Quite often I come across a post relating to speed with a given final gear ratio and gear of transmission. This is usually looked up on one of a variety of calculators. This is fine but I thought maybe some would like to know what that calculator is doing with the information you provide. So I will provide a fairly typical example and work it from a given engine speed.
Engine speed = 6,000 RPM
Transmission gear ratio = 1:1 (Third gear most older automatics)
Rear gear ratio = 3.73:1 (Fairly common)
Tire diameter = 26" tall (Again, fairly typical)
With only these specs we can determine the mph the vehicle is travelling.
We will start with the engine RPM as it relates to the transmission gear. A 1:1 ratio means that for every turn of the engine's crank, the output shaft of the transmission will also turn one revolution. If the transmission gear ratio were 2.52:1, the engine's crank would turn 2.52 revolutions per one revolution of the output shaft of the transmission.
So, given our 6,000 RPM engine speed and 1:1 transmission gear ratio, the output shaft of the transmission (and thus the driveshaft and pinion of the differential) are turning at the same 6,000 RPM.
Now, we move onto the rear gear ratio. It is similar to the transmission gear ratio in that given a ratio of 3.73:1, the pinion of the differential will turn 3.73 revolutions per one revolution of the ring gear which is attached to the axles.
With the 6,000 RPM the pinion is moving and 3.73:1 ratio, we can find what RPM the axles are turning by dividing the 6,000 RPM pinion speed by the 3.73 ratio between the pinion and the ring gear.
pinion RPM/ring gear RPM = gear ratio
6,000 RPM/ring gear RPM = 3.73/1
6,000 RPM = 3.73*ring gear RPM
6,000 RPM/3.73 = ring gear RPM
ring gear RPM ~= 1,609
Quite simply we find that at 6,000 RPM, our combinations axles are turning at around 1,609 RPM.
Now, using the tire height we can find the mph the vehicle is moving. First, we must find the circumference. Circumference of a circle is given by the formula:
Circumference = 2*pi*radius = pi*diameter
With our given tire diameter of 26", we can find the circumference of the tire:
Circumference = pi*26" ~= 81.68"
USing our axle RPM and this circumference we can find how fast the vehicle is moving. For every revolution of the tire, the vehicle moves forward the circumference of the tire. So if the tire is turning at around 1,609 RPM and circumference of the tire is 81.68", we can find at what speed the vehicle is moving:
Speed in in/min = axle speed * movement(tire circ.)/rotation
Speed in in/min = 1,609 RPM * 81.68"/rotation
Speed = 131,423 in/min
With some simple math we can change the units of the speed we just calculated to something more familiar.
12 inches/foot
131,423/12 = 10,952 feet
5,280 feet/mile
10,952/5,280 = 2.0742 miles
60 minutes/hour
2.0742 miles/min * 60 minutes = 124.5 mph
We have found that at a 6,000 RPM engine speed with a 1:1 transmission gear, 3.73:1 rear gear, and 26" tall tire, our vehicle is moving at around 124.5 mph. Obviously, this is not an exact number due to slipping in the transmission and torque converter. But, it gives you a good idea of how gearing, tire height and engine speed effect vehicle speed and vice versa. The problems that were worked can be done with any combination and also done in reverse starting with a mph and working to find an engine speed if you would like to know that instead or any of the other variables.
In conclusion, I hope this helps more people understand the basic math that is used to find what the right drivetrain combination is and maybe find it useful in putting together their projects.
-Daniel
Engine speed = 6,000 RPM
Transmission gear ratio = 1:1 (Third gear most older automatics)
Rear gear ratio = 3.73:1 (Fairly common)
Tire diameter = 26" tall (Again, fairly typical)
With only these specs we can determine the mph the vehicle is travelling.
We will start with the engine RPM as it relates to the transmission gear. A 1:1 ratio means that for every turn of the engine's crank, the output shaft of the transmission will also turn one revolution. If the transmission gear ratio were 2.52:1, the engine's crank would turn 2.52 revolutions per one revolution of the output shaft of the transmission.
So, given our 6,000 RPM engine speed and 1:1 transmission gear ratio, the output shaft of the transmission (and thus the driveshaft and pinion of the differential) are turning at the same 6,000 RPM.
Now, we move onto the rear gear ratio. It is similar to the transmission gear ratio in that given a ratio of 3.73:1, the pinion of the differential will turn 3.73 revolutions per one revolution of the ring gear which is attached to the axles.
With the 6,000 RPM the pinion is moving and 3.73:1 ratio, we can find what RPM the axles are turning by dividing the 6,000 RPM pinion speed by the 3.73 ratio between the pinion and the ring gear.
pinion RPM/ring gear RPM = gear ratio
6,000 RPM/ring gear RPM = 3.73/1
6,000 RPM = 3.73*ring gear RPM
6,000 RPM/3.73 = ring gear RPM
ring gear RPM ~= 1,609
Quite simply we find that at 6,000 RPM, our combinations axles are turning at around 1,609 RPM.
Now, using the tire height we can find the mph the vehicle is moving. First, we must find the circumference. Circumference of a circle is given by the formula:
Circumference = 2*pi*radius = pi*diameter
With our given tire diameter of 26", we can find the circumference of the tire:
Circumference = pi*26" ~= 81.68"
USing our axle RPM and this circumference we can find how fast the vehicle is moving. For every revolution of the tire, the vehicle moves forward the circumference of the tire. So if the tire is turning at around 1,609 RPM and circumference of the tire is 81.68", we can find at what speed the vehicle is moving:
Speed in in/min = axle speed * movement(tire circ.)/rotation
Speed in in/min = 1,609 RPM * 81.68"/rotation
Speed = 131,423 in/min
With some simple math we can change the units of the speed we just calculated to something more familiar.
12 inches/foot
131,423/12 = 10,952 feet
5,280 feet/mile
10,952/5,280 = 2.0742 miles
60 minutes/hour
2.0742 miles/min * 60 minutes = 124.5 mph
We have found that at a 6,000 RPM engine speed with a 1:1 transmission gear, 3.73:1 rear gear, and 26" tall tire, our vehicle is moving at around 124.5 mph. Obviously, this is not an exact number due to slipping in the transmission and torque converter. But, it gives you a good idea of how gearing, tire height and engine speed effect vehicle speed and vice versa. The problems that were worked can be done with any combination and also done in reverse starting with a mph and working to find an engine speed if you would like to know that instead or any of the other variables.
In conclusion, I hope this helps more people understand the basic math that is used to find what the right drivetrain combination is and maybe find it useful in putting together their projects.
-Daniel
