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 (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.
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Data availability
The data analyzed during this study are included in this published article (Supplementary material).
Abbreviations
- F–V :
-
Force–velocity
- F 0 :
-
Estimated maximum isometric force
- F opt :
-
Optimal force
- Linearhyb :
-
Linear equation included in the hybrid equation
- Linear0–100 :
-
Linear equation applied to all force–velocity data
- Linear45–100 :
-
Linear equation applied to force–velocity data above 45% of maximum isometric force
- P max :
-
Maximum muscle power
- R 2 :
-
Coefficient of determination
- SEE:
-
Standard error of the estimate
- S FV :
-
Slope of the linear force–velocity relationship
- V 0 :
-
Estimated maximum unloaded velocity
- V opt :
-
Optimal velocity
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Acknowledgements
We thank all the participants for their collaboration and involvement in the experiments.
Funding
This work was supported by the Ministerio de Economía y Competitividad of the Spanish Government (MINECO/FEDER, EU) under Grants DEP2015-69386-R and BES-2016-077199; Biomedical Research Networking Center on Frailty and Healthy Aging (CIBERFES) and FEDER funds from the European Union under Grants CB16/10/00477 and CB16/10/00314; and Ministerio de Educación, Cultura y Deporte of the Spanish Government under Grant FPU014/05106.
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JA, IA, and LMA: conceived and designed research. JA, FP, CR, HG, JS, and PJC: conducted experiments. JA, FP, CR, and PJC: analyzed data. JA, FP, IA, and LMA: wrote the manuscript. All authors read and approved the manuscript.
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Communicated by Olivier Seynnes.
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Alcazar, J., Pareja-Blanco, F., Rodriguez-Lopez, C. et al. 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 122, 2305–2313 (2022). https://doi.org/10.1007/s00421-022-05006-1
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DOI: https://doi.org/10.1007/s00421-022-05006-1