Mechanical work and efficiency of 5 + 5 m shuttle running

Abstract

Purpose

Acceleration and deceleration phases characterise shuttle running (SR) compared to constant speed running (CR); mechanical work is thus expected to be larger in the former compared to the latter, at the same average speed (v _mean). The aim of this study was to measure total mechanical work (W ⁺_tot , J kg⁻¹ m⁻¹) during SR as the sum of internal (W ⁺_int ) and external (W ⁺_ext ) work and to calculate the efficiency of SR.

Methods

Twenty males were requested to perform shuttle runs over a distance of 5 + 5 m at different speeds (slow, moderate and fast) to record kinematic data. Metabolic data were also recorded (at fast speed only) to calculate energy cost (C, J kg⁻¹ m⁻¹) and mechanical efficiency (eff⁺ = W ⁺_tot C ⁻¹) of SR.

Results

Work parameters significantly increased with speed (P < 0.001): W ⁺_ext  = 1.388 + 0.337 v _mean; W ⁺_int  = −1.002 + 0.853 v _mean; W ⁺_tot  = 1.329 v _mean. At the fastest speed C was 27.4 ± 2.6 J kg⁻¹ m⁻¹ (i.e. about 7 times larger than in CR) and eff⁺ was 16.2 ± 2.0 %.

Conclusions

W ⁺_ext is larger in SR than in CR (2.5 vs. 1.4 J kg⁻¹ m⁻¹ in the range of investigated speeds: 2–3.5 m s⁻¹) and W ⁺_int , at fast speed, is about half of W ⁺_tot . eff⁺ is lower in SR (16 %) than in CR (50–60 % at comparable speeds) and this can be attributed to a lower elastic energy reutilization due to the acceleration/deceleration phases over this short shuttle distance.

Appendix

In any form of locomotion, the metabolic request to achieve a given speed can be fully understood if all components of the mechanical work are investigated.

In shuttle running, differently from constant linear running, braking can be substantial, thus the analysis of mechanical work performed by muscles to move BCoM (W _ext) and limbs (W _int) have to consider both positive and negative work in order to better estimate the metabolic demand. As a general rule, the metabolic cost of locomotion can be calculated as:

$$C = \frac{{W_{\text{ext}}^{ + } }}{{eff_{\text{ext}}^{ + } }} + \frac{{W_{\text{ext}}^{ - } }}{{eff_{\text{ext}}^{ - } }} + \frac{{W_{\text{int}}^{ + } }}{{eff_{\text{int}}^{ + } }} + \frac{{W_{\text{int}}^{ - } }}{{eff_{\text{int}}^{ - } }},$$

(3)

where + and − indicate positive and negative values. On the level, positive (W ⁺) and negative (W ⁻) work are by definition the same (whereas positive and negative power could be different because of a possible different duration of the positive and negative work phases). When this is the case, a multiple linear regression is unable to estimate different positive (eff⁺) and negative (eff⁻) values of efficiency.

It thus could be convenient to rearrange this equation by imposing W ⁺ = W ⁻ and a constrained ratio between eff⁺ and eff⁻ (e.g. 1:5) compatibly with previously reported data from muscle physiology (Wooledge et al. 1985) and reasonable for all activation levels. Accordingly, Eq. 3 turns into:

$$C = \frac{{W_{\text{ext}}^{ + } }}{{\frac{{{\text{eff}}_{\text{ext}}^{ + } {\text{eff}}_{\text{ext}}^{ - } }}{{{\text{eff}}_{\text{ext}}^{ + } + {\text{eff}}_{\text{ext}}^{ - } }}}} + \frac{{W_{\text{int}}^{ + } }}{{\frac{{{\text{eff}}_{\text{int}}^{ + } {\text{eff}}_{\text{int}}^{ - } }}{{{\text{eff}}_{\text{int}}^{ + } + {\text{eff}}_{\text{int}}^{ - } }}}} = \frac{6}{5}\left( {\frac{{W_{\text{ext}}^{ + } }}{{{\text{eff}}_{\text{ext}}^{ + } }} + \frac{{W_{\text{int}}^{ + } }}{{{\text{eff}}_{\text{int}}^{ + } }}} \right).$$

(4)

When considering individual data of W ⁺_ext , W ⁺_int and C measured in this study at maximal speed, eff ⁺_ext  = 12.7 % and eff ⁺_int  = 65.1 %; conversely, when considering individual values of W ⁺_ext and W ⁺_int measured in this study and a previous metabolic estimate at the same speed (C = −12.82 + 11.94 v _mean, Zamparo et al. 2015), the multiple regression analysis resulted in eff ⁺_ext  = 23.5 % (IC 95 % 20.8–27.0 %) and eff ⁺_int  = 18.9 % (IC 95 % 16.2–22.7 %) R ² = 0.982, P < 0.001. The apparent inconsistency in the obtained values of efficiencies by using these two methods and the fact that some of these estimates are far out of the physiological range can be explained by the limits data impose to statistics. Since the measured W ⁺_ext and W ⁺_int values are very similar when taken at the highest speeds only, this makes the regression unable to infer reliable estimates of eff ⁺_ext and eff ⁺_int . Formally, it corresponds to moving from (z = a · x + b · y) to (z = a · x + b · x), where x and y stand for W ⁺_ext and W ⁺_int , respectively, and a and b stand for 1/eff ⁺_ext and 1/eff ⁺_int respectively. The second equation (z = a · x + b · x) is equivalent to (z = c · x) with (c = a + b); however, infinite pairs of a and b values can produce the same value of c. Due to this uncertainty, a further assumption could be made: i.e. that eff ⁺_ext  = eff ⁺_int  = eff ⁺_tot ; in this case Eq. 4 can be simplified as:

$$C = \frac{6}{5}\left( {\frac{{ W_{\text{ext}}^{ + } + W_{\text{int}}^{ + } }}{{ {\text{eff}}_{\text{tot}}^{ + } }}} \right).$$

(5)

Equation 5 corresponds to turning (z = a · x + b · y) into (z = c (x + y)) where c (1/eff ⁺_tot ) is the sum of a (1/eff ⁺_ext ) and b (1/eff ⁺_int ). Being a and b reciprocal values of the unknown parameters, if c can be obtained as the arithmetic mean of a and b, eff ⁺_tot can be calculated as the harmonic mean of eff ⁺_ext and eff ⁺_int : 21.3 % (from eff ⁺_ext 12.7 and eff ⁺_int 65.1 %, maximal speed tests only) and 21.0 % (from eff ⁺_ext 23.5 and eff ⁺_int 18.9 %, whole tests including data from Zamparo et al. 2015). These total efficiency estimates comfortably sit within the physiological range and suggest that no elastic energy storage/release occurs in 5 + 5 m shuttle runs.

Since eff ⁺_ext and eff ⁺_int prove to be similar (e.g. 23.5 and 18.9 in the whole range of speeds), and considering that W ⁺_tot  = W ⁺_ext  + W ⁺_int , Eq. 5 can be further simplified as:

$$C = \frac{{6 W_{\text{tot}}^{ + } }}{{5 {\text{eff}}_{\text{tot}}^{ + } }}.$$

(6)

Equation 6 can be used to estimate the total efficiency for positive work, i.e.:

$${\text{eff}}_{\text{tot}}^{ + } = \frac{6}{5}\frac{{ W_{\text{tot}}^{ + } }}{ C},$$

(7)

and we need to recollect that this estimate, though indicating just the total positive mechanical work, corrects the efficiency according to the presence of negative work (the 6/5 coefficient). But this is still a prediction at a given level of exercise and all the previous provided figures for efficiency dealt with the mean values across all shuttle run speeds. When studying this activity at different speeds, efficiency values are shown to decrease from 24.0 %, at 2.0 m s⁻¹, to 16.1 %, at 3.5 m s⁻¹, (28.8 %, at 2.0 m s⁻¹, to 19.3 %, at 3.5 m s⁻¹ when the 6/5 coefficient is introduced) as shown in Fig. 4, which can also be expressed formally as

$${\text{eff}}_{\text{tot}}^{ + } { = }\frac{{ 1.595 v_{\text{mean}} }}{{11.94 v_{\text{mean}} - 12.82}},$$

(8)

by considering that W ⁺_tot  = 1.329 v _mean (see “Results”) and C = −12.82 + 11.94 v _mean (from Zamparo et al. 2015). As reported in the “Discussion”, the decrease of efficiency can be partially explained by the increase of muscle contraction speed, as a consequence of the Hill force/speed relationship.