Gear Ratio Doesn't Matter

[Paul Copioli]

I just thought I'd stir up the pot a bit to start the school year. when dealing with drive trains, gear ratio does not matter.

Over the past 2 years, a lot of questions regarding drive trains and gear ratio have been asked. Very rarely do I see mention to the wheel diameter wrt gear ratio. Gear ratio doe NOT matter.

Here's the proof:

Problem: Robot with mass, m; climbing up an incline, theta; on Earth (I threw that in for you NASA guys).

Assumptions:
1. 2 wheel drive (only to simplify the example)
2. Bearing friction and gear train friction negligible.
3. Coefficient of friction high enough to sustain wheel force.

Symbols:

g - Earth's acceleration due to gravity (9.807 m/s^2)
Fat - Total force at wheels (N)
Fa - Force at each wheel (N)
Rw - Wheel radius (m)
Tw - Torque at wheel output (N-m)
GR - Gear ratio = input speed/output speed (usually >1)
Tm - Motor torque (N-m)
Wm - Rotational motor speed (rad/s)
Vl - Linear robot speed (m/s)

1. Total force at the wheels

Fat > m*g*sin(theta)

2. Force at each wheel

Fa > {m*g*sin(theta)}/2

3. Torque at wheel output

Tw = Fa*Rw

4. Torque at motor (or drill motor housing output):

Tm = Fa*Rw/GR --> M*g*sin(theta)*Rw/(2*GR)

Motor torque depends on the wheel/gear ratio combination.

5. Rotational Motor Speed/ Linear robot speed

Wm = Vl*GR/Rw --> Vl = Wm*Rw/GR

Motor speed is inversely proportional to Rw/GR

As we can see from the two equations, a balance between speed and push/pull force must be met. In order to increase pulling force, we need to reduce the torque seen at the motor by reducing Rw/GR. In order to increase linear speed based on a limited motor speed, we need to increase Rw/GR.

For the reasons stated above, many teams are going to a gear shifting design (led by 45 and others). This enables you to optimize for both situations a little better. My stand on the shift v. no-shift is to pick the single best Rw/GR ratio and not shift, but depending on future games my position may change at any time.

I hope this clears up the significance (or insignificance) of gear ratio when dealing with a drive train. So for those of you with a 2:1 gear ratio using the drill motors in low speed, don't think you did anything wrong; but if your wheel diameter is 20 inches with a 2:1 ratio, then you might be in trouble.

I hope this sparks some interesting debate.

-Paul