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

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The tire's place in fuel economy

Automotive gasoline, diesel oil and/or battery energy generate engine power that is used to accelerate our cars while overcoming opposing forces from aerodynamic drag, rolling tires and driveline friction. Fuel energy is ultimately consumed not only in vehicle linear motion, but also in overcoming the inertia of rotating components. These spinning components create a dynamic response, known as an ‘effective mass’, that seemingly increases the overall bulk of an accelerating vehicle. This virtual mass, added to vehicle weight as an equivalent non-rotating constituent, adversely affects the acceleration performance and fuel economy of all road cars.

This effect took on great importance midway through my career at Firestone. One of the Detroit Big Three car companies in need of CAFE relief challenged its suppliers by offering a purchase premium of 50 cents/lb for weight reduction in a host of vehicle components – but double that amount for a rotating part that would still meet specifications. This period was the beginning of vehicle and component downsizing in the US automotive fleet to improve CAFE ratings.

Inherent in the OEM challenge was the tacit acknowledgement that a weight reduction in a translating, rotating tire is more important in improving vehicle fuel economy than in an identical non-rotating spare. While the tire in the trunk is an added weight burden, the same tire on the road not only expends fuel energy in overcoming rolling friction, but also suffers rotational energy losses associated with vehicle acceleration and deceleration. Engineers, of course, know that the resistance of a tire to a change in velocity depends not only on its dimensions and mass, but also on the distribution of its mass (including the wheel and brake rotor or drum) about the axle. The physical property governing this response is known as the mass moment of inertia.

These concepts were borne out about five years ago with results reported in a popular US automotive magazine that tested a 2010 VW Golf fitted with the same brand and line of plus-sized UHP tires ranging from 195/65R15 to 235/35R19 – all with same ODs. Counterintuitively to most auto enthusiasts’ perception, 0-60mph times increased by three-tenths of a second, while fuel economy worsened by almost 10%, as the tire aspect ratio decreased from the 65 series to the 35. In other words, the 15in wheels outperformed their trendier – but heavier – counterparts by relatively wide margins in two important performance attributes. There were no deviations from this trend among the five aspect ratios tested.

In light of these findings, several of my recent vehicle dynamics students undertook as their semester project the determination of the mass moments of inertia of various tire-wheel combinations. Fortunately, their day jobs kept them busy at Goodyear with access to state of-the-art inertia measuring equipment.

Calculations based on first principles and curve fits of experimental data were made for various PCR and TBR tires, alloy and steel wheels, and selected off-highway and aircraft tires. Results, for example for the plus-sized PCR tires, indicate that vehicle effective mass increases from 2.6% (195/65R15)
to 4.2% (235/35R17) due to tire-wheel rotational effects; the latter figure is equivalent to an additional vehicle weight of 59kg. Reductions in PCR tread mass from new to worn proved to be more important in reducing inertia resistance than a change in wheel material from steel to alloy. Similarly, a single 445/50R22.5 wide-base tire has a 7% decreased moment of inertia compared with comparable 295/75R22.5 tires in dual fitment.

Tire rotational inertia will measurably affect fuel consumption in stop-and-go driving, but is less of a consideration in highway use with minimal throttle or braking. In any case, one does not conserve battery energy by equipping an EV designed for urban use with plus-sized tires – an effect not to be confused with larger-diameter tires, which promote lower rolling friction.

Correspondingly, the overall moment of inertia properties of a vehicle about its principal axes determine how externally applied forces and moments affect vehicle roll, pitch and yaw motions. A century ago the British and German military began developing test rigs for measuring the principal inertias of WWI aircraft. Not until after WWII did the US auto industry develop similar equipment for road cars – a topic for future discussion.

 

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