Wheel Inertia explained.

I just bought a prius! This thing is fricken awesome.

Anyways... the first thing I noticed when I got to this forum, when researching aftermarket wheels, was the misconception about fuel economy and wheel weight/size. I haven't been able to find a thread that clarifies all of this accurately. Are there any other engineers here?

Misconception Number 1: MPG is dependent only on wheel weight.
Ok, in multiple threads, I read multiple expert opinions that mpg is dependent only on wheel weight, because it increases the "rotational weight" or something like that. Yes, it does increase the rotational weight, but wheel size has an effect on rotational weight also.

Misconception Number 2: MPG is dependent on rim size, irrespective of wheel weight.
In one thread, someone expressed their opinion that 17" wheels will be less fuel efficient, regardless of if they're lighter than 15" wheels. Okay, that's not always true.


So what exactly effects fuel economy?

Wheel Inertia!
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Let's look at the total kinetic energy of the prius. Hopefully you understand my variable usage.

E_in = energy put toward moving the car forward
E_out = kinetic energy of the car and wheels
m_car = weight of the car
v_car = velocity of the car
m_wheel = weight of a single wheel
r_wheel = radius of the center of mass of the wheel
w_wheel = angular velocity of the wheel
I_wheel = moment of inertia of each wheel. For simplicity, we will use the inertia equation for a hollow cylinder. Obviously, real wheels have a slightly more complex moment of inertia, but this is a valid assumption for our purposes.
I_wheel = m_wheel * r_wheel^2


E_in = E_out
E_in = 1/2 * m_car * v_car^2 + 4 * (1/2 * I_wheel * w_wheel^2)
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So as we see, the energy required to move the car is directly proportional to a property of the wheel, I_wheel, or the wheel inertia. And the wheel inertia, is dependent on wheel weight and the radius of the wheel's center of mass.

So if you're concerned about fuel economy, get wheels with low inertia, i.e. wheels with most of the weight distributed close to the center of the wheel. Large rims tend to have most of the weight distributed further from the center of the wheel.

 

No, I should have been more clear. This is the radius at which most of the weight of the wheel lies, which would be at the outer surface of the rim where the tire mates. Now, this is a gross oversimplification of the inertia calculation for an actual wheel (we are considering the wheel a hollow cylinder), but it is still valid for understanding how inertia of wheels works. Calculating the actual inertia of a wheel would require calculus.

I would assume so. Is it measurable though? That is a question that would be difficult to answer without some formal calculations. A wider wheel would have more inertia (assuming m_wheel increases and r_wheel doesn't decrease), so it would be hard to determine whether the decrease in fuel economy was from aerodynamic effects or inertial effects if you put wider wheels on. Most likely inertial effects would dominate.
I would guess aerodynamic effects would be negligible, since the the drag force is proportional to the projected frontal area of the car. Wider wheels would be a very small increase in the projected frontal area (as compared to the whole projected frontal area of the car)

 

What? Do you not understand how inertia works? Rotational inertia is dependent on the radius squared. For a distributed mass it's the sum (integral) of each point mass X the radius squared.

For example: A 10 lb hoop with a 10 inch radius has an inertia of 1000 lb-in^2, and a 11 lb hoop with an 11 inch radius has an inertia of 1331 lb-in^2. That's 33.1% more, from just a 1 lb and 1 inch displacement! That's not negligible at all during acceleration.

Taller? Yes. Because taller tires are run at a lower pressure, and deform more, increasing rolling resistance.

Wider? No, unless the rubber is softer. Rolling resistance is dependent on the deformation of the tire. If anything, wider tires are run at a higher pressure and have less rolling resistance. I'm sure you know enough physics to know that tire width alone has no effect on friction and rolling resistance.

Aerodynamic drag is almost negligible at non highway speeds, which is where inertial effects would be the most significant, during accelerating and stopping, so I don't really see your point here.

And you don't really explain why prius forum members who go with larger aftermarket rims get worse gas mileage. Your argument is rolling resistance, but low profile tires are run at a higher pressure and have less rolling resistance.

Please do your homework before you try to "debunk" my post, and maybe give a disclaimer when you're giving your opinion instead of taking the ignorant "I'm right" approach.

 

I am not ego stroking what-so-ever. I am just trying to share some information, and clarify some misconceptions that I've seen on this forum. Clearly, no one likes to hear that everything they are saying is a misconception.

I'm not ignoring this at all.



None of you explain, at all, why people are seeing losses in fuel economy with larger rims. So far, your arguments have been rolling resistance and air friction...

Rolling Resistance: Wheels with large rims and low profile tires are run at higher pressure, which decreases their rolling resistance because the tires deforms less while rolling. Contact friction is unchanged with wider wheels, because contact friction is independent of contact area. So rolling resistance can't be the explanation for lower fuel economy with larger rims.

Air Drag: The drag force is dependent on the frontal projected area of the car. This means the only area of the wheel that contributes to air friction is the lower 1/4 of the tire you see when you look under the car from the front, and any bit of the wheels that sticks out from the sides. And there is no way that increasing the frontal projected area of a car by a max of 5% (with larger wheels), causes a 5 mpg (10%) drop in fuel economy alone.