Gear Train
From BEAM Robotics Wiki
A gear train is a set of two or more gears arranged to transfer rotational torque from one element of a mechanical system to another. The gears are typically arranged so as to provide a speed to torque conversion (commonly known as "gear reduction" or "speed reduction") from a higher speed motor to a slower but more forceful output. However they can and are sometimes designed to for the opposite purpose. That is to converter a strong but slow torque into a higher speed rotation. Such is the case of the gear train of a ten speed bicycle.
Typically a gear train consists of:
- Driving gear - attached to the input shaft
- Driven gears / Motor gear - attached to the output shaft
- Idler gears - interposed between the driving and driven gear.
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[edit] Simple Gear Train
Gear reduction is probably the most common use of a gear train. As an example in figure 1 is a simple gear train. Let's say the first gear (#1), which is typically the smallest gear (also known as the pinion) in the gear train has 13 teeth, while a second, larger gear (#2, known as the idler gear) has 26 teeth. The gear ratio is therefore 13/26 or 1/2 (typically written as 1:2).
Consequently, for every one revolution of the pinion, the second gear has made 1/2 turn, or 0.5 revolutions. In practical terms, this means that the second gear turns more slowly.
Next suppose a third even larger gear (#3) has 39 teeth and is also meshed with the second gear. The gear ratio between the second and third gear is thus 26/39 = 2/3, giving us a ratio of 2:3, meaning that for every revolution of the second gear the third gear has has made 2/3 of a turn or 0.666... revolutions,
To determine the final ratio we need to multiply the first ratio (1/2) by the second ratio (2/3). Thus 1/2 x 2/3 = 1/3, giving us a total reduction of 1:3.
Therefore the input shaft must make three complete revolutions in order to make the output shaft perform one complete revolution.
Those with a sharp eye for math may have noticed that if we had taken the 13 teeth on the first gear, and divided that number by the 39 teeth on the third gear, we would have ended up with the same result. That is 13/39 = 1/3 giving us a ratio of 1:3.
In point of fact, in any gear train made up of gears chained together in series, no matter how many intermediate (idler) gears there are between the first gear and the last gear, the gear ratio depends only on the number of teeth on the first and last gears.
Now since, theoretically, any gear ratio that might be needed can be achieved using only two gears, it might well be asked, "why would a gear train include more than two gears?"
For one reason, if a gear train consists of an even number of gears (and two is an even number) the driven gear will turn in the opposite direction of the driving gear. But if a gear train has an odd number of gears, then the driving and driven gears will turn in the same direction. So if it is important that the driving and the driven gears turn in the same direction, then a third gear (in this case called a "reverse idler gear") would be added to eliminate the reversing effect.
A second reason would be to transmit rotation between distant shafts in situations where it would not be practical to simply make the distant gears large enough to bring them together. Larger gears occupy more space, and their mass and rotational inertia increase dramatically.
It should be noted though, that adding gears increases both the friction and the play or slop in the gear train. This being the case, if the distance between the shafts is too great, then using "pulleys and a belt" or (sprockets and a chain" instead of Gears might be a better way to transmit torque over the distance.
[edit] Compound Gear Train
Slightly more complex than the gear train above, this one consists of two or more gear pairs. At least one shaft of this gear train will have have two gears that are keyed together so that they share a common motion. These two gears become a single compound gear.
For the example in figure 2 let's say that as with the prior example the first gear (#1) again has 13 teeth, and the second gear (2) has 26 teeth. This being the case the ratio of the first gear pair remains 13/26 or 1:2. Let's also assume the same values for the second gear pair. Since gear 2a and 2b share a common motion we have to multiply the ratio of the first pair (1 & 2a) times the ratio of the second pair (2b & 3) to find the overall ratio of this compound gear train. In this case 1/2 * 1/2 or 1/4. Therefore the ratio of this gear train is 1:4, which means that the input shaft must turn four complete revolutions in order to make the output shaft complete one revolution.
When layered out properly, this arrangement can often make for a somewhat more compact gearbox. In particular, if a very large gear ratio is required, this type of gear train can separate this ratio into more manageable factors. For example, an analog watch display requires a gear ratio of 1:3600 between the second hand and the hour hand, so it makes sense to split this up into two factors of 1:60, e.g., as in the previous example, in order to avoid having a gear with over 10,000 teeth (apart from letting the intermediate gear drive the minute hand).
[edit] Reverted gear train
When the gears in a compound gear train are selected so as the centers of the first gear pair (1 & 2a) are the same distance apart as the centers of the second gear pair (2b & 3); then the shafts of the driving gear and the driven gear can be located along the same center line (See Figure 3).
This arrangement is called a "revered Gear train" and makes for the most compact type of gearbox.
There is another configuration made possible by using a reverted gear train. Since the input and output shafts share a common center line it, is perfectly feasible to build a gearbox with the shaft of the driving gear passing through the hollow center of the shaft of the driven gear (as shown in figure 4). This arrangement is commonly used in mechanical clocks and timers.
[edit] Epicyclic Gear Train
Epicyclic gearing (also called "planetary gearing") is the type of gear train found in the most common rechargeable screwdrivers. It is a four element, three step gear system that consists of...
(#1) A central or Sun gear
(#2) One or more Planet gears. mounted on...
(#3) A Planet gear carrier
(#4) An outer ring gear, called an annulus, which has its teeth facing inward.
The sun gear, meshes with one or more planet gears, which are mounted to, and usually turn freely on the planet gear carrier. In most cases, the planet gear carrier rotates with the planet gears as they revolve around the sun gear. Finally, the planet gears also mesh with, and move along the inner teeth of the inside of the annulus.
As with the previously described gear trains, the ratio of input rotation to output rotation is dependent (in part) on the number of teeth on each of the gears. But there is more to it than that, and as might be expected, calculating the ratio is not as intuitive as it is with the other gear trains. Even if you don't have to deal with the possibility of the planet gears being compound gears, there is still the fact that rotation can be applied too, and taken off of, a epicyclic gear train in any one of several different ways. Each of which affects the ratio differently.
Generally in epicyclic gearing systems, one of the three stages is held stationary. One of the two remaining stages would then be used as the input, providing power to the system while the remaining stage becomes the output, transmitting power from the system to the device or mechanism to which motion is to be imparted.
If we use a rechargeable screwdriver as an example, then typically the annulus is the element of the gear train that is fixed in place so that it cannot move. The sun gear usually serves as the input, while the planet gears basically act as idlers. And finally, the planet gear carrier serve as the output. But remember, this is only one of the several possible configurations that can be used, Other application may well use a similar epicyclic gear train, but a in quite different manner.
In the following tables; formulas are provided by which the relative speed of each element, per one revolution of the driving element, can be determined for a given configuration. The letters in each formulas denote the number of teeth on the member designated by that letter.
The first table and diagram are for Simple Epicyclic Gearing and the second table and diagram are for Compound Epicyclic Gearing
NOTE: You can > CLICK HERE to download these diagrams and tables as a PDF file <.
[edit] Calculating the Ratio of a Epicyclic Gear Train Using Simple Planet Gears
Turning again to the rechargeable screwdriver as the example, we can use the formula ( S / ( A + S ) ) (as presented for the planet carrier in line 5 of Table 1) to determine the ratio of its gear train (See:Figure 6).
Using this formula, we divide the number of teeth on the sun gear ( 8 ), by the number of teeth on the annulus ( 58 ) added to the number of teeth on the sun gear like this:
( 8 / ( 58 + 8 ) ), giving us a result of 0.1212, which when calculated, {(1 / 0.1212 = 8.2508}, gives us a ratio of just a hair over 1 : 8.25. This means that to get one complete rotaion on the output side, the input side needs to make (roughly) eight and one quarter rotations.
Estimating 25,000 RPMs from the motor, we multiply that number by the result of the above calculation,
25,000 x 0.1212, giving us a result of 3030 RPMs for the planet carrier, which serves as the output.
Buy that can't be right, you say. It is way to fast. and you are right. However, we are not finished yet. You see, most rechargeable screwdriver use a three stage gear train. That is, there are three identical epicyclic gear configurations stacked one on top of the other. What this means is that we have to multiply our first result (3025 RPMs) by our calculated ratio ( 0.1212 ). So ( 3030 x 0.1212 ) gives us a secondary result of 367.236 RPMs. We then multiply that result by our calculated ratio ( 367.236 x 0.1212 ) giving us 44.5090, or just a smidge over 44.5 RPMs as our final output, which is just about right.
[edit] Calculating the Ratio of a Epicyclic Gear Train Using Compound Planet Gears
The gear ratio of an epicyclic gear train can be increased by using compound planet gears. This has the same effect as increasing the inside diameter of the annulus, along with the number of teeth on the inside of the annulus. And it does this without having to increase the outside diameter of the annulus. If for example, the compound planets have twice as many teeth on their larger diameter, than they do on their smaller diameter; then the effect will be like doubling the size of, and number of teeth on the inside of the annulus, leaving the outer diameter of the annulus unchanged.
Let’s say for the sake of this example, that we have a epicyclic gear train that is similar to the one used in the rechargeable screwdriver we used as the previous example. The sun gear( S ) again has 8 teeth and the annulus ( A ) has 58 teeth. However, instead of using simple planet gears with 24 teeth, this gearbox uses compound planet gears having 32 teeth on Ps (the larger diameter half which meshes with the sun gear), and 16 teeth on Pa (the smaller diameter half which meshes with the annulus).
Using the formula "PsA / (PaS + PsA)" (as presented for the planet carrier in line 5 of Table 2) we calculate the gear ratio by placing the proper numeric values in place of the letters that represent the appropriate gear. So S = 8, Ps = 32. This gives us the following(...).
(16 x 8) / ((16 x 8) + (32 x 58) = 128 / (128 + 1856) = 128 / 1984 = 0.0645, which when caluculated (1 / 0.0645 = 15.5039) gives us a ratio of just over 15.5 : 1. This means that for the output shaft tomake one ful rotation, the input has to make fifteen and a half rotations.
Now, assuming that as with the earlier example, the motor rotates at 25000 RPMs, the next step is to multiply that number by the result of the above calculation, giving us...
25000 x 0.0645 = 1612.5 RPMs at the first stage output.
If as with the rechargeable screwdriver, the gearbox contains three identical epicyclic gear trains stacked one on top of the other, then we have to repeat our multiplication two more times...
1612.5 x 0.0645 = 104.0063 RPMs
104.0063 x 0.0645 = 6.7084 RPMs
So we find that we have 6.7084 RPMs as the rotational rate at the output of our gearbox. This is just a little over six and a half times slower than the 44.5 RPMs produced by the epicyclic gearing used in the rechargeable screwdriver. To achieve a similar result using simple planet gears, the annulus would have to be doubled in size.
[edit] Final Notes About the Tables and Diagrams
In the above diagrams and tables, only one planet gear is shown or considered. Since any additional panet gears only parallel the effect of a single planet gear the additional planet gears can be ignored when calculating the gear ratio.
This being the case, one might be tempted to ask the question “If only one single planet gear is need to achieve the desired gear ratio, then what is the purpose or benefit to using more than one of them?”
The answer is, to distribute the mechanical stress across the various additional areas where the planet gears mesh with the sun gear and with the annulus. The more planet gears used in an epicyclical gear train, the less mechanical stress] is experienced by the gear teeth at each point of engagement, hence the greater the level of overall torque the, gear train can handle.
However, when deciding how many planet gears to incorporate into an epicyclical gear train, a designer must also consider other effects the additional gears will have on the overall gear train. These effects include, added weight, increased operating friction, and a greater level of inertia to be overcome at start–up and when stopping. Whether or not any of these effects represent any significant problems at the scale of a typical BEAM robot is open to discussion.
Obviously, this has not been an in depth treatise on epicyclic gear trains. It isn’t intended to be. It’s purpose is only to introduce the basic concepts, and wet your appetite for more. And there is so much more for the interested reader to learn. To aid in this, several external references are listed below. Be sure to check them out.
[edit] Internal References
[edit] External References
- Understanding Gear Reduction By Brett "Buzz" Dawson, Team DaVinci Robotics
[edit] Epicyclic Gear Train References
- Wikipedia Article: Epicyclic gearing
- Epicyclic Gears, Some Theoretical Considerations (PDF)
- Advanced Gear Analysis, Epicyclic Gearing
- Planetary Gears Tutorial
- Planetary gear ratio calculations
- Epicyclic gearing as a transmission
- Epicyclic gearing in an automatic transmission
- Planetary Gear System Video
[edit] Video Tutorials