A nonlinear frequency-dependent spring-mass model for estimating loading caused by rhythmic human jumping

Rory E White*, John H G Macdonald, Nicholas A Alexander

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
39 Downloads (Pure)


An empirical nonlinear, frequency-dependant, spring-mass system is conjectured for modelling human rhythmic jumping. This model is vital for correctly estimating human-structure dynamic interactions. An experimental study was employed to evaluate the leg mechanics and dynamic loading of a human jumper. Testing was performed over a large range of prescribed jumping frequencies. Subjects performed rhythmic jumps on a force plate and they were monitored by a motion capture system from which the displacement of the centre of mass was identified. Least squares system identification was utilised to determine the parameters of the spring-mass model for human rhythmic jumping. A nonlinear stiffness, rather than a conventional linear spring, is proposed to better capture the observed behaviour during periodic jumping. Force-displacement curves of each subject, during the contact phase of rhythmic jumping, were explored. These display an array of both classical Duffing’s type nonlinear softening and hardening spring stiffnesses over the range of jumping frequencies. The coefficients of the Duffing’s type model are observed to be highly sensitive to jumping frequency. A Poincaré section (phase-space) representation is used to visualise the jumping attractor’s topology. Thus, an experimental bifurcation analysis is performed suggesting the presence of both period doubling and fold bifurcations. These describe the transition from observed period-2 to period-1 jumping and coexisting low/high amplitude jumping behaviour. This study presents a framework for characterising the nonlinear loading of a human performing rhythmic jumping from direct measurements of force and displacement.
Original languageEnglish
Article number112229
Number of pages15
JournalEngineering Structures
Early online date25 May 2021
Publication statusPublished - 15 Aug 2021

Bibliographical note

Funding Information:
The authors greatly acknowledge the assistance of Alex Nunn in performing the experimental study and preliminary data processing, Dr Matt Dietz for consultation of data processing techniques, Dr Luiza Dihoru for assistance in use of Qualysis Motioncapture software and Dr Tony Horseman for clarification of electrical sensors and time synchronisation. Rory White is funded by an EPSRC Doctoral Training Partnership studentship.

Publisher Copyright:
© 2021 Elsevier Ltd


  • human-induced loading
  • rhythmic jumping
  • nonlinear dynamics
  • biomechanics


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