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)