# Rolling motion mechanics & ideal economical gearbox ratios

• Schtoogie
In summary: VW engine is only rated for 3400 RPM at 70 mph in 5th gear.The Subaru's higher gear ratio might be necessary because the VW has a shorter final drive?In summary, the Subaru EE20 Diesel motor has a torque curve that peaks around 3600 RPM, which is about the peak of the power band. However, the overdrive gear on the stock 6 speed transmission is set for a higher engine speed, which results in high drag and little chance of ever exceeding 80 miles per hour. The final drive ratio on the Subaru diesel is actually shorter than the final drive ratio on a gasoline engine, and an electric supercharger might spread the torque out more evenly.
Schtoogie
Greetings, mechanical engineers of Physicsforums,

I have been struggling with a particular problem calculating an ideal motor RPM for economical highway cruising.

The motor I am looking at is the Subaru EE20 Diesel motor. Its torque curve can be found here: https://subdiesel.files.wordpress.com/2011/05/powergraphsubaru.png

I am considering this engine for a swap into my car for its great low-RPM grunt and economy. But with the stock 6 speed transmission, in top gear 75 mph comes at 3600 RPM -- just about the peak of the power band, and at around 290 N-m, or 80% peak torque of the motor. I have calculated the peak thrust at the wheels (from 1800-2400 motor RPM) to be around 1000 Newtons, while drag at 75 miles per hour (33.5 m/s) is around 700 Newtons. So clearly the motor running at that RPM for that speed (33.5 m/s) is sufficient to overcome drag on a perfectly flat plain. So why the heck is the overdrive gear set for such a high engine speed? Surely the 110 kW (148 hp) of power at that speed far exceeds what is required to move the car, isn't it? The car I would be swapping it into weighs around 16.5 kN, or 3700 pounds. This is a rough estimate for my own weight and a bunch of gear in the back. Am I missing something?

Additional note: The final drive of the car is 4.444:1. I calculated that with a 3:1 final drive, I could travel at 73 mph at 2400 RPM, which is close enough for my needs... or is it? Again, I feel like I am missing something. I know diesel power bands are far narrower than petrol ones, but I have trouble accepting that Subaru would design a transmission to basically never exceed 80 miles per hour. The CVT, which I would much rather have considering the narrow power band, has ratios closely matching the 6MT as well and has the potential to achieve a far greater gear spread, so I am not sure what to think from their perspective.

It's not just the transmission, it's the combination of your gearbox and the rather short final drive ratio of 4.444:1. That's drag racer type ratios, intended to make quick starts off the line for short dashes, fuel economy be danged. What's good for pulling stumps may not cut it for long-distance economy cruising on the highway.

IMO, you want to keep your new Subaru diesel operating in that band of 1800-2400 RPM as much as possible, because the torque falls off the cliff rather quickly outside this band.

Take a look at the data specifications for the same car fitted with diesel = CI, and gasoline = SI, engines. The final drive ratio is usually lower with CI used than when when SI is used. That compensates for the lower peak RPM of the CI engine. As turbochargers and common rail injection systems have improved, the performance of CI has moved closer to SI.

Increasing the rolling radius of the tyres can give a significant improvement when converting from SI to CI. But it is safer to lower the final drive ratio than to increase tyre rolling radius, since tyre rolling radius has unfortunate brake performance implications.

Right, this is why I was thinking that a quick and easy way to get a big economy boost would be to swap the differential from a 4.444:1 to a 3:1 or lower (which may need to be machined, I need to look into this more). This would keep my highway cruising speed in 6th gear between 55 mph and 72 mph at that 1800-2400 RPM torque plateau, which is what I want. Oddly enough, the final drive on the diesel is actually SHORTER than the final drive on a gas -- 4.444 for the 6MT diesel, 4.111 for the CVT diesel, and 3.900 for the CVT gas.

I just did some very rough calculations accelerating in 1st gear (3.454:1) and with a final drive of 3:1 based on average torque from 1000-3600 RPM, which is about 265 N-m. It yielded around 7 m/s^2 or about 0.7g with stock wheels. That's still pretty drivable in the city, but I am not sure how badly this would impact off-road performance, namely the ability to get out of mud or even to pull a trailer.

I wonder if the addition of an electric supercharger would spread the peak torque out more, namely have it come on earlier? The alternator for the diesel is rated at 170 amps, so I'm sure it could put out the juice.

Another odd thing: VW and Subaru's diesel drivetrains have similar gearing (Subaru is shorter on all but 5th and 6th), so for a comparison of highway cruising RPM I calculated about 3400 RPM highway at 70 mph. But after watching some enthusiast hypermiling videos, they are cruising at closer to 2000 RPM at these speeds in the very same car! I pulled the gear ratios and tire width directly from the VW site for the model that was being tested! What gives?

Watching 0-60 mph videos for the Subaru diesel, they are in 3rd gear at 60 mph and around 3500 RPM. The gearing is 1.062 with a final drive of 4.444. This calculates to over 5700 RPM. Am I missing something?...

It's still not clear what your ultimate goal is with this car. Do you want to make an economical car for cruising on the highway, or do you want a car which performs well off-road and/or towing something around? You can have one or the other, but I don't think you'll be able to come up with both simultaneously.

You should post the gear ratios for all the speeds in your 6-speed tranny. Shouldn't this box have an overdrive in 6th gear, maybe even in 5th gear, too?

My goal is 45 MPG at 75 MPH, but I will take a hit there if it means the difference between getting across town in a snow storm or not.

The gear ratios are, for the 6MT:
1: 3.454
2: 1.888
3: 1.062
4: 0.785
5: 0.634
6: 0.557
R: 3.636
F: 4.444

For the CVT it is 3.505:1 to 0.582:1 with a final drive of 4.111. They are geared very similarly and in fact the 6MT is more economical for cruising.

I don't see anything about an overdrive for either 5th or 6th gears. Assuming these numbers posted are the only ones in the box, the engine should be revving a lot higher than it is in the few videos I can find online. I am very confused.

Schtoogie said:
My goal is 45 MPG at 75 MPH, but I will take a hit there if it means the difference between getting across town in a snow storm or not.

The gear ratios are, for the 6MT:
1: 3.454
2: 1.888
3: 1.062
4: 0.785
5: 0.634
6: 0.557
R: 3.636
F: 4.444

For the CVT it is 3.505:1 to 0.582:1 with a final drive of 4.111. They are geared very similarly and in fact the 6MT is more economical for cruising.

I don't see anything about an overdrive for either 5th or 6th gears. Assuming these numbers posted are the only ones in the box, the engine should be revving a lot higher than it is in the few videos I can find online. I am very confused.
An overdrive is where the transmission gear ratio is less than 1:1. For the 6MT box, the 4th, 5th, and 6th gears are all overdrives.

## 1. What is rolling motion mechanics?

Rolling motion mechanics is the study of the movement of objects that roll, such as wheels or gears. It involves understanding the forces and energies involved in the motion, as well as how different components interact with each other.

## 2. How does rolling motion differ from sliding motion?

Rolling motion is when an object moves along a surface by rotating on an axis, while sliding motion is when an object moves along a surface without rotating. Rolling motion is more efficient and requires less energy than sliding motion, as there is less friction involved.

## 3. What is an ideal economical gearbox ratio?

An ideal economical gearbox ratio is the optimal ratio of the speed of the input shaft to the speed of the output shaft in a gearbox. This ratio is important in determining the efficiency of the gearbox and ensuring that the desired output speed is achieved without wasting energy.

## 4. How do you calculate the ideal economical gearbox ratio?

The ideal economical gearbox ratio can be calculated by dividing the speed of the input shaft by the speed of the output shaft. This ratio can be adjusted by changing the sizes of the gears in the gearbox to achieve the desired output speed.

## 5. What are some factors that can affect the ideal economical gearbox ratio?

Some factors that can affect the ideal economical gearbox ratio include the size and shape of the gears, the materials used to make the gears, the amount of friction between the gears, and the amount of power being transmitted through the gearbox. Other factors such as temperature and external forces can also impact the ideal ratio.

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