Thursday, September 22, 2011

What I know about gears, RPM and speed

I was always curious about cars maximum speed, not the one shown in the speedometer, but the real one. So, after some research, this came up to me that the speed is related to 3 simple things.

- RPM: The speed the engine can turn itself in revolutions per minute.
- The gears ratio: How the engine speed can be transferred to tires
- The tires's size: Yes, size matters!

So, lets do the magic.

1 - Gears: The gear is a toothed wheel that when coupled to another gear can alter the relation between the speed of a driving mechanism. This relation about teeth and speed, can be expressed in the following formula:

  V1 is the speed at the first gear.
  V2 is the speed at the second gear.
  G1 is the number of teeth in the first gear.
  G2 is the number of teeth in the second gear.

Looking at the example below we can see another gears property. Coupled gears work in inverse rotational directions, when gear A is turning clockwise, gear B is turning counter clockwise. If the same rotational direction is needed another gear can be added to the mechanism or pulleys can be used instead.

Using the same example, we can determine what will be the speed at both gears if one of them starts turning. Notice gear A has 12 teeth and gear B has 18 teeth. Applying the formula given above, let's suppose that gear A suddenly starts turning at the speed of 30 RPM, then:

A small gear coupled to a bigger one will decrease the speed of its counterpart (and increase the power of it), which brings us to the following question: What if the gear B was the one that started turning at 30 RPM? So, lets see what happens:

When this speed is applied to gear B, the gear A will start turning at 45 RPM. Therefore, a bigger gear coupled to a small one will increase the speed of the mechanism. Two important things: Gears in the same axis have the same RPM no matters the size of them. When the speed is increased, the power is decreased and vice-versa.

2 - Ratio: Another way of calculate the speed in each gear is use the ratio between the first and second gear. As you will discover, all the car gears are expressed using this number. This can be easily done applying this formula:

  R is the gear ratio
  G1 is the number of teeth in gear A
  G2 is the number of teeth in gear B
  V1 is speed at gear A (in RPM)
  V2 is speed at gear B (in RPM)

3 - Tires. There is a very good article about tire codes at Wikipedia. I got the picture below from there which I will use to do a little summary about the parts that is related to our job.

From the picture above we are interested in 3 things:

a - Nominal width: This is the width of the tire in millimeters, using the example above the tire's width is 215mm.

b - Ratio of the height to width: The height of the tire is a percentage from its width. Using the example above, this will be 65% of the width: 0.65 x 215 = 139.75mm.

c - Rim diameter code: This is the diameter of the wheel, this is expressed in inches. As all the calculations are done using the metric system, we are going to convert it to meters. Again using the example, the wheel diameter is 15 inches, or 381 millimeters.

We are going to use all these number to discover one simple thing, the circumference of the whole wheel (wheel + tire). From the picture above we can clearly see that the mounted wheel diameter is the wheel size + 2 times the tire height. So, the full diameter will be 381 + 2 * 139.75 = 660.5 mm. Now we just multiply by π and we get the circumference: 660.5 mm * 3.1416 = 2075.03 mm, or approximately 2.07 meters.

3 - Remembering: Let's make a summary about we got until now:

a - We know the engine speed is measured in RPM (a common RPM speed for cruise cars is about 5000).

b - We know that car have gears and these gears have ratios, some of them increase the speed of the engine and some of them decrease it. Some of them are fixed gears (called differentials) and some of them can be changed while you are driving. This is done using the clutch on manual cars, and they are automatically changed on automatic cars.

c - We know that gears on the same axis will turn at the same speed, doesn't matter their sizes.

4 - Speed:  How about after all the gearing the very last gear is attached in the same axis with the wheel? It will just be a matter of calculate all the speed transmission among the gears to discover the RPM at the last gear and as the last gear is in the same axis with the wheel, every time the gear turns, the whole will turns as well, or, every time the last gear turns, the car moves 2.07 meters using the tires from our example (I told you size matters!). Supposing that after all the coupling the last gear is turning at the speed of 10 RPM, the car will be moving at 20.7 meters per minute, or 3.45 meters per second or 12.42 km/h or 7.72 mph.

5 - Common ratios: After a little research in the net I found some gear ratios regarding commercial bikes (B), cars (C) and SUVs (S). See below the table.

VehicleRPM Start Differential Final Differential
Gear ratio
(C) Volkswagen Golf57503.0871.003.362.09 1.471.191.150.98
Speed (km/h) with default tire: 1.81m60.1697.31137.60169.96175.93206.45
(B) Suzuki GSX-R 1000108001.5532.472.687 2.0521.6811.451.3041.208
Speed (km/h) with default tire: 2.00m125.76164.64201233.04259.08279.72
(S) Ranger 3.0 Electronic38003.541.
Speed (km/h) with default tire: 2.19m34.5661.6395.92140.99195.92-
(B) BMW F650 GS Dakar65001.9462.93752.751.751.311.050.88-
Speed (km/h) with default tire: 2.18m54.0288.94118.77148.33176.84-
(B) BMW K 1200 S102501.5592.822.521.841.451.281.141.01
Speed (km/h) with default tire: 1.94m107.67147.48187.17211.96238.04268.65

To calculate the maximum hypothetical speed at any gear just do the math. For instance, to calculate the VW Golf speed at the 3rd gear with default tires:

The ratio is including fixed gears in its calculation (differentials), we just need to multiply them together. Notice these values are expressed as maximum possible values and will rarely be obtained due to external factors, like friction and engine efficiency, which can literally blow if this is kept on the maximum RPM for long periods.

Have fun!

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