The Pulley-Belt System

WAWY FKP, UMP September 2013

Pulley and belt are mechanical “transmission” elements. As a system, they transmit force-torque, velocity, tangential acceleration, angular acceleration, angular velocity and angular position between pulleys. Figure 1.0 below is a picture of a pulley-belt system.

Figure 1: A picture of an actual pulley-belt system application.

The purpose of a pulley-belt system is to manipulate the “motion”. By manipulating, we mean changing the motion magnitude and direction. An example of changing direction is shown in Figure 2. Referring to Figure 2, the magnitude of the displacement, velocity and acceleration of mass A is the same as mass B but the direction is different. Mass A is moving horizontally while mass B is moving vertically.

Figure 2: An example of motion changing direction using a pulley system.

Using a pulley-belt system, we can also change the motion magnitude without changing direction. By carefully choosing the right pulley size and configurations, we can change displacement, speed and acceleration of a pulley-belt system.

Am

Bm

1.0 Facts about pulley-belt system

Figure 3 below is a schematic of a pulley-belt system.

Figure 3: A simple pulley-belt system

Here are some facts about the system:

The power or energy flow is constant, or is assumed constant (assume no energy loss due to heat, for example). The principle “energy is not created nor destroyed” is still applicable here.

The motion at the contact points between the pulley and the belt are equal. They are moving together because we assume no slipping occurs between the belt and pulley. Referring to Figure 4 below, point A is a shared point between the pulley and the belt. Thus, they have the same velocity and tangential acceleration.

Figure 4: Point A is a shared point between the pulley and the belt.

Velocity and tangential acceleration magnitudes are equal along the belts because the belts are assumed to be inextensible (and also incompressible!). If the speed is different on the belts, then the belt will stretch or compress. Notice that it is the magnitude that is equals not the direction as can be seen from the Figure 5.

B

B

B

A

A

A

ArBr

AV

BV

TBa

TAa

NBa

NAa

Belt

Pulley

r

A

AV

r

A

AV

r

A

AV

Belt

Pulley

Belt Pulley

Figure 5: Speed and acceleration is the same at any point on the belt.

The magnitude of the normal acceleration can be different between the pulleys as shown in Figure 6. However, both pointed towards the centre of rotation.

Figure 6: Normal acceleration is not the same between the pulleys.

2.0 Angular Position

One revolution of pulley A will not make one revolution of pulley B since the radius of the pulley is not equal. Consider the system in Figure 7 below.

Figure 7: Angular displacement of a pulley-belt system

Whenever pulley A rotates A amount, the belt is moving with AS amount of displacement. The same

amount of displacement is rotated by pulley B. Thus, we have:

ArBr

AV

BV

TBa

TAa

BA VV

BA VV

ArBr

NBa

NAa

Belt

Pulley

B

Ar

Br

AAA rS

BA SS

BA SS

A

BBB rS

BBAA

BA

BBB

AAA

rr

SS

rS

rS

From the above relation, the angular position between pulley A and pulley B is given below.

A

B

B

A

r

r

Notice the inverse relationship between the radius and the angular displacement. The bigger pulley is “rotationally” smaller.

3.0 Angular Velocity

Since the velocity along the belt is the same, relation between angular velocities can be derived as shown below.

A

B

B

A

BBAA

BA

r

r

rr

VV

Notice the inverse relationship between the radius and the angular velocity. The bigger pulley is “rotationally” slower.

4.0 Angular Acceleration

TANGENTIAL angular acceleration between pulley A and pulley B is the same.

A

B

B

A

BBAA

BA

r

r

rr

aa

Notice the inverse relationship between the radius and the angular acceleration. The bigger pulley is “rotationally” less accelerating.

5.0 Normal Acceleration

The normal acceleration between the pulleys is different because the size of the pulley is different.

BBNB

AANA

ra

ra

2

2

7.0 Summary

“Motion” Relationship Remarks

Angular Position A

B

B

A

r

r

Smaller turns for the bigger

pulley.

Angular Velocity A

B

B

A

r

r

The bigger pulley is slower.

Angular Acceleration A

B

B

A

r

r

The bigger puller is accelerating less.

Velocity BA VV Same magnitude at contact point.

Tangential Acceleration

BBAA

BTAT

rr

aa

Same magnitude at contact point.

Normal Acceleration

BBBN

AAAN

BNAN

ra

ra

aa

2

2

Depend of the radius of the pulley and the angular velocity.

Problem 01

Starting from rest when s = 0, pulley A is given an angular acceleration α = 6 rad/s

2, where θ is in radians. Determine the

speed of block B when it has risen 6 m. The pulley has an inner hub D which is fixed to C and turns with it.

Given:

2

rad/s 6

m 6

m 0

A

fB

iB

s

s

What to determine?

The speed of mass B, BV when m 6fBs .

Observations

There are 6 rigid bodies: (1) the ground (2) pulley A (3) the belt (4) the pulley C and D (5) the cord DB and (6) the mass B.

Pulley A is accelerating which causes pulley C and D to accelerate which then accelerate the mass B.

The displacement of mass B is equals to the rotational displacement of pulley D.

The acceleration of mass B is equals to the tangential acceleration of pulley D.

Solution

We need to relate the acceleration of mass B with the given angular acceleration of pulley A. First, determine the angular acceleration of pulley C.

A

C

A

C

A

C

C

A

r

r

r

r

Pulley D angular acceleration is equals to pulley C accelerations. Then, the tangential acceleration of pulley D can be determined.

DA

C

A

DDDT

A

C

A

CD

rr

rra

r

r

Tangential acceleration of pulley D is equals to the acceleration of mass B. Furthermore, it is given

that AA 6 .

AD

C

A

B

DA

C

A

DTB

rr

ra

rr

raa

6

Now, we have to relate A with the vertical displacement of mass B. From our observation, the

displacement of mass B is equals to the rotational displacement of pulley D.

DDB rs

Now we need to relate D with A .

DA

C

A

C

A

C

C

A

r

r

r

r

Replacing into previous equation, we have:

AD

C

A

A

C

A

DDD

rr

sr

r

rrrs

Replacing into Ba , we have:

srr

srr

r

rr

r

ra

AD

C

D

C

A

AD

C

A

B 666

Finally, we can solve the motion of mass B.

2

m6

0

2

0

2

0

m6

0

m/s 7.14

)18(62

662

B

sB

V

V

ssds

V

adsvdv

adsvdv

B

Problem 02

A mill in a textile plant uses the belt-and-pulley arrangement shown to transmit power. When t = 0 and electric motor is turning pulley A with

an angular velocity of A = 5 rad/s. If this pulley is subjected to a

constant angular acceleration 2 rad/s2, determine the angular velocity

of pulley B after B turns 6 revolutions. The hub D is rigidly connected to pulley C and it turns with it.

Given:

2

rad/s 2

rad/s 5

A

iA

What to determine?

The angular velocity of pulley B, when rev 6B .

Observations

There are 6 rigid bodies: (1) the ground – three pins at the centre of each pulley (2) pulley A (3) the belt linking pulley A and pulley C (4) pulley C and hub D (5) the belt linking hub D and pulley B and (6) pulley B.

Pulley A is not accelerating meaning that it rotates at constant angular velocity. This allows us to use constant angular acceleration equations.

The angular velocity of pulley C is lower than pulley A because its radius is bigger. Similarly, the angular velocity of pulley B is lower than pulley C because the radius of hub D is smaller than pulley C. In other words, we expect the angular velocity of pulley B is lower than 5 rad/s.

On the other hand, we have an opposite situation with the angular displacement. When pulley B is rotating 6 revolutions, we expect hub D will rotate more than 6 revolutions because of its smaller radius. Likewise, pulley A will rotate more than pulley C. We expect pulley A to rotate more than 6 revolutions.

Solution

Once pulley B is rotating at 6 rev, we can determine the amount of revolution pulley A is rotating. Once we know pulley A revolution, we can use constant angular acceleration equation to determine the angular velocity of pulley A. Once the pulley A angular velocity is known, we can determine angular velocity of pulley C and hub D and then pulley B.

DC

B

D

B

D

B

D

B

r

r

r

r

D

B

rad 12rev

rad 2rev 6

Now, the relationship between A :

rad 78.17123

4

5.4

5

C

A

B

D

B

A

C

A

C

A

C

A

A

C

r

r

r

r

r

r

r

r

Now, we can use constant angular acceleration equation on pulley A.

rad/s 76.15

)78.17)(2(250222

0

2

A

AAAA

Now, let’s determine the angular velocity of pulley C.

rad/s 18.1476.155

5.4

A

C

A

C

C

A

A

C

r

r

r

r

SinceC D

, we can now determine the angular velocity of pulley B.

rad/s 64.1018.144

3

D

B

D

B

B

D

D

B

r

r

r

r