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The generalized plane strain problem of the contact of layered elastic solids is reduced to an integral equation using Green’s function approach. Approximate numerical solutions are obtained by replacing the integral equation by a matrix inversion when the trapezoidal rule is used to represent the integral. Results for determining the actual contact pressure at the center of the contact zone and size of contact zone for a wide range of layer thicknesses are presented for two most practical cases, (i) when the indenter is rigid, and (ii) when the indenter is elastic having a modulus of elasticity equal to that of the substrate of the indented body. When the layer is softer than the substrate it is found that the actual contact pressure distribution is very closely determined by a weighted sum of elliptic and parabolic functions. For a substrate softer than the layer the pressures substantially deviate from an elliptical or parabolic behavior, for the cases when the layer thickness is finite. The analysis checks with the Hertzian solution in the extreme cases when the layer thickness either tends to zero or approaches infinity.

An analytical formulation for the roller motion in a cylindrical roller bearing is presented in terms of the classical differential equations of motion. Roller-race interaction is analyzed in detail and the resulting normal force and moment vectors are determined. Elastohydrodynamic traction models are considered in determining the roller-race tractive forces and moments. Formulation for the roller end and race flange interaction during skewing of the roller is also considered. Roller-cage interactions are assumed to be either hydrodynamic or fully metallic. Simple relationships are used to determine the churning and drag losses.

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An analytical formulation for the generalized ball, cage, and race motion in a ball bearing is presented in terms of the classical differential equations of motion. Ball-race interaction is analyzed in detail and the resulting force and moment vectors are determined. The ball-cage and race-cage interactions are considered to be either hydrodynamic or metallic and a critical film thickness defines the transition between the two regimes. Simplified treatments for the drag and churning losses are also included to complete a rigorous analytical development for the real-time simulation of the dynamic performance of ball bearings.