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

Alcázar, JuliánPareja Blanco, FernandoRodríguez-López, CarlosGutierrez‐Reguero, HéctorSánchez-Valdepeñas Mateos-Aparicio, JuanCornejo Daza, Pedro JesúsAra, IgnacioAlegre, Luis M.2026-07-232026-07-232022-10-01Alcazar J, Pareja-Blanco F, Rodriguez-Lopez C, Gutierrez-Reguero H, Sanchez-Valdepeñas J, Cornejo-Daza PJ, Ara I, Alegre LM. A novel equation that incorporates the linear and hyperbolic nature of the force-velocity relationship in lower and upper limb exercises. Eur J Appl Physiol. 2022 Oct;122(10):2305-2313. doi: 10.1007/s00421-022-05006-1. Epub 2022 Jul 21. PMID: 35864344.10.1007/s00421-022-05006-1https://hdl.handle.net/10433/27268The 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 (F0), maximum muscle power (Pmax), and maximum unloaded velocity (V0) were calculated using a hybrid (linear and hyperbolic) equation and three different linear equations: one derived from the hybrid equation (linearhyb), one applied to data from 0 to 100% of F0 (linear0-100), and one applied to data from 45 to 100% of F0 (linear45-100). The hybrid equation presented the best fit to the recorded data (R2 = 0.996 and 0.998). Compared to the results derived from the hybrid equation in the LP, significant differences were observed in F0 derived from linear0-100; V0 derived from linearhyb, linear0-100 and linear45-100; and Pmax derived from linearhyb and linear45-100 (all p < 0.05). For the BP, compared to the hybrid equation, significant differences were found in F0 derived from linear0-100; and V0 and Pmax derived from linearhyb, linear0-100 and linear45-100 (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 F0, and overcame the limitations shown by linear equations while providing relevant results.application/pdfenSpringerLoad–velocityMaximum isometric forceMaximum unloaded velocityMuscle mechanicsMuscle powerTorque–velocityA novel equation that incorporates the linear and hyperbolic nature of the force–velocity relationship in lower and upper limb exercisesjournal articlerestricted access