Magic formula

Abstract: Abstract An account is given of the latest version 3 of the Magic Formula tyre model. The model provides a set of mathematical formulae from which the forces and moment acting from road to tyre can be calculated at longitudinal, lateral and camber slip conditions, which may occur simultaneously. The model aims at an accurate description of measured steady-state tyre behaviour. The coefficients of the basic formula represent typifying quantities of the tyre characteristic. By selecting proper values, the characteristics for either side force, aligning torque or fore and aft force can be obtained. The new version of the model contains physically based formulations to avoid the introduction of correction factors. Double-sided, possibly non-symmetric pure slip curves are employed as the basis for combined slip calculations. Suggestions are given to estimate the driving part of the longitudinal slip curve and to represent the characteristic at rolling backwards.

Abstract: The tyre force and moment generating properties connected with the vehicle's horizontal motions are considered. Knowledge of tyre properties is necessary to properly design vehicle components and advanced control systems. For this purpose, mathematical models of the tyre are being used in vehicle simulation models. The steady-state empirical ‘Magic Formula tyre model’ is discussed. The aligning torque description is based on the concepts of pneumatic trail and residual torque. This facilitates its combined slip description. Following Michelin, weighting functions have been introduced to model the combined slip force generation. A full set of equations of the steady-state part of the model of the new version ‘Delft Tyre 97’ is presented. The non-steady state behaviour of the tyre is of importance in rapid transient maneuvres, when cornering on uneven roads and for the analysis of oscillatory braking and steering properties. A relatively simple model for longitudinal and lateral transient responses...

Abstract: 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.

Abstract: SUMMARY Tyre behavior plays an important role in vehicle dynamics research. Knowledge of tyre properties is necessary to properly design vehicle components and advance control system. For that purpose mathematical models of the tyre are being used in vehicle simulation models. The Magic Formula Tyre Model is a semi-empirical tyre model which describes tyre behavior quite accurately. The Magic Formula Tyre Model needs a set of parameters to describe the tyre properties; the determination of these parameters is dealt with in this paper. A new method based on genetic techniques is used to determine these parameters. The main advantages of the method are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. So to start the search, it is not necessary to know a set of starting values of the Magic Formula parameters. The comparison between analytical optimization methods and the method proposed is discussed in this paper.