre property files) for different tyre pressures. It will allow evaluating tyre behaviour for pressures not in the measurement programme and it also leads to a reduction in the total number of measurements required. • To make the description of tyre dynamics consistent between MF-Swift and MF-Tyre. For example when including the dynamics of the tyre belt, the path dependent (basic) tyre relaxation behaviour should remain unchanged. In addition, various other enhancements have been made, for example in the description of rolling resistance and overturning moment. This paper aims to introduce the MF-Tyre/MF-Swift 6.1 model and accompanying equations. Due to space restrictions and the extent of the changes that were made in comparison with the MF-tyre 5.2 model, turn slip extensions [1] will not be discussed.