Gear Trains

In this tutorial, you will learn how gears are used in combination to create gear trains, and what different forms this combination may take. In short, you will learn about the reasons why a gear drive is used, a measure of their performance characteristics, and the different types of gear trains used in machinery.


  1. What are Gear Trains?
  2. Why Gear Trains?
  3. Speed and Torque Ratios
  4. Simple Gear Trains
  5. Compound Gear Trains
  6. Reverted Gear Trains
  7. Epicyclic Gear Trains
  8. What are Idler Gears?

What are Gear Trains?

Gear Trains refer to mechanical systems which are formed by mounting gears on a defined frame so that the teeth may engage. The gears are of the same module and are used to transmit power provided through shafts.

  • The Gear Teeth are designed to ensure the pitch circles of the gears roll on each other without slipping and provide a smooth transmission of rotation between gears.
  • The gears have an involute tooth which results in a smooth and continuous transfer of power without any interference.
  • The nature of the gear train used depends upon the velocity ratio needed and the relative position of the axes of the shafts.
  • The gear train may be made up of any type of gears, may it be spur gears, bevel gears, helical gears, or worm gears.

Why Gear Trains?

Mechanical drives in general are used to provide the interface between actuators like motors and the driving components of the machines. Mechanical drives include not only gear drives, but also systems like belt drives. However, gear trains have some features which make them a lucrative option for most applications.

  • Gear train drives are of positive drive type and the velocity ratio remains constant.
  • The center distances between the shafts are relatively small and hence the construction of the drives is relatively compact.
  • Gears trains transmit large power, which is beyond the range of belts.
  • They can transmit motion at a very low velocity which is not possible with belt drives.
  • The efficiency of gear drives is very high, even as large as 99% when including spur gears.
  • Gear trains included in gearboxes can provide a very large range of velocity ratios due to the simple shifting of the gears in mesh.
  • Gear drives however are found to be lacking in one aspect. They have higher manufacturing and maintenance costs. They require timely lubrication and cleanliness and precision in manufacture.


Speed and Torque Ratios

The Gear Teeth of the Gear Train is designed so that the number of teeth on each gear is proportional to the radius of the pitch circle. The pitch circles of the meshing gears are free to roll without any slipping.

  • The speed ratio for the gear pair in mesh is common for both the gears as the velocity at the point of contact is the same and is given by the following equations
    v= rA ωA= rB ωB
    The input gear A has the radius rA while the output gear has the radius rB, and ω represents the angular velocity.
  • Equating the velocity terms and including the proportion of the gear teeth.
    \(\frac{ω_A}{ω_B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}\)
    N represents the number of teeth on each gear.
  • The Mechanical Advantage of the pair of meshing gear for which the input gear has with the output gear is given by M A = NB/NA.
  • When the output gear B has more teeth than the input gear, there is an amplification of input torque. If the opposite were to be true, there is a definite reduction in the input torque with an increase in speed.

Simple Gear Trains

A Simple Gear Train is one in which there is only one gear on each shaft. In design processes, these gears are represented by pitch circles. The input driver is called the drive gear, while the gear which outputs the power is called the driven gears or simply the followers.
The following figure shows a Simple Gear Train

Simple Gear Train

In the above figure, we see the driver gear A and the driven gear B.
NA = Speed of Gear A
NB = Speed of Gear B
TA = Teeth on Gear A
TB = Teeth on Gear B
The speed ratio is the ratio of the speed of the driver gear to the driven gear.
Speed Ratio = \(\frac{N_A}{N_B} = \frac{T_B}{T_A}\)
The ratio of the driven gear to the driver gear is called the Train Value of the gear train.
Train Value = \(\frac{N_B}{N_A} = \frac{T_A}{T_B}\)
Often the distance between the gears is too large. The motion from one gear to another can be transmitted by either providing a larger gear or by using intermediary gears. If the number of these intermediaries is even, the motion is reversed, while if the number is odd, the motion remains unchanged.

Compound Gear Trains

When there is more than one gear on any shaft of the train, the system is classified as a Compound Gear Train. When the distance between the driver and the driven gears has to be connected by intermediate gears and a greater speed ratio is required then the advantage of intermediate gears is increased by providing compound gear. This is a result of the gears on the common shaft having the same angular velocity.
The following diagram shows a compound gear train.

compound gear train"

In the above figure, we see the driver gear A and the driven gear D.
NA = Speed of Gear A
NB = Speed of Gear B
NC = Speed of Gear C
ND = Speed of Gear D
TA = Teeth on Gear A
TB = Teeth on Gear B
TC = Teeth on Gear C
TD = Teeth on Gear D

The speed ratio is the ratio of the speed of the driver gear to the driven gear.
Speed Ratio = \(\frac{N_A}{N_B} × \frac{N_C}{N_D} = \frac{T_B}{T_A} × \frac{T_D}{T_C}\)
As NB = NC
Speed Ratio = \(\frac{N_A}{N_D} = \frac{T_B}{T_A} × \frac{T_D}{T_C}\)

The compound train has the advantage over a simple gear train in that it has a much larger speed reduction from the first shaft to the end shaft which can be obtained via gears.


Reverted Gear Trains

When the axes of the input shaft on which the driver and the output shaft on which the driven gear is mounted are co-axial, then the gear train is known as a reverted gear train. In a reverted gear train, the direction of rotation of the driver and driven gears are the same.
The following figure shows a Reverted gear train.

Reverted gear train

In the above figure, we see the driver gear A and the driven gear B.
NA = Speed of Gear A
NB = Speed of Gear B
NC = Speed of Gear C
ND = Speed of Gear D
TA = Teeth on Gear A
TB = Teeth on Gear B
TC = Teeth on Gear C
TD = Teeth on Gear D
The speed ratio is the ratio of the speed of the driver gear to the driven gear.
Speed Ratio = \(\frac{N_A}{N_B} × \frac{N_C}{N_D} = \frac{T_B}{T_A} × \frac{T_D}{T_C}\)
As NB = NC
Speed Ratio = \(\frac{N_A}{N_D} = \frac{T_B}{T_A} × \frac{T_D}{T_C}\)
Reverted gear trains are used in automotive transmission systems, back gears for lathes, speed reducers in industrial machinery, and clock mechanisms.

Epicyclic Gear Trains

An epicyclic gear train is often called a planetary gear system. It consists of gears that are mounted such as to allow revolution around a central gear. Often, a carrier or a latch joins the centers of the two gears. The central gear is called the sun gear while the ones which revolve around the sun gear are called the planetary gears, hence the name of the system.

  • Any point on the pitch circle of the planet gears creates an epicycloid locus. The planetary and sun gears roll on their pitch circles without slip.
  • Planetary gears systems often include an internal gear attached at the perimeter which becomes a part of the system. This is often called the ring gear.
  • The axes of all gears used are usually parallel but can be placed at an angle depending on the application.
  • Planetary gear systems can have any one of the sun, planetary, or ring gears fixed depending on the application.
  • More complicated structures include compound planetary gears which may have meshed planets where two or more planetary gears mesh together, stepped planets that have more than one planetary gear on the same shaft, or multi-stage structures where there are two or more planet sets.
  • For planetary gear systems, the fundamental equation of gear ratio is as follows
    (R-1)ωc = Rωr – ωs
    Where R = \(-\frac{N_r}{N_s}\)
    Nr = Number of Teeth on Ring Gear
    Ns = Number of Teeth on Sun Gear
    ω = Angular Velocity of the gear

What are Idler Gears?

An idler gear, commonly referred to as just an idler is a simple gear that is inserted in between two or more other gear wheels to either change the direction of rotation of the output shaft or simply reduce the size of the input or output shaft while achieving the desired distance between the shaft.

  • An idler gear does not affect the gear ratio of the gear train, regardless of its size. The velocity of the input gear is directly transferred to the idler gear, with no other form of actuation.
  • An idler gear, when used as a single unit, can be used to reverse the direction of the input shaft. This is a common application in the reverse gear of automobiles.
  • Idler gears also transfer rotation between distant shafts in cases where it is difficult to make direct meshing gears that are too large for manufacturing.

Key Points to Remember

Here is the list of key points we need to remember about “Gear Trains”.

  • Gears are confined into mechanisms called gear trains which transfers power between the input and output shafts while causing changes in speed, torque, or direction.
  • Speed and Torque ratios are derived from the number of teeth on the gears of the gear trains and define the transfer characteristics of the gear trains.
  • A simple gear train constitutes of driven and driver gears which are meshed directly or indirectly with each gear on one shaft.
  • A compound gear train uses multiple gears on the same shaft to transfer power between shafts.
  • A reverted gear train has input and output shafts located coaxially and allows for compact and direct transfer of power.
  • An epicyclic gear train uses a system of sun, planetary and ring gears to provide a large range of velocity ratios to handle the transfer of powers.
  • Idler gears are used in gear trains to reduces the sizes of the driver and driven gears and also handle the changes in speed direction between the input and output shafts.

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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