## How to match the gears in a gearbox?

Hello Guys, today I’ll show you how to setup the gearbox relations for each gear.

Let’s start establishing some hypothesis, the engine’s curve, top speed and climb ability.

Let’s suppose that:

- The car uses 205/60 15 tires, it will give us 0.304m as dynamic radius;
- 2.11m of frontal area and a drag coefficient of 0.31;
- The car’s weight is 1250 kg (something like a Honda Civic);
- Car’s max torque is 190Nm @4600rpm;
- The car needs to climb a 60% ramp with 320 kg more than his normal weight (load);

We have an important relation between torque (in Nm) and power (in KW): (Where T is engine’s torque and N is engine rpm)

Before start to set car’s gears we need to know about the forces involved in displacement process, I will talk a little bit about each one of them:

1) Rolling resistance force: Like the name says, it’s a reaction force between tire and ground. It’s a product between a rolling resistance coefficient (that depends on tire and ground) and the normal force in wheel. It’s a simple equation:

2) Aerodynamic resistance: In lower speeds it’s totally inexpressive, but is fundamental to determine car’s top speed. It’s the product between air’s density, car’s frontal area, drag coefficient and the square of car’s speed.

3) Inertia: In higher speeds it’s inexpressive too (for this analysis), but so significant when the car needs to start a movement. It’s the product between rotational inertia coefficient, vehicle’s mass and acceleration.

4) Slope: the most known force, product between mass, gravity and angle’s sine (sen).

Now we know this forces and the relation between power and torque, let’s check the engine’s torque (red) and power (blue) curve:

Now we can start to calculate the gears, beginning with the 5^{th}, I would like my car can reach at least 210km/h and I’ll suppose the car get this speed in engine’s max power point, we call it optimum project. We can setup this speed to occur before or after this point, it will give us more top speed or more acceleration.

The equation is so simple, we sum the aerodynamic resistance and rolling resistance and call it “resistant torque”, this value must to be equal engine’s torque multiplied by gear and differential relation:

Resistant Torque = Engine’s Torque (@6000rpm) x (5^{th} gear x differential)

Let’s choose the 5^{th} gear relation 0.7:1, so the unique “variable” we don’t know is the differential, if we see the parenthesis like just one thing we will find a value for relation and dividing it by 0.7 we have the differential. For this case it’s 4.58:1.

The top speed formula (in m/s) is: (raioDinamico = Dynamic radius, N = rpm),and for this case it’s 215km/h. Next problem now it’s to set 1^{st} gear, defined by our ramp maximum inclination.

Our resistant torque now is rolling resistance plus inertia and slope, in engine’s maximum torque (4650rpm). Solving this relation we have 2.82:1 for first gear.

Now we have first and last gear defined, let’s find the 3 remaining gears. This is the easiest part, there are basically two ways to get it, first one it geometrical method, it consists to setup an equal gap between the gears and it’s used to set up trucks gearboxes relations because of the high number of gears they have. For passenger cars we use progressive method, it gives more acceleration in first gears and more speed in last ones. In progressive method we setup the gears like this: n(i)= [n(i+1)*(2*n(i-1)-n(i))]/n(i-1)

My relations for this gearbox were:

Now we have all gearbox defined and it’s possible to look at some interesting graphics, first of all we have a comparison between resistant torque in each gear and engine’s torque curve (in red). Notice that last curve touches engine’s torque curve at 6000rpm, setting our maximum speed, exactly like we wanted to.

We can see the speed map too, that shows the relation between vehicle’s speed in actual gear and the engine’s RPM:

We could also calculate trative force and show vehicle performance map, but it’s for another post. That’s all for today, if any of you want to see the calculation I did, I can provide you a maple file with it, or the same thing in pdf for who don’t have this software.

Thanks for reading,

Rafael Basilio

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