A lot of mechanisms are characterized by rack-and-pinion-type motion. A rack-and-pinion-type mechanism is any mechanism where there’s a coupling between one body’s linear motion (the rack) and another body’s rotational motion (the pinion). If you rotate the pinion a certain number of degrees, the rack will move a corresponding number of centimeters. Examples of rack-and-pinion-type mechanisms:
- Actual racks and pinions
- A winch pulling a rope (like in Andymark’s climber in a box)
- A linear game-piece transport based on cords/belts
- A robot driving in a straight line, without slip (probably also holds for driving on a curve but I haven’t checked)
Rack-and-pinion-type mechanisms are characterized by the fact that there’s a linear relationship between angular and linear displacement, related by the pinion’s radius (same holds for velocity and acceleration):
Linear displacement of rack [certain length units] = pinion radius [same length units] * angular displacement of pinion [radians]
We are usually interested in the behavior of the rack and not the pinion (what’s important is that the robot climbs up, a winch pulling on a rope is just a means to an end). Because of the rack-and-pinion relation, when the pinion is driven by a gearbox and a motor, something interesting happens. All important inputs and outputs (linear velocity, force capacity, motor current draw, motor rotational velocity) are only dependent on the ratio between the pinion radius and the gear ratio, instead of being dependent on each of them independently as we might expect in other mechanisms.
This happens because, if you develop the equations, you will only find that the pinion radius (R) and the gear ratio (mG) together, either as R/mG or as mG/R. I call this the Z-ratio, for no particular reason, i.e. Z = R/mG.
Simply put, if we double the pinion radius and double the gear ratio, we will see no difference in the behavior of our system! This means that once we choose a good Z-ratio (what ratio is “good” depends on the mechanism and on our needs), we can choose as large or as small a pinion radius as we want, and we’ll immediately find the right gear ratio, and of course the opposite is also true.
We discovered this this season when designing our climb, and the discovery allowed us to use a small winch and a 20:1 gear ratio instead of what we did in the past, which was using a large winch and a 50:1 gear ratio. Saved us a lot of space, weight and effort.
Note, according to the physics, the radius or the gear ratio don’t matter individually, but this is where theory meets reality and physics meets engineering. There are important issues to consider when choosing the pinion radius.
A small radius may be good because:
- It allows for a smaller gear ratio, which might mean less stages. Less stages mean more efficiency.
- The torque on the output shaft is smaller for the same Z-ratio (even though the motor torque is identical), meaning less chance of failure in the gears.
- Smaller mechanism (less space, less weight).
A large radius may be good because:
- It means a larger gear ratio, which means more internal friction. If there’s enough internal friction, it might be enough to hold the rack in place without power, which might save on current and/or spare a locking mechanism.
- In winch-and-cable mechanisms, with enough spooling, the cable might start spooling on top of itself, which a. increases the effective radius and changes the Z-ratio (usually for the worse) and b. risks entanglement. The larger the radius, the less this is a risk, since more cable length will be spooled with every turn of the winch.
Of course, the advantages of small radii are the disadvantages of large radii and vice versa.
