Scaling of tyre model parameters as a function of road surface roughness

The development and adoption of new autonomous and active safety features in modern passenger cars has resulted in an increasing demand for accurate and efficient mathematical tyre models. Such models may operate within digital twins of the vehicle to predict brake distances, for example, or used in a more traditional manner to simulate the vehicle’s behaviour at the design stage. In all cases, tyre model parameterisation is essential, and this typically involves testing on flat track machines – a process that ensures good repeatability and tightly controlled testing conditions. However, the surface used on such machines is hardly representative of the variety of the real road surfaces that the vehicle may encounter in operation. This causes large discrepancies between tyre models fitted using flat track data and the forces observed on real roads. Up to now, engineers employed mostly ad-hoc methods to scale model parameters so that they better describe tyre behaviour on real surfaces.

The aim of this work is to explore the physical mechanisms that give rise to the differences in tyre behaviour between different surfaces and to further propose a systematic approach for the scaling that must be applied when moving from any surface to another. Key elements of tyre behaviour that are considered include the hysteretic friction of rubber and its dependency on road surface roughness, as well as the mechanism by which surface roughness influences the shear stiffness of tyres. Since the methods developed in this work are aimed at practical application, special attention has been paid on their computational efficiency and in this respect machine learning techniques have been applied in conjunction with physical modelling to provide practically relevant scaling algorithms.

First, a neural network representation of Persson's flash temperature model was employed to predict the hysteretic contribution of friction on asphalt surfaces with self-affine fractal characteristics in real-time applications. This allows the efficient calculation of the coefficient of friction between a tyre and the road surface, given the operating conditions, the tread material properties, and the power spectral density of road roughness. This is important because it allows the update of friction in a computationally efficient manner and therefore presents an opportunity for carrying out vehicle simulations on several road surfaces, without the requirement for tyre testing on every surface.

The change in the cornering/braking stiffness was theorised to be due to the influence of road roughness on the effective shear stiffness of the tread. On this basis, an enhanced brush-type model is presented to predict the effect of road roughness on the linear range of tyre operation. The model considers three separate stiffness terms including inflation, bending and tread stiffness. The latter is modelled using a finite element model with a rubber-road boundary comprising randomly distributed macro-asperity contact areas, calculated using Persson’s rubber contact theory.

A Magic Formula model was developed to include the artificial neural network and the effective shear stiffness dependency on the road roughness. Data from three tyres tested on both sandpaper and asphalt surfaces is presented to validate the enhanced model predictions when moving from sandpaper to asphalt. Both the predicted friction and the cornering stiffness on asphalt fall within the 95% confidence interval of the measured asphalt characteristics.