Simulation of centroids of non-circular wheels with internal and external rolling from arcs of symmetrical curves

Abstract

The article considers the design of noncircular wheels, which serve as centroids in the design of gears. Centroids consist of congruent arcs of a given symmetric curve. The number of these arcs, that is the elements of the centroid, is determined by the type of gearing (internal or external). In external gearing, the number of elements of both centroids can be arbitrary, starting with one element. In the case of internal gearing, the number of elements of the internal centroid must be one less than the number of elements of the external centroid. If the number of elements is the same, then the centroids coincide. Rolling centroids one by one occurs in the absence of sliding. This is possible provided that the lengths of the arcs of the individual elements of both centroids are equal to each other. The construction of a centroid is carried out in a polar coordinate system. Both centroids are formed by rotating its element, that is the arc of the curve, at a given angle around the pole. The magnitude of the angle depends on the number of elements of the centroid. When rolling one centroid on the other, the pole of the moving centroid must describe the circle. In this case, the rolling of a moving centroid on a stationary one can be replaced by the rotational motion of both centroids around the fixed centers (poles). The point of contact of the centroids during their rotation is on the segment connecting the centers of rotation and which is called the center-to-center distance. This point for non-circular wheels when they rotate makes a certain movement along the specified segment, and for round wheels remains stationary. The length of the arc of an element of one centroid is determined by the magnitude of the central angle on which it rests. The same applies to the element of the second centroid. If the lengths of the arcs of the elements of the centroid are equal, then the values of the corresponding angles are not equal and are in a certain functional dependence. Finding this dependence is reduced to the integration of the expression obtained on the basis of the equality of the differentials of the arcs of the corresponding centroid elements. This expression may not be integrated for all curves from which the arcs of the original or leading centroid are formed. If the expression cannot be integrated, then the construction of the driven centroid must be carried out by numerical methods. The article considers a curve based on the hyperbolic cosine, for which the obtained expression is integrated. The parametric equations of the curves of which the arcs of both the leading and the driven centroids consist are given. It is shown that for a centroid with a given ratio of elements the intercenter distance is determined unambiguously. Centroid drawings with different number of elements for internal and external gearing are constructed.