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

[Joe Johnson]

silly title, but it hooked me.

Of course you are right. Ratio is always in relationship to the size of the moment arm at the other end. But for a given size wheel (say 6 inches) the ratio is VERY important.

Joe J.

P.S. Dr. Joe's Rule of Thumb: The approximate ratio most robots should have between the armature of the drill motor and the wheel is 10 times the diameter of the wheel (measured in inches). If you are climbing ramps, pushing heavy objects, racing to the other end of a gym, or in some other special case, things may change, but for just flat floor tooling about this formula is a pretty good place to start. JJ

[Paul Copioli]

Dr. Joe,

I thought for sure someone (especially you) would get hooked by that title. It is such an absurd title. I'm glad you responded.

-Paul

P.S. - I'm still working on that White Paper I promised you about reflected inertia and spring balancers.

One last thing for anyone who is reading this:

If you are in the Detroit are and want to see a lot of industrial robots being built, tested, and running; drop me a line and I can arrange a personalized tour of FANUC Robotics.