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MRA Nondimensional Tire Model
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MRA Nondimensional Tire Model

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.

Normalized Lateral Force.  Click to enlarge. Normalized Aligning Torque.  Click to enlarge.

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 combined conditions
  • 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 test conditions

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 tire modeling.

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