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A novel equation that incorporates the linear and hyperbolic nature of the force–velocity relationship in lower and upper limb exercises

Abstract:

The purpose of this study is to provide a force–velocity (F–V) equation that combines a linear and a hyperbolic region, and to compare its derived results to those obtained from linear equations. A total of 10 cross-training athletes and 14 recreationally resistance-trained young men were assessed in the unilateral leg press (LP) and bilateral bench press (BP) exercises, respectively. F–V data were recorded using a force plate and a linear encoder. Estimated maximum isometric force (F<inf>0</inf>), maximum muscle power (P<inf>max</inf>), and maximum unloaded velocity (V<inf>0</inf>) were calculated using a hybrid (linear and hyperbolic) equation and three different linear equations: one derived from the hybrid equation (linear<inf>hyb</inf>), one applied to data from 0 to 100% of F<inf>0</inf> (linear<inf>0–100</inf>), and one applied to data from 45 to 100% of F<inf>0</inf> (linear<inf>45–100</inf>). The hybrid equation presented the best fit to the recorded data (R<sup>2</sup> = 0.996 and 0.998). Compared to the results derived from the hybrid equation in the LP, significant differences were observed in F<inf>0</inf> derived from linear<inf>0-100</inf>; V<inf>0</inf> derived from linear<inf>hyb</inf>, linear<inf>0–100</inf> and linear<inf>45–100</inf>; and P<inf>max</inf> derived from linear<inf>hyb</inf> and linear<inf>45–100</inf> (all p < 0.05). For the BP, compared to the hybrid equation, significant differences were found in F<inf>0</inf> derived from linear<inf>0–100</inf>; and V<inf>0</inf> and P<inf>max</inf> derived from linear<inf>hyb</inf>, linear<inf>0–100</inf> and linear<inf>45–100</inf> (all p < 0.05). An F–V equation combining a linear and a hyperbolic region showed to fit adequately recorded F–V data from ~ 20 to 100% of F<inf>0</inf>, and overcame the limitations shown by linear equations while providing relevant results.