6. Conclusions The Reynolds equation is essential for the prediction of the slipper's dynamic behavior. In this work, the Reynolds equation for the slipper bearing is derived from the equilibrium equation and continuity equation, and its velocity boundary condition is clarified for the first time. Also, the comparison of the Reynolds equation for the slipper bearing is presented in detail between the present work and previous studies. Based on the analytical results, the following conclusions may be drawn:
(1) The present Reynolds equation for the slipper bearing without the spinning motion (see Eq. (23)) is identical in form to that in the previous studies, but they are different from each other in the velocity boundary condition.
(2) For the slipper bearing, the velocity boundary condition for the Reynolds equation is determined by the slipper's macro motion and spinning motion on the swash plate. The slipper does experience a translation in an elliptical manner rather than a plane rotation on the swash plate.
(3) The assumption of the slipper's plane rotation on the swash plate in the previous studies will cause a significant relative error of up to 0.37 for the boundary velocity of the slipper bearing in a commercial axial piston machine.
(4) The Reynolds equation for the slipper bearing with the spinning motion is derived in this work (see Eq. (22)). It is found that the spinning speed of the slipper only influences the hydrodynamic effect associated with the tangential velocity component in the Reynolds equation.
