A Delay-Programmed Kinetic-Chain Model for Racket Strokes with a Two-Parameter Mechanical Stretch-Shortening-Cycle Extension

Efficient racket acceleration in tennis is commonly attributed to the kinetic chain, but a mechanical model must specify how energy stored at successive joints is released and transferred to the racket. We formulate a minimal planar body–arm–racket model in which each joint stores elastic energy in a latched posture and releases it after an explicit delay. In this framework, optimized delay produces a proximal-to-distal sequence and a superadditive increase in racket-tip speed from 16.6 m s−1 for one active joint to 30.2 m s−1 for two joints and 43.1 m s−1 for three joints, thereby providing a direct mechanical realization of the kinetic chain. The remaining question is how the stretch–shortening cycle (SSC), normally described through complex muscle–tendon physiology, can be incorporated into the same model. We address this by representing SSC action as a joint-level stiffness-modulation block governed by two effective active parameters: a rebound stiffness gain α and a velocity-dependent firing gain β. These parameters are athlete-dependent mechanical descriptors, not universal biological constants. For one illustrative parameter set, SSC engagement raises the peak speed to 46.7 m s−1 (approximately 8%), but this percentage changes with α and β. Because the size of an individual’s active SSC contribution is known to vary substantially from athlete to athlete, this percentage is reported only as an illustration and is not itself the result being claimed. The main contribution is a unified mechanical framework in which time-delayed kinetic-chain release and SSC-like enhancement can be optimized together for athlete-specific posture and stroke design, by adding the two parameters α and β on top of the already delay-optimized passive model without modifying its optimization architecture.