CONSTRUCTION OF CENTROIDS OF NON-CIRCULAR WEELS ON THE BASIS OF DEFORMATION OF AN ELLIPSE

Abstract

Cylindrical gears in which the centroids are circles are used in various mechanisms and machines to transmit rotational motion between parallel axes. These circles play the role of the initial data on the basis of which the gear teeth are designed. The distance between centroids is determined by the sum of their radii. When one centroid rotates, another centroid rotates in the absence of sliding. This means that the path (that is the length of the arc of a circle) that each centroid passes is equal. A characteristic indicator of such transmission is the gear ratio, which is determined by the ratio of the angular velocity of rotation of one centroid (circle) to another. The gear ratio can also be determined by the ratio of the radii of the centroids, that is circles. It is always constant. If the circles are equal, then the gear ratio is equal to one.
 Gears with non-circular wheels are used in various devices and mechanisms. To construct the original (leading) centroid, an equation in the polar coordinate system is used, which describes a whole group of characteristic closed curves. The integer n, which is included in the equation, affects the shape of the original centroid, which has n protrusions and depressions evenly spaced at equal intervals of the polar angle. The driven centroid, which is searched for by the obtained equations on the basis of its rolling on the leading without sliding, also has a similar shape with the number of m protrusions and depressions. The centroid pair is given by the numbers n for the leading curve and m for the driven curve. The combination of these two numbers gives different pairs of centroids. If n=m, then the master and slave centroids will be congruent. This case is advantageous in that both centroids have the same shape and are made according to the same scheme, which reduces the complexity of the work. At n=m=0 the centroids will be congruent circles, at n=m=1 - congruent ellipses with centers of rotation in the foci.
 An important feature of the developed approach is that the subinterval expression obtained on the basis of the equality of the corresponding arcs of both centroids can be integrated. This allows us to obtain the parametric equations of the driven centroid in the final form.