Nonlinear interaction between stroke length and stroke rate in elite swimming.

INTRODUCTION: Competitive freestyle swimming performance depends on the interaction between stroke rate (SR) and stroke length (SL), yet this relationship has generally been examined using linear approaches. This study investigated the nonlinear biomechanical interaction between SR and SL to identify the conditions associated with maximal swimming velocity. METHODS: Data were collected from fifty highly trained freestyle swimmers performing maximal-effort 50-m sprint trials. Swimming velocity was analyzed using polynomial regression, response-surface methodology, stationary-point analysis, and Hessian determinant testing to identify and verify optimal biomechanical interaction. RESULTS: The nonlinear interaction model explained 68% of the variance in swimming velocity and improved predictive performance by 20% compared with the linear model. Response-surface optimization identified a constrained biomechanical interaction corridor rather than a single optimal combination of SR and SL. Stationary-point analysis, confirmed by the Hessian determinant, demonstrated that this region represented a statistically significant local maximum, providing evidence that swimming velocity is governed by nonlinear rather than purely additive relationships between SR and SL. DISCUSSION: The findings demonstrate the value of integrating nonlinear regression, response-surface methodology, and mathematical optimization to characterize swimming biomechanics. The proposed analytical framework offers a reproducible approach for optimizing biomechanical performance and has practical applications in training prescription, performance monitoring, and future real-time coaching systems.