Abstract
The content of the paper is based on the mathematical construction of the parametric equation of the epi- and hypocycloid curve described by a circle point. The purpose of the paper is to present the equations of epi- and hypocycloids in a parametric form in relation to the epi- and hypocyclic mechanism in a form convenient for calculation; to present the results of computational experiments on constructing phase trajectories of motion of a moving point of an epi- and hypocycloid. A detailed analysis of the analytical model of epi- and hypocycloids circumscribed by a point of a circle (on a moving circle) has been made. The equations of epi- and hypocycloids are presented in parametric form as applied to the epi- and hypocyclic mechanism in a form convenient for calculation. The results of studies on the construction of phase trajectories of a moving point of an epi- and hypocycloid with an analysis of the obtained curves are presented. The analytical model of epi- and hypocycloids is of practical importance, since it allows designing geared linkage mechanisms formed by attaching two-wire Assur group of various modifications to the planetary mechanism, as the primary mechanism.
Highlights
The geared linkage mechanisms formed by fastening tightly to the connecting rod two-wire Assur groups of the second type (RRT: rotational - rotational - translational) of the satellite of the differential mechanism are well known
An analysis of works [3,4,5,6,7,8,9,10,11,12,13,14,15,16] showed that the geared linkage mechanism
Notations in fig. 1: O - the origin; P - the contact point of the satellite 2 with the fixed wheel 1; C - the axis of rotation of the gear wheel 2; A - hinge in the satellite 2 for the movable connection of the connecting rod 3; R and r - the radii of the gear wheels 1 and 2; AB - connecting rod; C3 - the center of gravity of the connecting rod
Summary
The geared linkage mechanisms formed by fastening tightly to the connecting rod two-wire Assur groups of the second type (RRT: rotational - rotational - translational) of the satellite of the differential mechanism are well known (see Fig. 179 in [1] and, which is one and the same, in task 16.36 [2]). 1: O - the origin; P - the contact point of the satellite 2 with the fixed wheel 1; C - the axis of rotation of the gear wheel 2; A - hinge in the satellite 2 for the movable connection of the connecting rod 3; R and r - the radii of the gear wheels 1 and 2; AB - connecting rod; C3 - the center of gravity of the connecting rod. If m is an integer, the curve consists of m branches that “go around” the fixed circle
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