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Who developed the Magic Formula Tyre Model widely used for the analysis of tire behavior?

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Cutting corners

In my last column I discussed what would be required to ‘fully test a tire’ and what useful information could come from this endeavor. However, every thought has an equal and opposite thought, which raises the questions: what is the minimum amount of testing required to generate useful results, and how can those results be used?

Tire testing will always be a very expensive pastime, so any cost-cutting ideas are broadly welcomed. This formed the basis of my ongoing PhD to develop a cost-effective and complete flat-track test procedure called GS2MF. The objective is to obtain all the required data using the least possible rig time, thereby minimizing costs. However, even when using GS2MF at least a few hours of flat-track time are required to obtain one complete data set used to build a fully populated Magic Formula 6.1 handling model with pressure sensitivity. That’s still a costly chunk of test time, so what if that’s not feasible? Can you semi-populate a tire model with a reduced data set to save money? Would that still be useful?

The law of diminishing returns can be applied here, whereby the more testing you do the less useful each additional piece of data is. So, halving the amount of testing will not usually halve the value of the data. With that in mind, in place of a full test procedure such as GS2MF, very minimal testing could be carried out and still give very useful results.

Conducting simple steering sweeps at around five loads will provide adequate data pertaining to what are arguably the most important tire performance attributes, namely cornering stiffness, aligning stiffness and peak grip, as well as load sensitivities. Add in extra sweeps at the middle load and at three camber angles, then again at three inflation pressures, and a reasonable approximation of the tire’s complete steering characteristics can be established from just 11 test sweeps.

Furthermore, statistical approaches exist that can estimate a tire’s longitudinal and combined performance from just the steering data. If there is budget for extra testing, the same 11 sweeps could be run longitudinally to measure the pure braking and driving performance. Then one can use a friction ellipse assumption to fill in the gaps and estimate the combined performance (steering while braking or accelerating).

With only this minimal data set, a perfectly reasonable tire model could be parameterized and used successfully in full-vehicle simulations.

A similar approach could be used to obtain the tire’s ride performance. Extensive footprint tests can be conducted very cheaply using a hydraulic press with a load cell and either carbon paper or normal paper and a hot wire. With this it is possible to build up a decent data set of footprint sizes and shapes comprising many different loads, camber angles and inflation pressures, without spending heavily.

This data is very important and fundamental to the parameterization of the FTire, CDTire or RMOD-K ride models, among others, where the mantra of ‘Get the footprint right and the rest will follow’ is often adhered to.

After this, fitting a regulator valve to the hydraulic press along with a distance sensor will enable measurement of the tire’s vertical force versus displacement. Using this system, an extensive data set of static vertical stiffnesses can be obtained cheaply, and this forms the basis for the next step of the ride model parameterization. Once this is complete, some testing on a drum rig will be required to gather dynamic cleat test data necessary to finish off the ride model. As with the handling model, this will not be a perfect ride model but will be perfectly reasonable and could be used effectively in full-vehicle simulations to generate valuable vehicle-level results.

Tire testing will always be expensive, but there are ways the costs can be minimized and useful results still obtained. Even with a substantial test budget, depending on the application it is often more beneficial to conduct a reduced test over a wide range of tires than an exhaustive test on just a few. This is due to the fact there are usually substantial variations between tire constructions, even within similar types and sizes of tires. An exploratory exercise such as the one described here could be informative, even for the established tire tester. I’d welcome your feedback!

July 12, 2017



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