How can we describe how a journal bearing works – specifically, how do we detemrine the torque required to rotate a shaft in a bearing and how the shaft stabilized in the houseing
PK Bot’s answer:
- ENGINEERING CONTEXT:
A journal bearing supports a rotating shaft by maintaining a thin lubricating film that both produces a retarding (viscous) torque and generates a pressure distribution that centers the shaft. Once mastered, you can design bearings that maximize load capacity and life by properly balancing viscous shear and pressure-generated stabilization. Critical engineering decisions include (a) what is the gap and how does that affect performance and (b) what material is used to line the bearing. You might enjoy this video:
-
PHYSICS:
The dominant effects are low-Reynolds-number viscous shear in the lubrication film (modeled by Couette flow) and pressure gradients generated by film thickness variations (modeled by a combination of Couette and Poiseuille flow) that produce a net force on the shaft. -
ASSUMPTIONS:
Assume steady, incompressible, low-Reynolds-number (lubrication approximation) flow in a narrow gap (gap thickness is small relative to the shaft radius) so that convection and unsteady effects are negligible; these are justified by the geometrical configuration and operating conditions of typical journal bearings. -
SOLUTION TECHNIQUE:
The viscous torque on a perfectly centered journal is computed by first solving the Stokes equation in the \theta direction for flow between concentric cylinders, giving the velocity
u_\theta(r)=\frac{\omega_1 R_1^2}{r}\frac{R_2^2 - r^2}{R_2^2-R_1^2}\,.
The shear stress is determined as
\tau_{r\theta}=\eta\,\frac{\partial u_\theta}{\partial r}\Big|_{r=R_1}\,.
Integrating over the cylindrical surface (per unit axial length) yields the torque
T'=2\pi R_1 \eta \frac{\partial u_\theta}{\partial r}\Big|_{r=R_1}\,.
For an off-center journal, the gap varies as
h(\theta)=R_2-R_1-d\cos(\theta-\alpha)\,,
and the Reynolds equation
\frac{\partial}{\partial \theta}\left(\frac{h^3}{12\eta}\frac{\partial p}{\partial \theta}\right)=\frac{1}{2}(U_2-U_1)\frac{\partial h}{\partial \theta}
can be solved (with appropriate boundary conditions) to show that the pressure deviation is
p(\theta)-p_{\mathrm{ref}}=6\eta\,\omega\,\frac{d}{R_2-R_1}\frac{R_1^2}{(R_2-R_1)^2}\sin(\theta-\alpha)\,.
This pressure variation produces an integrated force per unit length that acts perpendicular to the eccentricity – this is actually neutrally stable, not stable, this predicts what is called oil whirl. A solution of the fluid equations which includes the inertial terms is required to predict the stabilizing force, but Professor Kirby does not do that analysis in his textbook.
- REFLECTION:
The analysis shows that by integrating the spatially varying shear stress over the shaft surface we obtain the retarding torque required to overcome viscous resistance, and by solving the Reynolds equation for the pressure distribution we capture how the hydrodynamic pressure generates a force on the shaft. That analysis correctly describes how those forces scale with viscosity and gap spacing, but does not fully predict stability.
To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.1: Bearings Part 1 (HIGHLY RELEVANT)
- Chapter 11.1: Bearings Part 2 (HIGHLY RELEVANT)
- Chapter 11.1: Bearings Part 3 (HIGHLY RELEVANT)
- Chapter 11 How Stuff Works: How A Kingsbury tilt-pad thrust bearing works (HIGHLY RELEVANT)
- Chapter 4 Engineering Models: Hydrostatic Equation (RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.