Errors and Uncertainties

The greatest error during testing comes from the precision of the force balance system and ensuring the athletes adopt a consistent position during testing. It is important to wait for the wind to die down completely before the second calibration, which was monitored using a cotton tuft attached to the wind tunnel exit using clear, mylar tape. The use of the white board for tracing around the shadow side-on minimises the error associated with a consistent position, but it will be impossible for athletes to hold the exact same position during tests. Therefore comparisons were made for a number of both static and dynamic tests to determine the consistency of drag readings for an athlete or equipment in the same position. Static tests were carried out using a bike placed on the rig at a wind speed of 41kph, and the bike was removed and then replaced in the same position for a total of 7 tests. The results, shown in Table 5.3, indicate that the maximum difference in drag was 9.1g (1.36%).

Test Drag (kg) Difference from Lowest Drag (%)

1 0.6595 0 2 0.6614 0.29 3 0.6617 0.33 4 0.6686 1.36 5 0.6632 0.56 6 0.6644 0.74 7 0.6679 1.26 Mean 0.6638 SD 0.0034

Table 5.3: Drag for a stationary bike with 41kph wind speed

Dynamic tests were carried out using three different athletes, each adopting their own reference position for three, pedalling tests. After each test the athlete got off their bike and then back on again, and the white board was used to ensure athletes adopted the same position each time. The results, shown in Table 5.4, indicate that the maximum difference in drag for the same position for these three athletes is between 0.4 and 0.8%.

Measured Drag (kg)

Test Athlete 1 Athlete 2 Athlete 3

1 2.061 2.187 2.060

2 2.078 2.177 2.074

3 2.062 2.178 2.072

Max Difference in Drag (kg) 0.017 (0.8%) 0.009 (0.4%) 0.014 (0.7%)

Mean 2.067 2.181 2.069

SD 0.010 0.006 0.008

The precision of the force balance system was calculated by measuring the difference between the calibration readings before and after both static and dynamic tests for a total of six tests each. For the static tests the calibration was applied before and after placing and removing a 20kg weight to the middle of the rig. For the dynamic tests, a cyclist adopted their reference position on a road bike fixed to the cycle rig and the calibration was applied, after which point the cyclist began to pedal for 30 seconds and when they stopped the calibration factor was measured again, as shown in Figure 5.10.

Figure 5.10: LabVIEW display for dynamic testing without wind

For the static tests the maximum drift was 0.015kg and the average drift was 0.0058kg (Table 5.5). An elite cyclist has an approximate overall drag of 2kg, so a drift of 15g is less than 1% error.

Test Maximum Drift (kg)

1 0.0100 2 0.0150 3 0.0006 4 0.0009 5 0.0050 6 0.0030 Mean 0.0058 SD 0.0057

Table 5.5: Drift for static tests

For the dynamic tests the maximum drift was 0.016kg and the average drift was 0.015kg (Table 5.6). This error is slightly higher than for the static tests due to the increased impact on the load cell with every pedal stroke, which would not be present during static tests. However, this error is still less than 1% for a cyclist with an overall drag of 2kg. The drift was not a systematic error, as sometimes the drift would be positive but other times negative. There was no indication as to when a positive or negative drift would be recorded.

Test Maximum Drift (kg) 1 0.012 2 0.016 3 0.009 4 0.023 5 0.016 6 0.011 Mean 0.015 SD 0.005

Table 5.6: Drift for dynamic tests

The total error associated with athlete position and the force balance is therefore in the region of 2 to 2.5%. Additional errors are associated with the wind speed, air density, frontal area calculation and human error. The wind speed (in kph) is determined from the change in pressure between the pitot tube and atmosphetic pressure and can be calculated using Equation 5.2, where 'pis the manometer reading (mmH2O), t is the

ambient temperature (°C) and P is the ambient pressure (Pa).

V = 3.6_×7.5034_×('p(273 +t)) (P_×0.997)

0.5

(5.2) The manometer has an accuracy of ±0.5mm, the temperature an accuracy of 0.1°C, and the pressure an accuracy of 0.005Pa. Therefore the accuracy of the wind speed is 0.06kph (0.13% at 45kph) for an average temperature and pressure of 20°C and 1000mbar respectively, and manometer reading of 8mmH2O (typical

manometer reading for wind tunnel tests at the University of Canterbury) % δV V = -( δp p )2 +*δt_t+2+*δP_P +2 & . The air density is calculated byρ=_RTP , where P is the ambient pressure (Pa), R is the universal gas constant

(Jkg−1_K−1_{)and T is the ambient temperature (K). The accuracy of the temperature and pressure is 0.1°C}

(1.365K) and 0.005Pa respectively, giving an air density accuracy of 0.004kgm−3 _{(0.4%) at a temperature of}

20°C, pressure of 1000Pa, and air density of 1.188kgm−3

! δρ ρ = ,*_δT T +2 +*δP P +2$ .

The random error associated with the calculation of the frontal area was determined by using the digitizing method to calculate the frontal area for the same athlete in the same position three times. The maximum difference in frontal area calculation was 0.008m2_{(2.5%). There is also a systematic error with the calculation}

of the frontal area using the digitizing method due to perpective; the image is calibrated in the middle of the depth of field, typically at the hips or ribs of the rider, but for some riders their outline is nearer or farther from the camera. However, this systematic error is not significant for calculations or comparisons that use the drag area, CdA, calculated from drag measurements, rather than just the frontal area.

The uncertainty of the force balance, wind speed, air density, and frontal area then propagate to the calcula- tion of drag coefficient, as Cd= 1 F

2ρAV2,where F is the measured drag force (N),ρis the air density (kgm

−3

), A is the frontal area (m2_{) and V is the wind speed (ms}−1_{). The total error for the drag coefficient can}

therefore be calculated by:

δCd

Cd = -

*

δCd

Cd = ,

(2_×0.0013)2+ (0.025)2+ (0.004)2+ (0.025)2= 2.6%

It should be noted that this analysis does not include the systematic error in the determination of the frontal area, the variation of wind speed over the tunnel, or the blockage correction. These factors are not important if only trends are to be sought.