INTERNAL ROLLING OF NON-CIRCULAR CENTROIDS FORMED FROM THE ARCS OF LOGARITHMIC SPIRAL

Abstract

In the article the internal rolling of flat centroids on each other with simultaneous rotation around fixed centers is considered. A characteristic feature of the considered centroids is that the profile of each of them is formed by successive connection of identical arcs of the same logarithmic spiral. It is similar to the profile of a gear wheel. As in gears, such centroids can transmit rotational motion. Unlike gears the transmission of rotational motion occurs without sliding of the arcs in the contact area. This is due to the fact that the arc lengths of the tooth profiles are equal. In classical gears, an involute profile is used, which was once proposed by L. Euler [1]. Gears with such a profile are the most common. Other profiles are also known, for example, in Novikov gears, in which the tooth profile is circle or a curve close to a circle. During the operation of these gearing sliding occurs at the point of contact of the teeth, and in the Novikov gear it is less than in gears with involute profile. In these and other gears on both wheels there are circles that roll over each other without slipping. They are are called centroids or splines, whose diameters are the basis for calculating all geometric elements of the gearing. Accordingly and in our case, centroids can serve as the basis for designing a gear with involute or other tooth profile. In the article it is shown that such centroids can be formed with a given number of teeth in the form of a gear, so they can also serve as a gear transmission. The main advantage of such a transmission is the complete absence of sliding, which does not lead to friction of surfaces in the area of contact and their wear. The disadvantage is that the transmission ratio is not constant, it periodically changes periodically. However, for some cases this does not affect significantly on the operation of mechanisms (for example, clock [2] or counting devices). The mathematical description of the profiles of centroids is carried out. The possibility of constructing centroids with an arbitrary permissible number of teeth on each of them. The center distance depends on the number of teeth on each centroid and the angle at the top of the tooth. For the same number of teeth on both centroids they coincide. Pairs of centroids are constructed, and their intermediate positions are shown when one of them is rotated by a given angle. The angle of rotation of the second centroid is determined analytically and is a function of the angle of rotation of the first centroid.