Problem Statement
Determine the intensity of shear of an oil having viscosity = 1.2 poise and is used for lubrication in the clearance between a 10 cm diameter shaft and its journal bearing. The clearance is 1.0 mm and the shaft rotates at 200 r.p.m.
Given Data
- Viscosity, \(\mu = 1.2 \, \text{poise}\)
- Shaft Diameter, \(D = 10 \, \text{cm}\)
- Clearance (distance), \(dy = 1.0 \, \text{mm}\)
- Shaft Speed, \(N = 200 \, \text{r.p.m.}\)
Solution
1. Convert All Units to SI
To use Newton's law of viscosity, we must convert all given values to standard SI units (meters, seconds, N·s/m²).
2. Calculate the Tangential Velocity of the Shaft (du)
The rotational speed (r.p.m.) must be converted to the linear velocity at the surface of the shaft. This is the velocity that shears the oil.
3. Calculate the Shear Stress (τ)
We apply Newton's law of viscosity. The journal bearing is stationary, so the velocity change (\(du\)) occurs across the clearance (\(dy\)).
Substitute the values to find the shear stress:
The intensity of shear (shear stress) on the oil is \( \tau \approx 125.64 \, \text{N/m}^2 \).
Explanation of Shear in a Journal Bearing
A journal bearing supports a rotating shaft. A thin film of lubricating oil separates the shaft from the stationary bearing surface. As the shaft rotates, it drags the adjacent layer of oil with it. This creates a velocity gradient across the oil film, from the high speed at the shaft surface to zero at the stationary bearing surface.
This velocity difference causes the oil to be "sheared". The oil's internal resistance to this shearing action is its viscosity. The force per unit area required to overcome this resistance and maintain the rotation is the shear stress (\(\tau\)).
Physical Meaning
The calculated shear stress of 125.64 N/m² represents the frictional force that the oil exerts on the shaft and bearing surfaces per unit area. This value is critical for several engineering considerations:
- Power Loss: The shear stress directly relates to the frictional torque on the shaft. Overcoming this torque requires power, which is dissipated as heat in the oil. Engineers use this calculation to estimate the power loss due to friction in the bearing.
- Heat Generation: The energy lost to friction is converted into heat. The amount of heat generated must be managed by a cooling system to prevent the oil from overheating and losing its lubricating properties.
- Lubricant Selection: The choice of oil (its viscosity) is a trade-off. A higher viscosity oil can support greater loads but also generates more frictional heat and power loss. This calculation helps engineers select an appropriate lubricant for the operating speed and load conditions.
