An energy balance of front crawl

Abstract

With the aim of computing a complete energy balance of front crawl, the energy cost per unit distance (C= Ėv⁻¹, where Ė is the metabolic power and v is the speed) and the overall efficiency (η_o=W_tot/C, where W_tot is the mechanical work per unit distance) were calculated for subjects swimming with and without fins. In aquatic locomotion W_tot is given by the sum of: (1) W_int, the internal work, which was calculated from video analysis, (2) W_d, the work to overcome hydrodynamic resistance, which was calculated from measures of active drag, and (3) W_k, calculated from measures of Froude efficiency (η_F). In turn, η_F=W_d/(W_d+W_k) and was calculated by modelling the arm movement as that of a paddle wheel. When swimming at speeds from 1.0 to 1.4 m s⁻¹, η_F is about 0.5, power to overcome water resistance (active body drag × v) and power to give water kinetic energy increase from 50 to 100 W, and internal mechanical power from 10 to 30 W. In the same range of speeds Ė increases from 600 to 1,200 W and C from 600 to 800 J m⁻¹. The use of fins decreases total mechanical power and C by the same amount (10–15%) so that η_o (overall efficiency) is the same when swimming with or without fins [0.20 (0.03)]. The values of η_o are higher than previously reported for the front crawl, essentially because of the larger values of W_tot calculated in this study. This is so because the contribution of W_int to W_tot was taken into account, and because η_F was computed by also taking into account the contribution of the legs to forward propulsion.

Appendices

Appendix 1

In subjects swimming by using the leg kick, η_F can be assessed by means of methods usually applied to the study of undulating fish (Zamparo et al. 2002), e.g. from measures of the speed of the backward wave travelling along the body and of the forward speed (Alexander 1983; Daniel et al. 1992; Lighthill 1975). The mode of propulsion of the arm stroke is, in contrast, more similar to the one adopted by “rowing” animals that move in water by producing power strokes, during which an appendage is accelerated backwards, and recovery strokes, during which the appendage returns to its original position moving forward (Alexander 1983; Daniel et al. 1992). Rowing animals proceed in water with oscillating speed, i.e. they accelerate in the power stroke and decelerate in the recovery stroke. However, as a first approximation, these oscillations can be considered negligible and the body can be assumed to move forward at constant speed (v) while the appendages move forward and backward with a velocity u relative to the body. As indicated by Alexander (1983) the ratio v/u is proportional to the theoretical (or Froude) efficiency (η_F).

The ratio v/u is directly (or inversely) proportional to the theoretical efficiency in all fluid machines: pumps, turbines, propellers, fans, water wheels and paddle wheels. As an example, the theoretical efficiency of a water wheel can be calculated from the ratio of the tangential velocity of the paddles (the rim speed, u) to the velocity of the headwater stream (v). In this case u is less than v because only a fraction of the kinetic energy of the water (and hence of the power input) goes into shaft power output. On the other hand, the theoretical efficiency of a paddle wheel can be calculated from the ratio of the average velocity of the boat itself (v) to the tangential velocity of the blades (the rim speed, u). In this case v is less than u because only part of the shaft power input goes into “useful” motion, whereas the remaining fraction is wasted in giving “un-useful” energy to the water. Generally speaking, in all turbo-machines, the theoretical (Froude) efficiency depends (among the others) on the velocity components of the fluid and rotor at the inlet and outlet sections (Fox and McDonald 1992).

Humans can be considered as “fluid machines” that obtain the thrust necessary to proceed at a given speed (v) by using two engines: the legs and the arms. As indicated above, for subjects swimming by using the leg kick alone, the term u can be obtained from measures of the speed of the backward wave travelling along the body (as in the case of slender fish). The problem is how to calculate u in the arm stroke. The simplest way to do so is to model the movement of the upper limbs as that of a paddle wheel, a case in which the term u can be easily calculated/estimated from measures of rim speed.

The model presented in this paper is a simplified version of the model proposed by Martin et al. (1981) and considers the arm as a rigid segment rotating at constant angular velocity about the shoulder (see Fig. 5). The model assumes that the body is moving through the water with a constant speed v (metres per second), and that the arms rotate with a stroke rate SF (hertz). As proposed by Martin et al. (1981), it is also assumed that the two arms are 180° out of phase, and that one arm enters the water when the other completes its stroke. Hence, the angular position of the arm is α (α ranges from 0 to π), and the angular speed is: ω=α(t)=2πSF (constant through the cycle), where t is time. The efficiency of this form of propulsion can be calculated as the ratio of useful work rate to total work rate. The useful work rate is given by the product of the propelling force (F_p) times the swimmer’s velocity. In turn, F_p is the horizontal component of the force at the hand (F(α)):

Fig. 5

Schematic representation of the arm stroke modelled in analogy to the movement of a paddle wheel (see text for details)

$$ F_{{\text{P}}} (\alpha )\nu = F(\alpha )\;\sin \alpha \;\nu $$

From Newton’s second law (ΣF=mv̇), this force (the thrust) should be equal and opposite to the drag force:

$$ F_{{\text{p}}} (v) - F_{{\text{d}}} (v) = m\dot{v} $$

where m is the mass of the swimmer and v̇ his/her acceleration, that is assumed to be zero at constant speed (at steady state). Hence F_p(v)=F_d(v).

The total work rate can be calculated from the product of the moment about the shoulder (F(α) l) and ω:

$$ F(\alpha )l\omega = F(\alpha )l2\pi \;{\text{SF}} $$

The instantaneous efficiency is therefore:

$$ \eta {\left[ {\alpha {\left( t \right)}} \right]} = \frac{{F{\left( \alpha \right)}\sin \alpha \;\;v}} {{F{\left( \alpha \right)}2\pi \;{\text{SF}}\;l}} = \frac{{v\;\sin \alpha }} {{2\pi \;{\text{SF}}\;l}} $$

Over one “underwater cycle” (from 0 to π), the average efficiency is:

$$ \overline{\eta } = \frac{v} {{2\pi \;{\text{SF}}\;l}}{\left( {\frac{1} {\pi }} \right)}{\int\limits_{\alpha = {\text{0}}}^{\alpha = \pi } {{\left( {\sin \alpha } \right)}{\text{d}}\alpha = } }\frac{v} {{2\pi \;{\text{SF}}\;l}}{\left[ {\frac{2} {\pi }} \right]} $$

which is equivalent to Eq. 2a reported in the text.

In this last equation, the term (v/2π SF l) is the ratio of forward speed to rim speed (v/u), whereas the term 2/π indicates that only half a cycle (the pushing phase) is made underwater, whereas the recovery phase is made on air. At variance with swimming humans, paddle wheels and water wheels “rotate” outside water and exert a force (at the rim) which is always tangential to the direction of the water stream. In those two cases, the Froude efficiency is indeed given by the ratio v/u. The term 2/π (0.637) indicates that the maximal Froude efficiency of the arm stroke should be less than 1 (η_F=v/u 0.637) since the force exerted by the swimmer has both a horizontal (tangential to the water stream) and a vertical component (not useful for propulsion).

Appendix 2

To estimate the overall efficiency in the front crawl, the combined efficiency of the upper and lower limbs should be computed. This could be done by modelling the swimmer in analogy to a system with two engines/propellers working in parallel: the arms (A) and the legs (L). Since the two engines are not equal, each one will be characterized by its own efficiency (η), power output (PO) and power input (PI):

$$ \begin{array}{*{20}l} {{{\text{for legs }}} \hfill} & {{\eta _{{\text{L}}} = {\text{PO}}_{{\text{L}}} {\text{/PI}}_{{\text{L}}} } \hfill} \\ {{{\text{for arms }}} \hfill} & {{\eta _{{\text{A}}} = {\text{ PO}}_{{\text{A}}} {\text{/PI}}_{{\text{A}}} } \hfill} \\ {{{\text{since }}} \hfill} & {{\eta _{{{\text{AL}}}} = {\text{ PO}}_{{{\text{AL}}}} {\text{/PI}}_{{{\text{AL}}}} } \hfill} \\ {{} \hfill} & {{{\text{PO}}_{{{\text{AL}}}} = {\left( {{\text{PO}}_{{\text{A}}} {\text{ + PO}}_{{\text{L}}} } \right)}} \hfill} \\ {{} \hfill} & {{{\text{PI}}_{{{\text{AL}}}} = {\left( {{\text{PI}}_{{\text{A}}} {\text{ + PI}}_{{\text{L}}} } \right)}} \hfill} \\ {{{\text{then}}} \hfill} & {{\eta _{{{\text{AL}}}} = {\left[ {{\left( {{\text{PI}}_{{\text{L}}} {\text{/PI}}_{{{\text{AL}}}} } \right)}\eta _{{\text{L}}} } \right]} + {\left[ {{\left( {{\text{PI}}_{{\text{A}}} {\text{/PI}}_{{{\text{AL}}}} } \right)}\eta _{{\text{A}}} } \right]}} \hfill} \\ {{{\text{for}}} \hfill} & {{{\text{PI}}_{{\text{L}}} {\text{/PI}}_{{{\text{AL}}}} = {\text{ }}k} \hfill} \\ {{{\text{then}}} \hfill} & {{\eta _{{{\text{AL}}}} = \eta _{{\text{L}}} k + \eta _{{\text{A}}} {\left( {{\text{1}} - k} \right)}} \hfill} \\ \end{array} $$

In these equations, PI and PO stand for mechanical power input and output (the efficiency here is a Froude efficiency) and the effects of a positive/negative influence–interaction between the “two engines/propellers” are not taken into consideration. However, this factor is likely to be partially accounted for by the partitioning in the total propulsion of arms (1−k=0.9) and legs (k=0.1), as experimentally determined by several authors on the basis of independent methods (Bucher 1975; Deschodt et al. 1999; Hollander et al. 1988).