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 While nomograms have now been largely superseded as engineering tools, our analysis shows that they can still have some applications in workshops and for teaching purposes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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
Adams, D.P.: Nomographic synthesis of generator linkages. J. Eng. Ind. 82(1), 29–38 (1960). https://doi.org/10.1115/1.3662986
Aleksandrov, I.K.: Determining the limiting efficiency of a kinematic chain. Russ. Eng. Res. 31, 539–540 (2011). https://doi.org/10.3103/S1068798X11060037
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
Boulet, D., Doerfler, R., Marasco, J., Roschier, L.: pyNomo documentation (2020). http://lefakkomies.github.io/pynomo-doc/index.html
d’Ocagne, M.: Traité de nomographie. Gauthier-Villars, Paris (1899)
Doerfler, R.: On jargon – the lost art of nomography. UMAP J. 30(4), 457–493 (2009). https://www.comap.com/product/?idx=1048
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
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
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
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
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
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
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
Evesham, H.A.: The History and Development of Nomography. Docent Press, Mountain View (2010)
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
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
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
Hankins, T.L.: Blood, dirt, and nomograms: a particular history of graphs. Isis 90(1), 50–80 (1999). https://doi.org/10.1086/384241
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)
Hilbert, D.: Mathematische probleme. Archive für Mathematik und Physik 1(1), 44–63 (1901)
Hohenberg, R.: Detection and study of compressor-blade vibration. Exp. Mech. 7, 19A-24A (1967). https://doi.org/10.1007/BF02327002
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
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
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
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
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
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
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
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/
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
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
Warmus, M.: Nomographic Functions. Państwowe Wydawnictwo Naukowe, Warsaw (1959)
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
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
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
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-98499-1_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98498-4
Online ISBN: 978-3-030-98499-1
eBook Packages: EngineeringEngineering (R0)