The MRA Nondimensional Tire Model has a 40+ year history of
development. In 1960 at Cornell Aeronautical Laboratories, Hugo Radt and Bill
Milliken published the pioneering paper "Motions of Skidding Automobiles" which
included an early nondimensional treatment of tire data. Hugo Radt joined Bill
Milliken at MRA shortly after MRA was founded in 1976. MRA has continued the
development of the nondimensional technique ever since.
Unique to the nondimensional approach is the data compression achieved on the
measured tire data. This compression, based extensively on established tire
theory, results in data from various test conditions (such as several different
loads) to fall on a single curve. Any curve-fitting technique can then be used
on the resulting data -- we find the Pacejka "magic formula" works well. The
result is a vast collection of tire data which can be represented by a single
instance of the "magic formula" due to the nondimensional processing. Examples
of this for lateral force and aligning torque are shown below.
The essence of the nondimensional approach to tire data treatment is the
formation of dimensionless variables such as normalized lateral force,
normalized longitudinal force and normalized aligning torque as a function of
normalized slip angle, inclination angle and slip ratio. The approach is
analogous to the coefficient approach to aircraft aerodynamic data, which have
allowed a systematic approach to a complex set of nonlinear data.
The nondimensional technique accounts for the load sensitivity of cornering
stiffness, camber stiffness, friction coefficient, pneumatic trail and others.
The camber sensitivity of the tire/road friction coefficient, critical for the
proper modeling of wide racing tires, is accounted for.
The nondimensional technique is more than a curve fitting exercise; it is born
out of an investigation of the physical nature of tires. As such, the MRA
Nondimensional Tire Model expands tire data in a physically correct manner --
tire data measured at one friction coefficient can be expanded to simulate
operation on another surface with a different friction coefficient.
The benefits of using the MRA Nondimensional Tire Model are many:
Reduced tire test time and cost
Less tire wear during testing (fewer tires, more consistent data)
Improved tire model fidelity due to the inherent smoothing of the data for
Simple, accurate representation of tire data for vehicle simulations
Small number of coefficients used to describe complex data -- much fewer than
pure curve-fitting schemes
Mathematically consistent expansion, including straightforward and accurate
reconstruction of tire performance between test loads
Provides physically correct results on surface coefficients which differ from
The MRA Nondimensional Tire Model has been extensively used and validated
against experimental data through the years. We believe it combines the best of
theory and experiment into a sensible and powerful representation of tire
behavior for use in vehicle dynamics simulations.
Chapter 14 of Race Car Vehicle Dynamics is
devoted to a more detailed description of the MRA Nondimensional approach to
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