Problem Statement
A shaft 75mm in diameter is fixed axially and rotated inside a sleeve of diameter 75.2mm at 200rpm. The length of the shaft is 200mm. Determine the resisting torque exerted by the oil and the power required to rotate the shaft. The viscosity of oil is 5 NS/m2.
Solution
Given:
- Diameter of shaft (D) = 75mm = 0.075m
- Radius of shaft (r) = 0.075/2 = 0.0375m
- Length of shaft (L) = 200mm = 0.2m
- Speed of shaft (N) = 200rpm
- Dynamic viscosity of oil (μ) = 5 NS/m2
- Clearance (dr) = dy = (75.2 – 75)/2 mm = 0.1mm = 0.0001m
Calculations:
1. Angular velocity (ω):
ω = (2 × π × N) / 60 = (2 × π × 200) / 60 = 20.94 rad/s
2. Tangential velocity (u):
u = r × ω = 0.0375 × 20.94 = 0.785 m/s
3. Frictional force (F):
F = μ (du/dy) × (πDL)
Substituting values:
F = 5 × 0.785 / 0.0001 × (π × 0.075 × 0.2) = 1849.6 N
4. Torque (T):
T = F × r = 1849.6 × 0.0375 = 69.4 Nm
5. Power (P):
Using P = F × u:
P = 1849.6 × 0.785 = 1451.94 W = 1.45 kW
Alternatively, using P = T × ω:
P = 69.4 × 20.94 = 1451.94 W
Results:
- Resisting torque (T): 69.4 Nm
- Power required (P): 1.45 kW
Explanation
This problem involves calculating the resisting torque and power required to rotate a shaft inside a sleeve filled with oil. Here’s the detailed explanation:
- Angular velocity (ω): The shaft’s rotational speed is converted from RPM to rad/s using the standard formula.
- Tangential velocity (u): The tangential velocity at the shaft’s surface is calculated as the product of its radius and angular velocity.
- Frictional force (F): The oil’s dynamic viscosity, the velocity gradient (du/dy), and the surface area of the shaft are used to calculate the resisting force exerted by the oil.
- Torque (T): Torque is calculated as the product of the resisting force and the radius of the shaft.
- Power (P): The power required to rotate the shaft is calculated using two approaches: (a) force and tangential velocity, and (b) torque and angular velocity. Both methods yield the same result.
This problem demonstrates the application of fluid mechanics principles, including viscosity and shear stress, to determine forces, torques, and power in mechanical systems involving rotating shafts.
