Experiments Prove Equation for Bicycle Power

Photo of the first spider-based cycling power meter.

Validation of a Mathematical Model for Road Cycling Power

This important study was carried out to test whether existing mathematical models can accurately predict power during road cycling. It showed that the model can predict power with an accuracy of about 3 percent (standard error of 1.6 percent)

A number of resistance forces cause power consumption while a bicycle is in motion. These include aerodynamic drag, rolling resistance, friction in bearings, transmission losses, changes in potential energy and acceleration. Each of these resistance forces can be calculated based on experimentally determined constants, the combined mass of the bike and rider, the velocity of the bicycle, and environmental measurements. The resistance force, multiplied by the velocity, gives the power loss due to each factor. Using such an approach it is possible to predict how much power a cyclist requires to achieve a given speed.

Prior to this study, no tests had every actually measured the power produced by a cyclist during normal road cycling, together with the speed achieved. Such a comparison enables verification of the power calculation equation. The main reason was that power meters were not available. This study utilized an early commercial power meter, the SRM Training System.

Validation of Power Meter

The SRM power meter was compared with a cycle ergometer. This uses a frictiion belt passed over a drum, which moves a pendulum, to measure torque when cycling. It can be calibrated by hanging weights from the belt. Combined with measurement of cadence (speed of pedal-crank rotation) the power can be accurately measured.


A similar cycle ergometer to the one used in this study

The power measured by the SRM was found to be 97.698 percent of the power delivered to the ergometer. It was assumed that the SRM power was valid and that the chain drive efficiency was 97.698 percent. This was justified on the basis that this was the expected efficiency based on previous studies.

Determining parameters for mathematical model

The mathematical model included terms for:

  • Aerodynamic resistance to forward motion
  • Aerodynamic resistance to wheel rotation
  • Rolling resistance
  • Frictional losses in wheel bearings
  • Changes in potential energy (slope resistance)
  • Changes in kinetic energy (acceleration forces)
  • Frictional loss in drive chain

Aerodynamic resistance to forward motion was calculated in the normal way, using the equation: FA = ½ ρ Cd A vA2 .
ρ is the air density and vA is the velocity of the air relative to the bike. The drag area (CDA) was determined by measuring the drag force, air velocity and air density in the Texas A&M wind tunnel. Subjects pedaled at approximately 90 RPM and an electric motor was used to rotate the front wheel. This was done at a range of yaw angles to enable the drag area in any cross-wind conditions to be determined.

The aerodynamic resistance to wheel rotation was determined by using the SRM crank to rotate the wheels with the bicycle suspended above the ground. This gave the combined power to rotate the wheel, the bearings, and the chain. Independent measurements were made for rotating the hub with no wheel to determine how much of this was due to the bearings and chain. Since the wheel rotation resistance was assumed to be due to aerodynamic resistance, the resistance force was modeled as: FWR = ½ ρ FW vA2. The actual form of this equation was in terms of power but for consistency with other calculations on this site, the force form is used here. The factor FW is used in place of CDA to represent the incremental drag area of the spokes or disk. The rear wheel was a lens-shaped disk and the front wheel had an airfoil section rim with 24 oval spokes.

Note: Although this general approach is valid, it is incorrect to use the air velocity for wheel rotation. The speed of wheel rotation is related to the ground velocity, not the air velocity.

Rolling resistance was calculated in the normal way, as a ratio of the tangential force to the normal force. It was assumed that the rolling resistance factor CR did not vary with velocity. No assumption was made for small angles of slope. The angle of slope was therefore found from the arctangent of the gradient. The normal force is then the cosine of this angle multiplied by the combined weight of bike and rider (mg). Tires of 20 mm width were used, inflated to 9 atmospheres (130 psi). Rolling resistance was not measured, average values from previous studies were used.

The frictional losses in wheel bearings were not assumed to be constant with velocity, citing Dahn. Mai, Poland, and Jenkins (1991).

Note: Confusingly, they state that bearing friction was shown to be dependent on both load and rotational speed, but then go on to present an expression for bearing torque which is only a function of the rotational velocity.

The torque in each bearing pair was given by: T = 0.015 + 0.00005N. where N is the rotational velocity in RPM. This gives the total resistance force as PWG = 91 + 8.7vG

Changes in potential energy were found by multiplying the slope resistance, calculated in the normal way, by the distance traveled to give the work done.

Changes in kinetic energy were also calculated using standard equations for constant acceleration. The difference in kinetic energy between the beginning and end of each trial was recorded.

Experimental Validation

Subjects rode along a straight 470 m length of concrete. The accelerated to a steady speed before entering the test section and decelerated after leaving it. Trails were carried out in both directions and at 4 different speeds varrying between 7 ma and 12 m/s (15 and 27 mph).

Powers measured by the SRM had a mean of 172.8 W with a range of 29.4 W. The powers predicted by the model had a mean of 172.0 W with a range of 30.4 W. The results were highly correlated with a standard error of 2.7 W. About 1.6%.

Abstract:

This investigation sought to determine if cycling power could be accurately modeled. A mathematical model of cycling power was derived, and values for each model parameter were determined. A bicycle-mounted power measurement system was validated by comparison with a laboratory ergometer. Power was measured during road cycling, and the measured values were compared with the values predicted by the model. The measured values for power were highly correlated (R2 = .97) with, and were not different than, the modeled values. The standard error between the modeled and measured power (2.7 W) was very small. The model was also used to estimate the effects of changes in several model parameters on cycling velocity. Over the range of parameter values evaluated, velocity varied linearly (R2 > .99). The results demonstrated that cycling power can be accurately predicted by a mathematical model.

Reference:

Validation of a mathematical model for road cycling power

Martin, James C. (Motor Control Laboratory, Department of Kinesiology, University of Texas at Austin); Milliken, Douglas L.; Cobb, John E.; McFadden, Kevin L.; Coggan, Andrew R.

In: Journal of Applied Biomechanics, v 14, n 3, p 276-291, Aug 1998

Read the full paper…

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