Advances in the Modelling of Motorcycle Dynamics - ResearchGate

258 R.S. SHARP ET AL.extension for motorcycle tyres is relatively recent, with substantial changes beingnecessary to accommodate the completely different roles of sideslip and camberforces in the two cases. In each case, the “Magic Formula” is a set of equationsrelating load, slip ratio (longitudinal slip), slip angle and camber angle to longitudinalforce, sideforce and aligning moment (and possibly overturning moment), withconstraints on the parameters to prevent the behaviour from becoming unreasonablein any operating conditions. Only very limited parameter values can be found inthe literature, but a certain amount of relevant experimental data is available. Suchdata can be used for parameter identification.A complete set of parameter values for a given tyre will allow the calculation ofthe steady-state force and moment system for any realistic operating condition. Itis required here to determine such a full set of parameters for modern front and rearhigh performance motorcycle tyres, imposing the condition that the modelled tyreshave left/right symmetry. Test data used shows bias and it is necessary to ignoresuch bias and to omit certain offset terms from the “Magic Formula” relations, inorder to model the generic, rather than the particular. Significant published datacan be found in [2, 20–25]. Naturally, the older data refers to older tyres, while thenewer data relates to contemporary ones. The main sources relied upon here are[20, 23]. The other sources are used for checking purposes, as appropriate.4.1. LONGITUDINAL FORCES IN PURE LONGITUDINAL SLIPFrom Pacejka [23], with the simplifications explained above, the “Magic Formula”expressions for the pure longitudinal slip case are:d f z = (F z − F z0 )/F z0 (1)F x0 = D x sin[C x arctan{B x κ − E x (B x κ − arctan(B x κ))}] (2)D x = (p Dx1 + p Dx2 d f z )F z (3)E x = ( p Ex1 + p Ex2 d f z + p Ex3 d fz2 )· (1 − pEx4 sgn(κ)) (4)K xκ = F z (p Kx1 + p Kx2 d f z ) · exp(p Kx3 d f z ) (5)B x = K xκ /(C x D x ) (6)which must satisfy the constraints D x > 0 and E x < 1.Corresponding test results for a 160/70 ZR17 tyre are shown in [23]. The sequentialquadratic programming constrained optimisation routine “fmincon” wasemployed 1 to iteratively improve the elements of a starting vector of parameters1 Alternatively, for unconstrained optimization, the Nelder Mead Simplex routine “fminsearch”was employed. Also occasionally, it was necessary to “invent” data, outside the range of experimentalresults available, to force the identified parameters to give sensible predictions over a wide range ofoperating circumstances, a problem also referred to in [26]. Often, reasonably accurate starting valuesfor the parameters were needed to ensure convergence to the optimal solution. The methods need tobe judged by the results obtained.