Rolling-Element Bearing Heat Transfer—Part II: Housing, Shaft, and Bearing Raceway Partial Differential Equation Solutions

Part I of this three-part series presented a heat transfer rolling-element bearing model. The model is composed of solid conduction partial differential equations (PDEs), control volume formulation for lubricant temperatures, and heat partitioning. The model applies to systems with a shaft, housing, numerous bearings, gears, and various methods of lubrication. Part II, this work, presents a solution to the thermal conduction equations. The raceways are three-dimensional (3D), the shaft and housing models are two-dimensional (2D) and lumped in the third direction. This generalized method applies to ball, cylindrical, spherical, and tapered rolling-element bearings. Semi-analytic solutions are obtained by imposing integral transforms. This approach accounts for the axial and circumferential variations in the bearing load zone and rib heating, as well as the ability to link many bearings and gears within an assembly. The housing and shaft equations are radially lumped. The lumped fluxes account for internal and external convection and radiation, as well as conduction fluxes from contiguous bearings and gears. These equations are solved using a Fourier transform. The 3D bearing raceway solution uses a Fourier transform and a modified Hankel transform. Part III of this series presents additional results and experimental validation.