Machinery & Energetics

The article considers the internal rolling of flat centroids one by one with simultaneous rotation abount fixed centers. A characteristic feature of the considered centroids is that the profile of each of them is formed by a series 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 arcs in the contact zone. This is due to the fact that the lengths of the arcs of the tooth profiles are equal. In classical gears, an involute profile is used, which was once proposed by L. Euler. Gears with this profile are the most common. There are other profiles, for example, in Novikov's transmission, in which the tooth profile is a circle or a curve close to the circle. During the operation of these gears there is slippage at the point of contact of the teeth, and in Novikov gearing it is less than in gears with involute profile. In these and other gears on both wheels there are circles that roll one by one without slipping. They are called centroids or dividing circles, the diameters of which are the basis for calculating all the geometric elements of the gear. Accordingly, in our case, the centroids can serve as a basis for the design of gearing with an involute or other tooth profile. The article shows that such centroids can be formed with a given number of teeth in the form of a gear, so they can also act as a gear. The main advantage of this transmission is the complete absence of sliding, which does not lead to friction of surfaces in the contact zone and their wear. The disadvantage is that the gear ratio is not constant, it changes periodically. However, for some cases, this does not significantly affect the operation of mechanisms (clocks or counting devices). A mathematical description of centroid profiles is performed. The possibility of constructing a centroid with an arbitrary number of teeth on each of them is shown. The wheelbase depends on the number of teeth on each centroid and the angle at the top of the tooth. With the same number of teeth on both centroids, they coincide. A pair of centroids is 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.

logarithmic spiral, centroids, internal rolling, arc length, center-to-center distance