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 (F 0) and from the linearity at ≤45% of F 0 (both p < 0.05). Hyperbolic equations overestimated F 0 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 F 0, maximum muscle power, and velocity between 25% and 100% of F 0 (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 F 0, maximal muscle power and velocity between 25% and 100% of F 0.