How does tire pressure affect distance traveled by a bicycle coasting to a stop?

[tymi9]

Hi,
I have been wondering if there is some equation which will allow me to calculate the distance and, or time, necessary for my bike to come to a complete stop knowing only the initial speed and tire pressure, without applying brakes and with negligible air resistance?

 

[erobz]

Rolling resistance is what is going to stop you, which is probably inversely related to the tire pressure. I think it should also be a function of the load being carried by the tires.

 

[jim mcnamara]

https://www.omnicalculator.com/physics/rolling-resistance
You are interested in rolling resistance. The above link will help you get a feel for the parameters I think you may have not specified. Since you did not really indicate why you want the value other than curiosity this will definitely get you started.

 

[tymi9]

I am writing my physics paper on how my bike's tire pressure influences the distance it will take me to come to a complete stop. I figured out a way to accelerate to the same speed but can't seem to figure out what is the correlation formula between the pressure in my tires and the distance which will take me to a complete stop
.

 

[erobz]

I am writing my physics paper on how my bike's tire pressure influences the distance it will take me to come to a complete stop. I figured out a way to accelerate to the same speed but can't seem to figure out what is the correlation formula between the pressure in my tires and the distance which will take me to a complete stop
.

Well, what do you expect increasing/decreasing the tire pressure will do in relation to the stopping distance? Have you considered testing this idea by experimentation?

 

[jim mcnamara]

..moved to homework forum.

 

[tymi9]

Well, what do you expect increasing/decreasing the tire pressure will do in relation to the stopping distance? Have you considered testing this idea by experimentation?

That is my original idea but I need something accurate to compare it to, thus I'm looking for a formula which will be accurate

 

[berkeman]

That is my original idea but I need something accurate to compare it to, thus I'm looking for a formula which will be accurate

Well, now that you know the term "rolling resistance" for a wheel/tire, what have you found with your Google searching? I doubt there is a universal formula -- most likely you will need to do some measurement experiments and fit a function instead of finding a formula that you can apply...

 

[erobz]

That is my original idea but I need something accurate to compare it to, thus I'm looking for a formula which will be accurate

Well, certainly there must be a way to derive a formula (with some simple assumptions) using Newtons Laws. But a clear path forward is currently evading me.

 

[malawi_glenn]

Have you tried the wikipedia article?

Here are some more links i just found

https://www.sciencedirect.com/topics/engineering/rolling-resistance

https://www.researchgate.net/publication/299639874_ESTIMATION_of_the_ROLLING_RESISTANCE_of_TIRES

https://rpug.sspa.us/wp-content/uploads/2020/10/2.3-Wix.pdf

https://kth.diva-portal.org/smash/get/diva2:1678831/FULLTEXT01.pdf

https://liu.diva-portal.org/smash/get/diva2:1568991/FULLTEXT01.pdf (here they studied the dependence of temperature)

 

[haruspex]

Wrt the Wikipedia article, this statement looks useful:
"if the inflation pressure is not changed, then a 20% increase in load results in a 4% increase in Crr".
Increasing the load won't change the tyre pressure much; mostly it changes the shape, increasing the area of contact, which means a greater deformation. Likewise, changing the tyre pressure for the same load changes that area. So we can use this to infer how a tyre pressure drop increases Crr.

 

[erobz]

We've been talking about wheels and rolling a lot recently, and I still can't seem to produce a simple model ( whether a poor representation of reality or not ), say $c_{rr}$ is constant proportional to $N$.

Imagine a wheel of radius $R$ rolling to the right without slipping. But it is accelerating due to rolling resistance.

The rolling resistance should manifest as a torque that is accelerating the wheel?

$$ \circlearrowright^+ \sum \tau = f_{sf} R - c_{rr} m g R = I \frac{a_{cm}}{R} \tag{1} $$

$$ \rightarrow^+ \sum F = -f_{fr} + c_{rr} m g = m a_{cm} \tag{2} $$

So, if multiplty $2$ by $R$ and add $1 \to 2$ I end up with no forces ( hopefully I didn't make some silly algebra error )!

If I make the force of static friction in the same direction as the rolling resistance, same issue.

What are the issues here? I'm probably doing something silly.

 

[jrmichler]

There is a gas mileage site, Ecomodder.com, https://ecomodder.com/forum/, that has many discussions on the subject of tire pressure, rolling resistance, and gas mileage. Use their search window with search term tire pressure to get started.

The short version is that there is no equation that starts with basic principles, so you need to do your own experiments. Another variable that is as important as tire pressure is the tire rolling resistance due to hysteresis losses in the rubber. You will see references to LRR (low rolling resistance) tires, which are designed for low rolling resistance.

Car tires and bicycle tires are all pneumatic tires, so the principles are the same. The numbers will vary. An interesting experiment would compare the rolling resistance of a road bike (skinny tires) to a comfort bike (2" wide tires) to a fat tire bike (4" wide tires), with all of those at a range of tire pressures.

 

[hutchphd]

My guess is that the rolling resistance would be proportional to the size of the contact patch. I can think of several experiments involving weight of load and tire pressure that would exactly test that hypothesis. You need some heavy and light participants, a gauge and pump and a very small hill on a still day. Do an exact study of a limited hypothesis

 

[haruspex]

What are the issues here?

Your (2) assumes $C_{rr}mg$ is an actual force and acts horizontally. The rolling resistance torque arises from an altered distribution of the normal force. See the diagram at @malawi_glenn's linked Wikipedia article.

 

[haruspex]

My guess is that the rolling resistance would be proportional to the size of the contact patch.

As the normal force increases, the width and length of the patch increase, as does the angle the tire has to flex through.
I can't see that the increased length adds to the losses.

 

[haruspex]

Another variable that is as important as tire pressure is the tire rolling resistance due to hysteresis losses in the rubber.

Isn't that all the same thing? The lower pressure increases rolling resistance by increasing the flexing of the tyre.

 

[OmCheeto]

Beings that I used to be an avid bicyclist in my youth, I was really interested in the OP's question.
What I did not expect to find was something I never knew.
According to this article, there is a sweet spot for bicycle tire pressures which varies by surface, where resistance goes up with tire pressure after a certain point.

I digitized the data from one of their graphs as the original was a bit fuzzy.
The Roller Drum is apparently some lab device which is very smooth and is used to generate data which the OP is looking for.
Unfortunately, the real world is a bit lumpy and totally boogers what looks like a very beautiful curve.

Their explanation for this is that at higher pressures the wheels become so stiff that they start transferring energy to the bicycle and bicyclist at increasing rates. Or something to that effect. The last 3 paragraphs in the article try and describe in different ways.

 

[haruspex]

to the bicycle and bicyclist

except ... when it gets rough, the cyclist will tend to stand, reducing the energy lost in the bouncing.

 

[OmCheeto]

except ... when it gets rough, the cyclist will tend to stand, reducing the energy lost in the bouncing.

So how many variables are we up to now?
1. Tire pressure
2. Road surface roughness
3. Cyclist sitting or standing
4. Mass of the bicycle
5. Mass of the cyclist

I'm guessing whether or not a bicycle has shock absorbers will make a difference.
I'm also guessing that the rigidity of the rubber that makes up the tire will make a difference.
One article I read said that wider tires have lower rolling resistance.

 

[MRMMRM]

Maybe you just want to do somthing simple lke a regression line fit to some gathered data. The data points I choose ae just made up. "Distance to rolling stop" = 73.888ln(tire pressure) - 151.45.

Your regression might benefit from using an absolute tire pressure, instead of a typical gauge pressure.

 

[PeroK]

So how many variables are we up to now?
1. Tire pressure
2. Road surface roughness
3. Cyclist sitting or standing
4. Mass of the bicycle
5. Mass of the cyclist

I'm guessing whether or not a bicycle has shock absorbers will make a difference.
I'm also guessing that the rigidity of the rubber that makes up the tire will make a difference.
One article I read said that wider tires have lower rolling resistance.

And, we cannot ignore that other factors - especially air resistance - even if they are independent of tire pressure.

 

[malawi_glenn]

And, we cannot ignore that other factors - especially air resistance - even if they are independent of tire pressure.

I think friction from the axle is also quite important.
But both friction from axle and air resistance will be a systematic "error" so to say, will affect all data points more or less equal.

 

[erobz]

Your (2) assumes $C_{rr}mg$ is an actual force and acts horizontally. The rolling resistance torque arises from an altered distribution of the normal force. See the diagram at @malawi_glenn's linked Wikipedia article.

So, it's actually a torque from the offset normal force, and $c_{rr}$ scales the offset, so we can just use $R$ as the moment arm? So its manifests as a torque, but not a force in the direction of motion.

$$ \circlearrowright^+ \sum \tau = f_{sf} R - c_{rr} m g R = I \frac{a_{cm}}{R} \tag{1} $$

$$ \rightarrow^+ \sum F = -f_{fr} = m a_{cm} \tag{2} $$

Combining $(1)$ and $(2)$:

$$ a = \frac{ -c_{rr} m g R^2}{ I + m R^2} \tag{3} $$

This at least has the characteristics we should expect of a wheel coming to a stop under its own loading. The OP in principal could use a modifed version of this measuring the stopping distance $x$ at a certain tire pressure $P$, and make a $c_{rr}$ vs. $P$ plot?

 

[hutchphd]

I can't see that the increased length adds to the losses.

I think the friction from the tire is mostly "squirm" on the pavement. This basicly scales with the contact patch area, but depends upon wheel diameter and tire tread/surface roughness also. For a given wheel a larger patch is of necessity longer

I think friction from the axle is also quite important.

For this very low speed "roll to a stop" test this might be a factor. For normal cycling I think it not important at all (unless the bearing is faulty).
I have a reasonable portion of my life contemplating the wheels folling under me.

 

[malawi_glenn]

I think the friction from the tire is mostly "squirm" on the pavement. This basicly scales with the contact patch area, but depends upon wheel diameter and tire tread/surface roughness also. For a given wheel a larger patch is of necessity longer

For this very low speed "roll to a stop" test this might be a factor. For normal cycling I think it not important at all (unless the bearing is faulty).
I have a reasonable portion of my life contemplating the wheels folling under me.

I was considering this experiment yes

 

[mikeyyy]

As the normal force increases, the width and length of the patch increase, as does the angle the tire has to flex through.
I can't see that the increased length adds to the losses.

Id guess that the longer the flat patch pressed along the ground the sharper the angle between it and the unreformed tire that is about to be flattened as the tire rolls. This suggests wider tires are better, which pro cyclists seem to have discovered. This model makes the tire sidewall stiffness very important.

 

[phyzguy]

If you actually do the experiment, you will find that your initial assumption of, "with negligible air resistance" is a bad assumption. Unless you are moving very slowly, air resistance is the primary drag on a bicycle.

 

[haruspex]

Id guess that the longer the flat patch pressed along the ground the sharper the angle between it and the unreformed tire that is about to be flattened as the tire rolls.

Yes, but in my post I covered that separately. I don't think the additional area of contact resulting from the increased length matters.

 

[mikeyyy]

Yes, but in my post I covered that separately. I don't think the additional area of contact resulting from the increased length matters.

No, I hunk it’s the sharpness of the bend as the tire rolls onto the flat area. The longer the contact patch the sharper the bend. Which means sidewall properties are important.

 

[jbriggs444]

If you actually do the experiment, you will find that your initial assumption of, "with negligible air resistance" is a bad assumption. Unless you are moving very slowly, air resistance is the primary drag on a bicycle.

Once the force law is nailed down, there is a differential equation lurking here.

For purely linear drag (drag proportional to velocity), the time taken to come to a stop is infinite, but the distance taken to come to a stop should be finite -- the sum of a decaying geometric series.

[The time taken to halve the velocity is a constant and you never finish halving the velocity. But the distance traversed each time you halve the velocity is also halved]

For purely quadratic drag, both the time and distance taken to come to a stop should be infinite.

[The time taken to halve the velocity doubles each time you halve the velocity. So it still takes forever to slow down. This time the distance travelled for each halving is constant -- half the velocity for twice the time. So the total distance is infinite]

With a mix of linear drag and quadratic drag, there will be a rapid decay of velocity after which the linear drag will dominate. So an assumption of negligible air resistance after some point is not completely unreasonable [which point is already well understood by @phyzguy].

 

[berkeman]

No, I hunk it’s the sharpness of the bend as the tire rolls onto the flat area

Sorry, I'm a little slow on Jive. What's "hunk" in this context?