Tire Longitudinal Slip : tire road contact patch behavior and effective rolling radius - vehicle dynamics

 Contents

1. Effective Rolling Radius and Longitudinal wheel Speed
2. Quiz : what happens in the effective radius in braking and acceleration
• If any change in radius, it it longer or shorter in braking
3. Tyre longitudinal slip in acceleration
4. Tyre longitudinal slip in braking
5. Conclusion

Effective Rolling Radius and Wheel Speed

When a car stops, tire is loaded vertically by its weight. At that time, vertical wheel radius to the road surface is RL and unloaded circumferential radius is Ru. When a car starts on straight and level road, the longitudinal speed of wheel center Vx is neither ωRL nor ωRU. Real longitudinal speed of tire center will be somewhere between ωRL and ωRu. You can get the effective rolling radius if you count the number of wheel turns and measure the distance at a given constant speed. The effective rolling radius can be described as the ratio of  the vehicle speed to the angular velocity of wheel as shown in the equation above. Effective Radius Re can be expressed in terms of Ru and RL as shown as above in the right side. You can find the proof of this equation at the reference book shown in lower right of the picture or following brief explanation in the picture.

Referring to red line above, tyre circumferential speed slows down just before tire-road surface contact patch and become stable for entire tire footage and get faster and recover its speed right after tire footage.

Influence factors on Effective Rolling Radius

1. tire : structure, pressure, tread pattern shape, wear, temperature
2. other : road surface, reaction force, vehicle speed, brake force, traction force

Quiz

Which of the statement is right in braking?

1. The effective radius of wheel increases
2. The effective radius of wheel decreases
3. No change at all

Definition of Longitudinal Slip

following equation is the definition of longitudinal slip lambda.

we need instant value of tire slip because it will be used as a very important parameter for ABS(Antilocking Brake System) requiring solid decision within very short time.

So, we use velocity which is derivative of distance with respect to time rather than distance itself.

Every single term in the equation of slip changes with infinitesimal time interval △t going by.

slip speed Vss of tire element with respect to the road surface can be described as follows.

Other Names of λ 

tangential slip, circumferential slip

Book authors use different forms of slip equations on their own preference.

Free Rolling Wheel

In the free rolling condition, actual speed is equeal to the speed by the wheel effective radius(ω0=ω) and that speed is equal to the speed of a car

Longitudinal slip only happens when braking or acceleration not in the free rolling.

Longitudinal Slip in Acceleration

In the acceleration, tire element is compressed in the tire-road surface contact patch.

Therefore, tire element moves slower in the contact patch than in free rolling area.

In the acceleration,

Wheelspin

Referring to equation ①, In the extreme case in which engine torque is big enough to overcome tire frictionc wheelspin hsppens without any longitudinal movecment of wheel. Then, Vx is equal to zero and that makes λ ⟶ - ∞ (acceleration is greater than zero).

Tyre Footprint in Acceleration

This picture is exaggerated shape of tire deformation to help you understand the compression of tire element over tire-road contact patch. As I explained, in the acceleration, tire element is compressed in the tire-road surface contact patch.

1. Tire element experiences stretching in the interval d1. As a result, it has the largest arc length in the d1 interval.
2. Stretching diminishs gradually according to incremant of interval number until d6.
3. Then, from d7 compression starts and gradually grow until d14 .
4. Tire element enters its footprint region with minimized arc length and maximum compression.
5. In the tire footprint region,d14 is the most compressed element.
6. Then again compressing is reduced gradually over d15 and d16 with  longitudinal slip and stretching starts again with tyre element's departing from road surface.

Longitudinal Slip Graph in Acceleration

This graph shows the longitudinal friction coefficient as a function of longitudinal slip λ. μp is  peak friction coefficient, which is similar to static friction coefficient and μp is the kinetic friction coefficient. Traction force will be maximized at peak friction coefficient.

If traction force is big enough to overcome the peak friction force, wheelspin happens without longitudinal movement of the wheel.

Longitudinal Slip in Braking

In the braking, tire element is stretched in the tire-road surface contact patch.

Therefore, tire element moves faster in the contact patch than in free rolling area.

In the braking,

Wheelslide

Referring to equation ①, In the extreme case in which braking force is big enough to overcome tire friction, wheel slide happens with the wheel locked up. 

Then, Vbrake is equal to zero and that makes λ = 1 (acceleration is less than zero).

Tyre Footprint in Braking

This picture is exaggerated shape of tire deformation to help you understand the extension of tire element over tire-road contact patch. As I explained, in the braking, tire element is compressed in the tire-road surface contact patch.

1. Tire element experiences the biggest compression in the interval d13. As a result, it has the smallest arc length in the d13 interval.
2. Compression diminishes gradually according to decrement of interval number until d7.
3. Then, from d6 , extension starts and gradually grow until d16  counterclockwise.
4. Tire element enters its footprint region with maximized arc length and maximum extension.
5. In the tire footprint region, d16 is the most stretched element.
6. Then again stretching is reduced  gradually over d15 and half a d14 with  longitudinal slip and compression starts again with tyre element's departing from road surface.

Longitudinal Slip Graph in Braking

This graph shows the longitudinal friction coefficient as a function of longitudinal slip λ. μp is  peak friction coefficient, which is similar to static friction coefficient and μp is the kinetic friction coefficient. Braking force will be maximized at peak friction coefficient.

If braking force is big enough to overcome the peak friction force, wheel slide happens with the wheel locked up.

Instantaneous Center in Longitudinal slip

IC gets lower in braking along with the degree of longitudinal slip. You can understand the reason if you think IC approaches infinity with wheel locked up in heavy brake because all the elements of sliding wheel move parallel to the road surface and that makes IC approach infinity. IC gets higher along with the degree of longitudinal slip in acceleration. You can understand the reason if you think IC approaches to wheel center in case of  wheelspin in big acceleration well over the maximum friction force of tire. In that case, tire loses its traction force and just spin itself. Therefore, all the tire elements are rotating at the wheel center.

Concept of Slip and Effective Rolling Radius

This picture shows the rigid wheel with electric motor and brake system. The radius of the wheel is radius 0.5m. The picture shows the travel distance when it makes its one revolution(360°).

If this wheel makes its one revolution without longitudinal slip, it should reach the location ② with travel distance 3.14m.

But, in the case of bigger traction force than friction force, the wheel experiences the wheelspin at the expense of travel distance and reaches somewhere(location ①) between the origin and location ②. This resultant travel distance is the same one which smaller wheel(red color) makes by its one revolution without longitudinal slip. Therefore, we can say that the effective rolling radius is reduced in the acceleration. We have reached the same conclusion for effective rolling radius decided by IC(Instantaneous Center) discussed previously.

In the case of bigger braking force than friction force, the wheel experiences the wheel slide leading excessive travel distance and reaches somewhere(location ③) after overshooting the location②. This resultant travel distance is the same one which bigger wheel(blue color) makes by its one revolution without longitudinal slip. Therefore, we can say that the effective rolling radius is increased in the acceleration. We have reached the same conclusion for effective rolling radius decided by IC(Instantaneous Center) discussed previously.

Answer to the Quiz

Did you hit the answer to Quiz? The effective radius of wheel increases in the braking

Conclusion

1. Longitudinal slip is defined as velocity terms to get the instantaneous slip value.
2. Every single term in the equation of slip changes with infinitesimal time interval △t going by.
3. Braking has the extended and acceleration has the compressed tire element in the tire footprint.
4. Effective radius gets longer in braking and shorter in acceleration than in free rolling.
5. Excessive engine torque causes wheelspin without longitudinal wheel movement
6. Excessive brake torque causes wheel slide with wheel locked up
   
7. Wheelspin and wheel slide should improve because they waste engine torque and brake force.