Dr. Jagadeep Thota

MAE 364 Kinematics and Dynamics of Machines

Chapters 12 & 13: Gears & Gear Trains

MAE 364: Gears JT

Gears are used to transmit motion from a rotating shaft to

another which rotates, OR from a rotating shaft to a body

which translates and which can be considered as rotating

about an axis of infinity.

Gears are machine elements that transmit motion by means

of successively engaging teeth.

Gears are highly efficient due to primarily rolling contact

between the teeth.

Gears do not depend on friction and do best when friction is

minimized.

Gears

MAE 364: Gears JT

Spur Gear

They are the simplest and most common type of gears.

They transmit motion between parallel shafts.

They can be straight OR helical based on how the teeth are cut.

Straight Spur Gear Helical Spur Gear

The teeth are cut in a straight line. The teeth are cut in a more complex angle

Pictures from science.howstuffworks.com

MAE 364: Gears JT

Bevel Gear

Bevels gears permit transmission of motion between two shafts

angled relative to each other.

The top surface of the gear teeth can be extended to form a

cone.

Picture from science.howstuffworks.com

MAE 364: Gears JT

Worm Gear

This comprises of a worm (similar in shape to a screw thread)

which meshes with a worm gear (looks similar to a spur gear).

Power is always supplied to the worm.

They are used to produce high reduction in speeds in a compact

space.

Gear train (compound) with the worm gear

Picture from science.howstuffworks.com Picture from bidatools.com

MAE 364: Gears JT

These are a pair of gears which convert rotational motion into

linear motion.

The pinion (which looks like a spur gear) meshes with a linear

gear known as rack.

The rotational motion of the pinion in converted to a linear motion

of the rack.

Rack and Pinion

Car Steering Mechanism

Picture from iescjmechanisms.wikispaces.com Picture from knowtheworld.in

MAE 364: Gears JT

Gear Nomenclature

Picture from dieselpunks.org

MAE 364: Gears JT

Diametral pitch (P) is defined as the teeth per millimeter of

pitch diameter.

For a pair of mating (meshed) gears, the diametral pitch

should be equal.

This will result (theoretically) in a pure rolling contact

between the pair of gears (no slippage or sliding).

Diametral Pitch

𝑃 = 𝑁

𝐷 𝑖𝑛−1 𝑜𝑟 𝑚−1

Where, 𝑁 number of teeth on the gear D pitch diameter of the gear (in or m)

MAE 364: Gears JT

Derived In-class on White Board

The gears are represented in simplified diagrams as just

circles. The diameter of these circles represents the pitch

diameter (D).

Fundamental Law of Gearing

𝜔3

𝜔2= −

𝑁2

𝑁3 Fundamental Law of Gearing

Where, 𝜔3 speed of gear 3 which is meshing with gear 2 (rpm or rad/s) 𝜔2 speed of gear 2 which is in mesh with gear 3 (rpm or rad/s) 𝑁3 number of teeth on gear 3 𝑁2 number of teeth on gear 2

The negative sign in the above equation indicates that gear 3 will be rotating in the opposite direction to gear 2.

𝐺𝑒𝑎𝑟 𝑟𝑎𝑡𝑖𝑜 =

MAE 364: Gears JT

A combination of gears arranged for the purpose of transmitting torque and rotational motion from an input shaft to an output shaft is called a gear train.

Gear trains are used to transmit torque and rotary motion within machines AND/OR to change the rotational speed of various machine elements.

Gear trains generally reduce speed and increase torque.

Gear trains are also used when the gear ratio requirement cannot be met OR practically will lead to unacceptable design.

For example; a gear ratio of 10:1 is needed. This will result in one of the gear diameter to be 10 times larger than the other gear diameter. This will result in a poor design due to such big variation in gear sizes and wastage of material and space enclosing the huge gear.

Gear Train

MAE 364: Gears JT

Gear Train Classification

Gear Train

Ordinary Planetary

Simple Compound Epicyclic Differential

MAE 364: Gears JT

Two or more gears are in mesh such that they form a consecutive sequence

from input to output.

The outer most gears are the input and output gears.

The gears in between the output and input gears are known as the idler gears.

If there are even number of idle gears then the output gear will rotate in the

opposite direction to input gear.

If there are odd number of idle gears then the output and input gears will rotate in the same

direction.

Simple Gear Train

Output gear

Input gear

Idler gears

Picture from www.rlt.com

MAE 364: Gears JT

Example 1: Simple Gear Train

MAE 364: Gears JT

Determine the speed (and direction) of the output

gear if the input gear is rotating at 200 rpm

clockwise.

Example 2: Simple Gear Train

MAE 364: Gears JT

Solution:

Example 2: Simple Gear Train

- In order to calculate the magnitude for a simple gear train just pick input and output gears. - Then apply the fundamental law of gearing to this pair of gears:

𝜔𝐴

𝜔𝐺= −

𝑁𝐺

𝑁𝐴= −

32

24= −1.33

- Given, 𝜔𝐺 = 200 𝑟𝑝𝑚

𝜔𝐴 = −1.33 𝜔𝐺 = −1.33 200 𝑟𝑝𝑚 = −266.67 𝑟𝑝𝑚

∴ 𝜔𝐴 = 266.67 𝑟𝑝𝑚 𝐶𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒

- The negative sign in the above answer indicates the output gear rotates opposite to the input gear.

MAE 364: Gears JT

This gear train has four or more gears of which at least two

are attached to the same shaft.

In simple gear train, none of the gears share a shaft.

Preferred for large velocity ratios.

Compound Gear Train

Picture from curriculum.vexrobotics.com Picture from imgarcade.com

MAE 364: Gears JT

Find the speed (velocity) ratio of the shown compound gear

train.

Given, N2 = 9, N3 = 62, N4 = 20, N5 = 51, N6 = 15, N7 = 54

Example 3: Compound Gear Train

MAE 364: Gears JT

Solution: Going from input towards output:-

Example 3: Compound Gear Train

𝜔3

𝜔2= −

𝑁2

𝑁3

𝜔4

𝜔3= 1 (𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑 𝑔𝑒𝑎𝑟)

𝜔5

𝜔4= −

𝑁4

𝑁5

𝜔6

𝜔5= 1 (𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑 𝑔𝑒𝑎𝑟)

𝜔7

𝜔6=

𝑁6

𝑁7

No –ve sign since the internal teeth of gear 7 and the external teeth of gear 6 make an internal-external gear pair. Gears with such a pair rotate in the same directions

Now,

𝜔7

𝜔2=

𝜔7

𝜔6∙

𝜔6

𝜔5∙

𝜔5

𝜔4∙

𝜔4

𝜔3∙

𝜔3

𝜔2=

𝑁6

𝑁7∙ 1 ∙ −

𝑁4

𝑁5∙ 1 ∙ −

𝑁2

𝑁3=

𝑁6 ∙ 𝑁4 ∙ 𝑁2

𝑁7 ∙ 𝑁5 ∙ 𝑁3

𝜔7

𝜔2=

15𝑥20𝑥9

54𝑥51𝑥62= +0.0158 • Therefore, speed ratio of the given compound gear train is

+0.0158. • This means, the output gear (and shaft) will rotate in the

same direction as the input gear (shaft), and the output speed will be reduced by 0.0158 times the input speed.

∴ 𝜔7

𝜔2= +0.0158

MAE 364: Gears JT

In epicyclic gearing the axis of at least one gear, called the planet gear,

moves on a circular path relative to the base link.

It comprises of:

A centrally located sun gear (link 2) which has external teeth.

An outer ring gear which has internal teeth (link 5).

And one or more planet gears (link 4) whose external teeth mesh with the sun

gear and the ring gear.

An arm or crank (link 3) connecting the sun gear and planet gear.

Epicyclic (Planetary Gear Train)

MAE 364: Gears JT

For the epicyclic system shown:

Determine the unknown output speed?

Is the shown output rotation direction correct?

Example 4: Epicyclic Gearing

MAE 364: Gears JT

Solution:

Example 4: Epicyclic Gearing

Fixed Output Input

Gear C F Arm B A

Teeth 63 21 - 20 21

Rigid Motion 1 1 1 1 1

Relative Motion -1 -3 0 -3 2.86

Total Motion 0 -2 1 -2 3.86

𝐹𝑟𝑒𝑙 = −63

21∗ −1 = −3

𝐴𝑟𝑒𝑙 = +20

21∗ −3 = +2.86

𝜔𝑜𝑢𝑡

𝜔𝑖𝑛=

𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 𝑚𝑜𝑡𝑖𝑜𝑛

𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑝𝑢𝑡 𝑚𝑜𝑡𝑖𝑜𝑛=

1

3.86= 0.26

𝜔𝑜𝑢𝑡 = 0.26 ∗ 𝜔𝑖𝑛 = 0.26 ∗ 300 𝑟𝑝𝑚 = 77.78 𝑟𝑝𝑚

∴ 𝜔𝑜𝑢𝑡 = 77.78 𝑟𝑝𝑚 𝐶𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒

The output rotation direction shown in the figure is wrong

MAE 364: Gears JT

For the epicyclic system shown:

Determine the unknown output speed?

Is the shown output rotation direction correct?

Example 5: Epicyclic Gearing

MAE 364: Gears JT

Solution:

Example 5: Epicyclic Gearing

Fixed Input Output

Gear E D Arm B A C

Teeth 60 25 - 40 20 100

Rigid Motion 1 1 1 1 1 1

Relative Motion -1 -2.4 0 -2.4 4.8 -0.96

Total Motion 0 -1.4 1 -1.4 5.8 0.04

𝐷𝑟𝑒𝑙 = −60

25∗ −1 = −3

𝐴𝑟𝑒𝑙 = +40

20∗ −2.4 = +4.8

𝐶𝑟𝑒𝑙 = −40

100∗ −2.4 = −0.96

𝜔𝑜𝑢𝑡

𝜔𝑖𝑛=

𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 𝑚𝑜𝑡𝑖𝑜𝑛

𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑝𝑢𝑡 𝑚𝑜𝑡𝑖𝑜𝑛=

0.04

5.8=

1

145

𝜔𝑜𝑢𝑡 =1

145∗ 𝜔𝑖𝑛 =

1

145∗ 1800 𝑟𝑝𝑚 = 12.41 𝑟𝑝𝑚

∴ 𝜔𝑜𝑢𝑡 = 12.41 𝑟𝑝𝑚 𝐶𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒

The output rotation direction shown in the figure is correct.