Is a car's top speed the same on a wet road?

[electrodruid]

I'm a games programmer who is writing a simple car physics simulation, and I've had a disagreement with the designer about a car's behaviour - specifically, its top speed - in different conditions.

Imagine a car accelerating from a standing start to top speed along a perfectly straight, flat road. We know the car's mass, so we can calculate acceleration at any given moment using a = F/m ... To calculate the forces I'm assuming that the car is resisted by a drag force (proportional to the car's velocity squared), and rolling resistance (proportional to the car's velocity), and is accelerated forward by a tractive force. I have a function that returns a traction force for a given velocity. It's arrived at by slightly convoluted means and is meant to represent high torque in lower gears, decreasing as the car speeds up and the driver changes up through the gears, and it looks kind of like a graph of y = 1/x

So the optimum tractive force that can be applied at any given moment is proportional to the coefficient of friction between the tyres and the road, right? For good quality tyres on dry asphalt we can say that the coefficient is 1.0. Assuming we plug in sensible values, we get a decent-looking acceleration. Now we re-run the simulation when the road is wet and the friction coefficient drops to 0.7 ... I think that that will scale the usable traction force at any moment to 70% of what it could be in dry conditions (more than that would cause the car to wheelspin), which means not only a slower acceleration but also a slower top speed. The designer agrees that acceleration would be slower but thinks that given a long enough road the car could eventually reach the same top speed in the wet as it can in the dry.

Who is right? And if he is right, how can that be explained and simulated?

 

[Bystander]

Who is right?

Neither. You are less in error than he; rolling resistance on wet road is higher than on dry. All other things equal, horsepower output, the higher drag slows the car.

 

[jbriggs444]

The coefficient of static friction should act to limit the maximum tractive force that can be applied. Your tractive force should have a flat line maximum with a value that depends on the coefficient of static friction and vehicle mass.

 

[A.T.]

The designer agrees that acceleration would be slower but thinks that given a long enough road the car could eventually reach the same top speed in the wet as it can in the dry.

It depends on whether the maximal speed on the dry road is determined by max. engine power or static friction coefficient. If it's the later, then obviously any reduction of the coefficient will reduce the maximal speed. But if it's the max. power, then you can reduce the friction coefficient by a certain amount, without any consequences for maximal speed, until you reach a level where friction coefficient becomes the factor limiting maximal speed. The same argument applies to maximal acceleration.

If you are writing a simulation anyway, you can plug in different parameters and see when they affect top speed.

 

[russ_watters]

My suspicion is that for most real cars, the static friction coefficient stops being a driving factor by the time you get out of first gear.

 

[A.T.]

My suspicion is that for most real cars, the static friction coefficient stops being a driving factor by the time you get out of first gear.

In the first gear it's a limiting factor for acceleration. But for most normal cars on a dry or slightly wet road its probably not the limiting factor for speed. If the road is flooded, that creates it's own drag limiting the speed. I can imagine static friction coefficient limiting speed on ice and compressed snow.

 

[electrodruid]

Interesting. So to summarise, there's a threshold speed/gear above which the friction coefficient becomes effectively irrelevant and the car's top speed is instead limited by rolling resistance (or some additional drag force) instead? That raises some questions:

- How might you go about modelling that effect of the static friction? Presumably the point at which the limiting factor is drag rather than friction changes depending on the surface type (we're trying to model a bunch of different surfaces: wet/dry asphalt, dirt, mud, gravel, sand, snow, ice, etc), and presumably that has something to do with how much tractive force is available at a given gear/speed but unused because the friction coefficient limits how much force would be useful. But I'm having trouble visualising how that all fits together.

- Similarly, how might you model this drag/rolling resistance force? Is it like drag (i.e. proportional to velocity squared), or more like rolling resistance (i.e. proportional to velocity), or something else? And how might it differ for different surface conditions?

 

[jbriggs444]

Interesting. So to summarise, there's a threshold speed/gear above which the friction coefficient becomes effectively irrelevant and the car's top speed is instead limited by rolling resistance (or some additional drag force) instead?

The limit on top speed is the point at which thrust is equal to drag. It is the speed at which the graphs of drag-as-a-function-of-speed and thrust-as-a-function-of-speed intersect.

This point may occur in a region where thrust is friction-limited (e.g. a car driving on ice) or in a region where thrust is engine-limited (e.g. a ordinary car on dry pavement).

 

[A.T.]

the car's top speed is instead limited by rolling resistance (or some additional drag force) instead?

For most cars on a road it's mostly aerodynamic drag vs. propulsive power.

friction coefficient limits how much force would be useful

It limits how much propulsive force is possible

Similarly, how might you model this drag/rolling resistance force? Is it like drag (i.e. proportional to velocity squared), or more like rolling resistance (i.e. proportional to velocity), or something else?

The total drag is the sum of the individual drags.

 

[electrodruid]

It limits how much propulsive force is possible

Understood. Perhaps I need to clarify how I'm currently doing things. So I've calculated the propulsive and resistive forces for a car on a dry drag strip - friction coefficient of 1.0. So for a wet drag strip I'm just looking up those same numbers I calculated earlier, but multiplying the traction force for a given velocity by 0.7 (wet road friction coefficient) to simulate that limiting factor. Leaving the drag and rolling resistance forces the same, relative to velocity, and just running the simulation. What I'm gathering from this conversation is that this is probably the wrong approach since limiting the traction force in this way can limit the top speed (i.e. the speed at which the scaled-down tractive force is equal to the sum of the unscaled-down drag) in quite a dramatic way.

The total drag is the sum of the individual drags.

Understood. TotalDrag = Aerodynamic Drag + Rolling Resistance + Wet Surface Drag. I wasn't previously aware that a wet surface (presumably also mud, ice, gravel, etc) can impose a new individual drag force, so I'm trying to work out what kind of shape that drag might take. Is it constant, linear, quadratic...?

 

[jbriggs444]

Perhaps I need to clarify how I'm currently doing things. So I've calculated the propulsive and resistive forces for a car on a dry drag strip - friction coefficient of 1.0.

How does the friction coefficient of 1.0 figure into your calculations? From everything you say, it appears that you are including it incorrectly.

 

[A.T.]

but multiplying the traction force for a given velocity by 0.7

The coefficient of static friction gives you the maximal static friction possible, not the actual static friction. The traction force needed for a given velocity is determined by total drag-

 

[electrodruid]

How does the friction coefficient of 1.0 figure into your calculations? From everything you say, it appears that you are including it incorrectly.

For a given velocity v, my force calculation currently looks something like this:

F = (TractionForce(v) * frictionCoefficient) - ((DragCoefficient * v * v) + (RRCoefficient * v))

So with a friction coefficient of 1.0, i.e. a dry road, the available tractive force isn't scaled or limited in anyway. The car's acceleration is purely a result of the tractive and drag forces. For friction coefficients of < 1.0 the tractive force is limited, resulting in slower acceleration and a slower top speed.

 

[jbriggs444]

For a given velocity v, my force calculation currently looks something like this:

F = (TractionForce(v) * frictionCoefficient) - ((DragCoefficient * v * v) + (RRCoefficient * v))

So with a friction coefficient of 1.0, i.e. a dry road, the available tractive force isn't scaled or limited in anyway.

As suspected, that calculation is incorrect. The effect of the coefficient of static friction is to impose a maximum force equal to vehicle mass times the acceleration of gravity times the coefficient of friction. That is better modeled as:

Thrust = min(TractionForce(v), m * g * frictionCoefficient)

Edit: min, darnit.

 

[electrodruid]

Ah! Perfect! Thank you!

Okay, so what about the other part - the idea that there's some kind of change or addition to the drag forces on wet or slippery surfaces, so the top speed of the car is perhaps still affected somewhat compared to dry conditions (but not affected directly by the friction)? What might that look like, as a function of velocity?

 

[russ_watters]

You need to build a model that relates engine power to tractive force. If you do a forum search, we've had several threads describing how to do it.

 

[jbriggs444]

Note edit to post #14. It should have been min, not max.

 

[electrodruid]

Hehe. I always make that mistake first time around, too :)

 

[electrodruid]

You need to build a model that relates engine power to tractive force. If you do a forum search, we've had several threads describing how to do it.

My simulation doesn't really include engine power. For long, complicated, and not-terribly-relevant-to-my-question reasons, the tractive force is approximated from a sort of idealised mathematically (rather than physics-ally) constructed acceleration curve and the resistive forces. I don't really have usable values or calculates for engine RPM, torque, power, etc. Just a tractive force, some drag forces, the car's mass and a friction coefficient.

But I'm still not clear - assuming for a second that I did have a model that relates engine power to tractive force, or could build one, how would that help me to build a model for resitive forces of different surface types at different speeds?

 

[A.T.]

My simulation doesn't really include engine power.

As already said, this is what usually limits speed for normal cars on a dry road.

Just a tractive force, some drag forces, the car's mass and a friction coefficient.

Just make sure tractive force is never more than nomal force * static friction coefficient, unless you want to simulate sliding.

 

[russ_watters]

My simulation doesn't really include engine power. For long, complicated, and not-terribly-relevant-to-my-question reasons, the tractive force is approximated from a sort of idealised mathematically (rather than physics-ally) constructed acceleration curve and the resistive forces. I don't really have usable values or calculates for engine RPM, torque, power, etc. Just a tractive force, some drag forces, the car's mass and a friction coefficient.

Your first post said you were calculating the acceleration from the force. Now you are saying the opposite. Whichever the reality is, the accuracy of the simulation will depend on the accuracy of the input data and the math you use to analyze it. I'm not quite sure where to go from here without you explaining in more detail what you really have, really want and why. But I can't imagine a scenario where it would be possible to model a car's performance without knowing its engine output!

But I'm still not clear - assuming for a second that I did have a model that relates engine power to tractive force, or could build one, how would that help me to build a model for resitive forces of different surface types at different speeds?

It wouldn't: They are two separate analyses. Again, I'm not sure anymore what you have and want.

 

[electrodruid]

Neither. You are less in error than he; rolling resistance on wet road is higher than on dry. All other things equal, horsepower output, the higher drag slows the car.

Here's what I'm trying to get to the truth of. Bystander says that rolling resistance is higher on wet roads than on dry. Can anyone confirm or refute this idea? This is the only place I've heard that idea expressed. If it is true, can anyone shed any light on exactly HOW rolling resistance differs on wet vs. dry roads? And is this change in RR also applicable to other surfaces (dirt, mud, sand, gravel, ice, snow, etc)?

You need to build a model that relates engine power to tractive force. If you do a forum search, we've had several threads describing how to do it.

russ_watters, you appeared to write this post in response to one of my earlier posts asking for more information about this rolling resistance thing that Bystander was talking about. Perhaps we're still talking at cross-purposes, but if you're telling me that having engine power (as opposed to just working with the resulting tractive force) included in the model is an essential part of calculating the effect of rolling resistance on different road surfaces, you don't seem to have explained why that's the case.

I can get into an in-depth discussion about exactly how my simulation works, and why it works the way it does, but it will do nothing other than add confusion and distract from the issue I'm asking about. Taking jbriggs444's advice into account, my basic calculation looks like this:

Thrust = min(TractionForce(v), m * g * FrictionCoefficient)
F = Thrust - (DragCoefficient * v * v) - (RRCoefficient * v)
a = F / m
v = v + a

DragCoefficient is the cars width * height * Cd
RRCoefficient is some number such that the drag and rolling resistance forces are equal at 60mph
FrictionCoefficient is based on the tyre and surface type, with data taken from here: (http://hpwizard.com/tire-friction-coefficient.html)
TractionForce is a function that works by black-box hand-wavey magic, but is designed to give a reasonably realistic interpretation of the tractive force produced by a car engine and gearbox. In my simulation it's not calculated directly from engine power, but even if it was I'm failing to see what modelling the relationship between engine power and tractive force would help me to also model the changes in rolling resistance on different surfaces. If you can explain that correlation then I could see why it would be relevant to get more into the details of how I'm doing that calculation.

 

[drvrm]

In the present case the conditions of perfect rolling changes due to wet condition of the road- as the road gets wet rolling as well as part slipping will take place and at certain speed it may convert to slipping - any model should include this feature- the physical change is due to the liquid spreading itself in the microscopic hills and valleys of the contact -surfaces i'e' road and the tyre- rolling friction is independent of speed of the vehicle but slipping frictional coefficient will depend on speed. the physical picture may be made clear before a simulation experiment.though data from the race tracks may be available on the net!

 

[jbriggs444]

Here's what I'm trying to get to the truth of. Bystander says that rolling resistance is higher on wet roads than on dry. Can anyone confirm or refute this idea? This is the only place I've heard that idea expressed. If it is true, can anyone shed any light on exactly HOW rolling resistance differs on wet vs. dry roads? And is this change in RR also applicable to other surfaces (dirt, mud, sand, gravel, ice, snow, etc)?

Roughly speaking, rolling resistance relates to how well or poorly the tire will spring back on the back side compared to how much effort was expended to compress it on the front side. For perfect pavement, what matters is the tire and its inflation.

For wet pavement, you are squishing water out of the way on the front side and getting no corresponding rebound on the back. That increases rolling resistance. The same general principle applies to dirt, mud, gravel, and snow. They all "give" on the front side and do not return that effort on the back.

Ice might actually have reduced rolling resistance because tread squirm would not be resisted on the front side.

If you are after numbers, I have none to offer.

 

[drvrm]

Jones and Childers report coefficients of friction of about 0.7 for dry roads and 0.4 for wet roads. The tread design represents an "all weather" compromise. If you were an Indianapolis race driver, you would use "slick" racing tires with no tread. On dry surfaces you might get as high as 0.9 as a coefficient of friction, but driving them on wet roads would be dangerous since the wet road coefficient might be as low as 0.1 .

Many years of research and practice have led to tread designs for automobile tires which offer good traction in a wide variety of conditions. The tread designs channel water away from the bearing surfaces on wet roads to combat the tendency to hydroplane - a condition which allows your car to "ski' on the road surface because you have a layer of water lubricant under all parts of your tire.
one can see the following ref. for details;
http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/frictire.html

 

[russ_watters]

I've re-read the thread and you did seem to absorb everything else even though it was a bit confusing when you said you weren't considering engine power. That still concerns me (and doesn't make sense since you say you have a tractive force model based on gearing), but we can set it aside for now:

Here's what I'm trying to get to the truth of. Bystander says that rolling resistance is higher on wet roads than on dry. Can anyone confirm or refute this idea? This is the only place I've heard that idea expressed. If it is true, can anyone shed any light on exactly HOW rolling resistance differs on wet vs. dry roads? And is this change in RR also applicable to other surfaces (dirt, mud, sand, gravel, ice, snow, etc)?

While I generally agree with both Bystander and jbriggs, I'll add that rolling resistance is complicated and there may be specific instances where water will reduce it instead of increasing it. The resistance jbriggs describes, you can easily feel when you are driving on a very wet road (or through a puddle!). But for a very slightly damp road, the decrease in static friction may reduce rolling resistance by more than pushing the water away increases it. See the wiki:

Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body (such as a ball, tire, or wheel) rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy needed for deformation (or movement) of the wheel, roadbed, etc. is recovered when the pressure is removed. Two forms of this are hysteresis losses (see below), and permanent (plastic) deformation of the object or the surface (e.g. soil). Another cause of rolling resistance lies in the slippage between the wheel and the surface, which dissipates energy.

https://en.wikipedia.org/wiki/Rolling_resistance

Key-in on that last part: because the tire is deforming, it stretches, compresses and bends, which causes some of it to slip along the ground when flattening-out and being pulled back up. That's kinetic friction and may be reduced on a slightly damp road. But with more than just a slightly damp road, the water will increase the resistance.

I'll note that this simplistic rolling resistance estimator doesn't show a change for a wet asphalt road:
http://hpwizard.com/tire-friction-coefficient.html

And this one is targeted at bikers, but has quotes from a tire manufacturer:
http://cyclist.com.au/rain

Another confounding factor: water will cool the tire, which also increases rolling resistance.

The thing is though, rolling resistance is a pretty small fraction of the resistances on a car, particularly at high speed and high acceleration. From the wiki, it is on the order of 0.01, whereas we're discussing friction coeffients in the 0.7-1.0 range. You'd be hard pressed to make a model precise enough for that to matter.

For driving through a puddle, though, you might try an equation that relates friction coefficient to puddle depth. Say, 0.02 per mm. Of course, then there's hydroplaning...

 

[electrodruid]

Thanks, all. I've learned that the friction coefficient is used to set a limit on the car's tractive force rather than act as a multiplier for it, and that rolling resistance is... complicated, but does differ according to surface type (even if overall the effect is probably extremely subtle). I think on balance I got it more wrong than the designer did, but fortunately it means I'm now in a position to fix it