# Gear Train : Gear Ratio, Torque and Speed Calculations

A Gear train consist of two or more number of gears in series. It is used to increase or decrease the speed or torque of the output shaft. The term Gear Ratio is used to calculate speed and torque of output gear when torque is applied to the input gear.

For example, gear box is a type of gear train, that is used to increase engine torque and reduce speed at the car wheels. The increase in torque depends on at what gear you are driving your car. 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 remains constant. To achieve the condition of constant angular velocity, common normal at the point of contact between mating gear teeth always pass through the pitch point. Where Pitch point is the point of contact between mating gear pitch circles.

If angular velocity of of mating gear are constant we can conclude to following relation: Where ω1 and ω2: Angular Velocity in radian/sec for driver and driven gear respectively.

n1 and n2 = Gear Speed in RPM for driver and driven gear respectively.

d1 and d2 = driver and driven gear diameter respectively.

T1 and T2 = Number of Teeth on driver and driven gear respectively.

## Gear Ratio

Gear ratio is the ratio of the number of teeth of driven gear and driver gear. It is used to calculate the speed and torque of output shaft when input and output shafts are connected using a gear train.

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 and Speed

Power transmission through gear train affects rotational speed of output shaft. Therefore:

###### Speed of Output Shaft = Gear Ratio / Speed of input Shaft

As shown above, if the number of gears on the output shaft is greater than gears on the input shaft. This arrangement is also known as reduction gear Drive. In this arrangement Output shaft will have low speed compared to input shaft.

Whereas When we reverse the arrangement. In other words if number of gears on the output shaft is less than gears on input shaft. Output shaft speed will have high speed compared to input shaft.

## Gear Ratio and Torque

According to the law of gears. In a Gear Train Ratio of output torque to input torque is also a constant and equal to the Gear ratio. Therefore if input torque is known. Output torque can be calculated by multiplying input torque with gear ratio. ## Gear Train Types and their Calculation

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

1. Simple Gear Train
2. Compound Gear Train
3. Reverted Gear Train
4. Planetary Gear Train
##### 1) Gear Ratio Calculations For Simple Gear Train

A simple gear train can be of two gear train or multi gear train.

##### 1.1) Two Gear Train

Two Gear Train is a type of Simple gear train. It consists of two connected gears. For Example, As shown below in a two gear train. The Gear-1 is driver and Gear-2 is driven gear. When driver gear is rotated in a clockwise direction, driven gear will rotate in the anti-clockwise direction.

Question :  Calculate the Speed and torque of output shaft for a simple gear train. Where the number of teeth on driver and driven gear are 40 and 20 respectively. And Driver gear is rotating with 100 rpm and 10 N-m torque.

Number of teeth on driver Gear (T1) = 40

Number of teeth on driven Gear (T2) = 20

Speed of Driver Gear (n1) = 100 rpm

Torque acting on driver gear =  10 N-m

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

Output Gear Speed Calculation

Speed of Output Shaft/Gear = n1/GR = 100/0.5 = 200 rpm

Output Shaft/Gear Torque Calculation Torque generated by Driven gear = GR × Torque Generated by driver

= 0. 5  × 10 = N-m

##### 1.2) 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.

Question :  Calculate the gear ratio for multi gear train. Where number of teeth on driver, idler and driven gear are 40, 20 and 10 respectively.

Given Number of teeth

T1 = 40, T2 = 20, T2 = 10

Gear Ratio (GR) Calculation for Multi gear Train

Step-1 : Calculate Gear-Ratio between Gear-1 and  Gear-2 (Driver and Idler).

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

Step-2 : Calculate GR in between Gear-2 and  Gear-3 (Idler and Driven Gear).

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

Step-3 : By Multiplying gear-ratio between 1 to 2 and 2 to 3. We will get resultant GR between Driver and Driven Gear.

Resultant Multi Gear Train GR = 0.× 0.5 = 0.25

From the above calculated gear ratio we can calculate the speed and torque at output gear.

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

Compound Gears consist of more than one gear on single axis. Therefore gears on the same shaft rotate at the same speed and torque.

Question :  Calculate the gear ratio for compound gear train. Where the number of teeth on driver and driven gear are 40 and 10 respectively with one compound gear. Compound gear has one gear connected to driver with 30 teeth and the another gear connected to driven gear with 20 teeth.

Given number of teeth

T1 = 40, T2 = 30, T3 = 20, T4 = 10

Gear Ratio Calculation For Compound Gear

In above example Gear-2 and Gear-3  are on the same shaft.

Step-1 : Calculate Gear Ratio between Gear-1 and Gear-2

GR(1-2) = 30/40 = 0.75

Step-2 : Calculate GR between Gear-3 and Gear.

GR(3-4) = 10/20 = 0.5

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

Resultant Compound Gear GR = 0.375