Geodesic
Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 47 (2017), pp. 37-42 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

One-sheeted hyperboloids of revolution are basic surfaces (axoids) of the input and output details of hypoid gears occupying a prominent place in the large variety of gear trains. The main feature of the hypoid gear is the skew axes of rotation. Such arrangement of the axes allows one to provide the smoothness of movement, quietness of the operation, and higher load capacity of the mechanism. In this work, geometrical aspects of modeling the hypoid gear with perpendicular axes are considered, namely, (i) for parameters of the hyperboloids, the condition providing the existence of a common rectilinear generator passing through a common point of their striction lines has been found; (ii) using this condition, an original proof of the following well-known fact has been obtained: rolling of hyperboloids with tangency along a rectilinear generator when they rotate about their axis with the corresponding ratio of velocities is possible only if the imaginary axes of the rotating hyperbolas forming the hyperboloids are equal; (iii) it has been proved that, at given displacements of the axes and gearing ratio, parameters of the hyperboloids are determined uniquely.
Keywords: one-sheeted hyperboloids of revolution, displacement of axes, gearing ratio. 
@article{VTGU_2017_47_a3,
     author = {N. R. Shcherbakov and A. A. Shchegoleva},
     title = {Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {37--42},
     year = {2017},
     number = {47},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2017_47_a3/}
}
TY  - JOUR
AU  - N. R. Shcherbakov
AU  - A. A. Shchegoleva
TI  - Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2017
SP  - 37
EP  - 42
IS  - 47
UR  - http://geodesic.mathdoc.fr/item/VTGU_2017_47_a3/
LA  - ru
ID  - VTGU_2017_47_a3
ER  - 
%0 Journal Article
%A N. R. Shcherbakov
%A A. A. Shchegoleva
%T Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2017
%P 37-42
%N 47
%U http://geodesic.mathdoc.fr/item/VTGU_2017_47_a3/
%G ru
%F VTGU_2017_47_a3
N. R. Shcherbakov; A. A. Shchegoleva. Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 47 (2017), pp. 37-42. http://geodesic.mathdoc.fr/item/VTGU_2017_47_a3/

[1] Wildhaber E., Foundations of conical and hypoid gearing, Mashgiz, M., 1948

[2] Radzevich S. P., “Vector representation of gear pairs”, Teoriya mekhanizmov i mashin, 6:2 (2008), 74–81

[3] Nosko P., Shyshov V., Ratov D., Fill P., Lysenko A., “Helical gear train load capacity criterion”, Teka, 14:1 (2010), 182–190

[4] Rachkovskaya G. S., Kharabayev Yu. N., “Geometric modeling and computer graphics of kinematic ruled surfaces on the base of complex movingone axoid along another (one-sheet hyperboloid of revolution as fixed and moving axoids)”, Proceedings of the 17th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision 2009 (Plzen, Czech Republic, 2009), 31–34 http://wscg.zcu.cz/DL/wscg_DL.htm

[5] Hilbert D., Cohn-Vossen S., Anschauliche Geometrie, Verlag von Julius Springer, Berlin, 1938 | MR

[6] Delone B., Raykov D., Analytical geometry, v. 2, Gostekhizdat, M.–L., 1949, 516 pp.