Elmes2 Belt Drive

7/28/2019 Elmes2 Belt Drive

1/52

Mechanical Engineering

Udayana University

Flat Belt Drives


2/52


Udayana University

1. Introduction

The belts or ropes are used to transmit power from one shaft to another by

means of pulleys which rotate at the same speed or at different speeds. The

amount of power transmitted depends upon the following factors :

1. The velocity of the belt.

2. The tension under which the belt is placed on the pulleys.

3. The arc of contact between the belt and the smaller pulley.

4. The conditions under which the belt is used.

Note :

(a) The shafts should be properly in line to insure uniform tension across

the belt section.(b) The pulleys should not be too close together, in orderthat the arc of

contact on the smaller pulley may be as large as possible.


3/52


Udayana University

(c) The pulleys should not be so far apart as to cause the belt to weigh

heavily on the shafts, thus increasing the friction load on thebearings.

(d) A long belt tends to swing from side to side, causing the belt to run

out of the pulleys, which in turn develops crooked spots in the belt.

(e) The tight side of the belt should be at the bottom, so that whatever

sag is present on the loose side will increase the arc of contact at the

pulleys.

(f ) In order to obtain good results with flat belts, the maximum distance

between the shafts should not exceed 10 metres and the minimum

should not be less than 3.5 times the diameter of the larger pulley.


4/52


5/52


Udayana University

4. Types of BeltsThe following are the common types of belts :

Flat belt. The flat belt as shown in Fig. 1 (a), is mostly used in the

factories and workshops, where a moderate amount of power is to betransmitted, from one pulley to another when the two pulleys are not

more than 8 metres apart.

V- belt. The V-belt as shown in Fig.1 (b), is mostly used in the factories

and workshops, where a great amount of power is to be transmitted,

from one pulley to another, when the two pulleys are very near to eachother.

Circular belt or rope. The circular belt or rope as shown in Fig. 1 (c) is

mostly used in the factories and workshops, where a great amount of

power is to be transmitted, from one pulley to another, when the two

pulleys are more than 8 metres apart.

Figure 1 Types of belts


6/52


Udayana University

5. Material used for BeltsThe material used for belts and ropes must be strong, flexible, and durable. It must

have a high coefficient of friction. The belts, according to the material used, are

classified as follows:1. Leather belts. The best leather belts are made from 1.2 metres to 1.5 metres long

strips cut from either side of the back bone of the top grade steer hides. The hair

side of a belt should be in contact with the pulley surface as shown in Fig. 2. The

belts are specified according to the number of layers e.g. single, double or triple

ply and according to the thickness of hides used e.g. light, medium or heavy.

2. Cotton or fabric belts. Most of the fabric belts are made by folding convass orcotton duckto three or more layers (depending upon the thickness desired) and

stitching together. The cotton belts are cheaper and suitable in warm climates, in

damp atmospheres and in exposed positions. Since the cotton belts require little

attention, therefore these belts are mostly used in farm machinery, belt conveyor

etc.

Figure 2 Leather belts


7/52


Udayana University

3. Rubber belt. The rubber belts are made of layers of fabric

impregnated with rubber composition and have a thin layer of

rubber on the faces. These belts are very flexible but are quickly

destroyed if allowed to come into contact with heat, oil or grease.

One of the principle advantage of these belts is that they may be

easily made endless. These belts are found suitable for saw mills,

paper mills where they are exposed to moisture.

4. Balata belts. These belts are similar to rubber belts except that

balata gum is used in place ofrubber. These belts are acid proof and

water proof and it is not effected by animal oils or alkalies. The

balata belts should not be at temperatures above 40C because at

this temperature the balata begins to soften and becomes sticky.

The strength of balata belts is 25 per cent higher than rubber belts.


8/52


Udayana University

6. Working Stresses in BeltsThe ultimate strength of leather belt varies from 21 to 35 MPa and a factor of safety

may be taken as 8 to 10. However, the wear life of a belt is more important than

actual strength. It has been shown by experience that under average conditions anallowable stress of 2.8 MPa or less will give a reasonable belt life. An allowable

stress of 1.75 MPa may be expected to give a belt life of about 15 years.

7. Density of Belt MaterialsThe density of various belt materials are given in the following table.

Table 1. Density of belt materials

8. Belt SpeedWhen the speed of belt increases, the centrifugal force increases which tries to pull

the belt away from the pulley. This reduces the power transmitted by the belt. For the

efficient transmission of power, the belt speed 20 m/s to 22.5 m/s may be used.


9/52


Udayana University

9. Coefficient of Friction Between Belt and PulleyThe coefficient of friction between the belt and the pulley depends upon the

following factors:

1. The material of belt2. The material of pulley

3. The slip of belt

4. The speed of belt

According to C.G. Barth, the coefficient of friction () for oak tanned leather

belts on cast iron pulley, at the point of slipping, is given by the followingrelation, i.e.

where v = Speed of the belt in metres per minute.


10/52


Udayana University

The following table shows the values of coefficient of friction for various materials of

belt and pulley.

Table 1.2. Coefficient of friction between belt and pulley.


11/52


Udayana University

10. Standard Belt Thicknesses and Widths

The standard flat belt thicknesses are 5, 6.5, 8, 10 and 12 mm. The preferred values

of thicknesses are as follows:

a) 5 mm for nominal belt widths of 35 to 63 mm

b) 6.5 mm for nominal belt widths of 50 to 140 mm

c) 8 mm for nominal belt widths of 90 to 224 mm

d) 10 mm for nominal belt widths of 125 to 400 mm

e) 12 mm for nominal belt widths of 250 to 600 mm.

The standard values of nominal belt widths are in R10 series, starting from 25 mmup to 63 mm and in R 20 series starting from 71 mm up to 600 mm. Thus, the

standard widths will be 25, 32, 40, 50, 63, 71, 80, 90, 100, 112, 125, 140, 160, 180,

200, 224, 250, 280, 315, 355, 400, 450, 500, 560 and 600 mm.

11. Belt Joints

The various types of joints are :

1. Cemented joint (Fig. 3a)

2. Laced joint (known as straight-stitch raw hide laced joint), Fig. 3b dan 3c

3. Hinged joint (Fig. 3d)


12/52


Udayana University

Figure 3. Belt joints

h l


13/52


Udayana University

12. Types of Flat Belt Drives

The power from one pulley to another may be transmitted by any of the following

types of belt drives.

a) Open belt drive. The open belt drive, as shown in Fig. 4, is used with shafts

arrangedparallel and rotating in the same direction.

Fig. 4. Open belt drive.


14/52

M h i l E i i


15/52


Udayana University

c) Quarter turn belt drive. The quarter turn belt drive (also known as right angle belt

drive) as shown in Fig. 6 (a), is used with shafts arranged at right angles and

rotating in one definite direction. A quarter turn belt drive with a guide pulley, as

shown in Fig. 6 (b). In order to prevent the belt from leaving the pulley, the width

of the face of the pulley should be greater or equal to 1.4 b, where b is width of

belt.

Fig. 6

M h i l E i i


16/52


Udayana University

d) Belt drive with idler pulleys. A belt drive with an idler pulley (also known as jockey

pulley drive) as shown in Fig. 7 and 8, is used with shafts arranged parallel and when an

open belt drive can notbe used due to small angle of contact on the smaller pulley.

Fig. 7. Belt drive with single idler pulley.

Fig. 8. Belt drive with many idler pulleys.



17/52


Udayana University

e) Compound belt drive. A compound belt drive as shown in Fig. 9, is used when power

is transmitted from one shaft to another through a number of pulleys.

Fig. 9. Compound belt drive.



18/52


Udayana University

f. Stepped or cone pulley drive. A stepped or cone pulley drive, as shown in Fig. 10, is

usedfor changing the speed of the driven shaft while the main or driving shaft runs at

constant speed.

g. Fast and loose pulley drive. A fast and loose pulley drive, as shown in Fig. 11, is usedwhen the driven or machine shaft is to be started or stopped whenever desired

without interferring with the driving shaft.

Fig. 10. Stepped or cone pulley drive Fig. 11. Fast and loose pulley drive



19/52


Udayana University

Stepped or cone pulley drive



20/52


Udayana University

13. Velocity Ratio of a Belt Drive

It is the ratio between the velocities of the driver and the follower or driven.

Let d1 = Diameter of the driver,

d2 = Diameter of the follower,N1 = Speed of the driver in r.p.m.,

N2 = Speed of the follower in r.p.m.,

Length of the belt that passes over the driver, in one minute

= d1 N1Similarly, length of the belt that passes over the follower, in one minute

= d2 N2Since the length of belt that passes over the driver in one minute is equal to the

length of belt that passes over the follower in one minute, therefore

d1 N1= d2 N2and velocity ratio,

When thickness of the belt (t) is considered, then velocity ratio,



21/52


Udayana University

Notes :

1. The velocity ratio of a belt drive may also be obtained as discussed

below:

We know that the peripheral velocity of the belt on the driving pulley,

and peripheral velocity of the belt on the driven pulley,

When there is no slip, then 1 = 2.

2. In case of a compound belt drive as shown in Fig. 9, the velocity ratio is

given by



22/52


Udayana University

14. Slip of the Belt

If the forward motion of the driver without carrying the belt with it, this is called

slip of the belt and is generally expressed as a percentage.

The result of the belt slipping is to reduce the velocity ratio of the system. As the

slipping of the belt is a common phenomenon, thus the belt should never be used

where a definite velocity ratio is of importance (as in the case of hour, minute and

second arms in a watch).

Let s1 % = Slip between the driver and the belts2 % = Slip between the belt and follower

Velocity of the belt passing over the driver per second,



23/52


Udayana University

and velocity of the belt passing over the follower per second

Substituting the value of from equation (i), we have

If thickness of the belt (t) is considered, then



24/52


Udayana University

15. Creep of Belt

When the belt passes from the slack side to the tight side, a certain portion of

the belt extends and it contracts again when the belt passes from the tight side

to the slack side. Due to these changes of length, there is a relative motionbetween the belt and the pulley surfaces. This relative motion is termed as

creep. The total effect of creep is to reduce slightly the speed of the driven pulley

orfollower. Considering creep, the velocity ratio is given by

Where : 1and 2 = Stress in the belt on the tight and slack side respectively,

and E = Youngs modulus for the material of the belt.

Note: Since the effect of creep is very small, therefore it is generally neglected.



25/52

g g

Udayana University

Example 1.

An engine running at 150 r.p.m. drives a line shaft by means of a belt. The engine pulley is

750 mm diameter and the pulley on the line shaft is 450 mm. A 900 mm diameter pulley on

the line shaft drives a 150 mm diameter pulley keyed to a dynamo shaft. Fine the speed ofdynamo shaft, when 1. there is no slip, and 2. there is a slip of 2% at each drive.

Solution.

Given : N1 = 150 r.p.m. ; d1 = 750 mm ; d2 = 450 mm ; d3 = 900 mm ; d4 = 150 mm ; s1 = s2

= 2%. The arrangement of belt drive is shown in Fig. 12.

Let N4 = Speed of the dynamo shaft.

Fig. 12



26/52

g g

Udayana University

1. When there is no slip

2. When there is a slip of 2% at each drive



27/52

g g

Udayana University

16. Length of an Open Belt Drive

In an open belt drive, both the pulleys rotate in the same direction as shown in

Fig.13.

Fig. 13. Open belt drive.

Let r1 and r2 = Radii of the larger and smaller pulleys,

x = Distance between the centres of two pulleys (i.e. O1O2), and

L = Total length of the belt.



28/52

Udayana University

Let the belt leaves the larger pulley at E and G and the smaller pulley at F and H as

shown in Fig.13. Through O2 draw O2M parallel to FE. From the geometry of the

figure, we find that O2M will be perpendicular to O1E.

Let the angle MO2O1= radians. We know that the length of the belt,

L = Arc GJE + EF + Arc FKH + HG

= 2 (ArcJE + EF + Arc FK) ...(i)

From the geometry of the figure, we also find that

Since the angle is very small, therefore putting

Similarly,



29/52

Udayana University

Expanding this equation by binomial theorem, we have

Substituting the values of arcJE from equation (iii), arc FK from equation (iv) and EF

fromequation (v) in equation (i), we get



30/52

Udayana University

Substituting the value of from equation (ii), we get



31/52

Udayana University

17 Length of a Cross Belt Drive

In a cross belt drive, both the pulleys rotate in the opposite directions as shown in

Fig. 14. Let r1 and r2 = Radii of the larger and smaller pulleys, x = Distance between

the centres of two pulleys (i.e. O1

O2

), and L = Total length of the belt.

Let the belt leaves the larger pulley at E and G and the smaller pulley at F and H.

Through O2 draw O2M parallel to FE. From the geometry of the figure, we find that

O2M will be perpendicular to O1E.

Let the angle MO2O1= radians. We know that the length of the belt, L = Arc GJE + EF + Arc FKH + HG

= 2 (Arc JE + FE + Arc FK) ...(i)

Fig.14. Crossed belt drive.



32/52

Udayana University

From the geometry of the figure, we find that

Since the angle is very small, therefore putting

Similarly,

and



33/52

Udayana University

Expanding this equation by binomial theorem, we have

Substituting the values of arcJE from equation (iii), arc FK from equation (iv) and

EF from equation (v) in equation (i), we get,


d


34/52

Udayana University

Substituting the value of from equation (ii), we get


Ud U i i


35/52

Udayana University

18 Power Transmitted by a Belt

The driving pulley pulls the belt from one side and delivers it to the other side.

It is thus obvious that the tension on the former side (i.e. tight side) will be

greater than the latter side (i.e. slack side) as shown in Fig. 15.

T1, T2 = Tensions in the tight side and slack side of the belt respectively in newtonsR1, r2 = Radii of the driving and driven pulleys respectively in metres,

= Velocity of the belt in m/s.

The effective turning (driving) force at the circumference of the driven pulley

or follower is the difference between the two tensions (i.e. T1 T2).

Fig. 15. Power transmitted by a belt.


Ud U i it


36/52

Udayana University

Work done per second = (T1 T2) N-m/s

and power transmitted = (T1 T2) W ... ( 1 N-m/s = 1W)

A little consideration will show that torque exerted on the driving pulley is

(T1 T2) r1. Similarly, the torque exerted on the driven pulley is (T1 T2) r2.

19 Ratio of Driving Tensions for Flat Belt Drive

Consider a driven pulley rotating in the clockwise direction as shown in Fig. 16.

Let T1 = Tension in the belt on the tight side,

T2 = Tension in the belt on the slack side, and

= Angle of contact in radians (i.e. angle subtended by the arc AB, along

which the belt touches the pulley, at the centre).

Now consider a small portion of the belt PQ, subtending an angle at the

centre of the pulley as shown in Fig. 16.


Ud U i it


37/52

Udayana University

The belt PQ is in equilibrium under the following forces:

1. Tension T in the belt at P,

2. Tension (T + T) in the belt at Q,

3. Normal reaction RN, and4. Frictional force F = RN,

where is the coefficient of friction between the beltand pulley.

Fig. 16. Ratio of driving tensions for flat belt.


Udayana University


38/52

Udayana University

Resolving all the forces horizontally, we have

(i)

Since the angle is very small, therefore putting sin /2 = /2 in equation

(i), we have

(ii)

Now resolving the forces vertically, we have

(iii)

Since the angle is very small, therefore putting cos /2 = 1 in equation (iii),

we have

(iv)


Udayana University


39/52

Udayana University

Equating the values ofRN from equations (ii) and (iv), we get

Integrating the above equation between the limits T2 and T1 and from 0 to

, we have

(v)

The equation (v) can be expressed in terms of corresponding logarithm to

the base 10, i.e.


Udayana University


40/52

Udayana University

Notes :

1. While determining the angle of contact, it must be remembered that it is the

angle of contact at the smaller pulley, if both the pulleys are of the same material.

We know that

... (for open belt drive)

... (for cross-belt drive)

Angle of contact or lap,

...(for open belt drive)

... (for cross-belt drive)

2. When the pulleys are made of different material (i.e. when the coefficient of

friction of the pulleys or the angle of contact are different), then the design will

refer to the pulley for which . is small.


Udayana University


41/52

Udayana University

20 Centrifugal Tension

The tension caused by centrifugal force is called centrifugal tension. At

lower belt speeds (less than 10 m/s), the centrifugal tension is very small,

but at higher belt speeds (more than 10 m/s), its effect is considerable andthus should be taken into account. Consider a small portion PQ of the belt

subtending an angle d at the centre of the pulley, as shown in Fig. 18.

Fig. 18. Centrifugal tension.

m = Mass of belt per unit length in kg,

v = Linear velocity of belt in m/s,

r = Radius of pulley over which the belt runs in metres, andTC = Centrifugal tension acting tangentially at P and Q in

newtons.

We know that length of the belt PQ = r.d

and mass of the belt PQ = m.r.d

Centrifugal force acting on the belt PQ,


Udayana University


42/52

Udayana University

The centrifugal tension TC acting tangentially at P and Q keeps the belt in equilibrium.

Now resolving the forces (i.e. centrifugal force and centrifugal tension) horizontally, we

have

(i)

Since the angle d is very small, therefore putting in equation (i), we

have

Notes :

1. When centrifugal tension is taken into account, then total tension in the tight

side,

and total tension in the slack side,


Udayana University


43/52

Udayana University

2. Power transmitted,

...(in watts)

... (same as before)

Thus we see that the centrifugal tension has no effect on the power

transmitted.

3. The ratio of driving tensions may also be written as

where Tt1 = Maximum or total tension in the belt.


Udayana University


44/52

Udayana University

21 Maximum Tension in the Belt

A little consideration will show that the maximum tension in the belt (T )

is equal to the total tension in the tight side of the belt (Tt1).

Let = Maximum safe stress,

b = Width of the belt, and

t = Thickness of the belt.

We know that the maximum tension in the belt,

T = Maximum stress Cross-sectional area of belt = .b.t

When centrifugal tension is neglected, then

T (or Tt1) = T1, i.e. Tension in the tight side of the belt.

When centrifugal tension is considered, then

T (or Tt1) = T1 + TC


Udayana University


45/52

Udayana University

22 Condition for the Transmission of Maximum Power

The power transmitted by a belt,

P = (T1

T2

) v (i)where :

T1 = Tension in the tight side in newtons,

T2 = Tension in the slack side in newtons, and

= Velocity of the belt in m/s.

Ratio of driving tensions is

(ii)

Substituting the value ofT2in equation (i), we have

(iii)

where :


Udayana University


46/52

y y

We know that

T1 = TTC

where :T = Maximum tension to which the belt can be subjected in newtons, and

TC= Centrifugal tension in newtons.

Substituting the value ofT1 in equation (iii), we have

For maximum power, differentiate the above expression with respect to v and

equate to zero, i.e.

It shows that when the power transmitted is maximum, 1/3rd of the maximum

tension is absorbed as centrifugal tension.


Udayana University


47/52

y y

Notes :

1. We know that T1 = TTC and for maximum power,

2. From equation (iv), we find that the velocity of the belt for maximum

power,


48/52

Mechanical EngineeringUdayana University


49/52

Assuming that the belt material is perfectly elastic such that the length of

the belt remains constant, when it is at rest or in motion, therefore

increase in length on the tight side is equal to decrease in the length on

the slack side.

Thus, equating equations (i) and (ii), we have

(T1 T0) = (T0 T2)

or T1 T0 = T0 T2

(Neglecting centrifugal tension)

... (Considering centrifugal tension)



50/52

Example :

A leather belt 9 mm 250 mm is used to drive a cast iron pulley 900 mm in diameter at

336 r.p.m. If the active arc on the smaller pulley is 120 and the stress in tight side is 2

MPa, find the power capacity of the belt. The density of leather may be taken as 980

kg/m3, and the coefficient of friction of leather on cast iron is 0.35.

Solution :

Given: t = 9 mm = 0.009 m ; b = 250 mm = 0.25 m; d = 900 mm = 0.9 m ; N = 336 r.p.m ;

= 120 = 120 /180 = 2.1 rad ; = 2 MPa = 2 N/mm2 ; = 980 kg/m3 ; = 0.35

We know that the velocity of the belt,

and cross-sectional area of the belt,

a = b.t = 9 250 = 2250 mm2

Maximum or total tension in the tight side of the belt,

T = Tt1= .a = 2 2250 = 4500 N



51/52

We know that mass of the belt per metre length,

m = Area length density = b.t.l. = 0.25 0.009 1 980 kg/m

= 2.2 kg/m

Centrifugal tension,

TC= m.v2 = 2.2 (15.8)2 = 550 N

and tension in the tight side of the belt,

T1 = TTC= 4500 550 = 3950 N

Let T2 = Tension in the slack side of the belt.

We know that



52/52

And

We know that the power capacity of the belt,

P = (T1T2) v = (3950 1895) 15.8 = 32 470 W = 32.47 kW

Notes :The power capacity of the belt, when centrifugal tension is taken into account,

may also be obtained as discussed below :

1. We know that the maximum tension in the tight side of the belt,

Tt1 = T = 4500 NCentrifugal tension, TC = 550 N

and tension in the slack side of the belt, T2 = 1895 N

Total tension in the slack side of the belt,

Tt2 = T2 + TC = 1895 + 550 = 2445 N

We know that the power capacity of the belt,

P = (Tt1 Tt2) v = (4500 2445) 15.8 = 32 470 W = 32.47 kW Ans.

2. The value of total tension in the slack side of the belt (Tt2) may also be obtained by

using the relation as discussed in Art. 20, i.e.

Documents

Elmes2 Belt Drive