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Theory of Machinery

Gears and Gear Trains

Dr. Ahmed Shaaban

Dr. Mohammed Al-Hazmi

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GEAR TRAINS

Kinematics & DynamicsA gear train consists of several gears that are

meshed together to transmit the power from

input shaft to the output one. The gear works

as reducer or accelerator.

a) Ordinary gear trainsb) Planetary or Epicyclic gear trains

c) Any combination of a and b

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1-Kinematic Analysis of Gear Trains

Gear ratio, transmission ratio =i in/out = in/out

2-Kinematic Synthesis of Gear Trains

3-Dynamic Analysis of Gear Trains

Study the flow of power (or mainly, the torque)

through the individual shafts

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Kinematic Analysis of Ordinary Gear Train

The ratio of the speed of the driver to the speed of thefollower is known as velocity ratio or speed ratio.

The reciprocal of the speed ratio (or velocity) is known

as train value

Types of gear trainsa)Simple gear train

b)Compound gear train

c)Reverted gear traind)Epicyclic gear train

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Velocity Ratio of a Simple Gear Train

2

3

3

2.

Z

Z

N

NratioSpeed ==

Simple gear train in which

each shaft carries only onegear

2

4

4

2

3

4

2

3

4

3

3

2

.

.

Z

Z

N

Nratiospeed

Z

Z

Z

Z

N

N

N

NratioSpeed

==

==

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1

3

3

1

2

3

1

2

3

2

2

1

.

.

Z

Z

N

Nratiospeed

Z

Z

Z

Z

N

N

N

NratioSpeed

==

==

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Problem: A simple gear train consists of three gears, each mounted on

separate shaft. Gear 1 is the driver and Gear 3 is the follower. The

gear 1 rotates CW at speed of 750 rpm and the number of teeth of

gears 1,2 and 3 are 30,45 and 75 respectively. Find,

i) Speed ratio of the gear trainii) direction of rotation of the follower.

Given: N1=750 rpm, Z1=30i) Speed ratio = speed of driver/ speed of follower = Z3/Z1 =75/30=2.5

ii) Speed of the follower N3N1/N3=2.5

N3= 750/2.5 =300 rpm

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Problem: Two parallel shafts are connected with two gears. The number of

teeth of one gear is 38 and speed of its shaft is 420 rpm. If the speed ratio

is equal to 3 and circular pitch of the gear is 25 mm. Find,

i) Number of teeth and speed of the other shaft

ii) Center distance between shafts.

Solution: Z1=38 , N1=420 rpm , speed ratio i =3 & p= 25mm

i) Speed rat io = N1/N2= Z2/Z1 =3

N2=N1/3=420/3=140 rpm

Z2

=Z1

x3= 3x38= 114

ii) Center distance C= r1+r2 =(d1+d2)/2

p=xd1/Z1d1=Z1xp/ = 38x25/ =302.39 mm

Z2= d2/p

d2= Z2xp/ =114x25/ =907.18 mm

Center distance C = (302.39+907.18)/2= 604.78 mm

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GEAR TRAINS

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Compound Ordinary Gear Train

4

5

3

4

2

3

5

2

44

33

4

5

3

4

2

3

5

4

4

3

3

2

===

=

=

==

Z

Z

Z

Z

Z

Z

N

N

N

Ni

NN

NN

Z

Z

Z

Z

Z

Z

N

N

N

N

N

Ni

output

input

3

4

2

3

4

2

33

3

4

2

3

4

3

3

2

===

=

==

Z

Z

Z

Z

N

N

N

Ni

NN

Z

Z

Z

Z

N

N

N

Ni

output

input

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a) Compound gear train

b) Reverted gear train

3

4

1

2

4

1

32

3

4

1

2

4

3

2

1

ZZ

ZZ

NN

N

Ni

NN

Z

Z

Z

Z

N

N

N

Ni

output

input ===

=

==

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Problem 10-1: Find the direction and speed of the output

shaft if the input shaft rotates with 1250 rpm.

rpmN

N

Z

Z

Z

Z

N

N

N

Ni

output

input

.200

25.6

1250

25.6

25.62030

5075

1

4

3

4

1

2

4

1

===

=

====

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Problem 10-2: Find the direction and speed of the output shaft, if the

input shaft rotates with 1125 rpm.

rpmN

N

Z

Z

Z

Z

Z

Z

N

N

N

Ni

output

input

.60

75.18

1125

75.18

75.18284030

7012075

1

4

5

6

3

4

1

2

6

1

===

=

====