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<p>The dynamic forces induced during rhythmic human jumping are important in multiple domains, including structural safety, sports performance optimization, and biomechanical research. This work proposes a mathematical model to replicate the vertical dynamic forces induced during rhythmic human jumping on a rigid floor. The jumping motion of a person is modeled using a one-degree-of-freedom nonlinear self-sustained oscillator. This oscillator should be capable of capturing three key phenomena observed experimentally: (1) the vertical jumping force history is approximately periodic, (2) the presence of a stable limit cycle, and (3) self-sustained motion, meaning the jumper (oscillator) produces the required input energy to maintain its motion. The developed oscillator is in the form of a modified hybrid Van der Pol-Duffing (MHVD) system, incorporating two nonlinear inertia terms to fulfill these criteria. The force acting in the vertical direction is expressed through the restoring force of the MHVD oscillator. Analytical solutions for this oscillator are subsequently obtained using an energy-based approach and the Krylov–Bogoliubov method of perturbation. Moreover, the optimal model's parameters have been estimated using the genetic algorithm and experimental data of force signals. The stability of the model parameter is evaluated using bootstrapping with increasing ensemble sizes. Finally, the high coefficient of determination (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R^{2} = 0.97\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.97</mn> </mrow> </math></EquationSource> </InlineEquation>) indicates that the model-generated signals are in strong agreement with the experimental signals. The validated model enables engineers to predict human-induced loading accurately for vibration serviceability assessment of structures while providing sports scientists and biomechanics researchers with tools for performance analysis and movement optimization.</p>

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Modeling of dynamic rhythmic jumping force using a self-sustained oscillator

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Sunil Kumar Maurya,
  • Anil Kumar

摘要

The dynamic forces induced during rhythmic human jumping are important in multiple domains, including structural safety, sports performance optimization, and biomechanical research. This work proposes a mathematical model to replicate the vertical dynamic forces induced during rhythmic human jumping on a rigid floor. The jumping motion of a person is modeled using a one-degree-of-freedom nonlinear self-sustained oscillator. This oscillator should be capable of capturing three key phenomena observed experimentally: (1) the vertical jumping force history is approximately periodic, (2) the presence of a stable limit cycle, and (3) self-sustained motion, meaning the jumper (oscillator) produces the required input energy to maintain its motion. The developed oscillator is in the form of a modified hybrid Van der Pol-Duffing (MHVD) system, incorporating two nonlinear inertia terms to fulfill these criteria. The force acting in the vertical direction is expressed through the restoring force of the MHVD oscillator. Analytical solutions for this oscillator are subsequently obtained using an energy-based approach and the Krylov–Bogoliubov method of perturbation. Moreover, the optimal model's parameters have been estimated using the genetic algorithm and experimental data of force signals. The stability of the model parameter is evaluated using bootstrapping with increasing ensemble sizes. Finally, the high coefficient of determination ( \(R^{2} = 0.97\) R 2 = 0.97 ) indicates that the model-generated signals are in strong agreement with the experimental signals. The validated model enables engineers to predict human-induced loading accurately for vibration serviceability assessment of structures while providing sports scientists and biomechanics researchers with tools for performance analysis and movement optimization.