# The gear box: the point of view of the wheels

I found myself in trouble when I tried to implement a car simulator. The first trouble is to understand how the gear box works. I read a lot of documents around the WEB until every things have been cleared when I realized a very simple thing: “Engine and Wheels are not independent!”. Maybe this fact is well known but its consequences are not clear to all of us. Many things I am writing are maybe known to you but let me take the point of view of the wheel as opposite to the engine view. Using the wheel view I found all much easier.

First of all let me introduce the gear ratios as multipliers of the wheel RPM (revolutions per minute) to get the engine RPM. Well not exactly. There is a final multiplier to get the engine RPM from the wheel RPM: the “car final drive” better known as “differential”. For our purposes the differential is an other multiplier for the wheel RPM to get the engine RPM.

(1) RPMe=RPMw * Current_Gear_Ratio * Gear_Ratio_Final

Where:

RPMe is the engine revolutions per minute

RPMw is the wheel revolutions per minute (for us every wheel will have the same speed)

Current_Gear_Ratio: one of the possible “gear coefficient” you selected

Gear_Ratio_Final: the differential coefficient

Example of gear ratios are, for the Opel Agila 1.2, the following:

While the differential is 3.74

This mean for one revolution of the wheel using the first gear you have (MUST HAVE) 3.73*3.74 = 13.9502 revolutions per minute in the engine.

With the fifth gear you have 0.89*3.74=3.3286 revolutions in the engine.

On the other way round, if you have 13.9502 RPM at the engine, with the first gear, you must have 1 RPM at the wheel. So you can not have, without a clutch (or the gear in neutral), an immobile car and a running engine at the same time.

You can find easily on the WEB the gear coefficients for your own car.

As you maybe begin to understand I am starting from the wheel side and not from the engine side.

By the way if you know the wheel angular speed in radiant/second it is easy to get the Wheel RPM:

(2) RPMw = 60 * [wheel angular speed] / (2 * 3.14159)

Having the car speed in Kmh and knowing the wheel radius it is easy to get (assuming the tire does not slide) the wheel RPM as

(3) RPMw = [Car Speed] *1000/(2 * 3.14159 * Radius* 60)

Where:

• 1000 comes from meters to kilometres
• 60 comes from minutes in an hour
• 2 * 3.14159 * Radius comes from the wheel length given its radius(circumference)
• [Car Speed] is in kmh

Inserting the (3) in (1) you get:

(4) RPMe=[Car Speed] *1000* Current_Gear_Ratio * Gear_Ratio_Final/(2 * 3.14159 * Radius* 60)

It is worth to note that the RPM at the wheel in (3) does not depend from the current selected gear! You may say it is the car speed which, in turn, depends on the current gear. But this is the point I want to stress in this note. Let us start from the wheel situation and let us deduct the other parameters.

If you want you can go around with your car taking note of the gear, of the engine RPM, of the speed. With this data you can estimate the wheel radius (of course there are other easier methods to measure the wheel radius!). Using (3):

(5) Radius=[Car speed] *1000/(2 * 3.14159 * 60)/ RPMw

Using (1) you get the following:

(6) Radius=(Current_Gear_Ratio * Gear_Ratio_Final)* [Car speed] *1000/(2# * 3.14159 * 60)/ RPMe

For example I did register, using the AGILA’s first gear, the following:

Ok, I came out of the car and I did measure the tire radius directly. I obtained 26.6 cm. So the above estimate is about 10% wrong. No bad if you think the tachometer has a 5% overestimate.

Now let us came to the torque. From the Wikipedia “Torque is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist...In more basic terms, torque measures how hard something is rotated”.

Units for the Torque are Newton Meter (Nm)

Power (Watt) is torque times the rotation speed:

(7) Power= Torque* [rotation speed]

Where the [rotation speed] is given in radians/seconds

The Torque itself is a function of the engine RPM. It is difficult to find the Torque/RPM graph for a normal car (or at least it was so for me). What you find easily is the Torque maximum and the correspondent RPM. For example I found 110 Nm at 4000 RPM for the Opel Agila. It is also easy to find the max power and the correspondent RPM. 59 Kw at 5600 RPM for the Agila.

It is important to understand that the torque must go to zero at the maximum RPM (I assumed 7700 RPM for the Opel Agila) as well at zero RPM. The maximum speed I can go, with the first gear, is about 55 kmh. This means (from eq.4)

RPMe=55*1000* 3.73 * 3.74/(2 * 3.14159 *0.265* 60)=7680

I also found in a magazine, for the Agila, this value: 4300 RPM at 130 kmh (5th gear).

RPMe=130*1000* 0.89 * 3.74/(2 * 3.14159 *0.265* 60)=4331 confirmed!

To simulate the car I did inferred the Agila’s performance and I coded (using a look up table) the Torque as function of the engine RPM as in picture 1. In picture 1 you also find the power as taken from (7).

Now the central part of this note:

1) suppose you have a wheel speed (directly radians/second or from kmh)

2) from 1 you can compute the speed of the engine (RPM) (via the gear ratios)

3) from RPM you have the TorqueE= Torque at the Engine (via a function as in picture1)

4) from the TorqueE to the TorqueW, the torque at the Wheel (via gear ratios)

5) Optional compute the force at the wheel=TorqueW/Radius

The following table give an idea of the computation for two different speed and two different gear ratios for the Agila.

A: Speed in kmh

B: Gear

C: Gear Ratios (including the differential) as function of B, example B=1 C=3.73×3.74=13.95

D=A*1000/(2 * 3.14159 * 0.28* 60) (wheel speed in RPM)

E=C*D (RPM at the engine using gear ratios)

F=Torque(E) in Nm

G=F*E/60*2*3.14159/1000 (power at the engine in Kw) this is not necessary, just to show=I

H=F*C Torque at the wheel in Nm

I= H*D/60*2*3.14159/1000 (power at the wheel in Kw) just to prove it is=G

J=H/0.265 = Force at the wheel (N)

It is worth to note

G=I or the power at the engine is the same of the power at the wheel (friction is neglected)

At 60 kmh, with the first gear the torque is zero (maybe should be negative?)

Finally I would like to show the graph for the Torque at the Wheel, for different gears, as function of the wheel speed

If we call Z=1000/(2 * 3.14159 * 0.28* 60)

Torque at the wheel = [Gear ratio]*Torque([Gear Ratio]*[speed in kmh]*Z) )

This kind of pictures are never published because are useless to assess the car performance as they are derived from the Torque-Engine RPM function and from the gear ratios. One the other way round I think this picture is useful to show why we have to change the gear while driving and why, even if the first gear has more torque, you must goes with higher gears if you want more speed…