On the Analytical Approach to the Selection of a Centroid in a Flat Gearing

The theory of gearing, and kinematics in particular, has a long history, in which the names of great mathematicians such as Euler, Huygens, Chebyshev are inscribed. This theory is described in detail in many works, yet insufficient attention is paid to the simply formulated, but difficult to solve problem of selecting equations of curves describing centroids in a flat gearing, in which motion is transmitted with a constant ratio of angular velocities. The existing methods for analyzing the flat gearing are based on the premise about the existence of a center of engagement, the point at which the velocities of the links are equal, and through which the common normal of the centroid passes, that is, on the Willis theorem. This work is based on the assertion that the projections of the velocities of the common point on the common normal are the same. The paper presents the derivation of equations that the equations of curves must satisfy in order to fulfil the condition of constancy of the angular velocities. In general, it is necessary that three unknown functions satisfy four constraints: both curves have a common point (two restrictions), the normals to the curves are parallel at this point, the projections of velocities of the common point on the normal are identical and determined by the angular velocities of the links. As an example, the most common forms of curves are considered: the involute, the epicycloid, and the hypocycloid. It is shown that for the involute all the constraints are satisfied, while the transmission of rotation with a constant gear ratio is impossible when using the epicycloid and the hypocycloid. A variant is considered where the form of only one curve is given, and the form of the second curve is calculated proceeding from the condition of constancy of the gear ratio. As an example, equations are derived where the hypocycloid and the straight line are chosen as the first curve.

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