Answer :
### 22. Types of Various Flat Belt Drives
Flat belt drives are used extensively in mechanical systems to transmit power between shafts. There are various types of flat belt drives based on the application and mechanical setup:
a. Open Belt Drive
- Description: In this type, the driver and driven pulleys rotate in the same direction.
- Sketch:
```
Pulley 1 -----> Pulley 2
------>
```
b. Crossed Belt Drive
- Description: This configuration makes the driver and driven pulleys rotate in opposite directions. It is used when the shafts are required to rotate in opposite directions.
- Sketch:
```
Pulley 1 -----> Pulley 2
<------
```
c. Quarter-Turn Belt Drive
- Description: Used when the shafts are perpendicular to each other.
- Sketch:
```
Pulley 1
|
V _______ Pulley 2
```
d. Compound Belt Drive
- Description: It uses multiple shafts and is a combination of two or more belt drives.
- Sketch:
```
Pulley 1 -----> Pulley 2
------>
|
Pulley 3 -----> Pulley 4
<------
```
e. Belt Conveyor Drive
- Description: Commonly used in material handling equipment where a continuous movement of belt conveyor is required.
- Sketch:
```
------------ Pulley 1
Conveyor ------>
------------ Pulley 2
```
### 23. Factors that Control the Power Transmission Capacity of a Belt
a. Tension in the Belt
- Description: Higher tension allows more power transmission but can also increase the wear and tear on the belt.
b. Velocity of the Belt
- Description: The speed at which the belt moves affects power transmission; however, higher speeds may lead to increased friction losses.
c. Coefficient of Friction
- Description: The friction between the belt and pulleys determines how much force can be transmitted without slipping.
d. Belt Material
- Description: Strength and flexibility of the belt material influence its capacity to transmit power.
e. Belt Thickness and Width
- Description: Wider and thicker belts can carry more load.
f. Pulley Size
- Description: Large pulleys increase the contact area, reducing slippage and thus transmitting more power.
g. Angle of Contact
- Description: Larger angles of contact provide more frictional force, therefore, allowing more power to be transmitted.
### 24. Proof: Ratio of Driving Tensions on the Two Sides of a Pulley
Given:
[tex]\( T_1 \)[/tex] = Tension in the tight side of the belt,
[tex]\( T_2 \)[/tex] = Tension in the slack side of the belt,
[tex]\( \mu \)[/tex] = Coefficient of friction between the belt and the pulley,
[tex]\( \theta \)[/tex] = Angle of contact in radians.
To derive the relationship between [tex]\( T_1 \)[/tex] and [tex]\( T_2 \)[/tex]:
- Consider a small segment of the belt in contact with the pulley subtending a small angle [tex]\( d\theta \)[/tex]. Let the tension on one side of this segment be [tex]\( T \)[/tex] and on the other be [tex]\( T + dT \)[/tex].
- The normal reaction [tex]\( dN \)[/tex] due to the small segment is balanced by the tension difference [tex]\( dT \)[/tex], given by:
[tex]\[ dT = \mu \cdot dN \][/tex]
- The normal force [tex]\( dN \)[/tex] for the small segment can be expressed using the tension [tex]\( T \)[/tex] and angle [tex]\( d\theta \)[/tex] as:
[tex]\[ dN = T \cdot d\theta \][/tex]
- Substituting this into the tension equation:
[tex]\[ dT = \mu \cdot T \cdot d\theta \][/tex]
- Rearranging and integrating both sides from [tex]\( T_2 \)[/tex] to [tex]\( T_1 \)[/tex] and [tex]\( 0 \)[/tex] to [tex]\( \theta \)[/tex]:
[tex]\[ \frac{dT}{T} = \mu \cdot d\theta \][/tex]
[tex]\[ \int_{T_2}^{T_1} \frac{dT}{T} = \mu \int_0^\theta d\theta \][/tex]
[tex]\[ \ln\left(\frac{T_1}{T_2}\right) = \mu \cdot \theta \][/tex]
[tex]\[ \frac{T_1}{T_2} = e^{\mu \theta} \][/tex]
Thus, the ratio of the driving tensions on the two sides of a pulley is [tex]\( \frac{T_1}{T_2} = e^{\mu \theta} \)[/tex].
### 25. Defining the Life of a Bearing
The life of a bearing is defined as the number of revolutions or the number of hours at a particular speed, which the bearing can endure before the first evidence of material fatigue or failure appears. It is expressed as L10, which means that 90% of a group of identical bearings will complete or exceed the specified number of revolutions or hours under the same conditions. Bearing life can also be affected by lubrication, load, temperature, and other operating conditions.
Flat belt drives are used extensively in mechanical systems to transmit power between shafts. There are various types of flat belt drives based on the application and mechanical setup:
a. Open Belt Drive
- Description: In this type, the driver and driven pulleys rotate in the same direction.
- Sketch:
```
Pulley 1 -----> Pulley 2
------>
```
b. Crossed Belt Drive
- Description: This configuration makes the driver and driven pulleys rotate in opposite directions. It is used when the shafts are required to rotate in opposite directions.
- Sketch:
```
Pulley 1 -----> Pulley 2
<------
```
c. Quarter-Turn Belt Drive
- Description: Used when the shafts are perpendicular to each other.
- Sketch:
```
Pulley 1
|
V _______ Pulley 2
```
d. Compound Belt Drive
- Description: It uses multiple shafts and is a combination of two or more belt drives.
- Sketch:
```
Pulley 1 -----> Pulley 2
------>
|
Pulley 3 -----> Pulley 4
<------
```
e. Belt Conveyor Drive
- Description: Commonly used in material handling equipment where a continuous movement of belt conveyor is required.
- Sketch:
```
------------ Pulley 1
Conveyor ------>
------------ Pulley 2
```
### 23. Factors that Control the Power Transmission Capacity of a Belt
a. Tension in the Belt
- Description: Higher tension allows more power transmission but can also increase the wear and tear on the belt.
b. Velocity of the Belt
- Description: The speed at which the belt moves affects power transmission; however, higher speeds may lead to increased friction losses.
c. Coefficient of Friction
- Description: The friction between the belt and pulleys determines how much force can be transmitted without slipping.
d. Belt Material
- Description: Strength and flexibility of the belt material influence its capacity to transmit power.
e. Belt Thickness and Width
- Description: Wider and thicker belts can carry more load.
f. Pulley Size
- Description: Large pulleys increase the contact area, reducing slippage and thus transmitting more power.
g. Angle of Contact
- Description: Larger angles of contact provide more frictional force, therefore, allowing more power to be transmitted.
### 24. Proof: Ratio of Driving Tensions on the Two Sides of a Pulley
Given:
[tex]\( T_1 \)[/tex] = Tension in the tight side of the belt,
[tex]\( T_2 \)[/tex] = Tension in the slack side of the belt,
[tex]\( \mu \)[/tex] = Coefficient of friction between the belt and the pulley,
[tex]\( \theta \)[/tex] = Angle of contact in radians.
To derive the relationship between [tex]\( T_1 \)[/tex] and [tex]\( T_2 \)[/tex]:
- Consider a small segment of the belt in contact with the pulley subtending a small angle [tex]\( d\theta \)[/tex]. Let the tension on one side of this segment be [tex]\( T \)[/tex] and on the other be [tex]\( T + dT \)[/tex].
- The normal reaction [tex]\( dN \)[/tex] due to the small segment is balanced by the tension difference [tex]\( dT \)[/tex], given by:
[tex]\[ dT = \mu \cdot dN \][/tex]
- The normal force [tex]\( dN \)[/tex] for the small segment can be expressed using the tension [tex]\( T \)[/tex] and angle [tex]\( d\theta \)[/tex] as:
[tex]\[ dN = T \cdot d\theta \][/tex]
- Substituting this into the tension equation:
[tex]\[ dT = \mu \cdot T \cdot d\theta \][/tex]
- Rearranging and integrating both sides from [tex]\( T_2 \)[/tex] to [tex]\( T_1 \)[/tex] and [tex]\( 0 \)[/tex] to [tex]\( \theta \)[/tex]:
[tex]\[ \frac{dT}{T} = \mu \cdot d\theta \][/tex]
[tex]\[ \int_{T_2}^{T_1} \frac{dT}{T} = \mu \int_0^\theta d\theta \][/tex]
[tex]\[ \ln\left(\frac{T_1}{T_2}\right) = \mu \cdot \theta \][/tex]
[tex]\[ \frac{T_1}{T_2} = e^{\mu \theta} \][/tex]
Thus, the ratio of the driving tensions on the two sides of a pulley is [tex]\( \frac{T_1}{T_2} = e^{\mu \theta} \)[/tex].
### 25. Defining the Life of a Bearing
The life of a bearing is defined as the number of revolutions or the number of hours at a particular speed, which the bearing can endure before the first evidence of material fatigue or failure appears. It is expressed as L10, which means that 90% of a group of identical bearings will complete or exceed the specified number of revolutions or hours under the same conditions. Bearing life can also be affected by lubrication, load, temperature, and other operating conditions.
