Mechanisms for Generating an Ellipse
Abstract Generated abstract
The paper addresses the need to mechanize the accurate construction and marking of ellipses for machine-building and instrument-making applications, where existing mechanisms may be unreliable, incomplete in operation, or inconvenient for drafting use. Using exact synthesis methods, the authors present two plane mechanisms intended to generate a complete ellipse while being simpler and more readily adaptable to drafting machines than earlier spatial designs. For each mechanism, they describe the arrangement of guides, sliders, rods, screws, and articulated levers, and show geometrically that the coordinated motion produces the standard ellipse equation, with the ellipse parameters adjustable by screws.
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Mechanisms for Generating an Ellipse
E. I. Toropygin, V. M. Vladimirov
(Presented by the scientific seminar of the Department of Descriptive Geometry and Graphics)
A considerable part of the products of modern machine building and instrument making includes parts of elliptical cross-section. Thus, for example, gear wheels, parts of keyless joints, pump casings, conoids, cams, etc., have the figure of an ellipse in transverse section. The manufacture of such parts is based on the method of copying and partly on layout marking followed by mechanical processing.
The construction of an ellipse on the working drawing of a part with the aid of drafting instruments, and the marking of a copier and a template, are comparatively complex processes. Production practice imposes requirements as to the need to mechanize these operations.
At the present time a large number of mechanisms for constructing an ellipse are known, but many of them either do not provide for the generation of a complete ellipse with a single installation and adjustment, or are insufficiently reliable in operation, or have other shortcomings [1, 2]. Therefore, to the present time, searches are being carried out for the most successful designs, more fully satisfying certain requirements of modern practice. Searches are also being conducted in the field of approximate construction of an ellipse. A large amount of technical literature, especially foreign literature, is devoted to such studies [3, 4, 5, 6, etc.].
On the basis of methods of exact synthesis, we have developed two designs of plane mechanisms for generating an ellipse which, unlike the spatial-type designs previously developed by us [7], can easily be readjusted for use on drafting machines. In this case the design of the mechanisms will be considerably simplified.
The first mechanism consists of plate 8 (Fig. 1), carrying two mutually perpendicular guides \(m\) and \(n\). The plane-parallel displacement of plate 8, necessary for the operation of the mechanism, is ensured by the presence of four hinged levers \(l\). The fixed part of the mechanism is element 10, provided with two needles \(i\) at points \(o\) and \(c\). Rod 12, rotating about its axis, carries guides 5 and 7, along which, by means of two bushings, the central rod 6 moves with trunnion 4 in the middle and with two other bushings for guides 14 and 15 of rod \(2a\). Trunnion 4 is connected with link 11, which in turn is rigidly connected with slide \(F\). Slide \(F\) moves along guide \(n\) of plate 8. Rod \(2a\) has trunnion 2, connected with link 9. Link 9 is rigidly connected with slide \(P\), which moves along guide \(m\) of plate 8.
Fig. 1. Diagram of the first mechanism for generating an ellipse.
Fig. 2. Geometric interpretation of the generation of an ellipse.
Fig. 3. Diagram of the second mechanism for generating an ellipse.
The entire rotary system of the mechanism, consisting of guides 5 and 7 of rod 12 and, respectively, 14 and 15 of rod \(2a\), of rod \(2a\) itself, of the central rod 6, and of journals 4 and 2, rotates about rod 12 by means of head 1. The rotation of this system provides the required motion of plate 8, each point of which describes an ellipse. This is evident from triangles \(AEC\) and \(BDC\) (Fig. 2), the positions of whose points \(A\) and \(B\) in rectangular coordinates are determined, respectively, by the expressions
\[ y = b \sin \varphi, \]
\[ x = a \cos \varphi. \]
Solving these relations jointly, we obtain the equation of the ellipse
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
The writing element of the mechanism is fixed at point \(S\) of plate 8. The parameters \(a\) and \(b\) of the ellipse are adjusted by means of two screws 3 and 13.
The second mechanism (Fig. 3) consists of a rigidly connected T-shaped element having mutually perpendicular guides \(m\) and \(n\), along which sliders 3 and 4 move. Sliders 3 and 4 are rigidly connected with rods \(e\) and \(f\), which are hinged, respectively, to nuts \(g\) and \(h\). Nuts \(g\) and \(h\) are mounted on two independent screws 2 and 1, lying on one straight line and fixed axially at point \(P\). Rotation of screws 2 and 1 about the fixed point \(P\) provides the motion of the T-shaped element along an ellipse. The proof of this is analogous to the proof given for the first mechanism. In this case the rigidly interconnected guides \(m\) and \(n\) move only parallel to themselves, which is ensured by the presence of 4 articulated levers \(l\). The writing element is fixed at point \(s\) of guide \(n\). The mechanism is adjusted by means of screws 1 and 2.
REFERENCES
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E. I. Toropygin. Analysis of the theoretical accuracy of operation of a certain group of spatial mechanisms for generating an ellipse. Proceedings of the Tomsk Polytechnic Institute, vol. 104, 1959.
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E. I. Toropygin. Some questions of the history, classification, and analysis of the theoretical accuracy of operation of spatial mechanisms for generating ellipses. Proceedings of the Tomsk Polytechnic Institute, vol. 98, 1960.
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B. Goldberger. Näherungskonstruktion der Ellipse. Z. VDI, No. 26, 1959.
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B. Goldberger. Näherungskonstruktion für flache Ellipsen. Z. VDI, No. 14, 1961.
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E. I. Toropygin. Construction of elliptical compasses. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.