Error Estimate

Introduction	next	previous

Drag Measurement on HPVs

Error Estimate

There are two main sources for errors in the measurement:

slope of the path
wind

If we want to calculate the influence of these on our measurement results, we have to use a mathematical model for the drag force. A good model is

(see also the theory section). We will use the abbreviation

with

and

Constant Error Sources

For simplicity, we will begin with constant error sources: constant wind blowing from a constant direction and a constant slope in the path. The modified equation for the force is

where k is the tangens of the slope angle and w is the wind speed. There are two questions to answer from this equation:

What happens to the coefficients of a quadratic fit?
What happens when we average the force in both directions?

If we apply the quadratic fit explained in the theory section to the curve defined by the above formula, we obtain:

and

We can see, that the constant slope k has no influence on b_fit and thus on our measurement of C_d. Wind enters the equation linearly. This means, that if you run your tests in both directions, the effect of the wind averages out for b_fit. It is thus possible to obtain the correct value for C_d although there is a slope in the path and although there is wind.

In the expression for a_fit we find linear dependence on w and k, which average out in bidirectional tests. There is, however, a dependence on w², which does not average out. Consequently, wind leads to a systematic error in the measured rolling resistance.

Variable Error Sources

In this section we will investigate the effect of varying wind on the results. We can use statistical methods, because the speed of wind is not correlated to our experiments. An investigation of the effects of bumps in the path can not be done with statistical methods, because they appear always at the same location and are thus correlated to the experiment.

We can account for the variation in wind speed by having a separate value for each time the HPV's speed is measured: we use the set of values w(v) or w_i instead a single value w. Our equation for the force is now (slope effects are neglected this time):

or discretized for the points of measurement:

A similar calculation as for the case with constant error sources leads to

The fact that the wind speed is not correlated to our experiment (and averages to zero) can be formulated as

where the angle brackets denote an average over a set of coastdown tests. This leads to:

The average of the difference of w² values is clearly zero, which leads to the conclusion that the average of b_fit has no systematic errors:

Conclusion

The effects of a constant slope and any wind average out for b_fit and thus for the measured C_d, but wind introduces a systematic error for the measured rolling resistance C_r.

For practical measurements these statements must be modified. If a constant slope is too big, the HPV won't stop in one direction without braking and you cannot explore the same range of speeds as in the other direction. If you are only interested in aerodynamic drag, the wind effects should average out. But this does not say anything about the number of runs necessary to obtain reasonable accuracy. Because the errors in C_d are of statistical nature (non-systematic), statistical methods can be used to estimate the accuracy (which is done in the evaluation program). For practical purposes you will still need wind speeds below 1 or 2 m/s to achieve good accuracy.

Introduction next previous

Author: Christian Starkjohann <cs@hal.kph.tuwien.ac.at>