![]() The US military organisation TARDEC have published Matlab routines for handling the PAC2002 formulation model ( Goryca, 2010). To the authors' knowledge, Pirelli have also had (and may still retain) their own flavour of the Magic Formula model. The authors also considered modifications to the Magic Formula to deal with the complicated situation of combined slip. In this paper the authors describe how the Magic Formula has been tested at Michelin and ‘industrialised’ as a self-contained package for the pure lateral force model. The general acceptance of the Magic Formula is reinforced by the work carried out at Michelin and described in Bayle et al. Since Magic Formula version 6.0, so-called ‘turn-slip’ behaviour has been modelled to allow parking loads to be captured. ![]() ![]() It includes stationary operation by switching to a different model at speeds below walking speed in order to avoid the divide-by-zero problem at rest. MSC Software have a proprietary take on the Magic Formula that they label ‘Pac2002’ ( Kuiper and Van Oosten, 2007). The model name BNPS is in honour of Messrs Bakker, Nyborg and Pacejka who originated the Magic Formula and the S indicates the particular implementation developed by Smithers Scientific Services Inc. The BNPS model ( Schuring et al., 1993) is a particular version of the Magic Formula that automates the development of the coefficients working from measured test data. Other authors have developed systems closely based around the Magic Formula approach. Between version 3 and this version, a separate motorcycle version has been introduced (‘Pac-MC’ or ‘MF-MC’) and then merged with the basic formulation since version 6.0. At the time of writing, the latest version of the Magic Formula is 6.1.2. While not apparently claiming a monopoly on the term, it seems that ‘MF-Tyre’ has become associated specifically with the TNO implementation. TNO Automotive have become de facto custodians of the Magic Formula approach and have libraries for it and other tyre models that can be used directly with a number of popular software packages, including Matlab, Simpack, ADAMS, DADS and many others ( TNO, 2012). MF-Tyre 5.0 switched to a ‘normalised’ formulation that ostensibly gives improved numerical conditioning, and version 6.1 includes direct support for different tyre pressures, which had been hitherto absent. The tyre model in this paper also acquired a new name and was referred to as the ‘Delft Tyre 97’ version, which seems to have reverted over time to MF-Tyre 5.0 ‘Delft-Tyre’ has become an umbrella term to include not only the base MF-Tyre model but a modified version of it suitable for intermediate frequency events, known as SWIFT (short wavelength intermediate frequency tyre). The authors also discuss a relatively simple model for longitudinal and lateral transient responses restricted to relatively low time and path frequencies. The Magic Formula model is undergoing continual development, which is reflected in a further publication ( Pacejka and Besselink, 1997) where the model is not restricted to small values of slip and the wheel may also run backwards. ![]() Mention of Version 4 appears to have disappeared from the literature. (1989) and a later version Pacejka and Bakker, 1993), which is sometimes referred to as Version 3 of the Magic Formula. The tyre models discussed here are based on the formulations described in Bakker et al. The early version ( Bakker et al., 1986, 1989) is sometimes referred to as the ‘Monte Carlo version’ due to the conference location at which this model was presented in the 1989 paper. The Magic Formula is not a predictive tyre model but is used to empirically represent and interpolate previously measured tyre force and moment curves. Mike Blundell, Damian Harty, in The Multibody Systems Approach to Vehicle Dynamics (Second Edition), 2015 5.6.5 The ‘Magic Formula’ tyre modelįor smooth road handling work, the tyre model that is now most well established and has generally gained favour is based on the work by Pacejka and as mentioned earlier is referred to as the ‘ Magic Formula’.
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