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Nomograms: An Old Tool with New Applications

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Explorations in the History and Heritage of Machines and Mechanisms (HMM 2022)

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Abstract

In this paper, we consider the history of nomograms as a computational tool in mechanical engineering, together with their potential applications for teaching purposes, and summarize the mathematical methods used to derive them. Nomograms are graphical descriptions of a mathematical problem, such that the desired solution may be derived through a simple geometric construction, which usually requires nothing more than a straightedge. This way, a reasonably accurate solution to a complex problem can be quickly obtained even in adverse environmental conditions by low-skilled users; moreover, a nomogram can provide immediate insight on the relationship between the variables. Nomograms date back to the 1800s and have been used by engineers for decades, due to their convenience over manual computation, before computers became widespread. While nomograms have now been largely superseded as engineering tools, our analysis shows that they can still have some applications in workshops and for teaching purposes.

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Notes

  1. 1.

    In most nomograms of this type, the lines are vertical, for clarity. Here the lines are horizontal, due to lack of space; the y axis in the drawing plane is parallel to the scales.

References

  1. Adams, D.P.: Nomographic synthesis of generator linkages. J. Eng. Ind. 82(1), 29–38 (1960). https://doi.org/10.1115/1.3662986

    Article  Google Scholar 

  2. Aleksandrov, I.K.: Determining the limiting efficiency of a kinematic chain. Russ. Eng. Res. 31, 539–540 (2011). https://doi.org/10.3103/S1068798X11060037

    Article  Google Scholar 

  3. Antuma, H.J.: Triangular nomograms for symmetrical coupler curves. Mech. Mach. Theory 13(3), 251–268 (1978). https://doi.org/10.1016/0094-114X(78)90049-6

    Article  Google Scholar 

  4. Boulet, D., Doerfler, R., Marasco, J., Roschier, L.: pyNomo documentation (2020). http://lefakkomies.github.io/pynomo-doc/index.html

  5. d’Ocagne, M.: Traité de nomographie. Gauthier-Villars, Paris (1899)

    MATH  Google Scholar 

  6. Doerfler, R.: On jargon – the lost art of nomography. UMAP J. 30(4), 457–493 (2009). https://www.comap.com/product/?idx=1048

  7. Éidinov, M.S., Nyrko, V.A., Éidinov, R.M., Gashukov, V.S.: Torsional vibrations of a system with Hooke’s joint. Sov. Appl. Mech. 12, 291–298 (1976). https://doi.org/10.1007/BF00884975

    Article  Google Scholar 

  8. El-Shakery, S.A., Terauchi, Y.: A computer-aided method for optimum design of plate cam-size avoiding undercutting and separation phenomena–II: design nomograms. Mech. Mach. Theory 19(2), 235–241 (1984). https://doi.org/10.1016/0094-114X(84)90046-6

    Article  Google Scholar 

  9. Esmail, E.L.: Nomographs for synthesis of epicyclic-type automatic transmissions. Meccanica 48, 2037–2049 (2013). https://doi.org/10.1007/s11012-013-9721-z

    Article  MATH  Google Scholar 

  10. Esmail, E.L.: Configuration design of ten-speed automatic transmissions with twelve-link three-DOF Lepelletier gear mechanism. J. Mech. Sci. Technol. 30(1), 211–220 (2016). https://doi.org/10.1007/s12206-015-1225-4

    Article  Google Scholar 

  11. Esmail, E.L., Hussen, H.A.: Nomographs for kinematics, statics and power flow analysis of epicyclic gear trains. In: Proceedings of the ASME 2009 International Mechanical Engineering Congress and Exposition, vol. 13, pp. 631–640. ASME, Lake Buena Vista (2010). https://doi.org/10.1115/IMECE2009-10789

  12. Esmail, E.L., Pennestrì, E., Juber, A.H.: Power losses in two-degrees-of-freedom planetary gear trains: a critical analysis of Radzimovsky’s formulas. Mech. Mach. Theory 128, 191–204 (2018). https://doi.org/10.1016/j.mechmachtheory.2018.05.015

    Article  Google Scholar 

  13. Evesham, H.A.: Origins and development of nomography. IEEE Ann. Hist. Comput. 8(4), 324–333 (1986). https://doi.org/10.1109/MAHC.1986.10059

    Article  MathSciNet  MATH  Google Scholar 

  14. Evesham, H.A.: The History and Development of Nomography. Docent Press, Mountain View (2010)

    MATH  Google Scholar 

  15. de Freitas Avelar, A.H., Roschier, L., Fernandes Soares, L., Oliveira Ávila, P.H.S.: Analytical solutions and computational nomograms for maximum pressure angle for cam mechanisms for full and half cycloidal and harmonic motion curves. J. Mech. Eng. Sci. 235(15), 2725–2736 (2021). https://doi.org/10.1177/0954406220962823

  16. Glasser, L., Doerfler, R.: A brief introduction to nomography: graphical representation of mathematical relationships. Int. J. Math. Educ. Sci. Technol. 50(8), 1273–1284 (2019). https://doi.org/10.1080/0020739X.2018.1527406

    Article  Google Scholar 

  17. Grimes, D.A.: The nomogram epidemic: resurgence of a medical relic. Ann. Intern. Med. 149(4), 273–275 (2008). https://doi.org/10.7326/0003-4819-149-4-200808190-00010

    Article  Google Scholar 

  18. Hankins, T.L.: Blood, dirt, and nomograms: a particular history of graphs. Isis 90(1), 50–80 (1999). https://doi.org/10.1086/384241

    Article  Google Scholar 

  19. Hassaan, G.A.: Nomogram-based synthesis of complex planar mechanisms, part I: 6 bar-2 sliders mechanism. Int. J. Eng. Tech. 1(6), 29–35 (2015)

    Google Scholar 

  20. Hilbert, D.: Mathematische probleme. Archive für Mathematik und Physik 1(1), 44–63 (1901)

    MATH  Google Scholar 

  21. Hohenberg, R.: Detection and study of compressor-blade vibration. Exp. Mech. 7, 19A-24A (1967). https://doi.org/10.1007/BF02327002

    Article  Google Scholar 

  22. Hwang, W.M., Chen, K.H.: Triangular nomograms for symmetrical spherical non-Grashof double-rockers generating symmetrical coupler curves. Mech. Mach. Theory 42(7), 871–888 (2007). https://doi.org/10.1016/j.mechmachtheory.2006.05.008

    Article  MATH  Google Scholar 

  23. Kattan, M.W., Marasco, J.: What is a real nomogram? Semin. Oncol. 37(1), 23–26 (2010). https://doi.org/10.1053/j.seminoncol.2009.12.003

    Article  Google Scholar 

  24. Khoshnevis, S., Brothers, R.M., Diller, K.R.: Level of cutaneous blood flow depression during cryotherapy depends on applied temperature: criteria for protocol design. ASME J. Med. Diagn. 1(4), 041007 (2018). https://doi.org/10.1115/1.4041463

  25. Lu, D.M.: A triangular nomogram for spherical symmetric coupler curves and its application to mechanism design. J. Mech. Des. 121(2), 323–326 (1999). https://doi.org/10.1115/1.2829463

    Article  Google Scholar 

  26. Meyer zur Capellen, W.: Nomogramme für die Krümmung sphärischer und ebener Bahnkurven. Mech. Mach. Theory 18(3), 249–254 (1983). https://doi.org/10.1016/0094-114X(83)90098-8

  27. Miconi, D.: Vibration control in industrial plant: a methodological approach. J. Vib. Acoust. Stress Reliab. 109(4), 335–342 (1987). https://doi.org/10.1115/1.3269450

  28. Seireg, A.A., Houser, D.R.: Evaluation of dynamic factors for spur and helical gears. J. Eng. Ind. 92(2), 504–514 (1970). https://doi.org/10.1115/1.3427790

    Article  Google Scholar 

  29. Tournès, D.: Notes & debats—Pour une histoire du calcul graphique. Rev. d’Histoire des Math. 6(1), 127–161 (2000). http://www.numdam.org/item/RHM_2000__6_1_127_0/

  30. Tournès, D.: Du compas aux intégraphes: les instruments du calcul graphique. Repères-IREM 50, 63–84 (2003). https://publimath.univ-irem.fr/biblio/IWR03005.htm

  31. Tournès, D.: Calculating with hyperbolas and parabolas. In: Barbin, É., et al. (eds.) Let History into the Mathematics Classroom. History of Mathematics Education, pp. 101–114. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-319-57150-8_8

  32. Warmus, M.: Nomographic Functions. Państwowe Wydawnictwo Naukowe, Warsaw (1959)

    MATH  Google Scholar 

  33. Wellauer, E.J., Holloway, G.A.: Application of EHD oil film theory to industrial gear drives. J. Eng. Ind. 98(2), 626–631 (1976). https://doi.org/10.1115/1.3438951

    Article  Google Scholar 

  34. Wunderlich, W.: Nomogramme für die Wattsche Geradführung. Mech. Mach. Theory 15(1), 5–8 (1980). https://doi.org/10.1016/0094-114X(80)90028-2

    Article  Google Scholar 

  35. Zotov, N.M., Balakina, E.V.: Using the \(\varphi \) – s\(_{\rm x}\) nomogram in calculating the dynamics of a braked wheel. J. Mach. Manuf. Reliab. 36, 193–198 (2007). https://doi.org/10.3103/S1052618807020161

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Correspondence to M. Cocconcelli .

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Mottola, G., Cocconcelli, M. (2022). Nomograms: An Old Tool with New Applications. In: Ceccarelli, M., López-García, R. (eds) Explorations in the History and Heritage of Machines and Mechanisms. HMM 2022. History of Mechanism and Machine Science, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-030-98499-1_26

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  • DOI: https://doi.org/10.1007/978-3-030-98499-1_26

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