A caliper brake is required for the front wheels of a passenger car with a braking capacity of 450 Nm for each brake. Preliminary design estimates have set the brake geometry as ri = 100 mm, ro = 150 mm and θ = 40°. Pads with a coefficient of friction of 0.35 have been selected. Each pad is actuated by means of a hydraulic cylinder of nominal diameter 25.4mm. Determine the required actuating force, the average and the maximum contact pressures and the required hydraulic pressure for brake actuation.

(7)

T = Answer
 Nm

re = Answer
 m

F = Answer
 N

pmax = Answer
 N/m2

pav = Answer
 N/m2

Acylinder = Answer
 m2

phydraulic = Answer
 N/m2

To solve this problem, we'll need to apply several mechanical and hydraulic principles. Here are the steps and calculations required:

### Given Data
- Braking torque per brake, \( T = 450 \, \text{Nm} \)
- Inner radius of the brake, \( r_i = 100 \, \text{mm} = 0.1 \, \text{m} \)
- Outer radius of the brake, \( r_o = 150 \, \text{mm} = 0.15 \, \text{m} \)
- Angle of contact, \( \theta = 40^\circ = \frac{40 \times \pi}{180} \, \text{radians} \)
- Coefficient of friction, \( \mu = 0.35 \)
- Diameter of the hydraulic cylinder, \( d = 25.4 \, \text{mm} = 0.0254 \, \text{m} \)

### Calculations

#### 1. Effective Radius of the Brake
The effective radius \( r_e \) is the average of the inner and outer radii:
\[
r_e = \frac{r_i + r_o}{2} = \frac{0.1 + 0.15}{2} = 0.125 \, \text{m}
\]

#### 2. Actuating Force \( F \)
The braking torque \( T \) is related to the actuating force \( F \) and the effective radius \( r_e \) as follows:
\[
T = \mu \cdot F \cdot r_e
\]
Solving for \( F \):
\[
F = \frac{T}{\mu \cdot r_e} = \frac{450}{0.35 \cdot 0.125} = 10,285.71 \, \text{N}
\]

#### 3. Maximum Contact Pressure \( p_{\text{max}} \)
The maximum contact pressure can be estimated using:
\[
p_{\text{max}} = \frac{2F}{b \cdot (r_o - r_i)}
\]
Assuming \( b \) (pad width) is the difference between outer and inner radii \( r_o - r_i = 0.05 \, \text{m} \):
\[
p_{\text{max}} = \frac{2 \times 10,285.71}{0.05 \cdot 0.05} = 8,228,571.43 \, \text{N/m}^2
\]

#### 4. Average Contact Pressure \( p_{\text{av}} \)
The average contact pressure can be calculated using:
\[
p_{\text{av}} = \frac{F}{b \cdot (r_o - r_i)}
\]
\[
p_{\text{av}} = \frac{10,285.71}{0.05 \cdot 0.05} = 4,114,285.71 \, \text{N/m}^2
\]

#### 5. Cylinder Area \( A_{\text{cylinder}} \)
The area of the hydraulic cylinder is:
\[
A_{\text{cylinder}} = \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{0.0254}{2} \right)^2 = 5.067 \times 10^{-4} \, \text{m}^2
\]

#### 6. Required Hydraulic Pressure \( p_{\text{hydraulic}} \)
The hydraulic pressure is given by:
\[
p_{\text{hydraulic}} = \frac{F}{A_{\text{cylinder}}}
\]
\[
p_{\text{hydraulic}} = \frac{10,285.71}{5.067 \times 10^{-4}} = 20,295,422.45 \, \text{N/m}^2
\]

### Final Answers

- **Braking torque \( T \):** 450 Nm
- **Effective radius \( r_e \):** 0.125 m
- **Actuating force \( F \):** 10,285.71 N
- **Maximum contact pressure \( p_{\text{max}} \):** 8,228,571.43 N/m\(^2\)
- **Average contact pressure \( p_{\text{av}} \):** 4,114,285.71 N/m\(^2\)
- **Cylinder area \( A_{\text{cylinder}} \):** 5.067 \times 10^{-4} m\(^2\)
- **Hydraulic pressure \( p_{\text{hydraulic}} \):** 20,295,422.45 N/m\(^2\)

If you have any questions about these calculations, feel free to ask!

### Additional Questions

1. How does changing the coefficient of friction affect the required actuating force?
2. What impact does increasing the hydraulic cylinder diameter have on the hydraulic pressure?
3. How would a change in the angle of contact affect the braking torque?
4. Can we use different materials for the pads to increase the braking efficiency?
5. What safety factors should be considered in the design of the hydraulic system?
6. How does the effective radius change if the brake geometry is modified?
7. In what scenarios might the hydraulic pressure exceed safe operational limits?
8. How can the design be optimized for reduced wear on the brake pads?

### Tip

When designing brake systems, always consider the heat dissipation capability to avoid brake fade during prolonged use.

Explore the detailed calculations and analysis for designing a caliper brake system for a passenger car, including actuating force, contact pressures, and hydraulic pressure requirements. Understand the mechanical and hydraulic principles involved.

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