In mechanical engineering, the gear ratio represents the direct measure of the ratio between the rotational speeds of two or more interconnected gears. As a general rule, when you are dealing with two gear wheels, if the driving one (that is, the one that directly receives the rotating force from the engine) is larger than the driven one, the latter will turn faster and vice versa. This basic concept can be expressed with the formula Transmission Ratio = T2 / T1, where T1 is the number of teeth of the first gear and T2 the number of teeth of the second gear.
Steps
Method 1 of 2: Finding the Transmission Ratio of a Gear System
Two Gears
Step 1. Start by considering a two-wheel system
In order to determine the transmission ratio you must have at least two gears that are connected to each other and that form a "system". Usually the first wheel is called "driving", or conductor, and is connected to the crankshaft. Between these two gears there could be many others that transmit motion: these are called "referral".
For now, limit yourself to just two cogwheels. In order to find the transmission ratio, the gears must be interconnected, in other words the teeth must be "meshed" and the movement must be transferred from one wheel to another. As an example, let's consider a small driving wheel (G1) that moves a larger driven wheel (G2)
Step 2. Count the number of teeth on each gear
An easy way to calculate the gear ratio is to compare the number of teeth (the small protrusions on the circumference of each wheel). Start determining how many teeth there are on the motor gear. You can manually count them or check the information that is on the gear label itself.
For example, let's consider a drive wheel with 20 teeth.
Step 3. Count the number of teeth of the driven wheel
At this point you need to determine the exact number of teeth on the second wheel, exactly as you did in the previous step.
Let us consider a wheel driven with 30 teeth.
Step 4. Divide the two values together
Now that you know the number of teeth on each gear, you can easily find the gear ratio. Divide the number of teeth on the driven wheel by the number of teeth on the drive wheel. Depending on what your task requires, the answer could be expressed as a decimal number, a fraction, a ratio (i.e. x: y).
- In the example shown above, dividing the 30 teeth of the driven wheel by the 20 of the driving one gives: 30/20 = 1, 5. You can express this relationship as 3/2 or 1, 5: 1.
- This value indicates that the small motor gear must rotate one and a half times to make the driven gear rotate once. The result makes perfect sense, as the driven wheel is larger and turns slower.
More than Two Gears
Step 1. Consider a system with more than two gears
In this case you will have a number of cogwheels forming a long sequence of gears; you will not have to deal with just a driving wheel and a conduct. The first gear of the system is always considered the engine and the last duct; between them there is a series of intermediate gears called "return". Often the function of these is to change the direction of rotation or to connect two gear wheels which, if directly meshed, would make the system inefficient, bulky or non-reactive.
Now consider the two sprockets from the previous section but add a 7-tooth motor gear. The 30-tooth wheel remains driven while the 20-tooth wheel becomes a return wheel (in the previous example it was driving)
Step 2. Divide the number of teeth of the drive and driven wheels
The important thing to remember when working with a drive system that has more than two gears is that only the drive wheel and the driven wheel matter (usually the first and last wheel). In other words, the idler gears do not affect the final drive ratio for any reason. Once you have identified the drive and driven wheels, you can calculate the gear ratio exactly as in the previous section.
In this example, you need to find the gear ratio by dividing the number of teeth on the final wheel (30) by the number of teeth on the starting wheel (7), so: 30/7 = approximately 4, 3 (or 4, 3: 1 and so on). This means that the drive wheel has to turn 4.3 times to produce one full rotation of the driven wheel.
Step 3. If you wish, you can also calculate the various gear ratios between the intermediate gears
This is an easy problem to solve as well. in some practical cases. it is useful to know the transmission ratios of the idler wheels. To find this value, start with the motor gear and move towards the driven one. In other words, treat the first wheel of each pair as driving and the second as being driven. For each pair under consideration, divide the number of teeth on the "driven" wheel by the number of teeth on the "drive" wheel to calculate the intermediate gear ratios.
- In the example, the intermediate gear ratios are 20/7 = 2, 9 and 30/20 = 1, 5. Observe how none of these is equal to the value of the transmission ratios of the whole system (4, 3).
- However note that (20/7) x (30/20) = 4, 3. In general we can say that the product of the intermediate transmission ratios is equal to the transmission ratio of the entire system.
Method 2 of 2: Calculate the Rotation Speed
Step 1. Find the speed of rotation of the drive wheel
Using the concept of transmission ratio, you can imagine how quickly a driven gear rotates based on that "transmitted" by the motor gear. To get started, you need to find the speed of the first wheel. In most cases, speed is expressed in revolutions per minute (rpm), although you can use other units of measurement.
For example, consider the previous example in which a 7-tooth wheel moves a 30-tooth wheel. In this case, let's assume that the motor gear speed is 130 rpm. Thanks to this information, you are able to find the speed of the one conducted with a few steps
Step 2. Enter the data you have in the formula S1xT1 = S2xT2
In this equation S1 is the speed of rotation of the drive wheel, T1 is the number of its teeth, S2 is the speed of the driven wheel and T2 is the number of its teeth. Enter the numerical values you have, until the equation is expressed with a single unknown.
- Often, in these types of problems, you are asked to derive the value S2 even though you can obtain the value of any other unknown. Enter the data you know in the formula and you will have:
- 130 rpm x 7 = S2 x 30
Step 3. Fix the problem
To find the value of the remaining variable you just have to apply some basic algebra. Simplify the equation and isolate the unknown on one side of the equality sign and you will have the solution. Don't forget to express the result in the right unit of measurement - you may get a lower value if you don't.
- In the example, here are the steps for the solution:
- 130 rpm x 7 = S2 x 30
- 910 = S2 x 30
- 910/30 = S2
- 30, 33 rpm = S2
- In other words, if the drive wheel turns at 130 rpm, the driven wheel turns at 30.33 rpm. The result makes sense in reality because the driven wheel is larger and turns slower.
Advice
- In a speed reduction system (where the speed of the driven wheel is lower than that of the tractor) you will need an engine that generates optimum torque at high rpm.
- If you want to see the principles of the gear ratio in reality, take a bike ride! Notice how less effort you are to pedal uphill when using a small gear on the pedals and a large gear on the rear wheel. While it is much easier to spin the small cog with the push on the pedals, it will take a lot of rotations for the large rear cog to make a full rotation. This is inexpensive on flat routes because the speed will be reduced.
- The power required to move the driven gear is amplified or reduced by the transmission ratio. Once the gear ratio has been taken into account, the size of the motor must be determined according to the power needed to activate the load. A speed multiplication system (where the speed of the driven wheel is greater than the driving one) needs an engine that delivers optimum torque at low revs.
