Comparison of linear, hyperbolic and double-hyperbolic models to assess the force–velocity relationship in multi-joint exercises

Abstract

This study assessed the validity of linear, hyperbolic and double-hyperbolic models to fit measured force–velocity (F–V) data in multi-joint exercises and the influence of muscle excitation on the F–V relationship. The force–joint angle and F–V relationships were assessed in 10 cross-training athletes and 14 recreationally resistance-trained subjects in the unilateral leg press (LP) and bilateral bench press (BP) exercises, respectively. A force plate and a linear encoder were installed to register external force and velocity, respectively. Muscle excitation was assessed by surface EMG recording of the quadriceps femoris, biceps femoris and gluteus maximus muscles during the unilateral LP. Linear, Hill’s (hyperbolic) and Edman’s (double-hyperbolic) equations were fitted to the measured F–V data and compared. Measured F–V data were best fitted by double-hyperbolic models in both exercises (p < 0.05). F–V data deviated from the rectangular hyperbola above a breakpoint located at 90% of measured isometric force (F₀) and from the linearity at ≤45% of F₀ (both p < 0.05). Hyperbolic equations overestimated F₀ values by 13 ± 11% and 6 ± 6% in the LP and BP, respectively (p < 0.05). No differences were found between muscle excitation levels below and above the breakpoint (p > 0.05). Large associations between variables obtained from linear and double-hyperbolic models were noted for F₀, maximum muscle power, and velocity between 25% and 100% of F₀ (r = 0.70–0.99; all p < 0.05). The F–V relationship in multi-joint exercises was double-hyperbolic, which was unrelated with lower muscle excitation levels. However, linear models may be valid to assess F₀, maximal muscle power and velocity between 25% and 100% of F₀.

Keywords:

• Torque-velocity
• muscle power
• muscle mechanics
• maximal unloaded velocity
• biphasic force–velocity
• velocity-based training‌

Acknowledgements

We thank all the participants for their collaboration and involvement in the experiments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed here https://doi.org/10.1080/17461391.2020.1753816.

Additional information

Funding

This work was supported by the Ministerio de Economía y Competitividad of the Spanish Government (MINECO/FEDER, EU) [grant numbers DEP2015-69386-R and BES-2016-077199]; Biomedical Research Networking Center on Frailty and Healthy Aging (CIBERFES) and FEDER funds from the European Union [grant number CB16/10/00477]; and Ministerio de Educación, Cultura y Deporte of the Spanish Government [grant number FPU014/05106].