Skip to main content
SpringerLink
Log in

Impact of transition design on the accuracy of velodrome models

  • Original Article
  • Published:
Sports Engineering Aims and scope Submit manuscript

Abstract

This work outlines the creation of a model of a velodrome datum line constructed from two straights, two circular-arc bends, and four transition curves. Different transition curve options are outlined and contrasted with a model with no transition curves. The influence of velodrome geometry on the wheel speed, wheel acceleration, roll angle, roll rate, and power demand of a theoretical cyclist is presented. The results display similar wheel speeds and roll angles for different transition curves, with larger differences when compared to the no-transition-curve option. Greater differences are observed in the acceleration and roll rate, for which calculations are only possible when using a transition curve. Comparisons of the model with theodolite measurements of two velodromes show a mean minimum root mean square error of 0.0398 m. This work can be used to increase the accuracy of existing track-cycling analytic models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

′:

(Prime) derivative with respect to lap distance

\(\mathop {}\limits^{ \cdot }\) :

(Over-dot) derivative with respect to time

\(Y\) :

Track half width (m)

\(R\) :

Track mid-bend radius (m)

\(\kappa\) :

Curvature (/m)

\(\psi\) :

Curve tangential angle (rad)

\(a\) :

Scaling factor

\(\tau\) :

Curve parameter (m)

\({K}\) :

Transition curve curvature profile (/m)

\(({x}_{\text{T}},{y}_{\text{T}})\) :

Coordinates of the transition curve (m, m)

\(({x}_{\text{B}},{y}_{\text{B}})\) :

Coordinates of the bend (m, m)

\(({x}_{S},{y}_{S})\) :

Coordinates of the straight (m, m)

\(({x}_{\text{Tc}},{y}_{\text{Tc}})\) :

Coordinates of the transition curve centre of curvature (m, m)

\(({x}_{\text{Bc}},{y}_{\text{Bc}})\) :

Coordinates of the bend centre of curvature (m, m)

\({L}_{L}\) :

Length of the lap (m)

\({L}_{\text{T}}\) :

Length of the transition curve (m)

\({L}_{\text{B}}\) :

Length of the bend (m)

\({L}_{\text{S}}\) :

Length of the straight (m)

\(\beta\) :

Track bank angle (rad)

\({V}_{\text{W}}\) :

Velocity of the wheels (m/s)

\({V}_{\text{CG}}\) :

Velocity of the centre of gravity (m/s)

\({V}_{\text{A}}\) :

Velocity of the air relative to the cyclist (m/s)

\({a}_{\text{W}}\) :

Acceleration of the wheels (m/s2)

\({a}_{\text{CG}}\) :

Acceleration of the centre of gravity (m/s2)

\(\phi\) :

Roll angle (rad)

\({C}_{\text{D}}A\) :

Aerodynamic drag area (m2)

\({C}_{\text{RR}}\) :

Coefficient of rolling resistance

\({\mu }_{\text{s}}\) :

Coefficient of scrubbing resistance (/rad)

\(\eta\) :

Drivetrain efficiency

\(\rho\) :

Air density (kg/m3)

\({h}_{\text{CG}}\) :

Height of the centre of gravity (m)

\(m\) :

Combined cyclist/bicycle mass (kg)

\(g\) :

Gravitational acceleration (m/s2)

References

  1. Caddy O, Fitton W, Symons D, Purnell A, Gordon D (2017) The effects of forward rotation of posture on computer-simulated 4-km track cycling: Implications of Union Cycliste Internationale rule 1.3.013. Proc Inst Mech Eng Part P J Sports Eng Technol 231(1):3–13. https://doi.org/10.1177/1754337115619306

    Article  Google Scholar 

  2. Underwood L, Jermy M (2014) Determining optimal pacing strategy for the track cycling individual pursuit event with a fixed energy mathematical model. Sports Eng 17(4):183–196. https://doi.org/10.1007/s12283-014-0153-3

    Article  Google Scholar 

  3. Fitton B, Symons D (2018) A mathematical model for simulating cycling: applied to track cycling. Sports Eng 21(4):409–418. https://doi.org/10.1007/s12283-018-0283-0

    Article  Google Scholar 

  4. Fitzgerald S, Kelso R, Grimshaw P, Warr A (2019) Measurement of the air velocity and turbulence in a simulated track cycling team pursuit race. J Wind Eng Ind Aerodyn 190:322–330. https://doi.org/10.1016/j.jweia.2019.05.014

    Article  Google Scholar 

  5. Union Cycliste Internationale (2020) UCI cycling regulations: part III: track races. Version 12.06.2020. https://www.uci.org/inside-uci/constitutions-regulations/regulations

  6. Martin J, Gardner AS, Barras M, Martin D (2006) Modeling sprint cycling using field-derived parameters and forward integration. Med Sci Sports Exerc 38(3):592–597. https://doi.org/10.1249/01.mss.0000193560.34022.04

    Article  Google Scholar 

  7. Underwood L, Jermy M (2010) Mathematical model of track cycling: the individual pursuit. Procedia Eng 2(2):3217–3222. https://doi.org/10.1016/j.proeng.2010.04.135

    Article  Google Scholar 

  8. Lukes R, Carre M, Haake S (2006) Track cycling: an analytical model. Eng Sport 6:115–120. https://doi.org/10.1007/978-0-387-46050-5_21

    Article  Google Scholar 

  9. Benham G, Cohen C, Brunet E, Clanet C (2020) Brachistochrone on a velodrome. Proc R Soc A Math Phys Eng Sci 476(2238):20200153. https://doi.org/10.1098/rspa.2020.0153

    Article  MathSciNet  MATH  Google Scholar 

  10. Lukes R, Hart J, Haake S (2012) An analytical model for track cycling. Proc Inst Mech Eng Part P J Sports Eng Technol 226(2):143–151. https://doi.org/10.1177/1754337111433242

    Article  Google Scholar 

  11. Bos L, Slawinski MA, Slawinski RA, Stanoev T (2021) On modelling bicycle power for velodromes: Part II. Formulation for individual pursuits. [Preprint] Version 08.01.2021. https://arxiv.org/abs/2009.01162

  12. Bartels RH, Beatty JC, Barsky BA (1987) An introduction to splines for use in computer graphics and geometric modeling. Morgan Kaufmann, San Francisco

    MATH  Google Scholar 

  13. Brustad TF, Dalmo R (2020) Railway transition curves: a review of the state-of-the-art and future research. Infrastructures 5(5):43. https://doi.org/10.3390/infrastructures5050043

    Article  Google Scholar 

  14. Pendrill A-M, Eager D (2020) Velocity, acceleration, jerk, snap and vibration: forces in our bodies during a roller coaster ride. Phys Educ 55(6):065012. https://doi.org/10.1088/1361-6552/aba732

    Article  Google Scholar 

  15. Meek DS, Walton DJ (1989) The use of Cornu spirals in drawing planar curves of controlled curvature. J Comput Appl Math 25(1):69–78. https://doi.org/10.1016/0377-0427(89)90076-9

    Article  MathSciNet  MATH  Google Scholar 

  16. Nutbourne AW, McLellan PM, Kensit RML (1972) Curvature profiles for plane curves. Comput Aided Des 4(4):176–184. https://doi.org/10.1016/0010-4485(72)90072-3

    Article  Google Scholar 

  17. Gladstone H (2011) Making tracks: building the Olympic velodrome. https://www.cyclingweekly.com/news/making-tracks-building-the-olympic-velodrome-53916. Accessed 12 Nov 2020

  18. Martin J, Milliken D, Cobb J, McFadden K, Coggan A (1998) Validation of a mathematical model for road cycling power. J Appl Biomech 14(3):276–291. https://doi.org/10.1123/jab.14.3.276

    Article  Google Scholar 

  19. Elhassan IM, Ali AS (2011) Comparative study of accuracy in distance measurement using: optical and digital levels. J King Saud Univ Eng Sci 23(1):15–19. https://doi.org/10.1016/j.jksues.2010.10.001

    Article  Google Scholar 

  20. Gander W, Golub GH, Strebel R (1994) Least-squares fitting of circles and ellipses. BIT Numer Math 34(4):558–578. https://doi.org/10.1007/BF01934268

    Article  MathSciNet  MATH  Google Scholar 

  21. Hossain MI, Eager D, Walker PD (2020) Greyhound racing ideal trajectory path generation for straight to bend based on jerk rate minimization. Sci Rep 10(1):7088. https://doi.org/10.1038/s41598-020-63678-1

    Article  Google Scholar 

  22. Mahdavi F, Hossain MI, Hayati H, Eager D, Kennedy P (2018) Track shape, resulting dynamics and injury rates of greyhounds. In: ASME 2018 international mechanical engineering congress and exposition. Volume 13: design, reliability, safety, and risk. https://doi.org/10.1115/imece2018-87156

  23. World Athletics (2019) Track and field facilities manual 2019 edition—chapters 1–3. Version 01.11.2019. https://www.worldathletics.org/about-iaaf/documents/technical-information

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia

    Shaun Fitzgerald, Richard Kelso & Paul Grimshaw

  2. College of Health and Life Sciences, Hamad Bin Khalifa University, Doha, Qatar

    Paul Grimshaw

  3. Adelaide, Australia

    Andrew Warr

Authors
  1. Shaun Fitzgerald
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Kelso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul Grimshaw
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andrew Warr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaun Fitzgerald.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (PDF 136 KB)

Supplementary file 2 (M 17 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fitzgerald, S., Kelso, R., Grimshaw, P. et al. Impact of transition design on the accuracy of velodrome models. Sports Eng 24, 23 (2021). https://doi.org/10.1007/s12283-021-00360-3

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12283-021-00360-3

Keywords

Search

Navigation