# Gear Train : Gear Ratio, Torque and Speed Calculations

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Gear ratio is the ratio of output and input number of teeth on a shaft. Gear train consist of gears in series. They are used to increase or decrease the speed or torque of the output shaft. In this article we will discuss how to calculate gear ratio, speed and torque for various gear train.

To understand gear ratio we suggest you first read this article on Gear Terminology ( Various Terms used in Gears) and Various Types of Gears.

#### Law of Gearing Law of gearing states that the angular velocity ratio between mating gears is always constant. ω : Angular Velocity in radian/sec

n1 = Gear Speed in RPM

d = Gear and pinion diameter

T = Number of Teeth

#### Gear Ratio

Input gear where torque is applied is known as driver. Whereas output gear is known as driven gear. And gears used in between the driver and driven gears are known as idler gears.

Gear ratio is the ratio of the number of teeth of driven gear and driver gear. #### Gear Ratio and Speed

Power transmission through gear train affect rotational speed of output shaft. If number of gears on the output shaft is greater than gears on input shaft. Output shaft will have low speed. This arrangement is also known as reduction gear arrangement.

Whereas if number of gears on the output shaft is less than gears on input shaft. Output shaft speed will be very fast.

Therefore:

Output Gear Speed = Gear Ratio / Input Gear Speed

#### Gear Ratio and Torque

From the law of gears. Gear ratio is also equal to the ratio of output torque to input torque. Therefore output torque is calculated by multiplying input torque with gear ratio. #### Gear Train Types and their Calculation

A gear train consists of a series of number of gears to transfer power from one shaft to another. For example, power from the engine is transferred to the wheels through gear box.

A gear train can be classified into four types:

1. Simple Gear Train
2. Compound Gear Train
3. Reverted Gear Train
4. Planetary Gear Train
##### Two Gear Train
This is a simple gear train. One of the example of two gear train is shown in the image below. Where Gear-1 is driver and Gear-2 is driven gear. When driver gear is rotated in a clockwise direction, driven gear will rotate in anti-clockwise direction. Number of teeth on driver (T1) = 40

Number of teeth on driver (T2) = 20

Driver Gear Speed (n1) = 100 rpm

Driven Gear Speed = n2 rpm

Torque acting on driver gear =  10 N-m

Gear Ratio Calculation Gear Ratio = T2 / T1 = 20/40 = 0. 5

Gear Speed Calculation

n2 = n1/Gear Ratio = 100/0.5 = 200 rpm

Torque Calculation Torque generated by Driven gear = Gear Ratio X Torque Generated by driver

= 0. 5  X  10 = N-m

##### Multi Gear Train

Multi gear train consist of more than two gears to transfer motion from one shaft to another. Resultant gear ratio can be calculated by multiplying individual gear ratios. T1 = 40, T2 = 20, T2 = 10

Gear Ratio (GR) Calculation

Step-1 : Calculate gear-1 and  gear-2 GR.

GR(1-2) = 20/40 = 0.5

Step-2 : Calculate gear-2 and  gear-3 GR.

GR(2-3) = 10/20 = 0.5

Step-3 : Multiply GR-1 and GR-2

Resultant GR = 0.5 X 0.5 = 0.25

##### 2) Gear Ratio Calculations For Compound Gear Train

Compound Gears consist of more than one gear on single axis. Therefore gears on same shaft rotate at the same speed. T1 = 40, T2 = 30, T3 = 20, T4 = 10

Gear Ratio Calculation

In this example gear 2 and gear 3  are on same shaft.

Step-1 : Calculate gear-1 and  gear-2 GR.

GR(1-2) = 30/40

Step-2 : Calculate gear-3 and  gear-4 GR.

GR(3-4) = 10/20

Step-3 : Multiply GR(1-2) and GR(3-4)

Resultant GR = 0.375

##### 3) Reverted Gear Train Reverted gear trains are a type of compound gear trains in which input and output shafts are on the same axis. In this example, gear-1 and gear-3 are on same axis.

They are used to achieve high gear ratio within limited space. Riveted gear train gear ratio is calculated similar to compound gear train.

#### Conclusion

To sum up, gear ratio is used to calculate the resultant gear speed and torque. Value of gear ratio depends on the number of teeth on driver, idler and driven gear. To understand gear ratio we suggest you first read this article on Gear Terminology ( Various Terms used in Gears) and Various Types of Gears.

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