A vertical shaft passes through a bearing and is lubricated with an oil having a viscosity of
as shown in Fig.
Assume that the flow characteristics in the gap between the shaft and bearing are the same as those for laminar flow between infinite parallel plates with zero pressure gradient in the direction of flow. Estimate the torque required to overcome viscous resistance when the shaft is turning at 80 rev/min.
The Correct Answer and Explanation is:
To estimate the torque required to overcome viscous resistance in the bearing, we assume the lubricant behaves like a viscous fluid between two parallel plates. The shaft rotates, creating a shear flow in the thin gap of oil between the shaft and bearing. Let’s define the parameters required for the calculation:
Assumptions and Formula:
- Laminar Couette Flow (no pressure gradient, flow due only to shear from moving surface).
- Torque formula for a vertical shaft in a bearing:
T=2πμLR3ωhT = 2\pi \mu L R^3 \frac{\omega}{h}
Where:
- μ\mu = dynamic viscosity of oil (Pa·s),
- LL = length of shaft in contact with bearing (m),
- RR = radius of the shaft (m),
- ω\omega = angular velocity (rad/s),
- hh = thickness of the oil film (m).
Given Data (typical values, assuming not specified in your image):
- Shaft radius R=0.05R = 0.05 m,
- Shaft length L=0.2L = 0.2 m,
- Oil film thickness h=0.0005h = 0.0005 m,
- Dynamic viscosity μ=0.1\mu = 0.1 Pa·s,
- Rotational speed N=80N = 80 rev/min.
Convert NN to angular velocity ω\omega: ω=2πN60=2π⋅8060≈8.38 rad/s\omega = \frac{2\pi N}{60} = \frac{2\pi \cdot 80}{60} \approx 8.38 \text{ rad/s}
Torque Calculation:
T=2π⋅0.1⋅0.2⋅(0.05)3⋅8.380.0005T = 2\pi \cdot 0.1 \cdot 0.2 \cdot (0.05)^3 \cdot \frac{8.38}{0.0005} T=2π⋅0.1⋅0.2⋅1.25×10−4⋅16760T = 2\pi \cdot 0.1 \cdot 0.2 \cdot 1.25 \times 10^{-4} \cdot 16760 T≈2π⋅0.1⋅0.2⋅2.095=0.263 NmT \approx 2\pi \cdot 0.1 \cdot 0.2 \cdot 2.095 = 0.263 \text{ Nm}
Explanation
To estimate the torque required to overcome viscous resistance in a lubricated bearing, we consider the flow of fluid between the rotating shaft and the stationary bearing as analogous to laminar flow between two parallel plates. This assumption simplifies the complex fluid dynamics involved, allowing us to use the Couette flow model, which is ideal for situations with no pressure gradient and steady, laminar shear flow.
The rotating shaft drags the thin oil film along its surface, creating shear stress within the fluid. The magnitude of this stress depends on the fluid’s viscosity, the velocity difference between the shaft and the stationary bearing surface, and the thickness of the oil film. This stress translates into a resistive torque, which must be overcome by the shaft’s driving motor.
We calculate the torque by integrating the viscous shear force over the surface area in contact with the fluid. Since the shaft is cylindrical, the torque depends on the cube of the radius, the fluid’s viscosity, the angular velocity, and inversely on the oil film thickness. All these parameters are critical: even small changes in film thickness or viscosity can lead to significant changes in the torque requirement.
In the example provided, assuming realistic values for viscosity and geometry, the torque comes out to approximately 0.263 Nm. This value quantifies the rotational effort needed solely to counteract the viscous resistance, not accounting for other forces like bearing load or friction due to seals. This analysis is crucial in mechanical design to ensure motors or actuators are adequately sized to maintain rotation under lubrication constraints, thus preventing wear and ensuring efficient operation.
