Theory of Operation

Introduction	next	previous

Drag Measurement on HPVs

Theory of Operation

As already mentioned in the introduction, we define the drag as the force that slows down the HPV. This force can be measured by measuring the decrease of speed. Because we want to concentrate on rolling resistance and aerodynamic drag, we should eliminate all other forces on the HPV:

pedalling force: the measurement is done during a coastdown (without pedalling!)
wind force: wait for a day (or night) without perceptible wind
slope in the path: find a path as flat as possible with a slope as low and constant as possible

It is not a problem if the slope of your path is not exactly zero, because the resultant force averages out if you do the measurements in both directions. The same is true for a constant wind within certain limits. Wind enters the equation for the rolling resistance quadratically and can thus be ignored if the speed is low enough (below about 1 or 2 m/s). The aerodynamic drag should not depend on a constant wind (again, provided that you measure in both directions).

Processing the Input Data

The Data Logger measures the time needed for each wheel rotation and stores the data. This is equivalent to a speed measurement at equidistant points in space (mathematically: the function v(x) is known for a set of equidistant points x). Starting at

with the total mass m and the time t, we obtain

The last equation allows us to calculate the force as a function of x. This can be converted into a v-dependence F(v), because x(v) is known by inverting the known function v(x).

In practice, the derivative of v(x) is a very small quantity and thus subject to noise. Most of the noise is due to bumps in the test path, which can be reduced by taking an average over several measurements starting at different locations (avoiding a systematic error). Furthermore, the differentiation can be carried out as a local curve fit with a straight line. If a region of points around the actual coordinate is used, the result smoothes out. Doing all this we arrive at a relatively smooth function F(v).

For an error estimate the statistical error of all measured quantities is calculated using the formula (q being the measured quantity):

where the angle brackets denote averaging and N is the number of data points in the statistics. Because slope and wind effects average out only if coastdowns in both directions are averaged, each contribution to the statistics is an average of two runs in opposite directions.

Calculating the Coefficients of Drag

Assuming turbulent flow, the drag force can be modelled by

with

Doing a curve fit of this equation to the measured data, the coefficients C_d and C_r can be obtained. A quadratic fit with minimum sum of square deviations is defined by

with the coefficients (all sums go over the complete data set of N points indexed with i)

and

The density of air can be found in tables, normalized for a certain pressure and temperature. The actual value can be calculated from this utilizing the fact that the density is proportional to pressure and inverse proportional to temperature. From my table book [5]:

Introduction next previous

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