Understanding the Mechanics of Shifting

I've had my license for a few years now but I don't have a lot of experience. I've gotten a lot better but when I first started out I was probably the worst shifter ever. I don't have a mechanical background and the Eaton Fuller 18 speed was the first thing (out of anything) that I learned on.

I'm currently driving for an oilfield company. Up until now I really didn't care much about the Mechanics going on under the hood. As long as I could shift into the next gear with as little grinding as possible I was happy

Recently, I've become more interested in what's actually happening under the hood and possibly becoming a better shifter and ultimately driver. But I'm not sure if my current understanding is correct, hence this post.

I want to preface the following by saying that I'm looking for just the basic intuition and concepts here. I realize that I'm leaving a mountain of info out. If the exclusion of some piece of information is resulting in my misunderstanding please let me know.

Okay, so I've never really understood the mechanical reason for shifting and needing to shift certain gears at certain engine rpms and road speeds.

This is how I currently understand the process: you've got the input shaft which is connected to and spun by the engine (skipping over the clutch for now). On the end of the input shaft is a gear. This gear is in mesh with a gear on the countershaft of the transmission. The countershaft also has all of the other gears. These gears are in mesh with the gears on the output shaft.

The key point is that the gears are all different sizes. The 1st gear on the countershaft is small in diameter compared to the gear it is in mesh with on the ouptut shaft. So at a given engine rpm the smaller on the countershaft will have more revolutions than the larger gear it is driving. I should have mentioned that the gears on the output shaft are spinning freely of the shaft itself. There is another gear, the hub, which is connected to the output shaft. When shifting into the gear (let's say 1st) a sleeve connects the hub gear with the big first gear on the output shaft (again oversimplifying here). The big first gear is now driving the output shaft and it in turn is being driven by the smaller gear on the countershaft which is being driven by the engine.

Given that the output shaft big gear is being driven by the smaller countershaft gear we have lots of torque and this is what we need in the low gears to get the truck moving. The larger the radius of the gear the more torque we will have. But a larger radius means that the gear is not completing as many revolutions as a smaller gear would in the same amount of time. Therefore the truck has lots of torque but not a lot of speed.

Okay this comes to sort of the crux of my question. The truck is moving, but slowly because of the low rotational speed of the big gear. We want to go faster, but the big gear can only spin so fast. And so when we give the truck more gas increasing the engine rpms the truck does not speed up in direct proportion to the engine rpms because the smaller gear on the countershaft can only spin the big output shaft gear so fast. This may sound a little weird but there is diminishing returns on the speed of the output shaft given increasing speed (rpms) of the countershaft.

We need the truck to go faster. Given we already have forward momentum, we don't need as much torque as when the truck first started off. We need to trade speed for torque. We accomplish this by shifting to a larger gear on the countershaft that drives a smaller gear on the output shaft. A smaller gear on the output shaft means less torque but more rpms which results in a higher road speed.

If I'm double clutching is this why you increase the rpms let's say to 1600 (just as an example), shift into neutral, and then shift into the next highest gear at say 1200 rpms? We need to sync up the speeds of the lower larger gear and the higher smaller gear. So we get the larger gear spinning at the top of the rpms range and shift into the smaller gear at the bottom of the rpms range because this range basically compensates for different rotational speeds of the two gears?

And conversely when you are downshifting. Before downshifting, the gear you are in is in some way proportional to the road speed. The next lowest gear obviously has a bigger radius and thus is not turning as fast. So you brake a bit and let your rpms drop to the low range (again say 1200). This is going to slow the rotational speed of the gear you are in down. You shift into neutral. But the smaller gear that you just shifted out of is probably still rotating faster than the larger gear you are trying to shift into. To compensate for this we press the gas to increase the engine rpms. This spins the larger gear faster which helps to sync up with the faster rpms and road speed that the output shaft was just previously in.

I realize this question is extremely long but I'm hoping someone can see what I'm trying to ask here. Basically, where we set our engine rpms when shifting (up or down) is a result of trying to match up the rotational speed of the next gear with that of the output shaft. If I'm way off here can someone explain to me where I've gone wrong?

Thanks