Journal of Nonlinear Science

, Volume 27, Issue 6, pp 2037–2061 | Cite as

A New Twisting Somersault: 513XD



We present the mathematical framework of an athlete modelled as a system of coupled rigid bodies to simulate platform and springboard diving. Euler’s equations of motion are generalised to non-rigid bodies and are then used to innovate a new dive sequence that in principle can be performed by real-world athletes. We begin by assuming that shape changes are instantaneous so that the equations of motion simplify enough to be solved analytically, and then use this insight to present a new dive (513XD) consisting of 1.5 somersaults and five twists using realistic shape changes. Finally, we demonstrate the phenomenon of converting pure somersaulting motion into pure twisting motion by using a sequence of impulsive shape changes, which may have applications in other fields such as space aeronautics.


Nonrigid body dynamics Geometric phase Biomechanics Twisting somersault 

Mathematics Subject Classification

70E55 70E17 37J35 92C10 

Supplementary material

332_2017_9403_MOESM1_ESM.mp4 (442 kb)
Supplementary material 1 (mp4 441 KB)


  1. Ashbaugh, M.S., Chicone, C.C., Cushman, R.H.: The twisting tennis racket. J. Dyn. Differ. Equ. 3, 67–85 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. Bates, L., Cushman, R., Savev, E.: The rotation number and the herpolhode angle in Euler’s top. Zeitschrift füur Angewandte Mathematik und Physik (ZAMP) 56(2), 183–191 (2005)Google Scholar
  3. Batterman, C.: The Techniques of Springboard Diving. MIT Press, Cambridge (2003)Google Scholar
  4. Bharadwaj, S., Duignan, N., Dullin, H.R., Leung, K., Tong, W.: The diver with a rotor. Indag. Math. (2016). doi:10.1016/j.indag.2016.04.003
  5. Cabrera, A.: A generalized montgomery phase formula for rotating self-deforming bodies. J. Geom. Phys. 57, 1405–1420 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. Dullin, H.R., Tong, W.: Twisting somersault. SIAM J. Appl. Dyn. Syst. 15(4), 1806–1822 (2016)Google Scholar
  7. Fairbanks, A.R.: Teaching Springboard Diving. Prentice-Hall Inc., Englewood Cliffs (1963)Google Scholar
  8. Frohlich, C.: Do springboard divers violate angular momentum? Am. J. Phys. 47, 583–592 (1979)CrossRefGoogle Scholar
  9. Hanavan, E.P.: A Mathematical Model of the Human Body. USA Defense Technical Information Center, Accession Number AD0608463 (1964).
  10. Huber, J.: Springboard and Platform Diving. Human Kinetics (2015).
  11. Jensen, R.K.: Model for body segment parameters. In: Komi, P.V. (ed.) Biomechanics V-B, pp. 380–386. University Park Press, Baltimore (1976)Google Scholar
  12. Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn, vol. 1. Butterworth-Heinemann, London (1976)Google Scholar
  13. Montgomery, R.: How much does the rigid body rotate? A berry phase from the 18th century. Am. J. Phys. 59, 394–398 (1991)MathSciNetCrossRefGoogle Scholar
  14. Moriarty, P.: Springboard Diving. Ronald Press Company, NY (1959)Google Scholar
  15. O’Brien, R.: Springboard and Platform Diving, 2nd edn. Human Kinetics (2002)Google Scholar
  16. Still, S., Carter, C.A.: Springboard and Highboard Diving: Training, Techniques and Competition, Pelham pictorial sports instruction series, Pelham Books, London, illustrated ed. (1979)Google Scholar
  17. Tong, W.: Coupled Rigid Body Dynamics with Application to Diving, Ph.D. thesis, University of Sydney (2016)Google Scholar
  18. Yeadon, M.R.: The Mechanics of Twisting Somersault. PhD thesis, Loughborough University (1984)Google Scholar
  19. Yeadon, M.R.: The simulation of aerial movement—I: the determination of orientation angles from film data. J. Biomech. 23, 59–66 (1990)CrossRefGoogle Scholar
  20. Yeadon, M.R.: The simulation of aerial movement—II: a mathematical inertia model of the human body. J. Biomech. 23, 67–74 (1990)CrossRefGoogle Scholar
  21. Yeadon, M.R.: The simulation of aerial movement—III: the determination of the angular momentum of the human body. J. Biomech. 23, 75–83 (1990)CrossRefGoogle Scholar
  22. Yeadon, M.R.: The simulation of aerial movement—IV: a computer simulation model. J. Biomech. 23, 85–89 (1990)CrossRefGoogle Scholar
  23. Yeadon, M.R.: The biomechanics of twisting somersaults:Part I: rigid body motions. J. Sports Sci. 11, 187–198 (1993)CrossRefGoogle Scholar
  24. Yeadon, M.R.: The biomechanics of twisting somersaults Part II: contact twist. J. Sports Sci. 11, 199–208 (1993)CrossRefGoogle Scholar
  25. Yeadon, M.R.: The biomechanics of twisting somersaults Part III: aerial twist. J. Sports Sci. 11, 209–218 (1993)CrossRefGoogle Scholar
  26. Yeadon, M.R.: The biomechanics of twisting somersaults Part IV: partitioning performances using the tilt angle. J. Sports Sci. 11, 219–225 (1993)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of SydneySydneyAustralia

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