Gears and Gear Trains free study note for Diploma / BTech.

Posted on April 12, 2026June 28, 2026 by tilak.das007@gmail.com

1. What is Gears and Gear Trains ?

A gear is a rotating machine element with teeth that mesh with another toothed part to transmit power and motion between shafts. Gears are widely used where precise speed ratio, high efficiency, and reliability are required.

A gear train is a mechanical system formed by mounting two or more gears on a frame so that they mesh together and engage their teeth.

When you connect multiple gears in a series, it allows power and motion to be transmitted from one shaft to another, especially when the distance between the shafts is too large for a single pair of gears, or when a specific change in speed or rotational direction is required.

Table of Contents

Advantage of Gears	Dis-advantage of Gears
Positive drive (no slip) Constant velocity ratio High efficiency (≈ 95–99%) Compact design Suitable for high power transmission	Costly manufacturing Requires lubrication Noise at high speeds (especially spur gears)

2. Terminology of Gears

Important terms:

Pitch Circle: Imaginary circle that rolls without slipping
Pitch Diameter (D): Diameter of pitch circle
Module (m): $m = \frac{D}{T}$
Number of Teeth (T)
Circular Pitch (p): Distance between corresponding points on adjacent teeth
Pressure Angle (ϕ): Angle between line of action and tangent to pitch circle
Addendum: Height above pitch circle
Dedendum: Depth below pitch circle
Clearance: Gap between mating teeth

3. Types of Gears

(a) Spur Gears

A spur gear is a cylindrical gear with straight teeth cut parallel to the axis of rotation. It is the simplest and most commonly used type of gear for transmitting power between parallel shafts.

Key features

Produces more noise and vibration at high speeds than helical gears because the entire tooth engages at once.
Straight teeth parallel to the shaft axis.
Used to transmit rotary motion and power.
High efficiency (typically 95–99% under proper lubrication).
Easy to manufacture and relatively inexpensive.

gear trains

(b) Helical Gears

A helical gear is a cylindrical gear with teeth cut at an angle (helix angle) to the axis of rotation. Unlike spur gears, the teeth engage gradually, making helical gears smoother, quieter, and capable of carrying higher loads.

Key features

Suitable for high-speed and high-load applications.
Teeth are inclined at a helix angle (typically 15°–30°).
Used for transmitting power between parallel shafts (most common) and sometimes crossed shafts.
Teeth engage gradually, reducing shock and vibration.

(c) Bevel Gears

A bevel gear is a gear with conical-shaped teeth used to transmit power and motion between intersecting shafts, most commonly at an angle of 90°.

Key features

Available in several tooth designs for different applications.
Teeth are cut on a conical surface.
Used for intersecting shafts (typically 90°).
Changes the direction of power transmission.

(d) Worm and Worm Wheel

A worm and worm wheel is a gear drive consisting of a worm (a screw-like gear) and a worm wheel (a gear that meshes with the worm). It is used to transmit power between non-parallel, non-intersecting shafts, usually at 90°.

Key features

Shaft angle is typically 90°.
Suitable for high reduction ratios (e.g., 10:1 to 100:1 or more).
Smooth and quiet operation.
Can provide a self-locking effect in some designs.

4. Gear Ratio

Gear ratio determines speed relationship: $\text{Velocity Ratio} = \frac{N_1}{N_2} = \frac{T_2}{T_1}$

Where:

$N$ = speed (rpm)
$T$ = number of teeth

5. Gear Trains

A gear train is a combination of two or more gears used to transmit motion and power.

Types of Gear Trains

(a) Simple Gear Train

A simple gear train is the most basic type of gear transmission system used to transfer rotary motion and power from one shaft to another. In a simple gear train, each shaft carries only one gear, and all gears are mounted on separate shafts. It is widely used in machines where moderate speed reduction or speed increase is required.

Simple gear trains are commonly found in clocks, machine tools, conveyors, gear pumps, and many industrial machines because of their simple construction and reliable operation.

Construction of a simple gear train

A simple gear train consists of:

Driver Gear (Gear 1) – Receives power from the input shaft.
Idler Gear(s) – Intermediate gear(s) that transmit motion between the driver and driven gears.
Driven Gear (Gear 3) – Delivers power to the output shaft.

Working Principle

The driver gear receives rotational power from the prime mover.
As the driver rotates, it meshes with the idler gear.
The idler gear transmits motion to the driven gear.
The driven gear rotates at a speed determined by the gear ratio.

The idler gear only changes the direction of rotation and the distance between shafts. It does not affect the overall gear ratio.

Characteristics

Only one gear is mounted on each shaft.
Used between parallel shafts.
May contain one or more idler gears.
Easy to design and manufacture.
Suitable for moderate speed reduction or increase.
Simple maintenance.

Velocity Ratio

Let,

$N_1$ = Speed of driver
$N_2$ = Speed of driven gear
$T_1$ = Number of teeth on driver
$T_2$ = Number of teeth on driven gear

The velocity ratio is $\boxed{\frac{N_1}{N_2}=\frac{T_2}{T_1}}$

where

$N$ = Speed (rpm)
$T$ = Number of teeth

Gear Ratio

The gear ratio is $\boxed{GR=\frac{\text{Speed of Driver}}{\text{Speed of Driven}}=\frac{T_2}{T_1}}$

If

Driver has 20 teeth
Driven has 60 teeth

then $GR=\frac{60}{20}=3$

The output speed becomes one-third of the input speed.

Advantage of simple gear train	Disadvantage of simple gear train
Simple construction. Easy to manufacture. Low manufacturing cost. High mechanical efficiency. Easy maintenance. Reliable operation. Suitable for moderate speed changes.	Large speed reductions require many gears. Occupies more space for high reduction ratios. Each gear mesh introduces some power loss. Limited speed reduction compared with compound gear trains.

(b) Compound Gear Train

A compound gear train is a gear transmission system in which two or more gears are mounted rigidly on the same shaft. The gears on the same shaft rotate together at the same speed and in the same direction. Compound gear trains are used when large speed reduction or speed increase is required within a compact space.

Compared with a simple gear train, a compound gear train can achieve much higher gear ratios without requiring extremely large gears.

Construction of compound gear train

A typical compound gear train consists of:

Gear 1 (Driver) – Mounted on the input shaft.
Gear 2 – Meshed with Gear 1.
Gear 3 – Rigidly mounted on the same shaft as Gear 2.
Gear 4 (Driven) – Meshed with Gear 3 and mounted on the output shaft.

Since Gears 2 and 3 are fixed on the same shaft, they rotate together with the same angular velocity.

Working Principle

Gear 1 receives power from the prime mover.
Gear 1 drives Gear 2.
Since Gear 2 and Gear 3 are on the same shaft, Gear 3 rotates at the same speed as Gear 2.
Gear 3 then drives Gear 4.
Gear 4 delivers power to the output shaft.

Each pair of meshing gears contributes to the overall speed ratio.

Characteristics

Two or more gears are mounted on one shaft.
Provides very large speed reduction or speed increase.
Compact construction.
Used for transmitting power between parallel shafts.
More than one gear pair participates in power transmission.
Higher velocity ratios than simple gear trains.

Speed Relationship

Since Gear 2 and Gear 3 are fixed on the same shaft, $\boxed{N_2=N_3}$

where

$N_2$ = Speed of Gear 2
$N_3$ = Speed of Gear 3

Gear Ratio

The gear ratio is $\boxed{GR = \frac{T_2 \times T_4} {T_1 \times T_3}}$

Velocity Ratio

Let

$T_1$ = Teeth on Gear 1
$T_2$ = Teeth on Gear 2
$T_3$ = Teeth on Gear 3
$T_4$ = Teeth on Gear 4

Then

For Gear Pair 1 $\frac{N_1}{N_2}=\frac{T_2}{T_1}$ For Gear Pair 2 $\frac{N_3}{N_4}=\frac{T_4}{T_3}$ Since $N_2=N_3$ Overall velocity ratio $\boxed{\frac{N_1}{N_4} =\frac{T_2}{T_1} \times \frac{T_4}{T_3}}$

Advantage of compound gear train	Disadvantage of compound gear train
High speed reduction or increase. Compact arrangement. Large gear ratio without very large gears. Efficient power transmission. Suitable for heavy-duty applications. Requires less space than a simple gear train for the same reduction ratio. Can transmit higher torque.	More complex design. Higher manufacturing cost. More bearings and shafts. Increased friction due to multiple gear meshes. Requires accurate alignment and lubrication. Maintenance is more difficult than a simple gear train.

(c) Reverted Gear Train

A reverted gear train is a special type of compound gear train in which the axes of the first (driver) gear and the last (driven) gear are coaxial (lie on the same axis). It is widely used where a large speed reduction or increase is required while keeping the input and output shafts aligned.

Unlike a simple gear train, a reverted gear train contains at least four gears, with two gears mounted on the same intermediate shaft. The first and last gears rotate about the same center but are not directly connected.

Component of a reverted gear train

A reverted gear train consists of:

Gear 1 (Driver) – Mounted on the input shaft.
Gear 2 – Meshed with Gear 1.
Gear 3 – Mounted on the same shaft as Gear 2.
Gear 4 (Driven) – Meshed with Gear 3 and mounted on the same axis as Gear 1.

Working Principle

The input shaft rotates Gear 1.
Gear 1 drives Gear 2.
Since Gear 2 and Gear 3 are mounted on the same shaft, they rotate together at the same speed.
Gear 3 drives Gear 4.
The output is obtained from Gear 4.

Since the power passes through two gear meshes, the speed ratio is determined by both pairs of gears.

Characteristics

Input and output shafts are coaxial.
Provides large speed reduction or speed increase.
Compact arrangement.
Uses compound gearing.
Requires accurate alignment.

Condition for a Reverted Gear Train

For equal module gears,

Pitch circle condition

$D_1+D_2=D_3+D_4$

Since $D=mT$

where:

$D$ = Pitch diameter
$m$ = Module
$T$ = Number of teeth

Therefore, $\boxed{T_1+T_2=T_3+T_4}$

This is the basic condition of a reverted gear train.

Velocity Ratio

Let

$T_1$ = Teeth on Driver
$T_2$ = Teeth on Gear 2
$T_3$ = Teeth on Gear 3
$T_4$ = Teeth on Driven Gear

Then, $\text{Velocity Ratio} =\frac{N_1}{N_4} =\frac{T_2}{T_1}\times\frac{T_4}{T_3}$ where

$N_1$ = Speed of driver
$N_4$ = Speed of driven gear

Gear Ratio

$GR=\frac{\text{Speed of Driver}}{\text{Speed of Driven}} =\frac{T_2\times T_4}{T_1\times T_3}$

Advantage of reverted gear train	Dis-advantage of reverted gear train
Input and output shafts are on the same axis. Compact construction. High speed reduction possible. Efficient transmission. Suitable for precision mechanisms. Good load distribution.	More gears increase manufacturing cost. Requires precise alignment. More bearings and shafts than a simple gear train. Increased friction and maintenance. Design is comparatively complex.

(d) Epicyclic (Planetary) Gear Train

An epicyclic gear train, also known as a planetary gear train, is a special type of gear mechanism in which one or more gears rotate around the axis of another central gear while also rotating about their own axes. Unlike a simple gear train, the axes of some gears in an epicyclic gear train are not fixed.

This gear arrangement provides high torque transmission, compact size, multiple speed ratios, and high efficiency, making it one of the most widely used gear systems in automobiles, aerospace, industrial machinery, and robotics.

Construction of an Epicyclic Gear Train

An epicyclic gear train mainly consists of the following components:

A. Sun Gear

Located at the center.
Usually acts as the driving or driven gear.
Meshes with the planet gears.

B. Planet Gears

Small gears surrounding the sun gear.
Rotate on their own axes.
Revolve around the sun gear along with the carrier.
Usually two to six planet gears are used.

C. Planet Carrier (Arm)

Holds the planet gears.
Rotates about the central axis.
Transfers motion between gears.

D. Ring Gear (Annulus)

Internal gear having teeth on the inside.
Surrounds the planet gears.
Can be fixed, driven, or act as the output member.

3. Principle of Working

The operation depends on which member is fixed, which is the input, and which is the output.

The three principal members are:

Sun Gear
Ring Gear
Planet Carrier

Since there are three rotating members, numerous speed ratios can be obtained.

Example

Suppose:

Sun Gear = Driving member
Ring Gear = Fixed
Carrier = Output

Working:

Sun gear rotates.
Planet gears rotate around the sun.
Since the ring gear cannot rotate, the planets walk around the inside of the ring gear.
The carrier rotates at a reduced speed but with increased torque.

This arrangement is commonly used as a speed reducer.

Types of Epicyclic Gear Trains

A. Simple Planetary Gear Train

One sun gear
One ring gear
One carrier
One set of planet gears

Applications:

Automatic transmission
Gear reducers

B. Compound Planetary Gear Train

Planet gears mounted together.
More than one gear on the same shaft.
Larger reduction ratios.

Applications:

Heavy industrial gearboxes
Excavators

C. Differential Planetary Gear Train

Two outputs rotate independently.

Applications:

Automobile differential
Four-wheel-drive systems

6. Speed Ratio

The speed ratio depends upon:

Number of teeth
Fixed member
Input member
Output member

For a simple planetary gear train,

If

Sun Gear teeth = Zₛ
Ring Gear teeth = Zᵣ

Then one common relation is $\frac{\omega_s-\omega_c}{\omega_r-\omega_c} = -\frac{Z_r}{Z_s}$ Where

ωₛ = Speed of Sun Gear
ωᵣ = Speed of Ring Gear
ω𝚌 = Speed of Carrier

This equation is known as the fundamental equation of epicyclic gear trains.

6. Comparison of Simple, Compound, and Reverted Gear Trains

Feature	Simple Gear Train	Compound Gear Train	Reverted Gear Train
Gears on same shaft	No	Yes	Yes
Input & output shafts	Different axes	Different axes	Same axis
Speed reduction	Moderate	High	High
Construction	Simple	Moderate	More complex
Space required	More	Less	Very compact
Typical use	General power transmission	High reduction	Coaxial input/output systems

7. Applications of Gear Trains

Automobiles (gearbox, differential)
Clocks and watches
Industrial machinery
Robotics
Wind turbines

8. Advantages and Dis-advantages of Gear Trains

Advantage of Gear Trains	Dis-advantage of Gear Trains
Large speed variation possible Compact arrangement Accurate motion transmission High efficiency	Complex design Costly manufacturing Requires lubrication and maintenance

External Resources

NPTEL course for gear trains.

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