Understanding Gear Reduction
By Brett "Buzz" Dawson
     Team DaVinci Robotics

     First off, let me explain that gear reduction in the context of this help section refers to speed reduction in general whether it be by traditional gear, chain and sprocket, or belts. The goal of this section is to give anyone a basic understanding of what gear reduction is and how it can be used to help give an idea on how to implement it in a robot. Because there are different areas in a robot that could benefit from gear reduction we will focus on the most important one, the drive train. And, we will talk only about DC electric motors but the fundamental can be applied to other motors as well.

     The reason that we need to know about gear reduction is because the output speed of a motor is usually too fast for normal use. Most DC motors at normal operating voltages spin at well over 1,000 rpm (revolutions per minute) and some even as high at 50,000 rpm for brushless DC motors. If we had a motor than spun at say, 3,000 rpm, and we attached a 6 inch wheel to it then the wheel would theoretically be able to move the bot at almost 54 miles per hour! That is way too fast to control in an arena due to other considerations that wouldn't happen but we'll get into that later. So we need to reduce the rate at which the wheel spins so that we get a robot that we can at least control. Hint, the quick way of determining the speed of a wheel is to multiply the diameter (in inches) of the wheel by the rpm and divide the result by 336 (or for a really close figure use 336.13523981).

What is it and why does it work?

     Quite simply, gear reduction involves using gears/sprockets/pulleys of two different sizes to work together. Because they are of differing sizes they will have different circumferences (distance around the outer edge) and we can use this to our advantage. Let's take a look at what this circumference thing really means. To the left is a representation of a 4 inch diameter wheel. Click on the wheel to watch it as it moves through one complete revolution. You will see that the distance covered in one revolution is equal to the circumference of the wheel.

     Now, let's take a look at a wheel that is twice the size. Click on the 8 inch wheel to watch it as it goes through one complete revolution. What you will notice is that notice is that not only does the wheel have twice the diameter but it travels twice the distance in one revolution. Therefore the circumference is twice that of the 4 inch wheel. So, if the 4 inch wheel were to cover the same distance as the 8 inch wheel then it has to complete 2 full revolutions.

     So, is there a way to figure out the circumference of a wheel? You bet! Remember the magical number Pi? Pi is roughly equivalent to 3.14. So, to determine the circumference of the wheel we multiply 2 times Pi times the radius of the wheel (radius = half the wheel diameter). So, for our four inch wheel we have 2 x 3.14 x 2 = 12.5663706144... inches. (remember to keep your units of measure the same throughout your calculations).

      Okay, now that we know that they have different circumferences how do we apply it? Well, this time instead of rolling each along the ground we are going to roll them against each other. What you will notice is that the smaller wheel has to spin twice so that the arrows line back up whereas the big wheel only has to spin once. In other words the input has to spin 2 times to get the output to spin 1 time. Thus we get a 2 to 1 ratio more commonly written as 2:1. A configuration like this is referred to as a single stage reduction because there is only a single interaction between wheels.

      Now that we have the basic understanding, lets take it one step further, or one stage further as the case may be. In the example to the right we have added a second wheel/gear in between the two that we already had. It actually is two gears in one. The large gear interacts with the first input gear. It is physically attached to a smaller third gear so that the smaller one revolves at the same rate as the larger one. This smaller gear is then in turn interacting with the fourth output gear. Two keep things simple the second gear is twice the diameter of the first gear. The third gear is the same diameter as the first and the fourth is the same size as the second so that we have a 2:1 reduction between the first and second and a 2:1 between the third and fourth.

    So, with all that laid out we can begin. As we saw in the single stage reduction the small gear had to spin twice to get the larger gear to spin once. Well, the same holds true here but since the second gear is attached to the third gear the third gear also only spun once which means that the fourth gear only got turned half way. To spin the last gear the rest of the way the first gear needs to spin two more times for a total of four times. Therefore, when it is all said and done the input gear had to spin 4 times to get the output gear to spin 1 time, or 4 to 1, or 4:1. This example is known as a two stage reduction because there are two places where the gears interact (mesh). Remember that the second and third gear are the same piece of material and do not move independently of each other.

     Note: Reduction annotation is usually written with how many input rotations it takes to get one output rotation. Ex. 3.5 (input) : 1 (output)

    There are also multistaged reductions which involve many gears which can reductions of over 1000:1! It just depends upon the required output speed and torque (we'll get to that later). So, how can you determine what the gearing is of a set of gears/sprockets/pulleys? Well, gears, timing belts, and sprockets are easy because they are labeled according to how many teeth they have. If you have an input gear with 10 teeth on it and an output gear with 40 teeth then the 10 tooth gear will have to rotate 4 times (40/10) to get the 40 tooth gear to spin once. Therefore we have a 4:1 single stage reduction. V-Belt pulleys, on the other hand are based upon pulley diameter.

     Well, how do we determine the final reduction of a multistaged gearbox? It's really pretty easy. Multiply the reduction of the first set of gear times the reduction of the next set times the reduction of the next set and so on until you have included them all. That will give you the total gear reduction. So, if we had a three stage gearbox where the first gear set was reduced 4:1, the second set reduced 5:1, and the third set 6:1 then we would multiply 4 x 5 x 6 to get 120:1. (Or you could just look at the motor specs :-p) Now, let's use the motor that we talked about at the beginning and put this gearbox on it and then attach a wheel to the output shaft. Input rpm is 3000. With a 120:1 reduction we divide 3000 by 120 to get 125 rpm. If we attach a 6 inch wheel to that then our bot would move at 2.32 miles per hour. Hmmmmm.... That's a little slow for our taste so we'll have to come up with a gear box that gives us what we are looking for. So, let's determine what type of reduction we would need to achieve a target speed of 15 miles per hour for our bot (that is a quick bot!) First, we know that we are using 6 inch wheels and our motor spins at 3000 rpm and are target speed is 15 mph and our constant is 336. Plug them into this formula ((wheel size) x (motor rpm))/((target speed) x 336). If we plug in our numbers we would get (6 x 3000)/(15 x 336) = 3.57:1. It would be pretty hard to get that exact reduction but we can get close using a 10 tooth input sprocket or gear and a 35 or 36 tooth output sprocket or gear. But, also remember that the 3000 rpm is for an unloaded motor. Loaded motors will spin at a slower speed but determining that speed is beyond the scope of this help section.

     Now, there are those in the builders community who say that your reduction should be the same as your wheel size. i.e. If you are using a 6 inch wheel then you need 6:1 reduction or if you are using a 10 inch wheel then you need 10:1 reduction. I personally think that is over generalizing since different motor spin at different rates.

What are the advantages and disadvantages of gear reduction?

     Well, the two main disadvantages are #1 you lose speed and #2 you have added weight for the gear box. But, on the other hand, there are some great advantages to using gear reduction. First, you bring the bot down to a manageable speed. Second, the motor doesn't have to work as hard to spin the wheel which means it won't draw as much current from your batteries. And third, along those lines, the torque produced by the output is inversely proportional to the amount of reduction in the gear box. Say what? Basically, if you have a 4:1 gear box then the bot moves 1/4 as fast but has 4 times the torque! So you can have a 120 pound robot with the right gearing that will push a 400 pound load across the floor!

     The optimum configuration will give you greatest speed but still have enough torque to cause the wheels to break traction (peel out) before the motor stalls. That optimum configuration varies from bot to bot and is up to you to figure out how to best implement it with your own robot.


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Brett "Buzz" Dawson

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