Bnj 20303 Chapter 1-Belt Drive System

1

CHAPTER 1

BELT DRIVE SYSTEM

1.1 Introduction to Belt Drive System

Another method widely used in engineering practice in transmitting power between

shafts is belt drive system. Basic arrangement of a belt drive system consists of driver sheave,

driven sheave and belting material such as ropes, rubber bands or chain. Power is transmitted

from the driver sheave by means of a belt to the driven sheave.

A Belt is a looped strip of flexible material, used to mechanically link two or more

rotating shafts. They may be used as a source of motion, to efficiently transmit power, or to track

relative movement.

A pulley (also called a sheave or block) is a wheel with a groove between two flanges

around its circumference. The groove normally locates a rope, cable or belt. Pulleys are used to

change the direction of an applied force, transmit rotational motion, or realize a mechanical

advantage in either a linear or rotational system of motion.

2

1.1.1 Operation of belt drives

The operation of belt drives depends fully on the friction generated from contact surface

between the belt and sheaves. Thus, any slip occurrence on the contact surface will reduce the

efficiency of the power transmitted. Power transmitted from the driver to the driven sheaves also

depends on factor such as;

Speed of the belt itself.

Surface area of contact between the belt and sheaves.

Workspace constraint where belt drive is used. Most belt drive system will have

parallel shaft arrangement, but for a constraint workspace such as in a cars engine, an

idler sheaves maybe used.

3

1.1.2 Advantages of belt drive

They are simple.

They are economical.

Parallel shafts are not required.

Overload and jam protection are provided.

Noise and vibration are damped out.

Machinery life is prolonged because load fluctuations are cushioned (shock-absorbed).

They are lubrication-free. They require only low maintenance.

They are highly efficient (9098%, usually 95%).

Some misalignment is tolerable.

They are very economical when shafts are separated by large distances.

Clutch action may be obtained by relieving belt tension.

Variable speeds may be economically obtained by step or tapered pulleys.

1.1.3 Disadvantages of belt drive

The angular-velocity ratio is not necessarily constant or equal to the ratio of pulley

diameters, because of belt slip and stretch.

Heat buildup occurs.

Speed is limited to usually 7000 feet per minute (35 meters per second).

Power transmission is limited to 370 kilowatts (500 horsepower).

Operating temperatures are usually restricted to 31 to 185F (35 to 85C).

Some adjustment of center distance or use of an idler pulley is necessary for wear and

stretch compensation. A means of disassembly must be provided to install endless belts.

4

1.2 Choice of Belt Drive System

There are four general types of belts:

a) Flat belts

b) V-belts

c) Film belts

d) Timing belts

Each has its own special characteristics, limitations, advantages, and special-purpose

variations for different applications.

a) Flat belts

Flat belts, in the form of leather belting, served as the basic belt drive from the

beginning of the Industrial Revolution.

They can transmit large amounts of power at high speeds.

Flat belts find their widest application where high-speed motion, rather than

power, is the main concern.

Flat belts are very useful where large center distances and small pulleys are

involved.

They can engage pulleys on both inside and outside surfaces, and both endless

and jointed construction are available.

5

b) V-belts

V-belts are the basic power-transmission belt, providing the best combination of

traction, operating speed, bearing load, and service life.

The belts are typically endless, with a trapezoidal cross section which runs in a

pulley with a V-shaped groove.

The wedging action of the belt in the pulley groove allows V-belts to transmit

higher torque at less width and tension than flat belts. V-belts are far superior to

flat belts at small center distances and high reduction ratios.

V-belts require larger pulleys than flat belts because of their greater thickness.

Several individual belts running on the same pulley in separate grooves are often

used when the power to be transmitted exceeds that of a single belt. These are

called multiple-belt drives.

c) Film belts

Film belts are often classified as a variety of flat belt, but actually they are a

separate type.

Consisting of a very thin strip of material, usually plastic but sometimes rubber,

their widest application is in business machines, tape recorders, and other light-

duty service.

6

d) Timing belts

Timing belts have evenly spaced teeth on their bottom side which mesh with

grooves cut on the periphery of the pulleys to produce a positive, no-slip,

constant-speed drive.

They are often used to replace chains or gears, reducing noise and avoiding the

lubrication bath or oiling system requirement.

They have also found widespread application in miniature timing applications.

Timing belts, known also as synchronous or cogged belts, require the least tension

of all belt drives and are among the most efficient.

The choice of type of belt to be used is important so that the required power can be

delivered efficiently. Some of the factor to be considered in selecting a suitable belt type to be

used for a particular application is;

Speed of driver and driven sheaves.

Required velocity ratio

Distance between the driver sheave and driven sheave.

Workspace constraint

Total power to be transmitted

7

1.3 Belt Drive System Arrangement

Power from one shaft can be transmitted to another shaft using some of this arrangement;

a) Open Belt Drive Arrangement

b) Cross-Belt Drive Arrangement

c) Quarter Turn Belt Drive with Idler Sheave

d) Open Belt Drive with Idler Pulley

8

1.4 Velocity Ratio of Belt Drive System

Let say; 1d = diameter of driver sheave

2d = diameter of driven sheave

1N = Speed of driver sheave

2N = Speed of driven sheave

For one complete rotation of the sheave, length of belt per unit time used is;

Length of belt = d

Thus for a total of N rotation per unit time, length of belt used is;

Total length of belt = dN .. (1)

By using the assumptions that belt used is inelastic, thus the length of belt passing the driver and

driven sheave is the same. Also no slip occurs in the system thus total length of belt required for

driver and driven sheave is the same.

2211 NdNd .. (2)

Simplify equation (2) to get Velocity Ratio, n

1

2

2

1

d

d

N

Nn .. (3)

If the thickness, t of belt is considered, equation (3) becomes

td

td

N

Nn

1

2

2

1 .. (4)

Slack side

Tight side

Driven pulley Driver pulley

9

1.5 Slip in Belt Drive System

Effective speed (rad/s) at the driver sheave, v

v [Original speed] - [Change of speed due to slip]

1006060

11111 GNdNdv

(5)

Where 1G = Percentage slip at the driver sheave

2G = Percentage slips at the driven sheave

By assumption that the belt is inelastic at the no slip condition, thus;

But the effective speed at driven sheave is reduced due to slip at the driven sheaves surface of

contact. Thus;

100

22

Gvvv

10060

222 GvvNd

.. (6)

Put equation (5) into (6), thus results in

1001 21

1

2

2

1 GG

d

d

N

N .. (7)

If the thickness of belt, t is considered, thus equation (7) becomes

1001 21

1

2

2

1 GG

td

td

N

N .. (8)

Driver sheave Driven sheave v

d1 d2

N2

N1

Belt

= The effective speed of

driven sheave, 2v

Belt drive will slip if friction force on

the surface of contact between belt

and sheave is reduced. This will

cause the sheave to rotate without

holding the belt. Normally slip is

expressed in percentage slip.

Consider a belt drive system with slip

as figure beside.

The effective speed

of driver sheave, v

10

1.6 Angle of Lap

Angle of lap/contact is the enclosed angle in which the belt and surface of pulley are in

contact. Angle of lap is important to determine the right setting of driver sheave and driven

sheave that can produce effective power transmission.

Consider an open belt drive system below;

Let 1r = radius of the big pulley

2r = radius of the small pulley

= Angle of lap

X = Distance between centres of both pulley

From the geometry;

X

rr

OO

MEEO

OO

MO 21

21

1

21

1sin

. .. (9)

Thus, angle of lap,

180

2180

rad .....(10)

11

If considering a cross belt drive system as shown in figure below;

If angle of lap, is the same for both sheave, then from geometry;

X

rr

OO

MEEO

OO

MO 21

21

1

21

1sin

. (11)

And the angle of lap is

180

2180

rad .(12)

12

1.7 Overall Length of Belt Drive System

Figure above shows an open belt drive system.

Let 1r = radius of the driver sheave

2r = radius of the driven sheave

= angle of lap

X = Distance between centres of driver sheave with driven sheave

L = Overall length of belt drive

From the geometry, line MO2 is parallel with line EF and also

X

rr

OO

MEEO

OO

MO 21

21

1

21

1sin

Since, is small, thus sin . This leaves with

X

rr 21 rad. . (13)

Again, from the figure above,

Length of curve

21rJE . (14)

Line 22122

1

2

212 rrXMOOOMOEF

2

211

X

rrXEF . (15)

13

Use Binomial Theorem to expand equation (15), and the final equation is

X

rrXEF

2

2

21 .. (16)

Length of curve

22rFK .. (17)

Total length of belt is

openL = 2 [Length of curve JE + Length of line EF + Length of curve FK ]

2222 2

2

211 r

X

rrXrLopen . (18)

Now consider for a cross belt drive system as below figure. The same procedure previously, only

the difference is that location of line EF is parallel with line 2MO .

Thus total length of belt is;

crossL = 2 [Length of curve JE + Length of line EF + Length of curve FK ]

2222 2

2

211 r

X

rrXrLcross . (19)

14

1.8 Belt Tension in Belt Drive System

Belt tension in a belt drive system can be expressed in terms of ratio between tension on

the tight side and the slack side of the same belt when it was operating. Consider a pulley wheel

with a belt passing around it as shown below. In order for the belt to produce torque on the wheel

(whether or not it is rotating), there must be tension in both ends. If this was not so, the belt

would not be pressed against the wheel and it would slip on the wheel. The belt depends upon

friction between it and the wheel in order to grip and produce torque.

For the belt to produce torque on the wheel, the tension in one end must be greater than

the tension in the other end. Let T2 is larger than T1 and is the angle of lap. Now, consider an

elementary length of the belt on wheel. The tension in one end is T and the other end is dTT .

The angle made by the small length is d .

T2

T1

dTT T

15

1. First, resolve T radially and tangentially to the wheel.

2cos1

dTT , and for small angle , thus 10coscos o

TT 1 . (20)

2sin1

dTR , for small angle , thus sin

2

1

dTR .. (21)

2. Next, repeat for the other end to resolve dTT

dTTd

dTTT 2

cos2

.. (22)

22

sin2 d

dTTd

dTTR .. (23)

3. Ignoring the product of two small quantities, total reaction force is

TdRRRN 21 .. (24)

4. The resultant tangential force is

dTTTTR 12 .. (25)

5. As a summary, two important results we have obtained

TdRN And dTTR

T

1T

dTT

2T

16

6. Now, treat the small piece of belt as a small block about to slip on a flat surface

When the block just about to slip, force RT is equal to friction force, F

NR RFT

TdRdT N

dT

dT (26)

7. Integrating between limits of 0 and for angle and 1TT and 2TT for

force, thus we get;

12 lnln TT

eT

T

1

2

(27)

Equation (27) is the ratio of belt tension and is used ONLY for flat belt type.

RT

NR

17

The derivation for the belt tension ratio for Vee-Belt type can also be derived with the

same approach as previously. Consider a section of a Vee-belt with an included angle of 2 . The

wedging affect increases the reaction force between the sheave and the belt from R to 'R . Since

the friction force is increased, greater power can be transmitted before the belt slips.

Figure: Vee-Belt section

1. Resolving 'R vertically gives an upward force sin'R on each side of belt

sin2

' RR .. (28)

2. Previously, NRdT , but for vee-belt must use 'RdT

sin2

' RRdT .. (29)

3. Since there are two faces in contact with the wheel, the friction is doubled. Hence

sinsin2

22 'RR

RdT . (30)

4. Completing the derivation by integrating between limits as before, the results are

sin

1

2 eT

T (31)

Equation (31) is the belt tension ratio for the Vee-Belt type ONLY.

18

1.9 Maximum Power Transmitted by Pulley

The tension in a belt pulley increases with torque and power. The maximum power that a

pulley system can transmit is limited by the strength of the belt material. If this is a problem then

more than one belt should be used to share the load. If the belt does not break, then the

possibility of belt slipping exists and this depends upon the angle of lap and coefficient of

friction. If the coefficient of friction is the same for both wheels, then slippage will occur first on

the smaller wheel. The power at which the belt slips is not the absolute maximum power that

can be transmitted as more power can be transmitted with slippage occurring by using higher

wheel speed.

The friction between the belt and the wheel is further affected by centrifugal force which

tends to lift the belt off the wheel. This increases the likelihood of slipping. Friction between belt

and pulley can be increased by using a Vee-belt type instead of Flat Belt type since Vee-belt can

grip better.

1.9.1 Maximum Power with No Belt Slip

1. Power transmitted by a pulley is generally given by TvP where T is the belt tension and

v is the speed of pulley.

Tight side

Slack side

Driver Driven

N1

T2

T1

N2

To find the power transmitted for certain belt

type, use the belt tension ratio and substitute

into TvP .

For Flat type belt, maximum power; when

the belt starts to slip is

vTTP 12 Watt

ve

TP

112 . (32)

This is the maximum power that can be

transmitted with no slip occurring.

19

For Vee Belt type, the maximum power that can be transmitted with no slip occurring is

v

e

TP

sin

2

11

Watt . (33)

2.10 Effect of Centrifugal Force

Consider the element of belt on the wheel once again;

The length of the curved element is rd

Density of the belt material is

The cross sectional area of the belt is A

The volume is Ard

The mass of the elemental belt strip is Arddm

The centrifugal force is r

mvFC

2

1. Since we are dealing with elemental mass, thus the elemental centrifugal force acting on the

tiny mass can be written in

2

22

vAdr

vArd

r

dmvdFC

. (34)

2. The normal force NR pressing the element to the wheel derived earlier without

centrifugal effect is, TdRN

3. Now the normal force is reduced due to centrifugal force acting outward, so

T dTT

20

2vAdTdRN

2AvTdRN .. (35)

4. From relation NRdT , thus substitute it into equation (35)

2AvTddT

d

AvT

dT

2 (36)

5. Integrating both sides of equation (36) from 1T to 2T and angle from 00 to ,

02

2

1

dAvT

dTT

T

e

AvT

AvT

2

1

2

2 .. (37)

6. Let CTAv 2 that is the centrifugal force term, then

eTT

TT

C

C

1

2

.. (38)

We can see the effect of centrifugal force from equation (38). It shows that the tension on

the belt increase due to centrifugal effect. Centrifugal effect tends to lift the belt off the wheel,

thus increase the likelihood for slippage to occur.

Equation (38) represents the belt tension ratio for a Flat Belt type with the effect of

centrifugal force on the system. The effect of centrifugal force can be ignored when belt drive is

operating at low speed, but it must be taken care when it is operating at high speed. Also

centrifugal force effect must be included if mass per unit length of the belt is considered. Note

also that since the angle of lap is smallest on the small wheel, the belt always slips first on the

small wheel (if the coefficient of fiction is the same).

For Vee-belt type, the belt tension ratio when centrifugal effect is considered,

21

sin

1

2 eTT

TT

C

C

(39)

1.10.1 Maximum Power with Centrifugal Effect Included

Equation (32) and (33) earlier states that how a maximum power can be achieved with

condition no slip occurs (offset of centrifugal effect). However, due to the onset of centrifugal

effect, the belt tends to lift off the wheel and thus slippage is likely to occur. This will cause

power to decrease as the speed of belt drive increase. So, the problem here is how to maximize

the full power of the belt drive system by practically taking care of the centrifugal effect and

slippage that occur?

Modify equation (32), now take into consideration the effect of centrifugal force. It will

make the effective tension at tight side to be CTT 2 and slack side to be CTT 1 .

ve

TTP C

112 . (40)

22

Plot graph of power against speed for a given set of parameters as shown below.

The graph shows clearly that the power is increased as the speed increase but a point is

reached when the centrifugal force reduces the grip to such an extend that slippage reduces the

power. Further increase in speed reduces the power as the belt slips more, even though

practically, slips maybe start to occur at a point before reaching the critical speed. At very high

speed, there will be no more grips at all and power drops to zero.

At the peak point, gradient is zero. Differentiate Power with respect to velocity

0dv

dP

This will result with

CTT 32 .. (41)

And the critical velocity that gives maximum power is

2

1

2

3

A

TvP

... (42)

23

1.11 Initial Tension of Belt Drive System

Practically, setting of the tension on the driver and driven sheave is done when the belt drive is

not operating and still has some tension on it. This initial setting is called initial tension of the

belt. By setting the belt with some initial tension, it will increase the gripping ability of the belt

to the pulley. Let say;

AT = Initial tension of belt (N)

2T = Tension on the tight side of belt (N)

1T = Tension on the slack side of belt (N)

= Belts length constant

When power is transmitted, the tension on the tight side increase from AT to 2T while on the

slack side is reduced from AT to 1T . If the belt is assumed to obey Hookes Law and the length

of belt does not change, thus

Thus an equation can be developed that is

12 TTTT AA

For the case of centrifugal force effect is neglected, equation above becomes

2

21 TTTA

(43)

For the case where centrifugal effect is considered, then

2

221 CA

TTTT

(44)

Increase in length on

the tight side = Decrease in length on

the slack side

24

1.12 Creep on the Belt Drive System

When power is transmitted by a belt or rope, there is always a difference between the peripheral

speed of the driving pulley and that of driven pulley. Because of different tensions on the two

sides of the pulley, the stretch in the belt will be different. The portion of the belt, leaving the

follower and approaching the driver is stretched more than the portion of the belt, leaving the

driver and approaching the follower. These uneven extensions and contractions of the belt due to

varying tension will cause a relative motion of the belt on the pulley. This relative motion is

called creep of belt.

Consider one metre length of belt when unstressed. Because of tension T1 on the tight side, the

length of the belt is (1 + x1) metre, where x1 is the stretch. Similarly, due to tension T2 on the

slack side, the length of the belt is (1 + x2) metre, where x2 is the stretch. Obviously x1 is greater

than x2. A length (1 + x1) metre has approached the driver, but only (1 + x2) metre has moved off

the driver. Thus, the length of belt that leaves the driver pulley is less than that which has

approached it. But in the case of driven pulley, the length of belt leaving the driven pulley is

more than that, approaching it. Thus, there is some relative motion of the belt on the pulley, and

the belt is said to creep. The effect of creep in belt is to reduce the speed of the follower and

reduce the power output. Considering creep, the velocity ratio is given by

AE

TT

v

v 12

1

2 1

.(45)

Where A = Cross section of the belt (2m ) m2

E = Young Modulus of the belt material ( 2/ mN )

2v = Velocity of driven pulley (m/s)

1v = Velocity of driver pulley (m/s)