Contents

The Streamlines CIRRUS measurement system uses a gated energy balance across a period to calculate a Drag Coefficient × Area, CdA, or a Rolling Resistance Coefficient, Crr.

Crr values can be found from rolling drum tests, but actual rolling resistance values on real-world roads tend to be higher due to the additional energy loss from vibration of the bicycle-rider system.

The basic process consists of two steps:

1. Establishing a baseline CdA value using an estimated Crr value.
2. Measuring Crr from changes in road surface or bicycle changes, using the fixed CdA value established in step 1.

A fundamental question that arises from the process is the effect of the initial Crr estimate upon the reference CdA value, and the effect on further Crr measurements, if they were derived from an initial Crr estimate.

This paper will first outline the equations behind the CdA and Crr measurements, and then show that the following practical conclusions can be made:

CdA measurement

• Crr estimation has a small effect on the absolute value of CdA measured.
• Crr estimation has a negligible effect on the comparative value of CdA measured.

Crr measurement

• When comparing rolling resistance on smooth roads, only comparative Crr values can be measured.
• When comparing rolling resistance on rough roads (at least 2× energy loss), errors from Crr estimation are reduced, meaning absolute Crr values are less influenced by any error in the original estimate.

Basic energy equations

Energy balance

Where:

• Rider Work = Rider Power × Time
• Kinetic Work = −½ × Mass × (Final Speed² − Initial Speed²)
• Gravitational Work = −1 × Mass × Gravity × (Final Elevation − Initial Elevation)
• Rolling Resistance Work = −1 × Crr × Mass × Gravity × Distance
• Aero Work = −1 × CdA × Alpha

and Alpha is dynamic pressure integrated over the distance:

Note: Sign convention is positive propulsive, negative retarding. Rider Work will always be positive, Rolling Resistance Work always negative, and Kinetic and Gravitational will vary.

CdA equation using a fixed Crr

Measurement is based upon the conservation of energy.

Crr equation using a fixed CdA

Crr and CdA measurement in practice

The equations can also be rearranged into directly measured and indirectly measured quantities.

As a note, the directly measured variables are kinematic in nature and can be measured accurately over a very short duration. The indirectly measured variables are more entropy-related and usually require a measurement technique via energy loss.

At this point in time, it is worth mentioning that there are several techniques for isolating the rolling resistance work and the aerodynamic work. The separation of these variables was even done as early as a study of railway car drag in the 1840s. A commonly known version uses a constant rolling force and an aerodynamic force exactly proportional to air density × airspeed².

Streamlines does not use any method based on a strict speed and force relationship, as both its theoretical calculations and validated measurements show the underlying assumptions are not accurate enough for precise measurement in cycling.

Virtual Elevation (Chung method)

Virtual Elevation (VE), usually called the Chung method after Robert Chung, who developed it in the early 2000s, measures CdA with only a power meter and speed measurement. The single degree of freedom method (fixing CdA or Crr) with a looped course can give surprisingly accurate results on a calm day. It is covered here because it is the best-known field method, and Streamlines differs from it on three points.

It uses the same fundamental equations for the energy balance, but includes the following assumptions which Streamlines does not:

1. CdA is a constant value across airspeed.
2. Crr is a constant number across (a wide range of) ground speed.
3. The Virtual Elevation residual provides sufficient diagnostic information that repeat runs are unnecessary.details

Due to the third point above, repeat runs are seldom reported in VE testing, so there is a shortage of data to assess a VE CV%. One study that did repeat its runs, Dyer and Disley (2018), found a CV of 2–3% on CdA for VE in a velodrome, with a fixed (estimated) Crr.

For comparison, Streamlines customers generally achieve a CV of 0.5–1.2% on CdA outdoors in wind, and some sessions are under 0.3%. The figures come from different riders, venues and protocols, so they are a guide and not a head-to-head test.

As proponents of the Virtual Elevation claim it can measure CdA and Crr simultaneously, as well as produce standalone absolute values for each, Streamlines has produced a document addressing these claims and why we don't use these methods: Effect of Variable CdA on Combined CdA and Crr Solutions in Field Testing.

Gated energy balance

Streamlines uses an energy balance between two gates on a relatively straight and smooth road to measure the total energy loss.

The rider will traverse a segment between 600 m and 1200 m.

From this the following can be calculated:

• The CdA for the particular conditions of a traverse
• The CdA for a given airspeed and yaw angle with optional normalisation for head and chest position

In addition, after an initial baseline is established, the configuration can be changed in ways which affect rolling resistance.

• The energy difference between two configurations can be stated in terms of a Crr difference or rolling watts at speed.

This baseline can be re-verified after each individual test or after a number of changes, to establish if any drift has occurred.

If the rolling resistance on a rougher road is to be determined, the smooth-road CdA and aerodynamic characteristics can then be applied on the rougher road.

• The increased rolling resistance between the two surfaces can be measured.
• The increase from rolling-resistance-related configuration changes, such as tyre pressure or tyre type, can be measured.

In this mode, all changes in energy dissipation are assigned to rolling resistance; however, if the results are displayed as a wattage difference at speed, with the speed similar to the test speed, the differences are negligible.

Relation between CdA and Crr

One immediate question that arises is: “If rolling resistance is estimated, how can it then be measured?”

The short answer is that Streamlines states measured differences, and not absolute values. It can state with a high degree of certainty that one tyre is 3 watts faster than another or has a Crr decrease of 0.0003.

A very reasonable follow-up question is: “Does the original Crr estimate affect the difference measured?”

A highly unequal ratio

Since both CdA and Crr are variable, how can either be measured? Luckily, at racing speeds and on regular flat roads, aerodynamic forces are 5–10 times greater than rolling resistive forces.

The values below are for an 80 kg rider with an 8 kg bicycle, holding a steady riding position throughout.

Table 1. Rider power, road and air speed, the coefficients, and the power consumed by aerodynamic drag and by rolling resistance.

In the case above, that is 222 watts vs 28 watts, about 8× greater.

The table below shows the variation in CdA with Crr for the same conditions. The CdA and Crr variables always result in 250 watts.

Table 2. Pairs of CdA and Crr that each give 250 W at a road and air speed of 10.66 m/s, with their errors against CdA 0.300 and Crr 0.0030.

What this means in practice is:

At 40 km/h, a 10% error in Crr measurement is equivalent to a 1.1% error in CdA measurement.

Effect of CdA comparison with estimated Crr

Since Crr needs to be estimated, this section looks at the effect of an error in that estimate on the measured CdA.

Case study of error in Crr estimate

The following examples use a 10% error in the Crr estimate.

Table 3. The 250 W case at a road and air speed of 10.66 m/s, and the coefficients with a 10% error in Crr.

The 10% error results in a 1.2% CdA error, with CdA changing from 0.300 to 0.296.

Table 4 shows the change in CdA across speeds for the same mis-estimation of Crr:

Table 4. CdA at five speeds with Crr fixed at 0.0033. The error column is the change from the CdA at 10.66 m/s.

Across the ±2 m/s (±7 km/h) range the change is about ±0.5%. In practice, this speed error is much smaller than the Reynolds effect and is indistinguishable from it.

To conclude, the Crr estimation is a minor source of error in absolute measures of CdA; in this case a 10% Crr error results in a 1.2% error in the absolute CdA value.

A 10% error in Crr makes a negligible difference to relative CdA values. Both measurements are shifted by the same amount (0.004 in this example), so the difference between them is unchanged. As the previous example shows, that shift changes little with speed: about ±0.5% of CdA across ±2 m/s.

Effect of Crr comparison from initial Crr estimate

This section continues with the previous example data and the 10% Crr error.

Example case: rolling resistance changes on paved roads – small wattage differences

The table below lists steps of two watts with a constant road and air speed, and the expected effect on Crr. One set uses the correct CdA; the other uses the CdA that came from the 10% Crr error.

These 2 watt changes would be a typical magnitude for tyre pressure or tyre compound changes on relatively smooth roads.

Table 5. Crr from 2 W steps in power at a road and air speed of 10.66 m/s, with CdA held fixed: the coefficients with the correct CdA, and the alternate (Alt.) coefficients with the CdA from the 10% Crr error. Err is the difference between the two Crr values.

The initial Crr error of 0.0003 (0.003 vs 0.0033), which went into the CdA calculation, re-emerges; however, the lines are parallel, with the same 0.0003 offset at every power, showing that relative differences will be maintained.

The offset is the same at every power because every test is at the same speed as the baseline, so the fixed CdA mis-states the aerodynamic power by the same amount each time (2.8 watts here). If a test is ridden at a different speed from the baseline, the offset changes: 2 km/h away moves it by about 0.00003, or 0.3 watts. If Crr is also changing with speed, this error is additional to that change, which is why Streamlines recommends comparing at similar road speeds.

This means that the relative difference can be relied upon, but the absolute value cannot, since it is highly dependent on the initial Crr estimate.

Example case: rolling resistance changes on varied surfaces – large wattage differences

The table below shows larger wattage differences. These would be typical for comparing paved to unpaved or very rough roads such as cobbles.

Table 6. Crr from 50 W steps in power with CdA held fixed, as Table 5.

Here the percentage error reduces dramatically, since the original 10%, 0.0003 error is retained at larger magnitudes.

This shows that the initial Crr estimate has less effect on the high rolling resistance values seen from rougher roads. In this case, the absolute Crr values can be quite accurate.

Conclusion

The examples put numbers on the four conclusions stated at the start:

• A 10% error in the Crr estimate is about a 1% error in absolute CdA: 1.2% in the example at 10.66 m/s, and 1.1% at 40 km/h.
• At a given speed, it leaves the difference between two CdA values unchanged, and across ±2 m/s it moves CdA by about ±0.5%.
• On smooth roads, a Crr measured with the baseline CdA carries the estimate's error unchanged, 0.0003 in the example, so differences between tyres are kept and absolute values are not.
• On a rough road, the same 0.0003 is 1 to 4% of the value in Table 6, so the absolute Crr is much less influenced by the estimate.