All versions of the RKM incorporate a working chamber formed by smoothly conjugated circular arcs. The piston, shaped to conform to the chamber's walls, "jumps" from wall to wall, thus performing a rotary motion. The piston has an appropriately shaped aperture fitted with a gear structure, and this drives the power shaft (or two power shafts in some models).
The piston, its aperture and the working chamber of the RKMs represent, in their cross sections, multi-oval figures which, mathematically, are related to the class of figures of equal width. Those multi-ovals are non-analytical figures with a discontinuous second derivative of the contour line (the curvature). Hence, generally speaking, the trajectories of their centers of curvature are also non-analytical and, within the RKMs' geometry, have to have singular points.
In relation to the piston, the trajectory of the power shaft axis has corner points, which correspond to extreme positions of the piston as related to the working chamber. Those corners, which represent singular points of the power shaft's trajectory, cannot be avoided or rounded to provide for the kinematically closed functioning of the gear.
The reason why until now the geometry of curves of constant width could not be put to practical use in the gear design is that no conventional gear structure with the regular rolling on of the gears would permit the exact rolling-on of the singularities. The RKMs solve this problem by introducing the inversely conjugated gear system, which makes it possible to have singular trajectories of the axes of rolling-on gears and, thus, allows the transfer of the angular momentum during the passage of the piston through its stop positions.
In simple words, the gear mechanism introduces corrections to the piston's motion, correcting the axis of rotation as it leaves the stop positions, so as to create a smooth motion.
Read more about this topic: RKM Engine
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