Choosing the coefficients

Using the procedure described in Section 7.5 the coefficients of wind drag (Cd) and snow

resistance (μF) were calculated over different windows (lengths) of data (Figure 7.9) for Run

3. The data windows included all data above an increasing cut-off speed. Eventually when the cut-off speed was above 16ms-1 _{the solution became unstable because the number of data} points used was too small. The snow resistance coefficient was multiplied by ten so both coefficients could be viewed on the same graph.

Figure 7.9 shows a strong negative correlation between the optimised solutions of the two different coefficients making it very difficult to obtain a robust or accurate solution. The negative correlation is probably because both coefficients have a strong dependency on velocity as discussed in Sections 7.2.1 and 7.2.2. Systematic errors therefore could have been introduced into the data through the erroneous selection of model coefficients.

Any systematic errors in modelled forces could be exacerbated by the method used to model the ground reaction forces (as two forces acting nearly perpendicular to the ski bases). The athlete polling, crashing through the course gates or the ski flexing could produce ground reaction forces with components parallel to the ski base that do not fit the model and therefore could produce extreme statistical outliers. These extreme statistical outliers could cause, for example, the mean horizontal ground reaction forces acting on the athlete to be over-

estimated. This in turn could cause error in the selection of mean wind drag or snow resistance coefficients. Similarly the snow resistance force coefficient could be overestimated and the wind drag force coefficient underestimated. The model coefficients are interdependent because the calculation of forces is a closed loop constrained by the athlete‟s centre-of-mass trajectory.

Between 10ms-1 _{and 16ms}-1 _{on the horizontal axis, the solution appears to stabilise. The} athlete generally skied faster than 15ms-1 _{so it was decided to reduce the complexity of the} problem by assuming a fixed wind drag coefficient based on all data above 15ms-1_{. More} reasoning behind this choice is found in Section 7.2.1. The calculated wind drag coefficient chosen was Cd=0.52 (see Speed=15ms-1 in Figure 7.9) it was assumed to be constant and

independent of velocity. The analysis shows the wind drag coefficient could well have been as high as Cd=0.60 or as low as Cd=0.48 for the athlete.

Figure ‎7.9: Changes in optimised friction coefficients with velocity

After assuming the wind drag coefficient was constant over the majority of the racecourse it was then possible to investigate further the effects of velocity on the snow resistance. First, the data were ordered based on speed and selected large windows of data (>8 seconds) with different mean velocities. Snow resistance only was then solved for (μF, Equation 7.17).

It appeared from the data that snow resistance in giant slalom increased with velocity (Figure 7.10), but the relationship is not linear. There was not enough data to develop a more complex model and so a linear model was fitted to the section of the data between 3ms-1 _and 18ms-1_{. Below 3ms}-1 _{the athlete may have used his poles, which were not accounted for in the} force balance, and above 18ms-1 _{the reduced data window size increased the noise in the} motion.

The snow resistance model Equation 7.6 was the same as the linear model used to fit the athlete‟s data in Figure 7.10. The y-intercept in Figure 7.10 defined the static term for snow resistance (μF=0.042) and the slope of the line in Figure 7.10 defined the velocity dependant

term (μV=0.0061sm-1). If however a different wind drag coefficient had been chosen then the

snow resistance coefficients would also be different. Further analyses are therefore required to confirm these results.

Equation 7.6

Figure ‎7.10: Snow resistance coefficient versus Mean speed - fitted with a linear model

7.5.2.

Discussion about wind drag and snow resistance

How do the model coefficients for wind drag and snow resistance compare with previous research?

The snow resistance model (linear fit of Figure 7.10) predicts a total snow resistance coefficient (μ=μF+μVV) at slow speeds of around 0.04 but at high speeds the total coefficient

increases to 0.2, these values are within the range suggested in literature for hard and soft snow respectively (μ=0.04-0.2, (Shimbo, 1971). The model used to fit the data also agrees with data from Kaps (μ=0.060–0.171) for velocities between 0.6ms-1 _{and 16.6ms}-1 _{(Kaps, et} al., 1996), but it seems Kaps may have done his experiments on slightly softer snow as his values are slightly higher at slower speeds.

The snow resistance model coefficients used and therefore the calculated snow resistance agree with miniature force platform measurements of forces parallel to ski bases (Lüthi, et al., 2005). Lüthi showed, for his athlete and snow conditions skiing at 13m/s, between 7% and

10% of the ground reaction forces were parallel to the ski bases. The model predicted at 13m/s about 12% of the ground reaction forces should parallel to the ski bases (Figure 7.10). The difference might be because of softer snow conditions, different skis, different course setting and/or different wax used for ski preparation.

The athlete‟s wind drag area (CdAc) was calculated based on Cd=0.52 (Figure 7.11). While the

athlete was running both skis smoothly over the snow the wind drag area varies between 0.24m2 _{and 0.33m}2_{. This is in agreement with previous experiments conducted in wind} tunnels. It is not as low as values for the „egg posture‟ (CdAc=0.15). But it sits nicely between

semi squatting (CdAc=0.2) and upright poses (CdAc=0.35, (Barelle, et al., 2004).

Figure ‎7.11: Instantaneous wind drag area from FMC

How accurate is the fixed wind drag coefficient? The model incorporates the effect of changing posture cross sectional area on wind drag force. Because there was not sufficient data, the model does not incorporate the effects of changing velocity and characteristic posture length on Reynolds number and therefore drag coefficient (Cd).

Springs data (Spring, et al., 1988) can be re-arranged to show there may be a correlation between characteristic posture length (L) and drag coefficient (Cd). If Spring‟s measured wind

drag area is divide by the measured projected area a wind drag coefficient is obtained. Spring‟s data show moving from a crouched posture (long length parallel to the wind and higher Reynolds number) to an upright posture (short length parallel to the wind and lower Reynolds number) increases the wind drag coefficient by 20% to 40%. The optimisation used in this thesis selected an average value for the wind drag coefficient so the results may have underestimated the effect of stance changes on wind drag force from standing to crouching. To reduce the effect of stance changes on drag coefficient error some sort of mean characteristic length of the athlete‟s stance could be estimated by his silhouette as viewed perpendicular to the wind velocity. In the future the changing characteristic lengths (L) of the

athlete and the changing velocity (V) could be used to estimate changes in Reynolds number and the effects on the drag coefficient. However at this stage there was not have enough data to build such a detailed model and so it was decided that this approach was beyond the scope of this thesis.

The analyses of wind drag and snow resistance based on the FMC data have produced some interesting results. Because of the high correlation between the different optimised coefficients, the data were windowed by velocity using all data greater than 15ms-1 _{to predict} a constant coefficient of wind drag. A linear model was then used to predict the change in snow resistance coefficient with velocity. Neither assumption was perfect, but it was the best that could be done with the data available. In the future, if the analysis was repeated, 3D force plates could be used under the athlete‟s feet to separate snow resistance from wind drag. The athlete could ski through the course several times at different speeds and with different stance heights. This would hopefully provide enough data to improve the understanding of the dissipative forces in giant slalom skiing.

The results of this analysis are surprising: It appears that over the course snow resistance was generally larger than wind drag (Figure 7.12). This contradicts some previous research based on laboratory measurements of ski/snow friction (Barelle, et al., 2004). The snow resistance was higher than expected for the relatively hard and fast snow. The high snow resistance coefficients at high speeds were probably caused more by the displacement, compression and ploughing of the snow, especially while cornering. At slow speeds the traditionally Coulomb ski/snow sliding friction predominated.

Figure ‎7.12: Comparison of dissipative forces through the course

There is a small possibility that some error exists in these results, a consequence of statistical outliers in the motion capture data combined with analysis error, and exacerbated by the high

negative correlation between the dissipative force coefficients. In the future the results should be confirmed by additional data from multiple runs of multiple athletes on similar and steeper terrain with different snow conditions.