MATHEMATICAL MODEL

Symbols and Nomenclature

N = number of rowers
mB = mass of boat (including cox and/or other deadweight)
mR = combined mass of rowers
m1 = mean mass of rowers
u = boat speed at time t
t1 = duration of power stroke
t2 = duration of recovery phase
r = stroke rate (strokes per minute)

Equations of Motion

The derivation of the equations of motion used in the present study is given in Brearley and de Mestre (1996). (We have made several notational changes).

The equation of motion for the power stroke can be written as

...(1)

where FA(t) is the (axial) force exerted at the oarblades by the rowers, RT(t) is the total (aerodynamic plus hydrodynamic) resistance and H1(t) is the acceleration of the rowers' bodies during the power stroke.

The equation of motion for the recovery phase can be written as

...(2)

where H2(t) is the acceleration of the rowers' bodies during the recovery phase.

We assume that the relative motion of the rowers can be taken as the half-cycle of simple harmonic motion (SHM) as in Brearley and de Mestre (1996), and write

...(3)

and

...(4)

where n1=p/t1, and n2=p/t2. The average amplitude of the SHM of the centres of mass of the rowers' bodies, as, is assumed to vary with the height of the rower (or the cube root of the rower's mass).

...(5)

For a 57 kg rower this equation gives a (half) amplitude of as=0.315m, and for a 95 kg rower, as=0.374m. These values are in rough accord with the limits of 0.65-0.75m given for the total amplitudes in Dal Monte and Komor (1989, p. 57). Presumably their estimates were for (male) rowers with body weights which usually range from about 70 kg to 95 kg.

Rowing Force-Time Curves

The variation with time of the magnitude of the rowing force during the stroke is different for individual rowers and rowing styles. There seem to be three main styles of rowing, Dal Monte and Komor (1989).

The first style is characterised by a rapid increase of force during the initial phase of the power stroke, followed by a slow decrease towards the end of the pull. The third style is almost the opposite of the first; there is a slower increase of force at the start of the pull, the maximum force being attained in the final stage. The second style is more symmetric than the other two; the maximum force is attained near the middle of the power stroke. Following Millward (1987), we use a simple formula for the variation of the rowing force (in the axial direction). During the power stroke the variation of the rowing force is given by

...(6)

where Fmax is the maximum value of FA. For the recovery phase, FA is equal to zero. This model is closest to Style II, the symmetric rowing style shown in the figure above. Brearley and de Mestre (1996) used a similar model in their work, the only difference being that they used a first-power sine curve, rather than the sine-squared curve employed herein.

Power Stroke Duration

During a rowing race, both the stroke rate and the force exerted during the stroke can vary considerably. The duration of the power stroke has been shown to decrease slowly as the stroke rate increases, see for example, Celetano et al. (1974), Martin and Bernfield (1980) and Dal Monte and Komor (1989). These studies did not examine stroke rates above about r=40. (At the start of a race, stroke rates can be as high as r=48 strokes per minute). To estimate the duration t1 of the power stroke, we use the following equation, which was derived by fitting a quadratic equation to data from various sources.

...(7)

For a stroke rate of r=37.5, this equation gives t1=0.72 seconds. Brearley and de Mestre (1995) estimated t1=0.7 seconds from videos of top class rowers in an eight-oared shell.

Acceleration Phase

There are very few studies which show rowing force as a function of stroke rate. Schneider et al (1978) found that the peak rowing force exerted on the oar blade increased slowly, but quadratically, with stroke rate for stroke rates between r=24 and r=36. Their results are reproduced in Millward (1987, Figure 3, p. 97). In the present study, we will employ the following simple model of the stroke rate and the peak force exerted during the initial stages of a race. This model was cobbled together from discussions with rowers, coaches and boat-bulders, and from the present author's own observations.

We assume that single scullers will finish their acceleration phase, (that is the time during which their stroke rate decreases from a maximum to the final constant rate), at 350 metres; doubles will finish at 400 metres, quads and fours at 450metres, and eights at 500 metres. All classes begin with a stroke rate of 45. After their acceleration phase, crews will continue the race at a constant stroke rate: singles will continue at 38.5 spm, doubles at 39.0 spm, fours at 39.5 spm, and eights at 40.0 spm. The stroke rate decreases linearly with distance from where the acceleration phase ends for the particular class in question.

Furthermore, we assume that the peak force during the initial stages is given by:

F0max(r)=Fmax(1+0.5*(s-s0)^2/s0^2) ...(8)

where s is the distance travelled and s0 is the distance at which the acceleration phases finishes for the particular class. Fmax is the peak force that will be exerted at the end of the acceleration phase, that is, during the 'steady stroke-rate' phase.

Thus, for the initial stroke, the peak force is 50% higher than the force exerted during the steady stroke rate phase, which occurs a little after the end of the acceleration phase. For example, suppose that for a single sculler, the peak force for the last 1650m of the race is 300 N at a stroke rate of r=38.5. At the start of the first stroke, s=0.0, and F0max(r) = Fmax*1.5.

Equation (8) is little more than an educated guess based on scant data, however there is as yet little else in the literature to guide us. In particular, we are lacking good experimental data of the forces exerted at high stroke rates, and how these forces vary while the boat is accelerating from rest.

Hydrodynamic drag

The major forces resisting the motion of the boat are the hydrodynamic drag and the aerodynamic drag, the former being responsible for about 90% of the total. In the present study, the hydrodynamic drag is taken to be the sum of the viscous drag and the wave drag. The viscous drag is the sum of the skin friction (estimated using the 1957 ITTC line) and the form drag (estimated using the empirical formula of Scragg and Nelson (1993)). The steady, upright, calm-water wave resistance is estimated using Michell's (1898) thin ship theory. See also Tuck (1987).

For calculations we use water density r = 1000 kgm-3 and kinematic viscosity n = 1.14 centipoise.

Aerodynamic drag

Aerodynamic drag on the crew and the above-water portion of the hull is difficult to estimate. In his study of a single-scull, Millward (1987) uses a coefficient given in Hoerner (1965) for the drag of a standing man. For multiple crews, the shielding of the rearmost crew members adds further complications which we will ignore in the present study.

We assume that the aerodynamic drag is proportional to the rowers' frontal area (or the two-thirds power of their average mass). Based on Millward's estimate for the drag coefficient, we use the following formula for the air resistance of a single rower:

...(9)

For multiple crews we need to multiply Rair by the number of rowers.

Work and Power

There seems to be little agreement on definitions of work and power in the rowing biomechanics literature. Dal Monte and Komor (1989, p. 93) present a table summarising some of those that have been used and there are seven definitions of work and twelve definitions of power! In the present study, we define the useful work done to overcome the total resistance on the hull and rowers during a stroke as

...(10)

The mean power during the stroke is

...(11)