Machinery & Energetics

Constructing a closed profile in which curvilinear elements touch circle is important for the design of centroids of non-circular wheels. When rolling a polygon on such profile its center moves in a circle. If both centers (center of curvilinear profile and center of polygon) are stationary, then you can roll these figures while rotating around their centers. One centroid will be a polygon, the other will be a closed profile. The rolling of a flat figure in the form of an equilateral polygon on curvilinear profile is considered. The profile is periodic and is formed by a series connection of the arc of a symmetric curve so that its ends abut on the circle of a given radius. The equation of the curve from which the curvilinear profile is constructed is found provided that the center of the polygon, when rolling it on the profile, must also move in a circle. The equation of the curve from which the curvilinear profile is constructed is found provided that the center of the polygon, when rolling it on the profile, must also move in a circle. Rolling occurs without sliding, so the length of the arc of the curve is equal to the length of the side of the polygon. To find the equations of the profile curve, a firstorder differential equation is constructed and an analytical solution is obtained. The parametric equations of the curve are obtained in the polar coordinate system. The limits of changing the angular parameter for constructing a profile element that is part of the arc of the curve are found. According to the obtained equations, curvilinear profiles with different number of their elements are constructed. The mathematical relationship between the radius of the circle along which the center of the polygon moves when it is rolling and the radius of the circumscribed circle is established.

equilateral polygon, curvilinear profile, external rolling, differential equation, centroids