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DESIGN OF CYCLOIDS, HYPOCYCLOIDS AND EPICYCLOIDS CURVES WITH DYNAMIC GEOMETRY SOFTWARE. ENGINEERING APPLICATIONS

M. Florez, V. Carbonell, E. Martinez

Universidad Politecnica de Madrid (SPAIN)
There are numerous methods for analysis and synthesis of mechanisms based on geometrical constructions and it is necessary a deepen study of the curves described by a point and the relationship between the geometry of different parts. Many engineering studies are devoted to the study of curves of the tooth profile of gears as well as the coupler path of mechanisms. Then, Geometry plays an important role in many engineering applications, such as engines and mechanisms.
The study of curves dates from Ancient Greece, because the first mathematicians of History became interested in them. The Greeks were the first who studied the paths that describe planets in motion but they restricted their mathematics mainly to geometry, but even they were primarily concerned with figures which could be obtained from lines and circles (geometric locus). Conics were treated as plane sections of cones (solid locus) and other planar curves like cycloids and spirals were included in their studies although they couldnĀ“t be drawn from lines and circles. Indeed they were known as mechanical curves rather than geometrical curves.
In this paper we have focus the attention in drawing the mechanical curves most used in engineering by using dynamic geometry software; the different cycloid, hypocycloid, epicycloids have been drawn by using the Geogebra software. Some engineering applications of these mechanical curves, planetary gear trains, and the kinematic requirements have been also studied.