A Rigid-Body Pendulum Model for Plyometric Push-Up Biomechanics: Analytical Derivation and Numerical Quantification...

Conventional free-fall kinematic models applied to plyometric push-up assessment treat the upper body as a vertically translating point mass, ignoring the curvilinear trajectory imposed by the ankle pivot and systematically biasing flight-time and height estimates.

A planar rigid-body pendulum pivoting about the ankle axis was formulated via two independent derivation pathways (static moment equilibrium and a gravitational-torque coordinate approach), yielding effective pendulum length L = (M_W/M) × L_OS. Closed-form expressions for flight time, arc displacement, maximum height, and mean mechanical power were derived analytically from energy conservation and compared against free-fall predictions across seven pendulum arm lengths (L_OW = 0.50-2.00 m) and 500 initial hand velocities per length, using adaptive Gauss-Kronrod quadrature (relative tolerance 10^-10) with ODE cross-validation (maximum discrepancy < 2.5 × 10^-7 s).

Flight time equivalence (t^H = t^G) was formally established. The free-fall model overestimated flight time by up to 18.82% (Δt = 0.096 s; L_OW = 0.50 m, V_H,0 = 2.50 m/s) and maximum height by up to 28.43% (Δh = 0.087 m; L_OW = 0.50 m, t_flight = 0.50 s), with both errors growing nonlinearly with initial velocity. Overestimation in height was proportionally larger at shorter pendulum arm lengths (18.18% at t_flight = 0.30 s for L_OW = 0.50 m vs. 10.91% for L_OW = 1.00 m).

The pendulum model provides a physically consistent, analytically tractable framework for geometry-adjusted upper-body power assessment from four field-obtainable anthropometric inputs. These results reflect computational self-consistency; prospective experimental validation against force-plate kinematics is required before applied deployment. Prospective empirical validation against dual force-plate and motion-capture reference data is required to establish the model's accuracy boundaries under real push-up kinematics.