I thought I knew classical mechanics pretty well, but I must say I was pretty confused after reading the write-ups on rower energy dissipation by Atkinson and Van Holst. Well, this is how I calculate the energy balance. There are three energy sinks in the system:
- Drag. Energy dissipated by fluid friction along the boat. .
- Energy dissipated by the slip of the blade through the water. .
- Some energy may be dissipated in the rower’s body when he or she slows down at the end of the stroke or the end of the recovery
Figure 1 shows the result of a typical single stroke simulation. The system is in equilibrium, i.e. the hull speed at the start of the stroke is constant from stroke to stroke. The simulation was done for a single scull heavy rower rowing at an average velocity of 4.74 m/s, which corresponds to a 500m time of 1:45.5.
Following Van Holst, the force moving the mass of the rower with respect to the boat is given by
The power supplied by this force is equal to and is plotted in figure 2 (cf figure 4.6 here):
Positive values mean power supplied to the system by the rower, while negative values mean power absorbed from the system by the rower. This power cannot be reused and is thus lost for propulsion. Atkinson argues that the absorption of kinetic energy is done by different muscle groups than the generation, and this is reflected here by assuming it takes 1W to absorb 1W of kinetic energy, as depicted in the right hand side of the figure. (See also the discussion by Van Holst about energy consumed while walking down the stairs). In the simplified example of Van Holst, indeed some dissipation in the rower occurs. In my simulation results in figure 2, I see hardly any power dissipation, which is in agreement with the simulation results of Van Holst in the realistic system but in contrast with Atkinson’s findings. I guess a discussion on this is needed.
Figure 3 shows the energy invested into the system by the rower as a function of the time during the stroke. The blue line is the energy provided by (the area under the graph in figure 2). The black line, “puddle energy” is energy used to drag the blade through the water creating the puddle. The red line is the propulsive energy accelerating the boat at the oarlock, and the green line is the total energy.
Figure 4 shows the energy going into all the sinks (red line). Also depicted is the energy “temporarily stored” as kinetic energy (in excess of the system kinetic energy at the catch at t=0). The sum of these sinks plus storage should be equal to the energy delivered by the rower at each point in time. This is shown in figure 5:
The efficiency of the stroke is the ratio of energy dissipated in overcoming the hull friction vs. the total energy dissipated in the system.
The average power exerted by the rower is given by the total energy divided by the duration of the stroke (in seconds):
The typical efficiency values that I find are around 77-80%. This is significantly higher than Atkinson, probably due to the fact that I find negligible values of .
It is interesting to see that a significant portion of the rower energy is delivered through the kinetic energy of the rower (“through the footboard”). However, in real life, it is very hard for a rower to decouple these two, except for slight differences due to rowing style. One could wonder for example about the efficiency of different types of catch (“trunk swing” vs “leg push”), and perhaps I will try to model it in the future. In my model, this would be implemented as a different function of rower body of mass vs handle position.