Relevance of Damped Harmonic Oscillation for Modeling the Training Effects on Daily Physical Performance Capacity in Team Sport

in International Journal of Sports Physiology and Performance
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Positive and negative effects of training induce apparent oscillations of performance, suggesting that the delayed cumulative effects of training on daily performance capacity (DPC) are best fitted by sine waves damped over time.


To compare the criterion validity of the impulse-response (IR) model of Banister et al and the damped harmonic oscillation (DHO) model for quantifying the training load (TL)–DPC relationship.


Six female professional volleyball players (20.8 ± 2.4 y) were monitored using the session rating of perceived exertion (sRPE) for 9 mo to quantify TL. Countermovement-jump (CMJ) and 4-step-approach-CMJ (4sCMJ) performances were recorded once a month. Parameters of models were determined by minimizing residual-sum squares between predicted and real performances with a nonlinear regression.


DPC was best fitted by the DHO model rather than the IR model (CMJ, R2 = .80 ±.08 and.69 ±.20, respectively; 4sCMJ, R2 = .86 ± .09 and .67 ± .29, respectively). The damping parameter θ and the period T were positively correlated with age (ρ = 0.81, P < .05, and ρ = 0.86, P < .02, respectively).


The DHO model is a useful tool for modeling DPC as the sum of the delayed DPCs from the consecutive training and recovery days. DPC could be considered the expression of the individual process of accumulation and dissipation of fatigue induced by training. DHO-model parameters were correlated with age, which prompts one to postulate that expertise has a major influence on DPC. The DHO model will help coaches develop a greater understanding of training effects and make monitoring of the training process more effective.

The authors are with Faculty of Sport Sciences, University of Picardie, Amiens, France.

Address author correspondence to Stéphane Morin at