To determine the mass needed for a hydraulic accumulator to act under a pressure of 2 MPa, we can use the relationship between pressure, force, and area.
Step 1: Calculate the area of the ram
The area \( A \) of the ram can be calculated using the formula for the area of a circle:
\[ A = \pi \left(\frac{d}{2}\right)^2 \]
where \( d \) is the diameter of the ram.
Given:
- Diameter \( d = 450 , \text{mm} = 0.450 , \text{m} \)
Calculating the radius:
\[ r = \frac{d}{2} = \frac{0.450}{2} = 0.225 , \text{m} \]
Now, calculate the area:
\[ A = \pi (0.225)^2 = \pi (0.050625) \approx 0.159 , \text{m}^2 \]
Step 2: Calculate the force required using the pressure
The force \( F \) exerted by the hydraulic accumulator can be calculated using the formula:
\[ F = P \times A \]
where \( P \) is the pressure in pascals and \( A \) is the area in square meters.
Given:
- Pressure \( P = 2 , \text{MPa} = 2 \times 10^6 , \text{Pa} \)
Now, calculate the force:
\[ F = 2 \times 10^6 , \text{Pa} \times 0.159 , \text{m}^2 \approx 318,000 , \text{N} \]
Step 3: Calculate the additional mass needed
The net force exerted by the hydraulic system must balance the weight of the ram plus the additional weight (mass) contributing to the pressure.
The weight \( W \) of the ram and the additional mass \( m_{\text{additional}} \) can be represented as:
\[ W = (m_{\text{ram}} + m_{\text{additional}}) \times g \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)).
To solve for the additional mass needed, we set \( F = W \):
\[ 318,000 , \text{N} = (750, \text{kg} + m_{\text{additional}}) \times 9.81 , \text{m/s}^2 \]
Step 4: Solve for \( m_{\text{additional}} \)
Rearranging the equation:
\[ 750 + m_{\text{additional}} = \frac{318,000}{9.81} \]
Calculating the right side:
\[ \frac{318,000}{9.81} \approx 32,400.82 , \text{kg} \]
Now, solve for \( m_{\text{additional}} \):
\[ m_{\text{additional}} \approx 32,400.82 - 750 \approx 31,650.82 , \text{kg} \]
Conclusion
The additional mass needed for the hydraulic accumulator to act under a pressure of 2 MPa, considering the 750 kg mass of the ram, is approximately 31,650.82 kg.