A gear ratio is a direct measure of the ratio of the rotational speeds of two or more interlocking gears in mechanical engineering. In general, if the drive gear (the one directly receiving rotational force from the engine, motor, etc.) is larger than the driven gear, the latter will turn faster, and vice versa. This fundamental concept can be expressed using the formula Gear ratio = T2/T1, where T1 is the number of teeth on the first gear and T2 is the number of teeth on the second gear.
Method 1 Finding the Gear Ratio of a Gear Train
1. Begin with a two-speed train. To calculate a gear ratio, you must have at least two gears engaged with each other — this is referred to as a “gear train.” The first gear is typically a “drive gear” attached to the motor shaft, and the second is a “driven gear” attached to the load shaft. There may be any number of gears between these two to transfer power from the drive gear to the driven gear: these are known as “idler gears.”
For the time being, consider a gear train with only two gears. To be able to find a gear ratio, these gears must interact with each other — that is, their teeth must be meshed and one must turn the other. Assume you have one small drive gear (gear 1) turning a larger driven gear (gear 2). (gear 2).
2. The number of teeth on the drive gear should be counted. One simple method for determining the gear ratio of two interlocking gears is to compare the number of teeth (the small peg-like protrusions at the wheel’s edge) that they both have. Begin by determining the number of teeth on the drive gear. You can do this by manually counting or, in some cases, by looking for this information labelled on the gear itself.
Assume that the smaller drive gear in our system has 20 teeth for illustration purposes.
3. The number of teeth on the driven gear should be counted. Next, calculate the number of teeth on the driven gear in the same way you did for the drive gear.
Assume that the driven gear in our example has 30 teeth.
4. Multiply one tooth count by the other. You can find the gear ratio relatively easily now that you know how many teeth are on each gear. Subtract the driven gear teeth from the drive gear teeth. Depending on the nature of your assignment, you may be required to write your answer as a decimal, a fraction, or in ratio form (i.e., x : y).
In our example, 30/20 = 1.5 is obtained by dividing the 30 teeth of the driven gear by the 20 teeth of the drive gear. This can also be written as 3/2, 1.5 : 1, and so on.
This gear ratio means that the smaller driver gear must turn one and a half times in order for the larger driven gear to turn once. This makes sense because the driven gear will turn more slowly because it is larger.
5. Begin with a gear train that has more than two gears. A “gear train,” as the name implies, can be composed of a long series of gears, rather than just a single driver gear and a single driven gear. In these cases, the first gear remains the driver gear, the last gear remains the driven gear, and the middle gears are referred to as “idler gears.” When direct gearing would make them unwieldy or unavailable, these are frequently used to change the direction of rotation or connect two gears.
For illustration purposes, suppose the two-gear train described above is now driven by a small seven-toothed gear. In this case, the 30-toothed gear is still the driven gear, while the 20-toothed gear (which was previously the driver) is now an idler gear.
6. Divide the number of teeth on the drive and driven gears. When dealing with gear trains with more than two gears, keep in mind that only the driver and driven gears (usually the first and last ones) are important. In other words, the idler gears have no effect on the overall train’s gear ratio. Once you’ve determined your driver and driven gears, you can calculate the gear ratio exactly as before.
In our example, the gear ratio would be calculated by dividing the thirty teeth of the driven gear by the seven teeth of our new driver. 30/7 = approximately 4.3 (or 4.3 : 1, etc.) This means that the driver gear must turn approximately 4.3 times in order for the much larger driven gear to turn once.
7. Find the gear ratios for the intermediate gears if desired. You can also find the gear ratios involving the idler gears, which you may want to do in some cases. In these cases, begin with the drive gear and work your way to the load gear. In terms of the next gear, treat the preceding gear as if it were the drive gear. To calculate the intermediate gear ratios, divide the number of teeth on each “driven” gear by the number of teeth on the “drive” gear for each interlocking set of gears.
The intermediate gear ratios in our example are 20/7 = 2.9 and 30/20 = 1.5. It’s worth noting that neither of these is equal to the overall train’s gear ratio of 4.3.
However, keep in mind that (20/7) (30/20) = 4.3. In general, a gear train’s intermediate gear ratios will multiply together to equal the overall gear ratio.
Method 2 Making Ratio/Speed Calculations
1. Find out what the rotational speed of your drive gear is. Using the concept of gear ratios, it is simple to calculate how quickly a driven gear rotates based on the “input” speed of the drive gear. Begin by determining the rotational speed of your drive gear. Most gear calculations use rotations per minute (RPM), but other units of velocity will work as well.
Assume that in the above-mentioned gear train with a seven-toothed driver gear and a 30-toothed driven gear, the drive gear rotates at 130 RPMs. In the following steps, we’ll use this information to calculate the speed of the driven gear.
2. Enter your data into the formula S1 T1 = S2 T2. In this formula, S1 denotes the rotational speed of the drive gear, T1 denotes the teeth in the drive gear, and S2 and T2 denote the rotational speed and teeth of the driven gear, respectively. Fill in the variables until only one remains undefined.
In these types of problems, you’ll almost always be solving for S2, though you could solve for any of the variables. In our case, by plugging in the information we have, we get the following:
130 RPMs × 7 = S2 × 30
3. Solve. Finding your remaining variable is a simple matter of algebra. Simply simplify the rest of the equation and isolate the variable on one side of the equals sign to get your answer. Don’t forget to label it with the correct units — you could lose points in school if you don’t.
In our example, we can solve like this:
130 RPMs × 7 = S2 × 30
910 = S2 × 30
910/30 = S2
30.33 RPMs = S2
In other words, if the drive gear spins at 130 RPMs, the driven gear will spin at 30.33 RPMs. This makes sense — since the driven gear is much bigger, it will spin much slower.
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