Reading and rereading definitions over and over, I think I can make some clearer statements now on what things are an where the confusion comes from, and I know I made some incorrect (or at least unconventional) statements not so long ago. I think I should also make some statements about some better and worse terminology.
First, I’ll highlight the reasons that this gets really confusing:
From the get-go, you could reasonably define things any way you want. It’s ultimately arbitrary, as long as you apply your rules consistently. Ideally, you define things in a consistent way for clarity, but that has not actually been done.
The origins of all this are in simple machines and mechanical advantage, which is defined as output force:input force. However, for some reason along the way, people have decided to sometimes name things as output:input and sometimes as input:output. (Or driving:driven and driven:driving.) Switching orders in definitions (not in calculations, as those are dictated by the math) causes confusion. So much of the blame for the confusion should really fall on those who were inconsistent in the naming process.
As you’ll see in a comment below, there is absolutely no consistency with gear ratios. There is significant consistency within subsets of speech, though. Of course, there is more consistency in the non-technical subset than in the technical subset.
A lot more confusion shows up when people just write driving:driven or driven:driving instead of that they’re actually talking about. Those ratios are nonsensical on their own because there is no statement about what’s actually going into the ratios. Notice about how I specified output force:input force for mechanical advantage.
Now, where does this come from. With machines we have mechanical advantage being output force:input force. However, that is really tricky to work with outside of making measurements. The reason is that mechanical advantage also includes things like friction, the weights of parts of the machine themselves, etc. But we can get a good idea of what’s going on with ideal mechanical advantage, the mechanical advantage if friction, the machine’s weights, etc. can be ignored. Then, from conservation of energy, we get
IMA = ideal output force:input force = input distance:output distance
So we can see that, as IMA only depends on distances, it can be used quite readily with different set-ups to identify what is happening in the design. Sure, some force is lost, but as long as that is understood, the design can be communicated fairly well. Also, it means we can use it to examine the motion of parts of the machine since it’s based distances. Consider a machine that ideally multiplies the force by 4 (IMA), but due to friction it’s actually 3.5 (MA). That means the machine multiplies the distance or speed by 0.25; the output moves only a quarter as much as the input.
However, someone in what I consider questionable wisdom, decided to define speed (or velocity or movement) ratio as input distance:output distance. That’s the same as IMA. Why do I consider this questionable wisdom? Mechanical advantage is how a machine affects force, and theoretically (and for consistency) speed ratio is how a machine affects speed. But speed ratio isn’t how the machine affects the speed, but rather it’s the multiplicative inverse (reciprocal). With gears, this speed ratio is known as the gear ratio. Gear ratio could have been set to IMA with the speed ratio still being chosen the other way. But, alas, no. And now we have yet more confusion: a low-gear’s ratio is a high gear ratio and a high-gear’s ratio is a low gear ratio. I expect this is why I can find these ratios listed the other way, and not just on the VEX site.
But there is even more bad terminology. “Torque ratio” is labeled as the same thing. This sounds good and in some ways better, as this ratio is the idea torque ratio. But, due to friction and related losses, the actual ratio of the torques is less than this. It would have been less confusing if the torque ratio were more carefully defined like the force ratio (MA) instead of the ideal it never is (IMA).
So what we the majority in academia say is
idea mechanical advantage = speed ratio = gear ratio (for gears) = torque ratio (for gears and similar) = input distance:output distance = output (follower, driven) teeth:input (driver) teeth (for gears)
Before ending, I think there is one other important point to address. I’ve seen a lot of comments about mechanical advantage of a rack and pinion being undefined or nonsensical. That really isn’t true. The problem only arises when people decide to create a force:torque ratio. However, we know that if we multiply the pinion’s diameter by some factor, the ideal mechanical advantage gets divided by the same factor. What you really want to do is look at the torque from the motor and divide by its shaft’s diameter. We can then compare the force there to the force along the rack, and we can compare the diameter of the shaft to the diameter of the pinion to get the ideal mechanical advantage.