Designing non-circular wheels using a fourth-degree polynomial

This study’s object is the process of designing closed non-circular wheels with a given center-to-center distance with external rolling, provided that it occurs without mutual slip. Non-circular wheels serve as centroids in the design of cylindrical gear transmissions with a variable gear ratio. The gear ratio is characterized by the gear function. If the gear ratio is constant, then the centroids are circles. The gear ratio in this case is the ratio of the radii of these circles. Non-circular wheels can be designed according to a given gear function, which is determined by the kinematics of the mechanism’s links’ actuators. In this case, non-circular wheels can be non-closed. The study addresses another task related to designing non-circular wheels provided that they are closed. Various approaches can be used to this end. The current paper considers the use of a fourth-degree polynomial. In non-circular wheels, the radii, which are understood as the distances from the centers of rotation to the point of contact, are variable. The necessary conditions for constructing non-circular wheels are the constant value of the sum of these radii during the rotation of non-circular wheels, as well as the equality of the paths traveled, that is, the equality of the arcs that non-circular wheels pass during rotation. Pairs of wheels can have the same or different numbers of protrusions and depressions. This is explained by the use of a fourth-degree polynomial whose plot has an axis of symmetry. Accordingly, non-circular wheels or their protrusions also have an axis of symmetry. As an example, the construction of a leading centroid with one protrusion and one depression is given, for which the maximum and minimum distances from the center are 10 and 6.1 linear units, respectively. For this centroid, a trailing centroid has been constructed, and the center-to-center distance has been found, which is 16.79 linear units