In order to calculate gear ratio and speed, it is important to understand what a gear is. A gear is a toothed wheel that works together with another toothed wheel to alter the speed or direction of a driving mechanism (such as a motor).
Gears normally act in pairs; the driving gear and driven gear. The driving gear is the gear that is rotated by the prime mover, and the driven gear is the gear that is pushed (or driven) into rotation by the driving gear. A gear ratio is the relationship between the speed of the driving and driven gear in a gear set.
Calculating Gear Ratios for Gear Sets
When two meshed gears have a different number of teeth, a reduction in speed is produced. The gear ratio and speed for a gear set can be calculated by knowing the number of teeth on each gear and by knowing the speed in revolutions per minute (rpm) of each gear in the stage.
When you know the number of gear teeth:
When you know the gear set speed:
It is most common that the input speed (rpm) and gear ratio are known, and the above equation is solved for the output speed (rpm).
Gear ratio is typically written using the normal notation for ratio (e.g., 40:1 or 83.5:1). This format conveys that the input gear rotates a certain number of times for every one time the output gear makes a full rotation.
Calculating Gear Ratios For Gear Trains
It is common for more than two gears (multiple gear stages) to be used in a system. When there is more than one gear set in a system, it is called a gear train. The easiest way to calculate the gear ratio of a gear train is to look at each gear set individually, starting at the input and moving to the output.
Use the formulas above to calculate the gear ratio for each gear set in the gear train. Reference the following information to calculate total gear ratio in a gear train:
To calculate the gear ratio of the first stage, divide the number of teeth on the intermediate driven gear by the number of teeth on the input driving gear. For this example, 12 divided by 8 equals 1.5 (1.5:1)
You can then calculate the rpm of the driven gear by dividing the rpm of the driving gear by the ratio of the gear stage. In this example, 500 rpm divided by 1.5 equals 333 rpm for the driven gear in the first stage
Second Stage: Driving Gear = 8 teeth, Driven Gear = 12 teeth, Intermediate Shaft Speed = 333 rpm. The gear ratio of the second stage is calculate the same way as the first stage by dividing the the number on teeth of the driven gear by the number of teeth on the driving gear. In this example, the ratio of the second gear stage is the same as that of the first gear stage. 12 divided by 8 equals 1.5 (1.5:1).
The rpm of the driven gear in the second stage is calculated by using the rpm of the driven gear calculated above (which is the driving gear in the second stage) divided by the gear ratio of the second stage. For this example, 333 rpm divided by 1.5 equals 222 rpm for the driven gear of the second gear stage (as well as the output rpm of the entire gear train).
Now that we know gear ratio and speed (rpm) of each gear stage, we can calculate the gear ratio of the complete gear train by dividing the input rpm (500 rpm) by the output rpm (222 rpm), which equals a total gear ratio of 2.2 (2.2:1). You can also calculate the total gear ratio by multiplying the gear ratios of all the individual gear stages. 1.5 multiplied by 1.5 equals 2.2 (2.2:1)
Gear Ratios and Torque
Gear ratios also change the torque from the input to the output. Torque is a twisting or turning force. If the speed is reduced by the ratio, the torque is increased by the same ratio. The torque will also be affected by the system’s efficiency, but we will ignore that for this exercise.
In order to calculate torque, you must know the force and radius. Force is a push or pull and is often measured in pounds. Radius is the distance from the center of a circle/sphere to its perimeter. The radius of a gear is defined by the pitch circle. The pitch circle is the circle that can be drawn around each gear in a gear set that allows the two gears to roll:
orque is the product of force multiplied by radius. Therefore, if either the force or radius is changed, the torque will change.
Torque = Force x Radius
T = F x R
Why Gear Ratio Matters
One of the main benefits of a gearbox is it allows you to make adjustments to the speed and torque of a motor. It is important to purchase gearmotors from suppliers who offer a wide range of reduction ratios. A wide selection of available ratios allows for more opportunities to select an accurate speed reduction or torque for a specific application.