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Kinematic Pairs: Novel Kinds and Classification

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Recent Advances in Gearing

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Abstract

This section of the book deals with kinematic pairs, with those observed in gears in particular. The chapter begins with a brief overview of the earlier performed research in the field. Different kinds of kinematic pairs are distinguished, namely, point-contact kinematic pairs, line-contact kinematic pairs, and surface-to-surface-contact kinematic pairs. Contact geometry in kinematic pairs is taken into account aiming more in-detail analysis and the development of scientific classification of kinematic pairs. A novel classification of all possible kinds of kinematic pairs is developed.

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Notes

  1. 1.

    According to the definition of the term “kinematic pair,” only two rigid bodies in contact are considered in this text. No flexible bodies [belts, chains, and so forth (three bodies!!)] are considered here as they are not covered by the definition to the term “kinematic pair” (mechanisms with flexible bodies are often loosely referred to as “wrapping pair/lower pair”. Such a term is incorrect as mechanisms with flexible bodies have to be considered separately of kinematic pairs: such mechanisms are not kinematic pairs by nature). In mechanisms with a flexible body there is no relative motion of a pulley and a mating flexible body at points of their contact, that is, they are motionless in relation to one another. No contact is observed at the rest of points, at which the pulley and the flexible body travel in relation to each other.

    Two rigid bodies can make contact at a few points/lines (compound joints). Each of such a contact have to be considered as a separate point/line contact kinematic pair, as a conventional kinematic pair, and not in whole as a multiple-contact kinematic pair (see [3] and others).

  2. 2.

    Joseph-Louis Lagrange (January 25, 1736–April 10, 1813) – an Italian born [born Giuseppe Lodovico (Luigi) Lagrangia] famous French mathematician and mechanician

  3. 3.

    Augustin-Louis Cauchy (August 21, 1789–May 23, 1857) – a famous French mathematician

  4. 4.

    Jean Favard (August 28, 1902–January 21, 1965) – a French mathematician

  5. 5.

    For the first time ever, an equation of the indicatrix of conformity, CnfR (B1/B2), was published in:

    Pat. No. 1,185,749, USSR, A Method of Sculptured Surface Machining on a Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: October 24, 1983

    Pat. No. 1,249,787, USSR, A Method of Sculptured Surface Machining on a Multi-Axis NC Machine./S.P. Radzevich, Int. Cl. B23c 3/16, Filed: December 27, 1984

  6. 6.

    It is commonly adopted that “high-conformal point-contact kinematic pairs I” (as well as “high-conformal point-contact kinematic pairs II” below) cannot be composed of two convex functional surfaces, B1 and B2, by a convex and a flatten surfaces and so forth. This is correct to a certain extent. For example, one can imagine a convex functional surface, B1, of elliptical kind with a flatten functional surface, B2. If the principal radii of curvature, \( {R}_{1_{B.1}} \) and \( {R}_{2_{B.1}} \), of the functional surface, B1, approach an infinity (\( {R}_{1_{B.1}}\to \infty \) and \( {R}_{2_{B.1}}\to \infty \)), then any desirable degree of conformity of two convex functional surfaces, B1 and B2, can be attained. Kinematic pairs of this particular kind (as well as of similar kinds) are not covered in this research. The same is valid with respect to “convex-to-convex” contacts of the functional surfaces.

  7. 7.

    Kinematic pairs of the design under consideration can also be referred to as “true surface-to-surface contact kinematic pairs” (or just “TSSc−kinematic pairs,” for simplicity).

  8. 8.

    It is desirable to have these numbers verified by an independent researcher(s).

  9. 9.

    The results of a more in-detail investigation of all possible kinds of true point contact of two smooth, regular surfaces in the first order of tangency can be found in [5, 6, 16].

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Further Readings

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  • Ertel, A. M. (1939). Hydrodynamic theory of lubrication in new applications. Applied Mathematics and Mechanics, 3(2), 41–49.

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  • Khruschov, M. M. (1996). On the history of the development of the theory of Elastohydrodynamic lubrication. Friction and Wear, 17(5), 703–706.

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  • Krishna, R. K., & Sen, D. (2019). Second-order total freedom analysis of 3D objects in a single point contact. Mechanism and Machine Theory, 140, 10–30.

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  • Mohrenstein-Ertel, A., “Die Berechnung der hydrodynamischen Schmierung gekrümmter Oberflüchen unter hoher Belastung und Relativbewegung”, Fortschr.-Ber. VDI-Z, Reihe 1, Nr. 115, Düsseldorf, 1984, s. 85–89.

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  • Radzevich, S. P. (November 2005). A possibility of application of Plücker’s Conoid for mathematical modeling of contact of two smooth regular surfaces in the first order of tangency. Mathematical and Computer Modeling, 42(9–10), 999–1022.

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  • Radzevich, S. P. (2001). Fundamentals of surface generation (p. 592). Kiev, Rastan: Monograph. (In Russian).

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  • Radzevich, S. P. (2007). Kinematic geometry of surface machining. Boca Raton, FL: CRC Press. 508p.

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  • Radzevich, S. P. (2005). On Analytical Description of the Geometry of Contact of Surfaces in Higher Kinematic Pairs. Theory for Mechanisms and Machines, 3(2), 3–14. http://tmm.spbstu.ru.

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  • Zhivago, E.Ya., Fundamentals of the Theory of Kinematic Pairs, Sci.Dr. Thesis, 05.02.18: Theory of Mechanisms and Machine, SibSIU, Novokuznetsk, 2000, 261 pages.

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  1. Mechanical Engineering, Sterling Heights, MI, USA

    Stephen P. Radzevich

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    Stephen P. Radzevich

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Radzevich, S.P. (2022). Kinematic Pairs: Novel Kinds and Classification. In: Radzevich, S.P. (eds) Recent Advances in Gearing. Springer, Cham. https://doi.org/10.1007/978-3-030-64638-7_3

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  • DOI: https://doi.org/10.1007/978-3-030-64638-7_3

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