Why Does A Tyre Grip The Road? Part 4.2 - Lateral Grip
This is part 4.2 of our Why Does A Tyre Grip The Road series, today we will be looking further into Lateral grip. This post blew out a bit, so I decided to write it in two parts, it has been written, along with the rest of the series, assuming you have read the ones before it, but it's not necessary to read all of them, it just helps to provide a bit more colour. Read the first one on Rubber here, the second one on Grip here, the third one on Longitudinal Grip here, and the first part on Lateral Grip here.
Lateral Stiffness
As stated in Part 4.1, another tyre consideration that we need to take into account when designing our steering and suspension packages is the Lateral Stiffness, that is, a tyre's ability to hold shape whilst being subjected to lateral forces. We can see in the image below that a tyre will deform when a lateral force is applied.
A tyre deforming from a lateral force under different vertical loads. The dotted line shows the original tyre position [2]
The deformation of the tyre reshapes the contact patch, resizing and repositioning it from a centred ellipse to one a lot smaller and now at the fringes of the tyre. We can see this demonstrated below. On the left we can see the contact shear stress of a tyre travelling along longitudinally and on the right one turning a corner. Despite this being the same tyre under the same vertical load, we can see a few interesting things. The contact patch for the cornering tyre has reduced dramatically, resulting in an increase in shear stresses by almost 4 times, the centre of compressive stresses (darkest blue) is now well over to one side of the tyre, they have moved along the 1 axis, and we now have tensile forces at the opposite end of the contact patch.
Longitudinal (Left) and Transverse (Right) contact shear stress for a tyre with the normal load of 2000N and a friction coefficient of 0.4 [3]
Overturning Torque
The movement of the contact patch gives rise to a new moment, a moment that is applying a torque to the wheel's centre of rotation, an overturning moment.
We can see the overturning torque (Mx) in the image titled “A tyre deforming under different loads” above, it is the resulting force (Fz) multiplied with the distance between it and the center of the rim. As you can see, the position of the resulting force can be either positive or negative depending on the vertical load. Pacejka, H [2] states that under low loads the positive value “is expected to be connected with the belt that is stiff in torsion in the contact zone. This high stiffness opposes the flattening of the belt to remain in contact with the road when the belt is tilted through the action of the side force”. In other words, under low loads the contact patch is not being fully used, so the pneumatic scrub (the distance between the original centre of the contact patch with the new one) is on the left side. As the vertical force is increased and more of the contact patch is being used, the resulting force moves over to the other side, creating a moment that, to me anyway, makes more sense. We can see this behaviour depicted below, as the vertical force is increased the torque being produced flips over from positive to negative, we can also see that slip angle has a large effect on the overturning moment for large vertical loads.
Overturning moment vs slip angle for a range of vertical loads [4]
So why is this important? It’s important because as you can see above, the amount of torque being produced at high vertical loads is quite high, add to this wide tyres and we have a moment that could really unsettle our suspension setup. If we fail to take it into account, we wouldn’t be optimising our cars, meaning we would be leaving time on the table.
What can we do to affect a tyre's overturning moment?
That’s a good question, as it is affected by the contact patch, anything we do to either increase or decrease the contact patch will affect it. Increasing tyre pressures will reduce the contact patch, so we will flip over to negative sooner. The higher pressures will also create less lateral deformation, resulting in less overturning torque. A skinnier tyre, similar to a high inflation pressure, will have a smaller contact patch, its will be fully engaged earlier than a wider tyre, so it will also flip over sooner. Another one is camber, no doubt something you have realised I have left out so far, I’m going to give camber it’s own post, it’s complicated enough I think to deserve its own.
Self aligning torque
Overturning torque isn’t the only thing created from a deforming tyre, there is also a self aligning torque; the wheel wants to align itself with the direction of travel. A good analogue is “a weather vane aligning with the wind direction” [5], or a shopping trolley’s castor wheels. There are two causes of the self aligning torque, mechanical trail and pneumatic trail, these add together to create the total trail. Mechanical trail comes from the kingpin inclination and castor angle, it can be thought of as static, whilst the pneumatic trail, which comes from the tyre deforming, can be thought of as dynamic, it depends on the slip angle. We can see the pneumatic trail in the image above titled “Longitudinal (Left) and Transverse (Right) contact shear stress for a tyre with the normal load of 2000N and a friction coefficient of 0.4”. The image on the left shows a fairly even stress profile across the length of the contact patch, whereas on the right, we have a concentrated stress point which is not centrally located about the 1 axis, it’s this offset stress that creates the pneumatic tail.
If we look at the stress profile in the image below we can see this a little more clearly. The deforming of the rubber results in an asymmetrical stress profile, when we sum the stress profile and add in the resulting force to the drawing, we can see that it is behind the centreline. The position of the resulting force creates a moment about the centreline and with it the self aligning torque. The dynamic nature of the pneumatic trail comes from the varying position of the resulting force, which comes from varying slip angles, vertical loads, etc.
Asymmetrical forces across the contact patch sum to a resulting force that is behind the centre line, creating a moment about the centre line [6]
What is interesting about the aligning torque is that whilst it increases with slip angle, it starts to decrease far sooner than lateral force, see image below. This means that whilst the driver is turning, the force needed to overcome the aligning torque in building, at some point the force starts to decrease, the steering becomes lighter, it gives the driver the false sense that they have passed the point of maximum grip. But as you can see, for this tyre, peak aligning torque occurs at about 2.75°s of slip angle, but maximum lateral force occurs at around 5.5°s, the driver is only at about 2/3rds of their available grip despite feeling that they have passed max grip. Understanding this, as well as how the peaks shift with the slip angle due to changing conditions allows us to design a car that has some overlap of the two curves. The driver doesn’t have too much of a drop away in torque before peak grip, making it easier for them to push through the somewhat unnerving feeling.
Aligning torque and lateral force coefficient vs slip angle. The data used for the above graph is from [5] for a tyre with 1350lb of load on it. Refer to next image for tyre details
Just in case you thought designing a car wasn’t challenging enough, aligning torque isn’t only affected by the slip angles, it, like everything, is also affected by the vertical loads, see image below. One interesting thing about the below image is that at high slip angles the aligning torque can be negative. This means that if a driver passed this point, instead of the wheel trying to re-align itself with the driving direction, it will try to turn the wheel further. If you refer back to image title “Asymmetrical forces across the contact patch sum to a resulting force that is behind the centreline, creating a moment about the centreline” above, at high slip angles the grip is reduced, so the resulting force will move in front of the centerline, creating a moment that wants to create larger slip angles.
Not only does vertical load impact aligning torque, but so does tyre pressures. As you can see below, decreasing the tyre pressure results in a higher aligning torque. The reason for this is lower pressures result in a longer contact patch. If we refer to the aligning moment equation in the drop down titled “Click here to learn more about the affect the contact patch has on the cornering stiffness”, we can see that the aligning torque equation has half the contact length cubed within it, it is influenced by the contact length more than the lateral force and cornering stiffness, pretty interesting.
Aligning moment vs slip angle for a range of inflation pressures [7]
Induced Drag
Up to this point it would seem that increasing vertical load is always a good thing, whilst there might be diminishing returns, it always increases grip. Unfortunately there aren’t any free lunches in racing, increasing vertical loads have a downside, they increase our total rolling resistance.
Total rolling resistance is made up of two things, rolling resistance, the hysteresis within the tyre as it deforms to create the contact patch, and induced drag. Rolling resistance is fairly small and isn’t a phenomenon related to cornering specifically, we will therefore be talking about induced drag, the rolling resistance generated by the slip angle.
If we look at the image below we can see that the induced drag is calculated by
All we are doing here is resolving the centripetal force (Fy) along the velocity (V) direction. Just for interest sake, if we then do the same with the traction force (Fx), we can calculate the total trust or drag along the velocity direction
Net Rolling Resistance [5]
The two biggest things affecting induced drag is the slip angle and the vertical load, increasing either increases induced drag. The increased vertical load results in the tyre deforming more, this increase in deformation increases the contact patch which increases the drag. We can however offset this somewhat with increased tyre pressures. Slip angles impact is the interesting part. As you can see below, Milliken [5] has printed a graph from another paper [8] which shows the increase in net rolling resistance for a number of tyres at different slip angles. As you can see, the tyre's net rolling resistance has been normalised by dividing it by the vertical load. The total resistance increases quite quickly with increasing slip angles, any reduction in slip angle would help to reduce the drag experienced.
.jpg "Total rolling resistance vs slip angle")
Net Rolling Resistance vs Slip Angle [5]
Milliken [5] States that a Can-Am team used the above knowledge to their advantage. Across a number of events they slowly increased their tyre width, which increased their lateral force from a given slip angle, they were able to take corners with slightly less slip angle, reducing their drag and allowing them to use more of their horsepower for forward movement. What is interesting to me about this anecdote is that the increase in tyre width would have reduced the vertical load on the tyre (the area has increased P=F*A), which as we have seen should have meant there was less lateral force. But that isn’t the only impact increasing tyre widths have, they usually increase the steepness of the lateral force vs slip angle graph (cornering stiffness), which means they can get the same lateral force for a lower slip angle. The downside would have potentially been a lower maximum lateral force and a peak that breaks away quicker, but as the people in the Can-Am team were most likely professional drivers, they would have been able to take advantage of this tricker to drive tyre. The reduction in slip angle for the same lateral force must have reduced the induced drag resulting in a sizable impact on performance.
Unfortunately we don’t get any numbers for the above anecdote from Milliken [5], but they did give us a worked example which we can use to see induced drag's potential impact. I won’t go into the details specifically, but using the bicycle model, Milliken [5] is able to calculate the impact of induced drag on a car weighing 1868 lbs, travelling around a corner with a radius of 840 ft, banked at 9°s, and driven at 220mph. They calculated a total loss of horsepower of 132, which is 18.9% of the car's 700 available horsepower at that speed. 18.9% is being used to overcome the induced drag instead of pushing the car forward, pretty crazy. So what happens if like the Can-Am car the slip angle is reduced, what sort of improvements would we see? Well in Milliken’s [5] example, they have a front slip angle of 2.5°s and a rear slip angle of 1.4°s, so let’s reduce them both by 0.3°s and see what happens. This reduces the total drag from 225lbs to 175.5lbs, but more importantly it unleashes 30 horsepower, we have essentially untethered a ride on mower from behind us and chucked it under the bonnet. Not bad. It’s interesting to think about the progressive impact induced drag has on a tyre as it is being worn down, especially in something like an endurance race.
Relaxation Length
Another phenomenon that impacts how a car takes a corner, well it actually affects both lateral and longitudinal grip but this post is about lateral only, is the relaxation length of a tyre. The relaxation length is the amount of distance a tyre travels before the inputted force builds to 63% of its steady state value. That is, how long does it take after you turn the steering wheel before the tyre has reached 63% of what you have asked of it. Below we can see examples of what a typical relaxation length graph looks like on the left, and when compared instead to time on the right.
Relaxation Length [9]
The reason inputs are not instant is due to the visco-elastic properties of rubber. If we have a wheel rolling along and we then place a lateral force onto it, the rubber at the contact patch begins to deform, as new rubber is about to enter the deforming contact patch it needs to deform to match the already deformed rubber within the contact patch. It begins to deform before it enters the contact patch, this also happens in reverse as it leaves the contact patch. The amount of distance covered whilst it is getting ready to enter and after it leaves the contact patch before it is back to a relaxed state is the relaxation length.
Tyre’s manufactured to have a shorter relaxation length have a more direct feeling to them, they feel more like you are driving on rails. But the tyre's make up isn’t the only thing that impacts the relaxation length, we also have slip angle and vertical load. If we have a look below we can see, despite what I had assumed (if only there was a saying about assuming :)), increasing the slip angle or increasing the vertical load, essentially anything that increases the stress and with it the deformation of the tyre, decreases the relaxation length. I had assumed that with increasing levels of deformation there would be increasing relaxation lengths, as it would take longer for the rubber to reach the higher level of deformation. But it turns out that it is precisely the higher deformation and with it higher level of stress that forces the rubber to deform more quickly.
Relaxation length for a range of slip angles [10]
Vertical load’s impact on relaxation length [11]
Why Does A Tyre Grip The Road Series
References:
[1]Rainer W. Schlegelmilch, 1968, Link, retrieved on 15/04/22[2]Pacejka, H. 2012, Tyre and Vehicle Dynamics
[3]Behroozinia, P. 2018, ‘An investigation towards intelligent tyres using finite element analysis’, Link, retrieved on 25/03/22
[4]Lu, D et al. 2006, 'Modelling of tire overturning moment and loaded radius'
[5]Milliken, W. Milliken, D. 1995, ‘Race Car Vehicle Dynamics’
[6]Balkwill, J. 2017, 'Performance Vehicle Dynamics''
[7]Schmeitz, A. et al. 2005, ‘Extending the Magic Formula and SWIFT tyre models for inflation pressure changes’, Link retrieved on 06/04/22
[8]Gusakov, I., "Measuring Rolling Resistance of Tires in the Laboratory", Presented at the International Rubber Conference, Kiev, October 1978
[9] Zegelaar, P. 1998, "THE DYNAMIC RESPONSE OF TYRES TO BRAKE TORQUE VARIATIONS AND ROAD UNEVENNESSES", Link, retrieved on 10/04/22
[10]Wei, C. et al. 2016, "The effects of tyre material and structure properties on relaxation length using finite element method", Link, retrieved on 10/4/22
[11]Van der Jagt, P. 2000, "The Road to Virtual Vehicle Prototyping; new CAE-models for acceler-ated vehicle dynamics development. PhD-Thesis, Tech. Univ. Eindhoven, ISBN 90-386-2552-9 NUGI 834.
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