How to Calculate the Braking Momentum on a Wheel?

[erobz]

FYI, the phrase is "reining in". It originates from the notion of pulling on the reins to slow down a horse. The word "reign" would refer to rulership, most commonly by a king or queen.

Yeah, I get the proper spelling of words in my posts just about as well as I do physics. So, you can expect to see more "there, their, they're" type errors quite regularly.

 

[jbriggs444]

I not sure why I thought only external forces on the "free body" could do the "work" of stopping the vehicle.

There are two distinct notions of work that can be confusing. There are two different versions of the work-energy theorem. One to go with each.

1. Work = Force multiplied by the displacement of the center of mass. Sometimes called center-of-mass work.

The work-energy theorem for this one equates work with the change in the bulk kinetic energy due to the linear motion of the system being acted upon. $KE=\frac{1}{2}mv^2$ where $v$ is the velocity of the center of mass.

2. Work = force multiplied by the motion of the surface of the object being acted upon at the point where the force is applied.

The work-energy theorem for this one equates "work" with the mechanical energy transferred across the interface. This mechanical energy can manifest either as bulk kinetic energy in the object being acted upon or as internal motion (vibration, turbulence, deflection, rotation, etc).

For rigid, non-rotating objects or for point-like masses, the two notions of work are identical. However for extended non-rigid or rotating objects, they can differ.

 

[Orodruin]

The word "reign" would refer to rulership, most commonly by a king or queen.

By decree We will from now on make self references using the majestic plural.

 

[ROOT0X57B]

Ok so now, using all you have said to me, I have

In blue, not at scale, a layer of water

I suppose the brakes are at max power and no slip.
With $\mu_B$ and $\mu_S$ brakes and ground friction coefficients,
$l$ the tire width, $\rho$ the volumic mass of water and $C_x$ the fluid friction coefficient :
Moment/Torque of brakes force : $\mathcal{M}_\Delta(\overrightarrow{F_B}) = \mu_BmgR_\text{ext}$
Moment/Torque of water fluid friction : $\mathcal{M}_\Delta(\overrightarrow{F_E}) = -\frac{1}{2}\rho C_xhlv^2R_\text{ext}$
Moment/Torque of ground friction : $\displaystyle\mathcal{M}_\Delta(\overrightarrow{F_S}) = \mu_SmgR_\text{ext}$

Total moment of forces : $\displaystyle\mathcal{M}_\Delta(F) = \mu_BmgR_\text{ext} + \mu_SmgR_\text{ext} -\frac{1}{2}\rho C_xhlv^2R_\text{ext}$

I feel like I can maybe link $R_\text{ext}$ to stopping distance

Does it feel fair to you?

 

[erobz]

I feel like I can maybe link $R_{ext}$ to stopping distance

Is determining the stopping distance your objective?

The moment of the brake force is not correct. Why do you have ( presumably ) the mass of the vehicle in there? Also, the moment arm should be $r_b$, not $r_{ext}$ for that one.

Also, your sign convention is a bit mixed up, and the drag is going to be significantly more complicated

 

[ROOT0X57B]

Is determining the stopping distance your objective?

The moment of the brake force is not correct. Why do you have ( presumably ) the mass of the vehicle in there? Also, the moment arm should be $r_b$, not $r_{ext}$ for that one.

I have the mass because of @erobz formula :
$F_B = \mu_B * \frac{M}{4}\frac{R_\text{ext}}{R_B}g$
$m = M/4$
Also, $\mathcal{M}_\Delta(F_B) = F_B R_B$ so $\mathcal{M}_\Delta(F_B) = \mu_B * mgR_\text{ext}$

 

[erobz]

Yes, copy mistake, my bad (otherwise I wouldn't even have mentioned it)

I have the mass because of @erobz formula :
$F_B = \mu_B * \frac{M}{4}\frac{R_\text{ext}}{R_B}g$

Well, maybe slow down a bit on that. That relationship would be used just to put an upper bound on $\mu_b$ such that the tire won't slip on the road. Also, there was no drag in that model.

I would recommend starting with "no drag" just so you get an ideal of what you are up against.

 

[ROOT0X57B]

What do you call drag exactly ? The friction force with the ground ?

 

[erobz]

What do you call drag exactly ? The friction force with the ground ?

The drag force from the water. Ignore it for the time being and figure out the stopping distance of your vehicle on dry road for a parameter $\mu_b$ to get a sense for what you expect and how to tackle it. Its likely that the drag on the vehicle will dominate the drag on the tire...unless this is a reasonably deep stream of water, and the vehicle is not moving very fast. That being said, if it is reasonably deep, some buoyant forces are going to develop, and the static friction will significantly fall. There are many complications here.

 

[ROOT0X57B]

The point of my calculations is actually to find out the impact of water fluid friction on the stopping distance.
I only consider the water to apply a force on the tire : $-\frac{1}{2}\rho C_xhlv^2$

The water layer is really thin (1mm to 2cm)

 

[erobz]

The point of my calculations is actually to find out the impact of water fluid friction on the stopping distance.
I only consider the water to apply a force on the tire : $-\frac{1}{2}\rho C_xhlv^2$

The water layer is really thin (1mm to 2cm)

Well, I have a feeling the "wet road" is going to have a much more significant impact acting through $\mu_s$ (the static coefficient of friction) than the torque of the drag force from that tiny bit of water. Maybe I'm wrong.

Another thing, the minus sign in front of the drag is not exactly needed here. The force is negative, but the torque ( whether its positive or negative ) from it is going to be established by your convention when you apply Newtons Second Law.

I understand, that was your original motivation. but it is still a wise idea to solve the less complex model first of "dry road stopping" for some reference.

 

[ROOT0X57B]

Well, I didn't think about $\mu_S$ being modified because the road is wet
How could I model that ?

For the Second Law, I copied the moment expression without $R_\text{ext}$

 

[erobz]

Well, I didn't think about $\mu_S$ being modified because the road is wet
How could I model that ?

You might be trying to find some experimental results for that one.

 

[ROOT0X57B]

Like finding values of $\mu_S$ for wet and dry road?

 

[jbriggs444]

Surely if there is enough water to slow the vehicle down significantly when the brakes are already being applied, there is enough water so that hydroplaning is a factor that needs to be considered.

 

[ROOT0X57B]

Ok, finally I will have 3 models two make
1 - Dry road, no water, temperature not taken into account
2 - Wet road, water, temperature not taken into account
3 - Wet road, water, temperature taken into account

 

[ROOT0X57B]

Surely if there is enough water to slow the vehicle down significantly when the brakes are already being applied, there is enough water so that hydroplaning is a factor that needs to be considered.

I don't go fast enough I think (50kmh)

 

[jbriggs444]

I don't go fast enough I think (50kmh)

Think again. It is not just the possibility that the tires will be lifted out of contact with the pavement but the possibility that the normal force of tire on pavement will be significantly reduced.

 

[ROOT0X57B]

I will consider there is no aquaplaning then, to simplify equations
This is my project for engineering school entry exam (not supposed to be too complex though)

 

[erobz]

Well, something to consider. Depending on how much physics you know (or are expected to know), just figuring out the stopping distance on dry road is not quite as simple as you first might expect.

 

[ROOT0X57B]

Well, something to consider. Depending on how much physics you know (or are expected to know), just figuring out the stopping distance on dry road is not quite as simple as you first might expect.

Well, stopping distance on dry road would be a little bit of level I think, but could actually be given in some high ranked school entry test.
I want to keep it simple somehow, but I am expected to work on my project 2 hours a week for a year and half.

I also have to create something myself, this is why I am so obsessed with modeling the general situation.

 

[erobz]

Well, stopping distance on dry road would be a little bit of level I think, but could actually be given in some high ranked school entry test.
I want to keep it simple somehow, but I am expected to work on my project 2 hours a week for a year and half.

I also have to create something myself, this is why I am so obsessed with modeling the general situation.

This problem isn't 156 man hrs.

 

[hutchphd]

How does the height of the temperature layer and the heat generated by the friction with the ground affect the stopping distance?

With respect, this statement of the problem makes no sense to me. What is the "height of the temperature layer" ?

I know all of this may feel a bit confusing

We will do simpler
I have a disc of radius R spinning around its center axis
I now press an undeformable brake pad onto the disc
Thus, the pad is fixed while the disc keeps rotating, but there is friction between the disc and the pad
This friction obviously acts against the rotation of the disc
This is what torque I want to figure out.

So first solve the simple dry case (with help as necessary) forthe torque. That will be useful regardless.
What is the exact description from the engineering school for this exercise?? I personally think your choice is at best ill-formed and perhaps ill-conceived. Do you have other ideas and interests?

 

[ROOT0X57B]

With respect, this statement of the problem makes no sense to me. What is the "height of the temperature layer" ?

It's the height of the water layer, not temperature, my bad

Actually, one of the exams to be admitted to engineering school is what we call here "TIPE", it's a project of research within a given thematic (this year thematic is "the city"). We have to ask ourselves a question that could not be answered immediately, but not too complex either because we have to answer it in the end.
The research must be led in one of our fields of study (maths, physics, and computer science). Most importantly we have to produce something new (a physics model for me).

My question is "What is the impact of humidity and temperature of braking on the stopping distance of a car?"

So as said before, I planned to create three different models :
One with dry friction when not considering the temperature from braking (base case)
One with a water layer, no temperature, to get the impact of humidity
The last one with water and temperature considered, to get the impact of temperature

 

[erobz]

So as said before, I planned to create three different models :
One with dry friction when not considering the temperature from braking (base case)
One with a water layer, no temperature, to get the impact of humidity
The last one with water and temperature considered, to get the impact of temperature

Better check your experimental design. You're not going to get the impact of a parameter unless you hold the other parameters constant.

 

[ROOT0X57B]

Hmm, okay I guess?
I whished my final result could be a 3D graph :
For a chosen car (so mass, brake pressure...), to plot the stopping distance as a function of the height of water layer and braking temperature?

 

[erobz]

How are you going to get the braking temperature? You mean you are trying to model the temperature of the brakes?

 

[ROOT0X57B]

No, the temperature coming from the friction between the tire and the road

 

[erobz]

No, the temperature coming from the friction between the tire and the road

Thats a different type of friction all together. Rolling resistance is not static friction. I think you are being a bit ambitious for your experience level ( just my opinion ) it this topic.

 

[ROOT0X57B]

I did nothing for now about that so I am actually free to change up things
I thought only considering water could be a little bit less than expected so I had the idea of temperature.
Would you recommend moving to brakes temperature?