Gear Ratios for Dummies

1. Why Gears?

Imagine a car whose engine is directly connected to its drive wheels -- every revolution of the engine turns the drive wheels once. That is, 1 engine revolution per minute (RPM) results in 1 drive wheel revolution per minute.

Let’s consider how fast will that car will go when its engine runs at 1,000 RPM.

Wheels with 165-70/14 tires have a circumference of 72.55 inches, which should be close to my 1974 Spider’s original stock tire size. With the engine running at 1,000 RPM those wheels will be revolving at 1,000 RPM. We can calculate how fast those wheels are propelling the car by

Since inches per minute is not a common unit for expressing speed, we need to convert it to a more standard unit of measure, which in the US is MPH:

Pretty fast for a car with an engine running just a few hundred RPM above my Spider’s idle RPM.

(Aside: Some of you may have calculated an approximation in your head: 1,000 RPM * 6 feet per wheel revolution is 6,000 feet per minute, which is slightly over 1 mile per minute or 60+ MPH.)

What happens as engine RPM increases? In this simplified model any increase in engine RPM will result in a corresponding increase in wheel speed. At 4,000 engine RPM the wheels will be spinning 4 times as fast as at 1,000 RPM and the car will now be moving at 68.72 * 4 = 274.8 MPH. Tires surely smoking, probably shredded.

Ummm, I don’t think so! These speeds are clearly not realistic.

Thus, it's easy to conclude that engine RPM must be reduced to a lower, more manageable number of drive wheel revolutions per minute.

2. Reducing Engine RPM

The drivetrain connects the car’s engine to its drive wheels and is responsible for reducing engine RPM to a reasonable number of wheel revolutions per minute. The drivetrain’s overall gear ratio determines how much reduction is done. (The overall designation indicates that all of the drivetrain’s subcomponent gear ratios are combined into a single gear ratio. More on this topic later.)

Gear ratios, like other ratios, are typically denoted by two numbers separated by a colon ( : ), as in 4 : 1. For an overall gear ratio the first number represents the engine RPM; the second number represents the drive wheel RPM. Thus, an engine running at 4 RPM with a 4:1 overall gear ratio turns the drive wheels at 1 revolution per minute (1 WRPM). Since RPM is the typical measure of engine operation, in order to avoid confusion WRPM will be used to indicate Wheel RPM.

Real world gear ratios are rarely whole numbers, as they usually include fractional parts. For example, the attached illustration is of two gears with a gear ratio of 1.42 : 1.

In the illustration the gear on the left turns 1.42 times for every turn of the gear on the right. Connecting the (input) gear on the left to an engine and the (output) gear on the right to drive wheels has every 1.42 RPM producing 1 WRPM.


WRPM can be easily calculated by dividing the overall gear ratio into engine RPM.

For example, 3rd gear in my Spider has an overall gear ratio of 6.179 : 1. Running in 3rd gear at 3,000 RPM with a 4.56 rear axle ratio gives a speed of

which can be converted to MPH using tire circumference and the previously calculated conversion factor

3. How Many Gears?

One is not enough.

A single overall drivetrain gear ratio cannot effectively handle all driving situations. For example, try starting your car from a stop in 5th gear or going 70 MPH in 1st gear. (Please don’t try the latter!)

Automotive engineers specify a set of overall gear ratios to handle all the driving situations a car is expected to handle. 1st gear applies enough power to get a stopped car moving, 5th gear allows the engine to run efficiently at highway speeds, and intermediate 2nd, 3rd, and 4th gears accelerate and run efficiently at a variety of speeds.

Here are the overall gear ratios for my Spider:

As you can see above lower numbered transmission gears, such as 1st, have larger overall gear ratios than higher numbered transmission gears, such as 5th. The application of the mechanical advantage of levers results in the need for a variety of gear ratios. (Remember Archimedes’ famous declaration about levers: “Give me a place to stand, and I shall move the Earth with it.”)

Before diving into gears, let’s consider the challenge of pulling a nail out of a piece of wood. You have two claw hammers, one with a 6 inch handle and another with an 18 inch handle. You first try the 6 inch hammer and cannot extract the nail. Switching to the 18 inch handle allows you to extract the nail. The longer handle (lever) applies more force to the nail.

Gears are circular versions of levers. Let’s explore an example involving two gears, where a drive (input) gear rotates the driven (output) gear. In the nail extraction example above, using a longer hammer (lever) resulted in more force being applied to the nail. How then does one generate more force via a gear? That is, how is a gear made longer, as in a lever, to generate more force? By adjusting the number of teeth in the two mated gears, which taken together yields its gear ratio. By having the drive gear rotate many times to rotate the driven gear once, more force is applied to the wheels. (Torque is defined as twisting force; going forward the term torque will be used where appropriate.)

Consider the overall gear ratios for 1st and 3rd gears from the table above. 1st gear with a ratio of 15.066 : 1 applies more torque to the drive wheels than 3rd gear with its ratio of 6.179 : 1. 1st gear clearly provides more oomph.

There is a trade-off, however: torque versus acceleration. With more torque you get less acceleration. With more acceleration you get less torque. Consider cruising down the highway in 5th gear at 60 MPH. You approach a slower moving vehicle and decide to pass. You press harder on the gas pedal and the car begins to slowly accelerate above 60 MPH. Too slowly though. What do you do? You downshift to a lower transmission gear, one with more torque, that more quickly accelerates to a higher speed needed for passing. (Automatic transmissions do this for you when you stomp on the gas pedal; the transmission downshifts into a passing gear.)

4. Rear Axle Ratios

Most rear-wheel drivetrains are split into two separate sets of gears that work together to reduce engine revolutions, one set in the transmission and another in the rear axle.


Two rear axle ratios dominate gear ratio discussions for Alfa Romeos: 4.10 and 4.56. What’s the difference between them and why do we care?

Here are the overall gear ratios side-by-side for these two rear axle ratios:

Using the previous example of 165-70/14 tires at 3000 RPM, we computed a speed of 33.36 MPH in 3rd gear with a 4.56 rear axle gear ratio.

Plugging in the same tire diameter and RPM, but this time with a 4.10 rear axle ratio, the computed MPH is 37.09, almost 4 MPH more.

The essential difference:

For a given RPM a numerically smaller rear axle gear ratio results in a higher speed than a numerically larger rear axle ratio.

Lower RPM with higher speed yields better efficiency, more miles per gallon, and less engine effort.

Saying it differently, less effort is required of the engine to attain a given speed with a smaller rear axle gear ratio than with a larger one. Less effort means higher efficiency and lower fuel consumption. Although there’s also the possibility of a higher top speed, that’s not always reality due to other factors.

What’s the downside to a smaller rear axle gear ratio? A numerically smaller rear axle gear ratio results in slower acceleration to a given speed than a numerically larger one. Drivers concerned with 0-60 acceleration times may not be pleased with a smaller rear axle gear ratio.

5. Calculating 4.10 and 4.56

We need a bit more detail about a car’s rear axle to get an understanding of the rear axle ratio. Most rear-wheel cars, including most but not all Alfas, have their transmission right behind the engine and a drive or propellor shaft connecting the output of the transmission to the rear axle. At the end of the drive shaft is a pinion gear which connects to a ring gear in the rear axle. These gears serve two purposes: 1) they further reduce the number of revolutions from the transmission to the drive wheels and thereby produce additional mechanical advantage and 2) they turn the rotation of the driveshaft by 90 degrees to rotate the wheels.

You see how rear axle gears works in this video


The smaller of the two is the pinion gear; the larger is the ring gear.

A rear axle ratio is calculated by dividing the number of teeth in its ring gear by the number of teeth in its pinion gear. Specifically:


(When discussing rear axle ratios, the ":1" is typically omitted.)

A rear end ratio of [4.10, 4.56] means that the output of the transmission via its drive shaft revolves [4.10, 4.56] times for every revolution of the wheel.

Later model year Alfas sold in the US came with 4.10 rear end ratios for increased fuel economy and better emissions.

6. Short or Tall?

Shorter gears, taller gears. These terms appear regularly in the automotive press and here on the AlfaBB. What do they mean?

Short and tall refer to the output of a gear ratio by giving a relative indication of the output gear’s speed.

For cars, short and tall indicate how fast the rear axle is turning the wheels. As there is a range of rear axle gear ratios, taller means faster (as in higher number of WRPM) and shorter means slower (as in lower number of WRPM).

Taller means higher speed (MPH). Shorter means slower speed (MPH).

Let’s explore this concept by calculating the output of just the rear axle's gears. The simplest case is to compute the number of wheel revolutions for a drive shaft turning 1 revolution per minute:

These calculations show that the 4.10 rear axle turns the wheels faster than the 4.56 rear axle, so the 4.10 is the taller of the two. Correspondingly, the 4.56 rear axle turns the wheels slower than the 4.10, so the 4.56 is the shorter one. (Yes, the meanings are the opposite of numeric values of the gear ratios.)

These terms make sense when you consider that dividing RPM by a gear ratio yields a result that is inversely related to the gear ratio. Larger gear ratios result in smaller speeds (which means it’s a shorter gear). Smaller gear ratios result in bigger speeds (which means that it’s a taller gear).

There may be a spread of available rear axle gear ratios, depending on the model and the car manufacturer, ranging from 2 point something to 5 point something, so the range of speeds (and acceleration times) can be wider than those for just these two Alfa rear axle gear ratios.

There are many synonyms for these terms. Here are a few of them that reinforce the theme:

7. Tire Size and Speedometer Reading

What happens when you switch to a non-stock size tire? Your ride and handling may improve. Your perceived acceleration and MPG may also change, but those metrics depend on having an accurate speedometer. Jokes aside about Alfa speedometer accuracy, let’s compare speedometer readings for different tire sizes.

Here is a table showing speed in gear at 3,000 RPM with a 4.56 rear axle for three tire sizes:


Alfa Romeo’s engineers calibrated my Spider’s speedometer for 165HR14 tires. It's nice to see that 185/70-14 tires result in speeds close to those of the stock 165HR14 tires.

Close though is not spot on, so there are speed differences among the tires. Larger circumference tires result in higher speeds and smaller circumference tires result in lower speeds. Both situations result in speedometer error.

Summary: Running larger circumference than stock tires yields higher speeds in each gear than indicated by the speedometer. Driver beware!

THANK YOU!

Hi Bob,

That is clearly the most informative and useful thread that has been posted here in the last several months.

I have only skimmed through it so far, but plan to spend several hours reading and re-reading your most helpful, organized and well-written presentation.

Your contribution is a masterpiece in engineering tutorials, and you are to be commended.

MOLTE GRAZIE!

BobG said:

sniparoo...

Alfa Romeo’s engineers calibrated my Spider’s speedometer for 165-70/14 tires.

Click to expand...

Great posts! Allow me to be the first to pick nits.

The stock tire size on the '74 Spider is actually 165HR14, not 165/70/R14. These old tires with no aspect ratio are actually something like 82, not 70. The Vredestein Sprint Classic is still available in the OEM size:
https://www.cokertire.com/165hr14-vredestein-sprint-classic.html

As you can see the overall diameter is 24.5", which is what you really care about for gearing, speedo calibration, etc. Picking a few common sizes:

165HR14 (Stock size for a '74 Spider): 24.5"
185/70R14 (Stock 14" size for an S3 or S4 Spider): 24.2"
195/60R15 (Stock 15" size for S3 / S4 Spider): 24.21"
175/65R15: 23.98"
185/65R15: 24.47"
175/70R14: 23.65"
195/60R14: 23.21"

I had 175/70R14s on a Giulia Super with the 4.56 axle and the revs were way up there on the highway. I really don't see any reason not to go with a 4.10 on any street driven 2L or 1750.

-Jason

A very comprehensive write up. :nerd: <<< Which other 1 would I use? :whistling::whistling::whistling:

Just 1 question: When is the U.S. going to move over to the metric system of measurement?
12 inches in a foot and there's 3 feet to a yard, but how many yards per mile?
And lets not mention deviding an inch into fractions of itself. :cursing:
Funny how the inch does get broken down to thousands of an inch, tho. I wonder why that is.............. :wink2:

Here in Aus and maybe the same in other parts of the world, farmers often refer to rain fall in 'points', which are 100th of an an inch.

Hmmmmmmmmmmmmmmm. >

OK, I'll stop being a pest (for now).

More About 165HR14 Tires

jarrington said:

....
The stock tire size on the '74 Spider is actually 165HR14, not 165/70/R14. These old tires with no aspect ratio are actually something like 82, not 70. ...

As you can see the overall diameter is 24.5", which is what you really care about for gearing, speedo calibration, etc.

....
-Jason

Click to expand...

Having a tire's diameter is not sufficient to determine its actual circumference for speed calculations. Huh, you say. Read on for my findings.

Using 24.50" for a 165HR14 tire's diameter we compute its circumference by multiplying by pi and get 76.97". 76.97" seems too big. It's certainly larger than the circumferences of modern era 70 series tires.

Went back to the Michelin Classic Tire Catalog and carefully reread the XAS tire info on page 9.

To the right of the Overall diameter (mm) column is a Rolling circ. (mm) column. Circ is surely an abbreviation for circumference.

Michelin's data for their XAS 165 HR 14 tire is as follows:

• Overall diameter (mm) = 626mm (* pi = 1967 mm circumference)
  
  Note that this tire has a diameter of 24.65".​
• Rolling circ. (mm) = 1903 mm (74.92")

Wow, that's a 64 mm difference between theoretical and actual circumferences! 2.5"!

So, based on this data it seems that one cannot just use a tire's diameter to compute its actual rolling circumference. You need its rolling circumference from the manufacturer, or some formula to compute it.

A few other notes:

1. My computed speed in 3rd gear @ 3,000 RPM for a Michelin XAS 165HR14 tire is 34.44 MPH.
2. A tire's shape appears to be altered after mounting on a wheel. Perhaps a tire engineer can fill us in about the disparity between theoretical and actual circumferences.

Duk said:

Just 1 question: When is the U.S. going to move over to the metric system of measurement?

Click to expand...

Sigh, seems like never.

With some trepidation about taking this thread way beyond gear ratios ...


Sorry for the digression.

I suggest that any further discussion on the US and the metric system be moved to a different, topic specific thread.

Thanks