Chapter Five 2

Chapter five: Power transmission system mechanism 2011

Chapter five: Power transmission system mechanism

5.1. Belt drive

5.1.1. Introduction

Belts are used for transmitting power from one shaft to another shaft. The belts may be flat belts,

V-belts etc. other method of transmitting power are ropes and chains. The belts are and ropes are

running over the pulleys. The pulleys are mounted on the two shafts.

Belts, ropes and chains are used where the distance between the shafts is large. For small

distances gears are used. Belts and ropes are not having constant velocity ratios whereas chains

and gears have constant velocity ratios.

The flat belt drives are the following important types:

1. Open belt drive,

2. Cross belt drive and

3. Compound belt drive.

Selection of a belt drive depends upon:

(i) Center distance between the shafts,

(ii) Power to be determined,

(iii) Speed of the driving and driven shafts and

(iv) Speed reduction ratio

Though there are many types of belts, yet the following types are important:

(i) Flat belt –rectangular in section as shown in figure 5.1(a)

(ii) V-belt –trapezoidal in section as shown figure 5.1 (b)

(iii) Circular belt or rope – circular in section as shown figure 5.1 (c).

a. b. c.

Fig 5.1. Types of belt

Adam University/SoE/Agricultural eng dept/ AgMe 422 lecture notes by SKB Page 1


5.1.2. Open flat belt

An open belt drive consisting pulleys A and B are shown in figure 5.2. The pulley which is

keyed to the rotating shaft is known as the driver pulley whereas the pulley which is keyed to the

shaft which is to be rotated is known as driven pulley or follower pulley. Here pulley A is driver

pulley and Pulley B is driven pulley.

Fig 5.2. Open flat belt

When the shafts are arranged in parallel and rotating in the same direction, open belt drive as

shown in figure 5.2 is used. The driver pulley A pulls the belt from lower side and drivers it to

the upper side. Hence the tension in the lower side of the belt will be more than the tension in the

upper side. The lower side belt is known as tight side whereas the upper side belt is as slack side.

5.1.2.1. Velocity ratio of open belt drive

The ratio of the velocity of the follower (or driven) to the velocity of driver is known as velocity

ratio. Mathematically, it is given as:

Let N1= speed of the driver in r.p.m.

d1=Diameter of the driver

N2 , d2=Speed and diameter of the follower respectively

Let us consider the length of the belt that passes over the driver and the follower in one minute.

Length of belt passing over the driver in one minute

=circumference of the driver x number of revolution per minute.



πd1∗N1

Similarly, length of belt passing over the following in one minute

Length of belt passing over the driver in one minute

=circumference of the driver x number of revolution per minute.

πd2∗N 2

But the length of belt passing over the driver in one minute is equal to the length of belt passing

over follower in one minute,

πd1∗N1=πd2∗N 2

Or

N 2

N1=

d1

d2

Velocity ratio,

N 2

N1=

d1

d2 5.1

Effect of belt thickness on velocity ratio

If belt thickness is considered, then for exact analysis the mean diameter of rotation is used for

the calculation of the velocity ratio. The mean diameter of rotation will be equal to the diameter

of the driver (or follower) plus the thickness of belt. Hence the radius of rotation will become as

the radius of the driver (or follower) plus half the belt thickness. If there is no slip between the

belt and the pulleys, then the peripheral speed of the two pulleys should be the same. Also this

will be equal to the velocity of belt.

V=ω1rm 1=ω2 rm 2 (i)

Where ω1=Angular velocity of the driver=

2 πN 1

60

ω2=Angular velocity of thedriven=2 πN2

60

rm1=Meanradius of rotation of driver pulley=r 1+

12

rm2=Meanradius of driven pulley=r2+12

t=Thickness of belt Equation (i) becomes, as



ω1(r1+t2 )=ω2(r 2+

t2 )

ω2

ω1=

r 1+t2

r 2+t2

=2r 1+t2r 2+t

=d1+ td2+ t

Or

N 2

N1=

d1+td2+t 5.2

Alternate method

Let N1= r.p.m of driver pulley

N2= r.p.m of driven pulley (follower)

d1=Diameter of driver pulley =2 r1

d2=Diameter of driven pulley =2 r2

ω1= Angular velocity of driver pulley =

2 πN 1

60

ω2= Angular velocity of driven pulley =

2 πN 2

60

If there is no slip between the belt and the pulleys and also thickness of the belt negligible, then

peripheral speed of the two pulleys should be same. Also this will be equal to the velocity of belt.

Hence if V is the velocity of belt, then

V=ω1r1=ω2 r2

Or

2 πN1

60∗

d1

2=

2 πN 2

60∗

d2

2 (ω1=2 πN1

60∧r1=

d1

2etc)

Or N1d1=N2 d2

Or

N 2

N1=

d1

d2


N 2

N1


In the above equation is known as velocity ratio. Also from the above equation it is clear that

speed of the pulley is inversely proportional to its diameter.

Problem 5.1. Find the speed of the shaft which is driven with the help of a belt by an engine

running at 200 r.p.m. The diameter of the engine pulley is 51 cm and that of the shaft is 30 cm.

Solution:

Given

Speed of engine (driver), N1=200 r . p .m .

Diameter of the engine (driver) pulley d1=51 cm .

Diameter of the driven shaft d2=30 cm .

Let N2= speed of driven shaft

Using equation (5.1), we get,

N 2

N1=

d1

d2

N2=d1

d 2∗N1=

5130

∗200=340 r . p. m .

Problem 5.2. If in the above equation, the thickness of the belt is 10mm, then find the speed of

the shaft.

Solution:

Given

Thickness, t=10 mm=1cm

Speed of engine (driver), N1=200 r . p .m .

Diameter of the engine (driver) pulley d1=51 cm .

Diameter of the driven shaft d2=30 cm .

Let N2= speed of driven shaft

Using equation (5.2), we get,

N 2

N1=

d1+td2+t

N2=( d1+ tdd2+t )∗N 1=(51+1

30+1 )∗200=335 . 4 r . p . m .



5.1.2.2. Slip of the belt

When the driver pulley rotates, it carries the belt, due to a firm grip between its surface and the

belt. The firm grip between the pulley and the belt is obtained by friction. This firm grip is

known as frictional grip. But sometimes the frictional grip is not sufficient. This may cause some

forward motion of the driver pulley without carrying the belt with it. This means that there is a

relative motion between the driver pulley and the belt. The difference between the linear speeds

of the pulley rim and the belt is a measure of slip. Generally, the slip is expressed as a

percentage. In some cases, the belt moves faster in the forward direction, without carrying the

driven pulley with it. Hence in case of driven pulley, the forward motion of the belt is more than

that of driven pulley.

Let v=Velocity of belt, passing over the driver pulley, rim

N1= r.p.m of driver pulley

N2= r.p.m of driven pulley (follower)

s1=Slip between the driver and the belt in percentage

s2=Slip between the follower and the belt in percentage

The peripheral velocity of the driver pulley

=ω1 r1

¿2πN1

60∗

d1

2

¿πN1 d1

60m /s

Now due to slip between the driver pulley and the belt, the velocity of belt passing over the

driver pulley will decrease

Velocity of belt=

πN1d1

60−( πN 1d1

60 )∗ s1

100

Velocity of belt=πN1 d1

60 (1−s1

100 )



Now this belt is passing over the follower pulley (i.e. driven pulley). And as there is a slip

between the belt and the driven pulley, the velocity (i.e. peripheral velocity) of the follower

pulley will decrease.

Peripheral velocity of follower

=Velocity of belt−(Velocity of belt∗s2

100 )=Velocity of belt∗(1−

s2

100 )=

πN1d1

60 [1− s1

100 ] [1− s2

100 ] (i)

∵Velocity of belt=πN1 d1

60 (1−s1

100 )But peripheral velocity of follower pulley

=ω2 r2

¿2 πN 2

60∗

d2

2 (∵ω2=2 πN2

60∧r2=

d2

2 )=

πN2d2

60 (ii)

Equating the two values given in equations (i) and (ii), we get

πN 2 d2

60=

πN1 d1

60 [1−s1

100 ][1− s2

100 ]N2 d2=N1 d1 [1−

s1

100 ] [1− s2

100 ]N2 d2=N1 d1 [1−

s2

100−

s1

100+

s1 s2

10000 ]N2 d2=N1 d1 [1−

s1+s2

100 ] (negelectings1 s2

10000 )N2 d2=N1 d1 [1− s

100 ]Adam University/SoE/Agricultural eng dept/ AgMe 422 lecture notes by SKB Page 7


(Where s=s1+s2 i.e. total percentage of slip between driver and follower)

N 2

N1=

d1

d2[1− s

100 ]7.3

If the belt thickness (t) is taken in to account, then

N 2

N1=( d1+t

d2+t )(1− s100 )

7.4

5.1.2.3. Creep of the belt

In a belt drive, the belt is passing over the driver pulley and driven pulley (i.e. follower). When

the power is transmitted with the help of this belt, the belt is subjected to tensions. The part the

belt which leaves the follower and approaches the driver is known as the tight side of the belt and

is subjected to tension T1. But the part of the belt which leaves the driver and approaches the

follower is known as slack side of the belt and subjected to tension T2. The two tensions are not

equal in magnitude, i.e. T 1>T 2 . Hence the stretch in the belt due to different tensions the on the

two sides of the pulley will be different. A certain portion of the belt when passes from slack side

to tight side, extends and the same portion contracts again when the belt passes from the tight

side to slack side. Due to these changes in length, there exists a relative motion between the belt

and the pulleys surfaces. This relative motion is known as creep. The creep reduces slightly the

speed of the follower (i.e. driven pulley).

The velocity ratio when creep is considered is given by,

N 2

N1=

d1

d2∗[ E+√ f 2

E+√ f 1 ]5.5

Where E=Young’s modulus for the material of the belt

f 1= Stress in the belt on the tight side, and



f 2= Stress in the belt on the slack side.

Problem 5.3. A shaft running at 200 r.p.m. is to drive a parallel shaft at 300 r.p.m. The pulley on

the driving shaft is 60 cm diameter. Calculate the diameter of the pulley on the driven shaft:

(i) Neglecting belt thickness.

(ii) Taking belt thickness into account, which is 5mm thick.

(iii) Assuming in the later case a total slip of 4%.

Solution:

Given

N1=200 r . p . m . N2=300 r . p . m . d1=60 cm & t=5 mm=0 .5 cm

Total slip s=4%

(i) Neglecting belt thickness

Let d2=Diameter of the pulley on driven shaft

Using equation (5.1)

N 2

N1=

d1

d2→d2=

N1

N2∗d1=

200300

∗60=40 cm

(ii) Taking belt thickness into account only,

Using equation (5.2), we get

N 2

N1=

d1+td2+t

Or

(d2+t )=( d1+t )∗N1

N2

(d2+0 .5 )=(60+0 .5 )∗200300

=60.5∗23

d2=60 .5∗23−0 .5=121

3−0.5



d2=40 . 33−0.5=39.83 cm

(iii) Considering the belt thickness and total slip.


N 2

N1=( d1+t

d2+t )(1− s100 )

300200 =(60+0 .5

d2+0.5 )(1−4

100 )

(d2+0 .5 )=60 . 5∗0. 96∗200300

=38 .72

d2=38 .72−0 .5=38 .22 cm

Problem 5.4. The power is transmitted from a pulley to 1m diameter running at 200 r.p.m. to a

pulley 2.5m diameter by means of a belt. Find the speed lost by the driven pulley as a result of

the creep, if the stress on the tight and slack side of the belt is 1.44 N/mm2 and 0.49 N/mm2

respectively. The Young’s modulus for the material of the belt is 100 N/mm2.

Solution:

Given

Diameter of the driver pulley, d1=1 m

Speed of the driver pulley, N1=200 r . p .m .

Diameter of the driven pulley, d2=2 .5 m

Stress on tight side, f 1=1 .44 N /mm2

Stress on slack side, f 1=0. 49 N /mm2

Young’s modulus, E=100 N /mm2

Let N2= speed of the driven pulley.




N 2

N1=

d1

d2∗[ E+√ f 2

E+√ f 1 ]N 2

200= 1

2 . 5∗[100+√0 . 49

100+√1 . 44 ]= 12. 5 [100 . 7

101 . 12 ]

N2=2002 .5 [100 .7

101 .12 ]=79 .67 r . p.m .

If the creep is neglected, then from equation (5.1), we get

N 2

N1=

d1

d2

N2=d1

d2∗N1=

12. 5

∗200=80 .r . p .m .

Speed lost by driven pulley due to creep

=80−79 . 67=0 .33 r . p .m .

5.1.2.4. Cross belt drive

In an open belt drive both the pulleys rotate in the same direction, whereas in the cross belt drive

they rotate in the opposite direction. Hence the cross-belt drive is used when the shafts are

arranged in parallel and rotate in the opposite direction as shown in figure 5.3. the driver pulley

A pulls the belt from CD side and delivers it to EF. Hence the tension in the belt CD will be

more than that in the portion FE. And so the belt CD is known as light side and portion FE is

known as slack side.



Fig 5.3. Cross belt drive

5.1.2.5. Compound belt drive

When the power is to be transmitted from one shaft to another through a number of pulleys, then

a compound belt drive is used. Figure 5.4 shows a compound belt drive, in which the pulley 1

drives the pulley 2. But pulleys 2 and 3 are keyed (or fixed) to the same shaft, therefore the

pulley 1 also drives pulley 3. But pulley 3 drives pulley 4 as shown in figure 5.4.

Fig 5.4 Compound belt drive

Velocity ratio of compound belt drive

Let N1=Speed of pulley 1 in r.p.m.

d1=Diameter of pulley 1

N2 , d2=Speed and diameter of pulley 2 respectively

N3 , d3∧N 4 , d4= Speed and diameter of pulleys 3 and 4 respectively.

From equation (5.1), we know that the velocity ratio of pulleys 1 and 2 is given by



N 2

N1=

d1

d2 (i)

Similarly, the velocity ratio of pulleys 3 and 4 is given by

N 4

N3=

d3

d4 (ii)

Multiplying equation (i) and (ii), we get

N 2

N1∗

N4

N3=

d1

d2∗

d3

d4

But N2=N3 as the pulleys 2 and 3 are keyed to the same shaft

N 4

N1=

d1

d2∗

d3

d4 5.6

The above equation can be written as

Speed of last followerSpeed of first driver

=Pr oduct of dia . of driversPr oduct of dia . of followers 5.7

Problem. With the help of a belt, an engine running at 200 r.p.m., drives a line shaft. The

diameter of the pulley on the engine is 80 cm and the diameter of the pulley on the line shaft is

40cm. A 100 cm diameter pulley on the line shaft drives a 20 cm diameter pulley keyed on a

dynamo shaft. Find the speed of the shaft when:

(i) There is no slip

(ii) There is a slip of 2.5% at each drive

Solution:

Given

Engine speed N1=200 r.p.m.

Diameter of the pulley on the engine, d1=80 cm

Diameter of the follower pulley, d2=40 cm

Diameter of the driver pulley on the shaft line, d3=100 cm

Diameter of the follower pulley on the dynamo shaft, d4=20 cm

Slip at each drive, s1=s2=2. 5



Let N4= speed of the dynamo shaft.

(i) When there is no slip


N 4

N1=

d1

d2∗

d3

d4

Or N4=

d1

d2∗

d3

d4∗N 1=

8040

∗10020

∗200=2000 r . p . m .

(ii) When there is a slip of 2.5% at each drive.

In this case, we will have the equation

N 4

N1=

d1

d2∗

d3

d4(1−

s1

100 )(1− s2

100 )

N4=

d1

d2∗

d3

d4(1−

s1

100 )(1− s2

100 )∗N1

N4=8040

∗10020 (1−2. 5

100 )(1−2 .5100 )∗200

N4=2000(97 . 5100 )(97 . 5

100 )=1901 .25 r . p . m .

5.1.2.6. Length of belt

The length of a belt means the total length of the belt required to connect a driver and follower.

There are two cases to be considered. They are:

(i) Length of an open belt drive, and

(ii) Length of cross-belt drive.



5.1.2.6.1. Length of an open belt drive

If the distance between the centers of the two shafts and the diameter of pulleys are known, the

required length of the belt may be easily calculated.

Fig 5.6. Length of open belt drive

The total length of an open belt is equal to the length of the belt not in contact with either pulley

+ the length of belt in contact with the larger pulley + length of belt in contact with smaller

pulley.

Let x= Distance between the centers of the two pulleys (i.e. length AB)

r1=Radius of the larger pulley

r2=Radius of the smaller pulley

L= Total length of belt.

Both the pulleys in this case will rotate in the same direction as shows in figure 5.6.

The belt leaves the larger pulley at C and D and smaller pulley at E and F. Join C and D with A.

Also join E and F with B.

From B draw BN parallel to EC. But CE is tangent at C. Hence AC is perpendicular to CE,

which means ∠ ACE=900 or

π2 . As BN is parallel to CE, hence ∠ ANB=900

or

π2 radians.

Let the angle ABN=α radians. Then angle ABN=( π

2−α)

. But angle BAK=900 or

π2

.

hence angle KAC=α . similarly, it can be shown that angle MBE=α .

Now length AN=AC−CN =r1−r2

BN=√ AB2−AN2=√ x2−(r1−r2)2



Or BN=CE=√x2−(r1−r2 )2

Also from triangle ABN

sin α= ANAB

=r1−r2

x

Since α is very small, hence sin α=α

α=sin α=

r1−r2

x (i)

Now total length (L) of the belt is given by,

L=Arc DGC+CE+ Arc EHF+FD

L=2 [ Arc GC+CE+ Arc EH ] [∵CE=FD ]

L=2 [r1 ( π2+α)+√x2−(r1−r2 )2+r2( π

2−α)]

(ii)

L=2 [ π2 (r1+r2)+α (r1−r2)+√x2−(r1−r2 )2]

5.7A

The above equation gives the exact length of the open belt drive. An approximate relation for the

length of the belt can also be obtained as given below:

As√ x2−(r1−r2)2=x √1−( r1−r2

x )2

√ x2−(r1−r2)2=x [1−( r1−r2

x )2 ]

1/2

√ x2−(r1−r2)2=x [1−1

2 (r 1−r 2

x )2

+ .. .. . .. ..] ( From Binomial expansion )

√ x2−(r1−r2)2=x [1− (r1−r2 )

2 x2

2] ( Neglecting smaller terms )

Substituting these values in equation (ii), we get



L=2 [r1 ( π2+α)+x (1−

(r1−r2)2

2 x2 )+r 2( π2−α)]

L=2 [ π2 (r1+r2)+α ( r1−r2)+(x−

(r1−r2 )2

2 x ) ]L=π (r1+r2)+2 α (r1−r2 )+2 x−

(r1−r2)2

x

But from equation (i), we have

α=r1−r2

x

Substituting this value of α in the above equation, we get

L=π (r1+r2)+2∗(r 1−r 2

x ) (r1−r2 )+2 x−(r1−r2)2

x

L=π (r1+r2)+ 2x (r1−r2)2+2 x−

(r1−r2)2

x

L=π (r1+r2)+(r1−r2)2

x+2 x

5.8

The above equation gives the approximate length of open belt drive. From the above equation,

we observe that the length of the open belt depends upon the sum and difference of the radii.

5.1.2.6.2. Length of cross-belt drive

The cross-belt is shown in figure 5.7.

Let x=Distance between the centers of the two pulleys, length AB,

r1=Radius of the larger pulley,

r2=Radius of the smaller pulley,

L= Total length of the cross-belt.



Fig 5.7 Length of cross-belt

Both the pulleys and in this case will rotate in the opposite directions as shown in figure 5.7.

The belt leaves the larger pulley at C and D and smaller at E and F. From B, draw BN parallel to

FC. Now angle ACF =∠ ANB=900. Let angle ABN=α . then it can be shown that angle

KAC=α and also angleEBM=α .

Length AN=AC+CN=r1+FB (∵CN=FB )

AN=r1+r2 (∵FB=r2)

In triangle ABN, BN=√ AB2−AN 2=√ x2−(r1+r 2)2

And sin α= AN

AB=

r1+r2

x

sin α=α

sin α=α=

r1+r2

x (i)

Now total length (L) of the belt is given by,

L=Arc DGC+CF+ Arc FHE+ED

L=2 [ Arc GC+CF+ Arc HE ] [∵ED=CF ]

L=2 [r1 ( π2+α )+BN +r2 ( π

2+α )] [∵CF=BN ]

L=2 [r1 ( π2+α )+√x2−(r1+r2)2+r2 ( π

2+α )]

(ii)



L=2 [ π2 (r1+r2)+α (r1+r2 )+√ x2−(r1+r2 )2]

5.8A

The above equation gives the exact length of the cross-belt drive.

An approximate relation for the length of the belt can be obtained as given below:

As√ x2−(r1+r2 )2=x √1−( r1+r2

x )2

√ x2−(r1+r2 )2=x [1−( r1+r2

x )2]

1/2

√ x2−(r1+r2 )2=x [1−1

2 ( r1+r2

x )2

+.. .. . .. ..] ( From Binomial expansion )

√ x2−(r1+r2 )2=x [1− (r1+r 2)

2 x2

2] ( Neglecting smaller terms )

Substituting this values in equation (ii), we get

L=2 [r1 ( π2+α )+x (1−

(r1+r 2)2 x2 )+r2( π

2+α )]

L=2 [ π2 (r1+r2)+α (r1+r2 )+(x−

(r1+r2 )2

2 x )]L=π (r1+r2)+2α (r1+r2)+2 x−

(r1+r2)2

x

But from equation (i), we have

α=r1+r 2

x

Substituting this value of α in the above equation, we get

L=π (r1+r2)+2∗(r 1+r2

x )(r1+r 2)+2 x−(r1+r2)2

x

L=π (r1+r2)+ 2x (r1+r2 )2+2 x−

(r1+r2)2

x



L=π (r1+r2)+(r1+r2 )2

x+2 x

5.9

The above equation gives the approximate length of cross-belt drive. From the above equation,

we observe that the length of the cross-belt depends only upon the sum of the radii.

Hence for the belt connecting two stepped pulleys, the length of the crossed belt is constant if the

sum of the radii of the corresponding steps is constant. If (r 1+r2 ) is constant, (r 1−r 2) is not a

constant hence this rule does not apply for an open belt.

Problem: Two parallel shafts 6 m apart are to be connected by a belt running over pulleys of

diameter 60 cm and 40 cm respectively. Determine the exact and approximate lengths of the belt

required:

(i) If the belt is open

(ii) If the belt is crossed.

Solution:

Given

Distance between the center of shafts (or of pulleys), x=6m=600 cm

Diameter of larger pulley =60cm

Radius of larger pulley, r1=

602

=30 cm

Diameter of smaller pulley =40 cm

Radius of smaller pulley, r2=

402

=20 cm

Let L= required length of the belt.

(i) If the belt is open

Exact length. Exact length for the open belt is given by equation (5.7A), as

L=2 [ π2 (r1+r2)+α (r1−r2)+√x2−(r1−r2 )2]

Where α for open belt drive is given by

sin α=r1−r2

x



α=sin−1( r1−r2

x )=sin−1(30−20600 )

α=sin−1(10600 )=sin−1 0 . 01666=0. 9550

α=0 . 9550∗ π180

radians=0 . 01666 radians

The value of α is 0.9550 which is very small. Hence α in radians will be equal to sinα. Here is

also α in radian = 0.01666 and sinα = 1/60 = 0.01666.

Substituting this value of α and also values of r1 and r2 and x in the above equation, we get exact

length as

L=2 [π2 (30+20 )+0 . 01666 (30−20 )+√6002− (30−20 )2 ]L=2 [ 25 π+0. 1666+599 . 916 ]=2 [678 .62 ]=1357 . 24 cm

Approximate length, using equation (5.8), we get

L=π (r1+r2)+(r1−r2 )2

x+2 x

L=π (30+20 )+(30−20 )2

600+2∗600=1357 .16 cm

(ii) If belt is crossed

Exact length: exact length of cross-belt drive is given by equation (5.8A) as

L=2 [ π2 (r1+r2)+α (r1+r2 )+√ x2−(r1+r2 )2]



Where α is given by,

α=sin−1 r1+r2

x=sin−1(30+20

600 )=sin−1 0 . 08333=4 .780

α=4 .780∗ π180

radians=0 . 08343 radians .

Substituting this value of α in the above equation, we get exact length as

L=2 [π2 (30+20 )+0 .08343 (30+20 )+√6002−(30+20 )2 ]L=2 [ 25 π+4 . 1715+597 .913 ]=2 [680 . 6243 ]=1361 .24 cm

Approximate length, using equation (5.9), we get

L=π (r1+r2)+(r1+r2 )2

x+2 x

L=π (30+20 )+ (30+20 )2

600+2∗600=1361 .31 cm .

5.1.2.6.3. Ratio of belt tensions

Figure 5.9 shows a driver pulley A and driven pulley B rotating in the clock-wise direction.

Figure 5.10 shows only the driven pulley B. Consider the driven pulley B.

Fig 5.9. Pulley A and B fig 5.10 Driven pulley B

Let T 1= Tension in the belt on the tight side



T 2= Tension in the belt on the slack side

θ=Angle of contact, i.e. the angle subtended by the arc EF at the center of the

driven pulley,

μ= Co-efficient of friction between the belt and pulley.

The ratio of the two tensions may be found by considering an elemental piece of the belt MN

subtending an angle δ at the center of the pulley as shown in figure 5.10. The various forces

which keep the elemental piece MN in equilibrium are:

(i) Tension T in the belt at M acting tangentially,

(ii) Tensions T+δT in the belt at N acting tangentially,

(iii) Normal reaction R acting outward at P, where P is the middle point of MN,

(iv) Frictional force F=μR acting at right angles to R and in the opposite direction of

the motion of pulley.

Now anglePBM= δθ

2 . Also angle TPF= δθ

2

Resolving all the forces acting on the belt MN in the horizontal direction, we get

R=T sin δθ2

+(T+δT ) sin δθ2

Since the angle δ is very small, sin δθ

2 can be written as

δθ2 . Hence the above equation

becomes as

R=T δθ2

+(T+δT ) δθ2

R=T δθ2

+T δθ2

+δT δθ2

R=T δθ+δT δθ2

R=T δθ (Neglecting the small quantity δT∗δθ2 )

(i)

Now resolving all the forces vertically, we get

F=(T+δT ) cos δθ2

−T cos δθ2



Since δ is very small, hence cos δθ

2 reduces to unity i.e. 1. Hence the above equation becomes

as

F=(T+δT )−T=δT

Or μR=δT (∵F=μR )

Or R=δT

μ (ii)

Equating the two values of R given by the equations (i) and (ii), we get

T∗δθ=δTμ

Integrating the above equation between the limits T2 and T1, we get

∫T2

T 1 dTT

=∫ μ∗dθ=μ∫ dθ

Or

log e

T 1

T 2=μ∗θ

Or

T1

T2=eμ∗θ

5.10

In equation (5.10), should be taken in radians. Here is known as angle of contact. For an

open belt or a crossed belt the angle of contact is determined.

5.1.2.6.4. Angle of contact for open belt drive

With open belt drive, the belt will begin to slip on the smaller pulley, since the angle of lap is

smaller on this pulley than on the large pulley. The angle should be taken as the minimum



angle of contact. Hence in equation (5.10), the angle of contact of lap () at the smaller pulley

must be taken into consideration.

With the reference to figure 5.6 for open belt, the angle of lap on the smaller pulley is equal to

angle EBF which is equal to(1800−2α ) .

Angle of contact, θ=(1800−2 α ) 5.11

But the value α is given by,

α=r1−r2

x 5.12

Where r1=Radius of larger pulley,

r2=Radius of smaller pulley andx=Distance between the centers of two pulleys.

5.1.2.6.5. Angle of contact for crossed belt drive

For a crossed belt, the angle of lap on both the pulleys is same. With the reference to figure 5.7

for crossed belt, the angle of lap on the smaller pulley or larger pulley is equal to(1800+2 α ) .

Angle of contact, θ=(1800+2 α ) 5.13

But the value α is given by,

α=r1+r 2

x 5.14

Where r1=Radius of larger pulley,

r2=Radius of smaller pulley andx=Distance between the centers of two pulleys.

5.1.2.6.6. Power transmitted by belt

Let T 1= Tension in the tight side of the belt,

T 2= Tension in the slack side of the beltv=Velocity of the belt in m/s



The effective tension or force acting at the circumference of the driven pulley is the difference

between the two tensions (i.e.T 1−T 2 ).

Effective driving force =(T1−T 2)

Work done per second =force∗Velocity

=(T1−T 2)∗v Nm/ s

Power transmitted =(T1−T 2)∗vWatts 5.15

P=(T 1−T2 )∗v1000

kW5.16

Equation (5.15) gives the power in Watts, whereas equation (5.16) gives the power in kW. In the

case equation (5.16), the tensions T1 and T2 are taken in Newton’s.

Torque exerted on the driving pulley =( T1−T 2)∗r1 5.17

And Torque exerted on the driven pulley =(T1−T 2)∗r2 5.18

Example: A belt is running over a pulley of diameter 120 cm at 200 r.p.m. The angle of contact

is 1650 and co-efficient of friction between the belt and the pulley is 0.3. if the maximum tension

in the belt is 3000 N, find the power transmitted by the belt.

Solution:

Given

diameter of pulley, d=120 cm=1. 2m

Speed of the pulley, N=200r . p .m .

Angle of contact, θ=1650=1650∗ π

180radians . (∵10= π

180radian)

Co-efficient of friction, μ=0 .3

Maximum tension, T 1=3000 N

Velocity of belt, v=π dN

60= π∗1 . 2∗200

60=12 .56 m / s

Let T 2= Tension on the slack side of the belt.




T1

T2=eθ∗μ

T1

T2=e

165∗ π180∗0.3

=e0 .8635 =2. 3714

3000T2

=2. 3714

T 2=

30002 .3714

=1265 N

Power transmitted is given by equation (5.16) as,

P=(T 1−T2 )∗v1000

P=(3000−1265 )∗12 .561000

=21. 79 kW .

5.1.2.6.7. Centrifugal tension

The tension caused in the running belt by the centrifugal force is known as centrifugal tension.

Whenever a particle of miss m is rotated in a circular path radius r at a uniform velocity v, a

centrifugal force is acting radially outward and its magnitude is equal to

mv2

r, where m is mass

of the particle.

The centrifugal tension in the belt can be calculated by considering the forces acting on a

elemental length of the belt (i.e. length MN) subtending an angle δ at the center as shown in

figure 5.11.



Let v = velocity of belt in m/s

r = Radius pulley over which belt runs

M = Mass of elemental length of belt MN

m = Mass of belt per meter length

TC = Centrifugal tension acting at M and N tangentially

R = Centrifugal force acting radial outward

Fig 11. Centrifugal tension

The centrifugal force R acting radially outwards is balanced by the components of TC acting

radially inwards. Now elemental length of belt MN = r x δ.

Mass of belt MN

= (mass per meter length) x length of MN

M=m∗r∗δθ

Centrifugal force, R=M∗v2

r=m∗r∗δθ∗v2

r



Now resolving the forces horizontally, we get

T C sin δθ2

+T C sin δθ2

=R

Or 2 TC sin δθ

2=m∗r∗δθ∗v2

r

As the angle δ is very small, hence

sin δθ2

=δθ2

Then the above equation becomes as

2 TC∗δθ2

=m∗r∗δθ∗v2

r

T C=m∗v2

5.19

Note (i). From the above equation, it is clear that the centrifugal tension is independent of T1

and T2. It depends upon the velocity of the belt. For lower belt speed (i.e. belt speed less than 10

m/s) the centrifugal tension is very small and may be neglected.

(ii). When centrifugal tension is to be taken in to consideration then total tensions on tight side

and slack side of the belt is given as

For tight side 5.20

For tight side 5.21

(iii). Maximum tension (Tm) in the belt is equal to maximum safe stress in the belt multiplied

by cross-sectional area of the belt.


=T 1+T C

=T 2+T C

∵T m= f∗(b∗t )


5.22

Where Maximum safe stress in the belt

b=Widthof the belt

t=Thickness of the belt

Then if centrifugal tension is to be considered

If centrifugal tension is to be neglected

5.1.2.6.8. Maximum power transmitted by a belt

Let T 1= Tension in the tight side of the belt,

T 2= Tension in the slack side of the beltv=Velocity of the belt in m/s

Then the power transmitted is given by equation (5.15) as

P=(T1−T 2)∗v (i)

But from equation (5.10), we know that

T1

T2=eμ∗θ

Or T 2=

T 1

eμ∗θ

Substituting the value of T2 in equation (i) , we get

P=(T 1−T 1

eμ∗θ )∗v


f =

T m=T 1+T C−−−−−

T m=T 1+T C−−−−−


P=T 1(1− 1eμ∗θ )∗v

(ii)

Let (1−

1e μ∗θ )=k

The above equation becomes as

P=T 1∗k∗v or kT 1 v (iii)

Let Maximum tension, and

Centrifugal tension which equal to

Then

T max=T 1+T C

T 1=T max−T C

Substituting this of T1in equation (iii), we get

P=k (T max−T C)∗v

P=k (T max−mv2 )∗v ∵T C=mv2

P=k (T max v−mv3 ) (iv)

The power transmitted will be maximum, if

dPdv

=0

Hence differentiating equation (iv) with respect to v and equating to zero for maximum horse

power, we get

dPdv

=k (T max v−mv3)=0

Or T max−3 m∗v2=0

Or T max=3m∗v2(v)

Or v=√ T max

3 m 5.23


T max=

mv2T C=


Equation (5.23) gives the velocity of the belt at which maximum power is transmitted. From

equation (v),

T max=3 T C ∵T C=mv2

Hence when the power transmitted is maximum, centrifugal tension would be

13 of the

maximum tension. We also know that

T max=T 1+T C

T max=T 1+T max

3 (∵T C=T max

3 )T 1=T max−

T max

3

T 1=23

T max(vi)

Hence conditions for the transmission of maximum power are:

(i)T C=1

3T max

and

(ii)T 1=

23

T max

The maximum power transmitted is obtained by substituting the values of T1 from (vi) and v

from equation (5.23) in equation (ii).

Maximum power =

23 T max (1−

1eμ∗θ )∗√ T max

3 5.24

Example: A belt embraces the shorter pulley by an angle of 1650 and runs at a speed of 1700

m/min. Dimensions of the belt are: width = 20 cm and thickness = 8mm. Its density is 1 gm/cm3.

Determine the maximum power that can be transmitted at the above speed, if the maximum

permissible stress in the belt is not to exceed 250 N/cm2 and = 0.25.

Solution: Given

Angle of contact, θ=1650=165∗ π

180=2 . 88rad



Speed of belt, v=1700 m /min=1700

60=28 .33 m /s

Width of belt, b=20 cm

Thickness of belt, t=8mm=0 . 8cm

Density of belt, ρ=1 gm /cm= 1

1000kg /cm3

Maximum permissible stress, f =250 N /cm2

Value of μ=0 .25

Let us first find the maximum tension and centrifugal tensions in the belt. Using equation (5.22),

maximum tension (Tm) is given by:

T m=f∗( Areaof belt )=f ∗b∗tT m=250∗20∗0.8=4000 N .

Centrifugal tension (TC) is given by equation (5.19),

T C=m∗v2

Where m=mass of belt per meter length

m=ρ∗Volumeof belt of 1 mlengthm=ρ∗( areaof belt∗length)

m=ρ∗(b∗t∗L )

m= 11000

∗(20∗0 . 8∗100 ) kg (∵Length of 1 m=100 cm )m=1 . 6 kg

Substituting this value in the above equation (i), we get

T C=1.6∗28 .332=1284 N

But we know that,

T m=T 1+T CT 1=T m−T C=4000−1284=2716 N

Let T 2= Tension on the slack side of the belt. Using equation (5.10), we get

T1

T2=eμ∗θ=e0 . 25∗2. 88=2 .056



T 2=

T 1

2 .056=2716

2 . 056=1321 N

Now the maximum power transmitted is given by,

P=(T 1−T2 )∗v1000

=(2716−1321 )∗28. 331000

=39 . 52 kW .

Example: A belt of density 1 gm/cm3has a maximum permissible stress of 250 N/cm2. Determine

the maximum power that can be transmitted by a belt of 20 cm x 1.2 cm if the ratio of the tight

side to slack side tension is 2.

Solution: given

Width of belt, b=20 cm

Thickness of belt, t=1 .2cm

Density of belt, ρ=1 gm /cm= 1

1000kg /cm3

Maximum permissible stress, f =250 N /cm2

Ratio of the tension,

T1

T2=2.0

Let us find the mass of 1m length of the belt and also the maximum tension in the belt.

Let m=Mass of one meter length of belt

m=ρ∗Volumeof belt of 1 mlengthm=ρ∗( areaof belt∗length)

m=ρ∗(b∗t∗100 ) (∵Length of 1 m=100 cm )

m= 11000

∗(20∗1.2∗100 ) kg

m=2 .40 kgUsing equation (5.22),

T m=MaximumTension

T m=( Max . stress )∗Area of cross−sec tion of belt

T m=250∗b∗t=250∗20∗1 .2=6000 N .

Now for maximum power transmitted, the velocity of the belt is given by equation (5.23) as



v=√ T m

3 m=√60003∗2 . 4=28 . 86m /s

Now maximum power transmitted is given by equation,

P=(T 1−T2 )∗v1000 (i)

Let us find the values of T1 and T2

We know that T m=T 1+T C

Where T C= Centrifugal tension, and

T 1= Tight side tension

But for maximum power transmission,

T C=13

T m

Substituting this value in equation (ii), we get

T m=T 1+13

T m

T 1=T m− 13

T m= 23∗6000=4000 N

But

T1

T2=2

(Given)

T 2=

T 1

2=4000

2=2000 N

Substituting the values of T1, T2 and v in equation (i), we get

P=( 4000−2000 )∗28 .861000

=57 .18 kW .

5.2. Gears

5.2.1. Introduction

The motion from one shaft to another shaft may be transmitted with belts, ropes and chains.

These methods are mostly used when the two shafts are having long center distance. But if the

distance between the two shafts is small, then gears are used to transmit motion from one shaft to



another. In case of belts and ropes, the drive is not positive. There is slip and creep which

reduces velocity ratio. But gear drive is a positive and smooth drive, which transmits exact

velocity ratio. The gear is defined as toothed element which is used for transmitting rotary

motion from one shaft to another.

From small power transmission, the friction wheels as shown in figure 5.12 can be used. These

wheels are mounted on the two shafts, having sufficient rough surfaces and pressing against each

other.

Fig 5.12. Friction wheels

The friction wheel 1 is keyed to the rotating shaft whereas the friction wheel 2 is keyed on the

shaft wheel is to be rotated. When the friction wheel 1 rotates, it will rotate the friction wheel 2

in the opposite direction as shown if figure 5.12. There will be no slip between the two wheels

for small power transmission. Hence the same motion can be transmitted easily as explained

below:

For no slip of the two surfaces (i.e. no relative motion between the two surfaces of the friction

wheels), their tangential velocities at the contact surfaces should be same.

i.e. V 1=V 2

Or ω1 r1=ω2 r 2 (∵V 1=ω1 r1∧V 2=ω2 r2)

Or

2 πN1

60∗r1=

2πN 2

60∗r2 (∵ω=2πN

60 )



Or

N1

N 2=

r2

r1(or=

d2

d1)

5.25

Where N1=Speed of friction wheel 1 in r.p.m.

N2=Speed of friction wheel 2 in r.p.m.

r1=Radius of wheel 1, and

r2=Radius of wheel 2.

From equation (5.25), it is clear that if two wheels are rotating without slip then the speeds the

two wheels will be inversely proportional to their radii. In order to prevent the slip the two

surfaces, a number of projections (known as length) are provided on the periphery of the wheel

1, which will be fit in to the corresponding recesses will much with each other and slip between

them will be prevented.

Fig 5.13

The friction wheel with teeth cut on it is known as gear wheel or gear. The motion between the

two friction wheels is rolling whereas the motion between the gear is sliding.

Note (i). If the wheel 1 rotates in clockwise direction, the wheel 2 will rotate in anti clockwise

direction.



(ii). The friction wheels are rotating without slip and tooth gears are identical.

(iii). Friction wheels are used for small power transmission. For large power by friction

wheels, slip occurs. Hence definite motion can not be transmitted.

5.2.2. Classification gears

The following are the important classification of gears:

1. Classification based on the position axes of the shafts

The axes of two shafts between which motion is to be transmitted may be

(a) Parallel shafts

(b) Intersecting, and

(c) Non-parallel and non-intersecting

(a). parallel shafts

The following are the main types of gears to join parallel shafts

(i) Spur gears

(ii) Helical gears

(iii) Double helical gears

(i) Spur gear

The gears used to connect two parallel shafts and having straight teeth parallel to the axes of the

wheel as shown in figure 5.14 are known as spur gears. Figure 5.14 (a) the gears have external

teeth on the outer surfaces and the two shafts rotate in opposite direction. In figure 5.14 (b), the

internal teeth are formed over the outer wheel and external teeth are formed over the inner wheel.

The inner wheel having external gears (smaller gear) is known as pinion. The two shafts will

rotate in the same direction. In spur gears, the contact occurs across a line. Hence spur gears are

having line contact.



Fig 5.14 Spur gears

(ii) Helical gears

The gears used to connect two parallel shafts and having teeth inclined (or curved) to the axes of

the shafts as shown in the figure 5.15 are known as helical gears. In helical gears each teeth is

helical in shape. The two matting gears have the same helix angle, but have teeth opposite hands.

In helical gears, the contact occurs at a point of curved teeth at the beginning of engagement and

afterwards extends along a diagonal line across the teeth.



Figure 5.15 Helical gears

(iii) Double helical gears

A pair of helical gears secured together, one having a right-hand helix and the other a left-hand

helix is known as double helical gears as shown in figure 5.16. This pair is mounted on one shaft

and the similar other pair (having teeth of opposite hand) is mounted on other parallel shaft is to

be connected.

The double helical gear is known as herringbone gear if the left and the right inclinations of the

double helical gears meet at a common apex and there is no groove in between as shown in the

figure 5.16.

Fig 5.16 Double helical gear

(b) Intersecting shaft

The gears used to connect two intersecting shafts are known as bevel gears. If the teeth on the

gears are straight radial to the point of intersection of shaft axes then gears are known as straight

bevel. But if teeth are inclined then gears are known as helical bevel (or spiral bevel) as shown in

figure 5.17 and 5.18.



Fig 5.17 intersecting shaft

Fig 5.18 Bevel gears

(c). Non-parallel and non-intersecting shafts

The gears used to connect two non-parallel and non-intersecting shafts, are known as skew bevel

gears or skew spiral gears.

2. Classification based on the peripheral velocity of the gears

According to the peripheral velocity, the gears are classified as:

(a) Low velocity gears



(b) Medium velocity gears, and

(c) High velocity gears.

If the velocity of the gears is less than 2 m/s, they are known as low velocity gears. For medium

velocity gears, the velocity of the gears are between 3 m/s and 15 m/s. but if the velocity of gears

are more than 15 m/s, the gears are known as high velocity (or high speed) gears.

3. Classification based on the position of teeth on the wheel

According to the position of the teeth on the wheel, the gears are classified as:

(a) Straight

(b) Inclined and

(c) Curved

The spur gears have straight teeth, helical gears have inclined teeth (which are inclined to the

wheel rim surface) and spiral gears have curved teeth over the rim surface.

4. Classification based on type of gearing

(a) External gear

(b) Internal gear and

(c) Rack and pinion

If the gears of the two shafts mesh externally with each other, then the gears are known, external

gears as shown in figure 5.14(a). For external gears, the two shafts rotate in opposite direction.

The smaller wheel is known as pinion whereas the larger wheel is called spur wheel.

For internal gears, the two gears mesh internally with each other and the shafts rotate in the same

direction as shown in figure 5.14 (b). The smaller wheel is known pinion whereas the larger

wheel is known as annular wheel.

If the gear of a shaft meshes with the gears in a straight line (a wheel of infinite radius is known

as a straight line), then the gears are known as rack and pinion as shown in figure 5.19.



Figure 5.19 Rack and pinion

The straight line gear is known as rack (rack is shown in figure 5.19 (a)) and the circular wheel is

called pinion. The rack and pinion combination converts rotary motion in to linear motion or vice

–versa.

Fig 5.19 (a) Rack

5.2.3. Definition of the terms used in gears

Figure 5.20 shows the profile of a gear along with important terms, which are used in the study

of the gears.



Fig 5.20 the profile of a gear along with important terms

The terms are defined as:

(i) The pitch circle diameter: it is the diameter of a circle which by pure rolling action

would produce the same motion as the toothed gear wheel. It is also known as pitch

diameter.

(ii) Pitch point: it is the point of contact of two pitch circles of the mating gears.

(iii) Circular pitch (pc): it is the distance measured along the circumference of the pitch

circle from a point on one tooth to a corresponding point on the adjacent tooth. It will

be equal to the pitch circle circumference divided by the number of teeth on the

wheel. It is denoted by pc. hence

Circular pitch=πDT

or pc=πDT 5.26

Where D= diameter of pitch circle

T=Number of teeth

pc=Circular pitch



(iv) The diameteral pitch (pd): it is equal to the number of teeth per unit length of pitch

circle diameter. It is denoted by pd

pd=TD 5.27

If we multiply equation (5.26) and (5.27), we get

pc∗pd=π 5.28

(v) Module (m): it is defined as the length of the pitch circle diameter per tooth. It is

denoted by m. hence

m= DT 5.29

m is generally expressed in millimeter. Module is the reciprocal of diameteral pitch. Equation

(5.26) can be written as:

pc=π∗D

T=m∗π

5.29(a)

(vi) The addendum. It is the radial distance of the tooth above pitch circle. Its value is

generally one module. It is denoted by ‘a’.

(vii) The dedendum: it is the radial distance of the tooth below the pitch circle. Its value is

generally 1.157 modules or (1+ π

20 )module.

(viii) Addendum circle: it is the circle which passes through the top of the teeth. Diameter

of addendum circle= p . c . d+2 m .

(ix) Dedendum circle: it is the circle which passes through the bottom of the teeth.

Diameter of dedendum circle= p . c . d+2+1 .157 m .

(x) Face of the teeth: it is that part of tooth surface which is above the pitch circle.

(xi) Flank of the tooth: it is that part of the tooth surface which is below the pitch circle.

(xii) Path of contact: It is the curve traced by the point of contact of the two mating teeth

from the beginning to the end of engagement of the two teeth.

(xiii) Path of approach: It is the path of contact from the beginning of the engagement to

the pitch point of the two mating teeth.



(xiv) Path of recess: It is the path of contact from the end of the engagement of two mating

gears.

(xv) Pressure angle: It is the angle which the common normal to the two teeth at the point

of contact makes with the common tangent to the two pitch circle at the pitch point.

(xvi) Clearance: The radial height difference between the addendum and dedendum is

known as clearance. Hence, clearance1 .157 m−m=0 .157 m .

(xvii) Profile: The curve forming face and flank is known as profile.

(xviii) Pinion: It is the smaller and usually the driving gear of the pair mated gear.

(xix) Rack: A gear wheel of the infinite diameter is known as rack as shown in figure

5.19(a).

(xx) Gear ratio (G): the ratio of number of teeth on the gear to that on the pinion is known

as gear ratio. It is denoted by G. hence

G=Tt 5.30

Where T=Number of teeth on the gear

t= Number of teeth on the pinion and

G= Gear ratio.

(xxi) Velocity ratio (V.R): The ratio of the angular velocity of the follower to the angular

velocity of the driving gear is known as velocity ratio. It is denoted by V.R. Hence

velocity ratio,

V .R .= Angular velocity of followerAngular velocity of driver

V . R .=ω2

ω1 (ω2=2 πN2∧ω1=2 πN1)

V .R .=ω2

ω1

V .R .=d2

d1(∵πd1 N1=πd2 N2∵

d1

d2=

N2

N1)



V . R .=T 1

T 2(∵ pc=

πd1

T 1=

πd2

T2∵

d1

d2=

N 2

N1)

5.31

(xxii) Pressure line (or line of action): The common normal at the point of contact of the

mating gears is known as pressure line (or line of action). This is also the line from

the pitch point to the point of contact of two gears. Actually the forces is transmitted

from the driving tooth to the driven tooth along this line.

Example: Determine the number of teeth and speed of the driver if the driven gear has 60 teeth

of 8mm module and rotates at 240 r.p.m. The two spur gears have a velocity ratio of

14 . Also

calculate the pitch line velocity.

Solution:

Given

Number of teeth on driven gear, T 2=60

Speed of driven gear, N2=240 r . p . m .

Module, m=8 mm

Velocity ratio, V . R .= 1

4

Let T 1= No. of teeth on the driver gear,

N1=Speed of driver gear

Using the equation (5.31), we get

V . R .=N2

N1=

T 1

T 2

14=

N2

N1

N1=4∗N 2=4∗240=960 r . p. m .

Also V . R .=

T 1

T 2

T 1=V . R .∗T 2=4∗60=15.



Pitch line velocity (Vp) is given by

V p=ω1 r1 or ω2r2

V p=2πN1

60∗

d1

2

V p=2 πN1

60∗

mT1

2

V p=2 π∗96060

∗8∗152

V p=6031 . 8 mm/s=6 .0318 m /s

5.2.4. Law of gearing or condition for constant velocity ratio of gear wheels

Law of gearing states that the common normal to the two surfaces at the point of contact

intersects the line joining the centers of rotation of the two surfaces at a fixed point, which

divides the center distance inversely as the ratio of angular velocities.

Let the two curved bodies 1 and 2 are rotating about the centers A and B as shown figure 5.21.

The two bodies are contact at point C. the body 1 is rotating clockwise with angular velocity ω1

and the body 2 is rotating anticlockwise with angular velocityω2 . Let t-t is the common tangent

to the curves at point C and n-n is the common normal to the two surfaces at point C.

Let V 1= linear velocity of point C when point is assumed to be on the surface of the body 1. This

velocity will be to the line AC and will be equal to ω1∗AC .

V 2=Linear velocity of point C when point C is assumed to be on the surface of the body 2. This

velocity will be to the line BC and will be equal to ω2∗BC .

Join the two centers A and B which cuts the common normal at point P.



Fig 5.21

Let θ=Angle made by V 2 with common normal n-n.

φ=Angle made by V 1 with normal n-n.

From B draw BD on n-n and from A draw AE on n-n.

As V 2 is perpendicular BC hence BCV 2 =900. Now

∠DCB=180−∠BCV 2−θ=180−90−θ=90−θ

Similarly as V1is perpendicular to AC, hence ACV1=900. Now

∠ ACE=180−∠ ACV 1−φ=180−90−φ=90−φ

In BDC, DCB = 900 - DBC =

In AEC, ACE = 900 - ϕ CAE = ϕ

If the two surfaces are to remain in contact, one surface may slide relative to the other along a

common tangent t-t. But relative motion between the surfaces along the common normal n-n

must be zero to avoid separation of the two surfaces or the penetration of the two surfaces in to

each other.



Component of V1 along normal n-n

=V 1 cosφ


=V 2 cosθ

Relative motion along normal n-n

=V 1 cosφ−V 2 cosθ

For proper contact,

Relative motion along normal = zero

Or V 1 cos φ−V 2 cosθ=0

Or V 1 cos φ=V 2 cosθ

Or (ω1∗AC )cosφ=(ω2∗BC )cosθ

(∵V 1=ω1∗AC∧ω2∗BC )

Or (ω1∗AC )∗AE

AC=(ω2∗BC )∗BD

BC

(∵ In Δ ACE , cosφ= AEAC

∧in ΔBDC , cosθ=BDBC )

Or ω1∗AE=ω2∗BD

Or

ω1

ω2=BD

AE

ω1

ω2=BP

AP

5.32



The above equation shows that the common normal to the two surfaces at the point of contact

divides the line joining the centres of rotation in the inverse ratio of the angular velocities. But

this ratio of angular velocities must be constant for all positions of the wheel. This will be, if the

point P is fixed point (i.e. pitch point).

Thus for constant angular velocity ratio of the two surfaces, the common normal at the point of

contact must pass through the pitch (fixed point) on the line joining the center of rotation. This

statement is the law of gearing.

The s BDP and AEP are similar

BPAP

= DPPE

Substituting the value of in equation (5.32), we get

ω1

ω2= DP

PE

Or ω1∗PE=ω1∗DP

5.33

5.2.5. Velocity of sliding


BPAP


If the two surfaces are to remain in contact, one surface may slide relative to other along the

common tangent t-t of figure 5.21.

The velocity of sliding is the velocity of one surface relative to the other surface along the

common tangent at the point of contact. Referring to figure 5.21.

Component of V1 along tangent t-t

=V 1 sin φ


=−V 2 sin θ

(-ve sign is due to opposite direction)

Velocity of sliding = Relative velocity between two surface along tangent t-t

=V 1 sin φ−(−V 2 cosθ )

=V 1 sin φ+V 2 cosθ

=(ω1∗AC )sin φ+ (ω2∗BC )sin θ

(∵V 1=ω1∗AC∧V 2=ω2∗BC )

=(ω1∗AC )∗ EC

AC=(ω2∗BC )∗ DC

BC

(∵ In Δ ACE , sin φ= ECAC

∧in Δ BDC , sinθ= DCBC )

=ω1∗EC=ω2∗DC

=ω1∗( PC−PE )=ω2∗(DP+PC )

(∵EC=PC−PE∧DC=DP+PC )

=ω1∗PC−ω1∗PE+ω2∗DP+ω2∗PC



=ω1∗PC+ω2∗PC

(∵From equation(5 .33 ) ω1∗PE=ω2∗DP

Velocity of sliding=(ω1+ω2) PC5.34

The above equation shows that the velocity of sliding is equal to the product of the sum of the

angular velocities and the distance from the point of contact to the point of intersection of the

common normal and the lining joining the centers of rotation (i.e. pitch point).

5.2.6. Length of path of contact

Figure 5.22 shows the two gear wheels in contact, with A and B as centers. The pinion is

rotating in clockwise direction and is driving the wheel in anticlockwise direction. Hence the

pinion is the driver and the wheel is driven. The two pitch circles meeting at point P. The line

CD is the common tangent to the two base circles. This line is also known as line of action.

The point E is the intersection of the common tangent and addendum circle of the wheel. The

point F is the intersection of common tangent and addendum circle of pinion. The contact of two

teeth begins where the addendum circle of the wheel meets the common tangent (i.e. point E)

and ends where the addendum circle of the pinion meets the common tangent (i.e. point F). The

line EF gives the length of path contact. The length EP is known as path of approach whereas

length PF is known as path of recess.



Fig 5.22 Length of path of contact

Let r = Radius of pitch circle of pinion i.e. length PA

R = Radius of pitch circle of wheel i.e. length of BP

ra = Addendum circle radius of pinion i.e. length AF

Ra = Addendum circle radius of wheel i.e. length BE

From figure 5.22, we know that

Length of path of contact = path of approach + path of recess

Or EF=EP+PF

EF=( ED−PD )+(CF−CP )(i)

Let us now find the values of ED, PD, CF and CP.

In right angled triangle BPD,



PD=BPsin φ=R sin φ

BD=BPcos φ=R cos φ

In right angled triangle BED,

ED=√ ( BE )2− (BD )2=√Ra2−R2 cos2φ (∵BD=R cos φ )

In right angled triangle ACP,

CP=PA sin φ=r sin φ (∵PA=r )

CA=PA cosφ=r cosφ

In the right angled triangle FCA,

CF=√ ( AF )2− (CA )2

CF=√ra2−r2 cos2φ

Path of approach

EP=ED−PD

EP=√Ra2−R2cos2 φ−R sin φ

5.34 (b)

Path of recess

PF=CF−CP

PF=√ra2−r2cos2 φ−r sin φ

5.34 (c)

Length of path of contact,

EF=(√Ra2−R2cos2 φ−R sin φ)+√ra

2−r 2cos2 φ−r sin φ

EF=(√Ra2−R2cos2 φ−R sin φ)+√ra

2−r 2cos2 φ−r sin φ5.35



The above equation gives the length of path of contact. If dimensions of driven wheel are given

then path of approach can be calculated. Similarly is dimensions of the driving wheel (pinion)

are known, path of recess can be obtained.

5.2.7. Length of arc of contact

The arc of contact is the path traced by a point on the pitch circle from the beginning to the end

of engagement of a pair of teeth. In figure 5.23, the driving wheel both at the beginning of

engagement with the driven wheel tooth is shown as GH whereas at the end engagement it is

shown as FL. The arc of contact is P’PP” where the point P’ is on the driving wheel pitch circle

at the beginning of engagement and point P” is on the same pitch circle at the end of

engagement. This arc of contact is divided into two parts i.e. arc P’P and arc PP”. The arc P’P is

known as arc of approach and arc PP” is known arc of recess.

From figure 5.23, we know that,

Arc of approach

=ArcP' P= EPcosφ

Arc of recess= ArcPP = { { ital FP } over {cosφ} } } { ¿

Now length of arc of contact=ArcP ' P+ ArcPP } {¿

= EPcosφ

+ FPcosφ

=EP+FPcosφ



= EFcosφ

5.35(a)

= Lengthof contactcosφ

5.36

(∵EF=length of path of contact )

Fig 5.23 length of arc of contact

5.2.8. Number of pair of teeth in contact (or contact ratio)

The number of pairs of teeth in contact is defined as the ratio of length of arc of contact to the

circular pitch. Hence,

Number of pairs of teeth in contact (or contact ratio)



= Lenght of arc contactCircular pitch

5.36(a)

But from the equation (5.35(a)) length of arc of contact is equal to = EF

cosφ

And circular pitch from equation (5.29 (a)) is pc=π∗m

Where m = module

Number of pairs of teeth in contact

= EFcosφ

∗ 1πm

5.37

From the continuous transmission of motion, at least one tooth of one wheel must be in contact

with another tooth of the second wheel. Hence number of pairs of teeth in contact (which is also

known as contact ratio) must be greater than one. If contact ratio lies between 1 and 2, the

number of teeth in contact at any time will be more than one but will never be more than two. If

contact ratio is 1.6, this means that one pair of teeth is always in contact and second pair of teeth

is in contact 60% of the time. The gears will operate more quietly if contact ratio is large.

Example . Calculate:

(i) Length of path contact,

(ii) Arc of contact and

(iii) The contact ratio when a pinion having 23 teeth drives having teeth 57. The profile of

the gears is involute with pressure angle 200, module 8mm and addendum equal to

one module.

Solution:



Given

No. of teeth on the pinion, t=23

No. of teeth on gear, T=57

Pressure angle φ=200

Module, m=8 mm

Addendum =1mod ule=8mm

We know that the pitch circle radius of pinion is given by,

r=m∗t2

=8∗232

=92 mm

And pitch circle radius of gear is given by,

R=m∗T2

=8∗572

=228mm

Now the radius of addendum circle of pinion and gear are calculated

ra=Radius of addendumcircle of pinion

ra=r+addendum=92+8=100 mm

Ra=Radius of addendumcircle of gear

Ra=R+addendum=228+8=236 mm

(i) Length of path of contact

Now using equation (5.35) for the length of path of contact, we get

Length of path of contact

=√Ra2−R2 cos2 φ+√ra

2−r2cos2 φ−( R+r ) sin φ

=√2362−2282 cos2 200+√1002−922 cos2200−(228+92 ) sin 200



(ii) Arc of contact

Length of arc of contact is obtained from equation (5.36), as

Lengthof arc of contact= Lengthof path of contactcos φ

=39 .78cos200

=42 .33 mm

(iii) The contact ratio

The contact ratio is given by equation (5.36 a). hence using equation (5.36 a), we get

Contact ratio= Lengthof arc contactCircular pitch

Where circular pitch pc=π∗m=π∗8mm=25. 13 mm

And length of arc of contact = 42.33 mm

∵Contact ratio= Lengthof arc contactCircular pitch

=42 .33 mm25 .13 mm

=1 .68 say 2.

5.2.9. Helical gears

In article 5.2.2, it is already mentioned that the gears used to connect two parallel shafts and

having teeth inclined to the axes of shafts are known helical gears. Helical gears are already

shown in the figure 5.15. In the helical gears, the contact occurs at a point of curved teeth at the

beginning of the engagement and afterwards extends along a line across the teeth.

Figure 5.24 shows two helical gears. The gear 1 in figure 5.24 (a) is a left-handed helical gear, as

the helix slopes towards the left of viewer when this gear is viewed parallel to the axis of the

shaft. Similarly gear 2 in figure 5.24 (b) is a right-handed helical gear.


=98. 96+50.26−109. 44=39 . 78 mm


Fig 5.24 Helical gears

5.2.10. Important terms for helical gears

1. Helix angle. The angle, to which the teeth are inclined to the axis of a gear, is known as helix

angle. It is denoted by ϕ as shown in figure 5.25. Helix angle is also known as spiral angle of the

teeth.

2. Normal pitch (pn). The shortest distance between similar faces of the adjacent teeth is known

as normal pitch. It is denoted by pn. The normal pitch of two mating gears must be same.

3. Circular pitch (pc). The distance measured parallel to the axis between similar faces of

adjacent teeth is known as circular pitch. It is denoted by p or pc. it is also known as axial pitch

or transverse pitch. From figure 5.23, we have

pn=pc cos φ5.38



Fig 5.25. Important terms for helical gears

5.2.11. Spiral gears

The gears used to connect two non-parallel and non-intersecting shafts are known as spiral gears.

They are used where small power is to be transmitted. The spiral gears may be of the same hand

or opposite hand. There is a point of contact between two mating gears of spiral gears. The pitch

surfaces are cylindrical. The shortest distance between the two shafts gives the center distance

for a pair of spiral gears. Two mating spiral gears have a point contact. This is proved by having

two cylinders. When the axes of the two cylinders are parallel, they have a line contact. But if

one cylinder is rotated through same angle, so that their axes are no longer parallel, then they

have a point contact only.



Figure 5.26 shows two pairs of spiral gears (i.e. gear 1 and gear 2). Let us first define the shaft

angle. The shaft angle is the angle through which the axis of one shaft must be rotated in order to

bring it parallel to the axis of other shaft and the two shafts revolving in opposite directions. The

normal pitch of the teeth must be same for both wheels, but the circular pitch will be different.

Expression for center distance. The shortest distance between the two shafts is known as center

distance between the two shafts.

Let normal pitch

The spiral angle of the teeth for gear 1,

The spiral angle of the teeth for gear 1,

Circular pitch for gear 1,

Circular pitch for gear 2,

No. of teeth on gear 1 and 2,

Pitch circle diameter for gear 1 and 2,

Speed of gear 1 and 2,

Shaft angle,

C= Center distance

G=Gear ratio=T 1

T 2=

N1

N2

We know that, normal pitch in terms of helix angle (or spiral angle) and circular pitch is given

by,

pn= pc cos α


pn=

α=

β=

p2=

p1=

T 1∧T 2=

θ=

D1∧D2=

N1∧N2=


For gear 1, pn= p1 cos α ∵ p1=

pn

cos α

And for gear 2, pn=p2cos β ∵ p2=

pn

cos β

We also know that circular pitch in terms of pitch circle diameter and number of teeth on the

wheels, is given by

pc=πDT

5.39

For gear 1, we have

p1=πD1

Tor D1=

p1∗T 1

π

And for gear 2, we have

Fig 5.26 expression for center distance



p2=πD2

Tor D2=

p2∗T 2

π

The center distance (c) is given by

c=D1+D2

2=

p1∗T 1

π+

p2∗T 2

π2

c=p1∗T 1

2 π+

p2∗T 2

2 π=

pn

cos α∗

T 1

2 π+

pn

cos β∗

T 2

2 π

(∵ p1=pn

cos α; p2=

pn

cos β )c=

pn∗T 1

2 π [ 1cosα

+ 1cos β

∗T 2

T 1 ]c=

pn∗T 1

2 π [ 1cosα

+ 1cos β

∗G ] (∵ T 2

T 1=G)

c=pn∗T 1

2 π [ 1cosα

+ Gcos β ]

5.40

5.2.18. The efficiency of the spiral gears

Figure 5.27 shows two spiral gears A and B in mesh at point P. The gear A is driving the gear B.

Hence gear A is the driving gear and gear B is driven gear. These two gears are in contact along

the inclined surface CC (or along teeth line).



Fig 5.27 Two spiral gears in mesh

Let Fa = Axial force on gear A

Fb = Axial force on gear B

RN = Normal reaction at the point of contact

ϕ = Angle of friction i.e. angle made by the resultant reaction with the normal

R = Resultant reaction at the point of contact, and

= Shaft angle

θ=α+ β

Where α = Angle between axis of shaft (to which gear A is fixed) and inclined surface CC.

= Angle between axis of shaft (to which gear B is fixed) and inclined surface CC.

N-N = Line normal to CC.

(A) Idle case i.e. no friction



Let us first consider that there is no friction between the mating gears. Then force of friction (i.e.

* RN) will be zero. At the point of contact , the following forces will be acting on gear at A as

shown in figure 5.27 (a):

(i) The normal reaction RN which is normal to surface CC and towards the surface CC at

point of contact.

Fig 5.27 (a) fig 5.27 (b)

(ii) The axial force Fa

For the equilibrium of gear A in the direction of axial force, we have

Fa=RN cos α

Or FN=

Fa

cosα(i)

Now consider the equilibrium of gear B. The following forces will be acting on gear B at the

point of contact as shown in figure 5.27 (b):

(i) Normal reaction RN which is normal to surface CC and towards the surface CC at the

point contact.



(ii) The axial force Fb

For the equilibrium of gear B in the direction of axial force (Fb), we have

Fb=RN cos β

Or FN=

Fb

cos β(ii)

From equations (i) and (ii), we have

Fa

cos α=

Fb

cos β

(Note. The reaction of B and A is equal to reaction of A and B)

Fa=Fb

cos β∗cosα

5.41

(B) Considering friction

In actual case, there is a friction between the contact surfaces of the mating gears. Hence the

force of friction ( *FN) will be acting on the contact surfaces in the direction opposite to that of

relative sliding.

The resultant reaction (R) will not be mount to the surface but will inclined at angle of friction

(ϕ) with the normal to the surfaces. Hence the following forces will be acting on gear A at the

point of contact as shown in figure 5.27 (c):

(i) The normal reaction FN

(ii) Force of friction, *FN



(iii) (The sliding surface of gear A is having relative motion downward and hence force of

friction acts opposite to the direction of motion).

Fig 5.27 (c) Friction force

(iv) The axial force Fa*.

(Axial force is the force required with friction when Fb is constant)

The normal reaction RN and force of friction *FN combine into a single resultant force R,

which is inclined at an angle ϕ with the normal. For the equilibrium of gear A in the direction of

axial force (Fa*), we have

Fa¿=R cos (α−φ )

Or R=

Fa¿

cos (α−φ )(iii)

Now consider the equilibrium of gear B. The following forces will be acting on gear B at the

point of contact as shown in figure 5.27 (d):



Fig 5.27(d)

(i) The normal reaction, FN

(ii) Force of friction, *FN

(iii) The axial force, Fb.

The normal reaction RN and force of friction *FN combine into a single resultant force R,

which is inclined at an angle ϕ with the normal. For the equilibrium of gear B in the direction of

axial force (Fb), we have

Fb=R cos (α+φ )

Or R=

Fb

cos (α+φ )(iv)

From equations (iii) and (iv), we have

Fa¿

cos ( α−φ )=

Fb

cos ( β+φ )



Or

Fa¿=

Fb

cos ( β+φ )∗cos ( α−φ )

5.42

Efficiency of spiral gears

The efficiency of spiral gear is given by the ratio of the force Fa required without friction to that

required with friction when Fb is constant. Hence from equations (5.41) and (5.42), we have

η=Fa

Fa¿ =

Force without frictionForce with friction

η=( Fbcosαcos β )

Fb∗cos ( α−φ )cos ( β+φ )

η=cosα cos ( β+φ )cos β cos (α−φ )

5.43

But = - α, hence above equation becomes

η=cosα cos (θ−α+φ )cos (θ−α )cos (α−φ )

η=2cos α cos (θ−α+φ )

2 cos (θ−α ) cos (α−φ )

(Multiplying the numerator and denominator by 2)

η=cos [α + (θ−α+φ ) ]+cos [α−(θ−α+φ ) ]cos [ (θ−α )+ (α−φ ) ]+cos [(θ−α )−( α−φ ) ]

[∵2 cos A cos B=cos ( A+B )+cos ( A−B ) ]

η=cos (θ+φ )+cos (2 α−θ−φ )cos (θ−φ )+cos (θ−2 α−φ )



η=cos (θ+φ )+cos (2 α−θ−φ )cos (θ−φ )+cos (2α−θ−φ )

5.44

[∵cos (−A )=cos A ]

For a give value of and ϕ, the efficiency will be maximum, when cos (2α - - ϕ) is a

maximum. But maximum value of cos (2α - - ϕ) is 1

cos (2 α−θ−φ )=1=cos 00 [∵cos 00=1 ]

2 α−θ−φ

Or 2 α=θ+φ

Or α=θ+φ

25.45

Substituting the value of α in equation (5.44), we get the maximum efficiency.

ηmax=

cos (θ+φ )+cos (θ+φ−θ−φ )cos (θ−φ )+cos (θ+φ−θ−φ )

ηmax=cos (θ+φ )+cos0cos (θ−φ )+cos 0

ηmax=cos (θ+φ )+1cos (θ−φ )+1 [∵cos 00=1 ]

5.57

Example: The appropriate center distance between two meshing gears is 375 mm and gear ratio

is 2. The angle between the shaft is 500 and normal circular pitch is 19 mm. if the driving and

driven wheels are having same spiral angles and friction angle = 60, then determine:

(i) Number of teeth on each wheel,

(ii) The exact center distance and

(iii) Efficiency of the drive.



Solution:

Given

Approximate center distance, C = 375 mm; Gear ratio, G = 2; shaft angle, = 500; Normal pitch,

pn = 19 mm; friction angle, ϕ = 60.

Spiral angle of driving wheel = spiral angle driven wheel

Let α = Spiral angle of driving wheel

= Spiral angle of driven wheel

α=β

(given)

But shaft angle,

θ=α+ β

Or 50=α+β

α=β=50

2=250

(i) Number of teeth on driving wheel

Let T1 = Number of teeth on the driving wheel

T2 = Number of teeth on the driven wheel

The center distance (C) between two shaft is given by equation (5.51) as,

C=pn∗T1

2 π [ 1cos α

+ Gcos β ]

Or

375=19∗T 1

2 π [ 1cos 250 + 2

cos250 ]

[∵C=375 mm; α=β=250 ; pn=19 ]



375=19∗T 1

2 π [ 1+2cos 250 ]=19∗T1

2 π∗ 3

cos250

T 1 ==375∗2 π∗cos250

19∗3=37. 46 say 38

And

(ii) Exact center distance


C=pn∗T1

2 π [ 1cos α

+ Gcos β ]

C=19∗382 π [ 1

cos250+ 2

cos250 ]C=19∗38

2 π∗ 3

cos250=380 . 4 mm .

(iii) Efficiency of the drive

The efficiency of the spiral gears is given by equation (5.54)

η=cosα cos ( β+φ )cos β cos (α−φ )

η=cos250 cos (250+60 )cos250 cos (250−60) (∵α=β=250 ,φ=60)

η=cos310

cos190 =0 . 85710 . 9455

=0 .9065=90 .65 %


T 2=G∗T 1=2∗38=76


5.2.12. Gear trains

A combination of two or more gears, which are arranged in such a way that the power is

transmitted from a driving shaft to driven shaft, is known as a gear train. This term is generally

applied to mean more than two gears in mesh between the driving shaft and driven shaft. The

gear train may consist of spur, bevel or spiral gears.

Types of gear train

(i) Simple gear train and

(ii) Compound gear train

5.2.12.1. Simple gear train

If the axes of all gears (or the axes of the shafts on the gears are mounted) remain fixed relative

to each other, the gear train is known as simple gear train or ordinary gear train. In the case of a

simple gear train, each gear is on a separate shaft as shown figure 5.28 (a). in figure 5.28 (a),

there are only two gears. Each gear is mounted on the separate shaft. The combination of these

two gears is known as simple gear train. If the power is transmitted from gear 1 to gear 2, then

gear 1 is driver whereas gear 2 is driven or follower. These two gears rotate in opposite direction.



Fig 5.28 simple gear train

In figure 5.28 (b), there are three gears. The shafts 1 carries only one gear, shafts 2 also carries

only one gear and shaft three has only one gear. The combination of these gears, is known as

simple gear train, when they are arranged in such a way that power is transmitted from a driving

shaft to driven shaft. The shaft 1 is a driver whereas the shaft 3 is known as driven shaft or

follower. The shaft 2 is called intermediate shaft. It may be noted that when the number of

intermediate shafts are even, the motion of follower will be in the opposite direction of the

driver.

5.2.12.2. Compound gear train

In a compound gear train there are more than one gear on a shaft, generally intermediate shaft

have two gears rigidly fixed to the shaft so that these two gears have the same speed as they, are

mounted on same shaft.

Figure 5.29 shows a compound gear train in which gear 1 (or wheel 1) drives the gear 2 mounted

on the intermediate shaft. On the intermediate gear train shaft is mounted another gear 3 which



meshes with gear 4 mounted on the driven shaft. Gears 2 and 3 rotate at the same speed as they

are on the same shaft.

Fig 5.29 Compound gear train

5.2.12.3. Velocity ratio of gear trains

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

speed ratio of the gear train. Mathematically, it is written as

Velocity ratio (V .R . )= Speed of driverSpeed of driven

The reciprocal of the speed ratio (or velocity ratio) is known as train value of gear train.

5.2.12.3.1. Velocity ratio of simple gear train

First case



Figure 5.30 shows a simple gear train in which each shaft carries only one gear. The power is

transmitted from gear 1 to gear 2. Hence gear 1 is known as driver and gear 2 is known as driven

or follower. The driving gear (i.e. gear 1) is moving clockwise whereas the driven gear (i.e. gear

2) is moving anticlockwise. Hence in a simple gear train, the meshing gears always in opposite

direction.

Fig 5.30 Velocity ratio of simple gear train

Let N1=Speed of driver (i.e. gear 1)

N2=Speed of driven (i.e. gear 2)

T 1= No. of teeth on gear 1, and

T 2= No. of teeth on gear 2.

Speed ratio (velocity ratio) = Speed of driver

Speed of follower=

N 1

N 2

But the ratio of speeds of any pair of gears in mesh is the inverse of their number of teeth.

N1

N 2=

T2

T1

Speed ratio =

N1

N2=

T 2

T 1

The reverse of the speed ratio is known as the train value of the gear train.



Train value =

N2

N1=

T 1

T 2

Second case

Figure 5.31 shows a simple gear train with one intermediate shaft

Fig 5.31 Simple gear train with one intermediate shaft

Let T 1= No. of teeth on driver wheel

T 2= No. of teeth on intermediate gear

T 3=No. of teeth on follower gear.

N1 , N2 and N3 =Speed of driver, intermediate and follower respectively in r.p.m.

When the driver and intermediate gears are in mesh, we get

N 2

N1=

T1

T2 (i)

Similarly, when the intermediate gear and follower are in mesh, we get

N 3

N2=

T 2

T3 (ii)

Multiplying equations (i) and (ii), we get

N 2

N1∗

N3

N2=

T 1

T 2∗

T2

T3



Or

N 3

N1=

T 1

T3

Speed of followerSpeed of driver

= No . of teeth ondriverNo . of teethon follower

Or Speed ratio= Speedof driver

Speed of follower= No .of teethon follower

No .of teeth on driver 5.47

From equation (5.58), we see that the velocity ratio is independent of the no. of teeth on

intermediate wheels (or gears). It depends only upon the number of teeth on the driver and

follower. As the intermediate wheels do not affect the velocity ratio, hence they are known as

idler. These idlers are used only:

(i) To change the direction of rotation and

(ii) To bridge a gap when the distance between the driver and follower shafts is too great

for just two gears.

Equation (5.47) holds well even if there are any numbers of intermediate wheels (or gears). The

driver and follower will rotate in the same direction if the numbers of intermediate shafts are odd

whereas they will rotate in opposite direction if the numbers of intermediate shafts are even.

Example: A simple gear train consists of two gears only, each gear mounted on separate shafts.

The shafts are parallel. The gear 1 is driving the gear 2. The speed of first gear is 1000 r.p.m. The

number of teeth of gears 1 and 2 are 24 and 60 respectively. Determine:

(i) Speed ratio of the gear train,

(ii) Train value of the gear train,

(iii) Speed of the second gear

(iv) Direction of rotation of second gear if the first gear is rotating clockwise

Solution:

Given

N1 = 1000 r.p.m. ; T1 = 24; T2 = 60

(i) Speed ratio

Speed ratio of simple gear train is given by,

Speed ratio =

N1

N2=

T 2

T 1



Here the values of T2 and T1 are given

Speed ratio =

T 2

T 1=60

24=2 .5

Fig 5.32

(ii) Train value

Train value = 1

Speedratio= 1

2 .5=0 . 4

(iii) Speed of second gear

Speed ratio =

N1

N2

Or 2 .5=1000

N2

N2=

10002. 5

=400 r . p . m .

Example: A simple gear train consists of three gears, each mounted on separate shaft as shown

in figure 5.33. The shafts are parallel. The gear 1 is the driver and gear 3 is follower. The gear 1

is rotating clockwise at a speed of 750 r.p.m. The no. of teeth on gears 1, 2 and 3 are 30, 45 and

75 respectively. Find:

(i) Speed ratio of the gear train,

(ii) Direction of rotation and speed of rotation of follower.



Fig 5.33

Solution:

N1 = 750 r.p.m.; T1 = 30; T2 = 45 and T3 = 75

(i) Speed ratio


Speed ratio= Speed of driverSpeed of follower

= No .of teeth on followerNo .of teeth on driver

Speedratio=T3

T1=75

30=2.5

(ii) Direction and speed of rotation of follower

Speedratio= Speed of driverSpeed of follower

Speedof follower=Speed of driverSpeed ratio

=N1

2.5=750

2 .5=300 r . p .m .

Direction of rotation

In a simple gear train with intermediate gears, the driver and follower will rotate in the same

direction if the numbers of intermediate shafts (gears) are odd. Hence the number of intermediate

shaft is only one (i.e. number gear is odd), hence follower will rotate clockwise.

5.2.12.3.2. Velocity ratio of compound gear train

In a compound gear train as shown in figure 5.34, the intermediate shafts have two wheels (or

gears) rigidly fixed to it so that these two gears have the same speed. One of these two gears

meshes with the driver and the other with follower.



In figure 5.34, the gear 1 (or wheel 1) drives gear 2 mounted on the intermediate shaft. On the

intermediate shaft is mounted another gear 3 which meshes with gear 4 mounted on driven (or

follower) shaft. Gears 2 and 3 rotate at the same speed as they are mounted on the same

intermediate shaft.

Let T 1= No. of teeth on driver 1, T 2= No. of teeth on gear 2

T 3=No. of teeth on gear 3, T 4=No. of teeth on gear 4

N1=Speed of driver 1

N2 , N3 and N4 = Speed of gears 2, 3 and 4 respectively in r.p.m. The gear 1 drives the gear 2,

therefore,

N 2

N1=

T1

T2 (i)

Gear 3 drives the gear 4, therefore

N 4

N3=

T3

T 4 (ii)

Multiplying equations (i) and (ii), we get

N 2

N1∗

N4

N3=

T1

T 2∗

T 3

T 4

But N2=N3 as gears 2 and 3 are mounted on the same shaft. Hence the above equation

becomes are

N 4

N1=

T 1

T 2∗

T 3

T 4

As gear 1 drives gear 2 whereas gear 3 drives gear 4 hence gears 1 and 3 becomes as drivers

whereas gears 2 and 4 becomes as followers. Hence the above equation can be stated as


=Pr oduct of teeth on driversPr oduct of teeth on followers 5.48

Example: In a compound gear train shown in figure 5.35, the power is transmitted from a motor

shaft to output shaft. The motor shaft is connected to gear 1 whereas the output shaft is

connected to gear 4. The gears 2 and 3 are mounted on the same shaft. The motor is rotating at



1250 r.p.m. in the clockwise direction. Find the direction of the output shaft. The no. of teeth on

each gear is given below:

Gear 1 2 3 4

No. of teeth 30 75 20 50

Fig 5.35

Solution:

Given N1 = 1250 r.p.m; T1 = 30; T2 = 75; T3 = 20; T4 = 50

From figure 5.35 it is clear that gears 1 and 3 are driving gears (i.e. drivers) whereas gears 2 and

4 are driven gears or followers. The gears 2 and 3 are mounted on the same shaft and hence they

will rotate in the same direction. The gear 1 is rotating in clockwise direction. The gear 2 is in

mesh with gear 1. Hence gear 2 will rotate in opposite direction i.e. in anticlockwise direction.

Gear will rotate in the direction of gear 2 i.e. in anticlockwise direction. The gear 4 is in mesh

with gear 3. Hence gear 4 will rotate in opposite direction i.e. clockwise direction. Gear 4 is

connected output shaft.

Direction of rotation of output shaft = clockwise direction

Let N4= speed of output shaft (or speed of gear 4)

Using the relation,


=Pr oduct of teethon driversPr oduct of teethon followers

Or

N 4

N1=

T 1

T 2∗

T 3

T 4=75

30∗50

20=6 . 25

N4=

12506 . 25

=200 r . p . m .


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Chapter Five 2