INTRODUCTION

In Olympic rowing events, the ultimate measure of performance is the time taken to complete 2000 metres. To minimise this time, rowers must use efficiently a variety of external forces to overcome forces opposing the forward progress of their boat. What makes the problem both fascinating and challenging from an engineering and computer modelling perspective is the complexity of the system; the pulsatory form of propulsion, the motion of the rowers on sliding seats, and the complicated flows of air and water around the crew and boat, make for an intractable mathematical problem.

Millward (1987) developed a simple model of the motion of a single-scull rowing hull, but ignored the motion of the rower within the boat. A similar model was used by Charlton, Denman and Millward (1987) to investigate the feasibility of a human-powered hydrofoil. In their model of the rowing stroke, Brearley and de Mestre (1996) included the motion of the rowers in a coxed eight, and were able to derive familiar aspects such as the variation in speed during a stroke. They then used their model to investigate an energy storing system devised to improve the performance of rowers.

In the present study, we assume that the rowing stroke is comprised of two separate phases. During the power stroke, the rowers push on their footrests and, while sliding towards the bow of the boat, pull on the oarhandles. During the recovery phase, oarblades are removed from the water and are moved towards the bow of the boat while the rowers slide sternwards.

The force exerted on an oar handle and its variation during the power stroke is different for each rower. Although there are published data of these variations, for example Wing and Woodburn (1995), difficulties arise when we try to convert the force applied at the oarhandle to the ultimate propulsive force which acts near the centre of the oarblade and in a longitudinal (or axial) direction. Celetano et al (1974) showed that there are substantial mechanical losses at the oarlocks that can be difficult to estimate. For experienced rowers the losses are around 22%, for inexperienced rowers the losses can be of the order of 40% or more. We can avoid having to account for these losses by specifying the peak value and the shape of the rowing force tending to propel the boat forwards. In the present study, as in Millward (1987) and Brearley and de Mestre (1996), we will use a simple mathematical function to model the variation of the rowing force throughout the power stoke. For the recovery phase we assume that the rowing force is identically equal to zero. The effect of different rowing styles and different real force-time curve shapes will be examined in a future paper.

To account for the motion of the rowers' bodies in the boat, we follow Brearley and de Mestre (1996) and assume that the rowers' displacement around some mean position can be reasonably described by a half of a cycle of simple harmonic motion.

Both Millward's (1987) and Brearley and de Mestre's (1996) models assumed a constant stroke rate for the duration of the entire race; in the present model, we allow variations in stroke rate. In particular, we assume that the stroke rate decreases from a maximum at the beginning of the race, to a value at which it then remains constant for the rest of the race. This requires estimates of variations in force for different stroke rates and the duration of the power stroke for different rates; empirical formulae gleaned from a variety of sources are used to estimate both. We also require estimates of how long this 'acceleration' phase lasts. In the present model, we use values gathered from observations of real races and from discussions with rowers, boat-builders and others.

The main forces opposing the forward motion of the boat are the hydrodynamic drag and the aerodynamic drag, the former accounting for roughly 90% of the total drag. Both of the aforementioned models used experimental data for the hydrodynamic forces acting on the boat hull: Millward used data from experiments an a single-scull shell; Brearley and de Mestre used Wellicome's (1967) experimental data for an eight-oared shell in their work. The present model requires as input the actual offsets of the hull. From these we estimate the skin friction using the 1957 ITTC line. Form drag is estimated using a simple empirical formula developed by Scragg and Nelson (1993) for rowing shells, and the wave resistance is estimated using Michell's (1898) thin ship theory. See also Tuck (1987). The advantage of the present model over earlier mathematical models is that we can easily assess the effect on rowing performance of changes to the shape of the hull and other fundamental dimensions without having to resort to experiments.

Aerodynamic drag on the crew and the above-water portion of the hull is estimated using a simple empirical formula. The shielding of the rearmost rowers by the rowers at the front of multiple crew boats is an extremely complicated flow problem. Jackson (1995) included an estimate for the shielding effect in multiple crew Olympic kayaks, but it is unclear how he derived his drag coefficients. For multiple crew rowing events, the rowers' motion adds further complications. In the present study, we will ignore shielding effects.

To verify that the model reasonably predicts the final time of a real race, we will specify the hull shape and rowers' particulars and then vary the peak rowing force until the final predicted time matches the time taken for world class events. If the required peak force is in agreement with measurements taken from top-class rowers, then we can claim some confidence that the model is giving reasonable overall estimates.