Superposition of rotations in mechanisms

Abstract

The prerequisite for consideration of the superposition of rotations in mechanisms is the principle of summation of rectilinear uniform movements. The purpose of the work is to establish how things are in a similar situation when adding rotational synchronous or multiple motions. The objectives of the research are the analytical description and construction of hodographs of combined rotational mechanical motions. The practical aspect of the topic is determined by the fact that in many mechanisms, such as a multi-inert oscillator, planetary gears, cutter drives in machines for cleaning pipes of large diameters, etc. summation of rotational motions is realized, and the shape of the hodograph is useful information in the design of such devices. It was found that, just as the result of the addition of two uniform rectilinear motions is also a uniform rectilinear motion, the result of the addition of two uniform unidirectional circular motions is also a uniform circular motion. When two uniform oppositely directed circular motions are added, the hodograph is an ellipse. In a particular case the ellipse can degenerate into a straight line segment. The characteristics of the elliptical hodograph are determined, including major and minor semiaxes, roots of the characteristic equation, eccentricity, foci, angle of inclination of the axes. The inverse problem – determination of the rotation speeds by the type of the hodograph – is solved. When two asynchronous rotations are added, the occurrence of hodographs in the form of a snail is possible, which is similar to Pascal's snail. The results obtained can be used in the design of mechanisms that implement complex rotational motions.