published in Radfahren 2/1990, pp. 44 - 46

Translated by Damon
Rinard and John Allen from the original German at:

http://www.lustaufzukunft.de/pivit/aero/formel.html.

(Numbers in parentheses refer to the pertinent bibliography)

Other articles by Rainer Pivit published in "Radfahren" magazine:

In science and technology it is usual to "illustrate" nature and technology using mathematical constructs. Abstract formulas describe all of the forces which affect the bicycle, those the bicycle rider welcomes - strong tail winds or a downslope - as well as ones that make him groan - for instance ascending for hours with a heavily laden bicycle in sweltering heat.

The force resisting the motion of the
bicycle *F _{total}* consists of the sum of rolling friction

F_{total}
= (F_{roll} + F_{slope}
+ F_{accel} + F_{wind})/η |
|||

where | η : |
drivetrain efficiency, dimensionless. | |

The individual retarding forces are described as follows: | |||

F_{roll}
= c_{r} m g |
where | c_{r} : |
coefficient of rolling resistance, dimensionless |

m : |
total mass of the vehicle with driver in kg | ||

g : |
acceleration due to gravity ยป
9.81 m/s^{2} |
||

Values for c_{r}
for typical bicycle tires and surfaces range between
0.0015 and 0.015. |
|||

F_{slope}
= s m g |
where | s : |
upward slope, dimensionless |

F_{accel}
= a m |
where | a : |
acceleration in m/s^{2} |

F_{wind}
= r c_{w} A v_{wind}^{2}
/2 |
where | r : |
density of air in kg/m^{3} |

c_{w} : |
coefficient of wind resistance, dimensionless | ||

A : |
frontal area in m^{2} |
||

v_{wind }: |
wind velocity in m/s | ||

The power required to overcome the total drag is: | |||

P = F_{total}
v |
where | v : |
velocity in m/s |

The formula for air resistance
applies strictly only if there is no wind. If there is wind, the vector sum of wind
due to motion of the bicycle and true wind is to be taken
instead of *v*; however *c _{w}* and

The bicycle's drivetrain efficiency *η* amounts
to about 96% max. A derailleur gear system reduces the efficiency
only slightly by an additional 1 to 2%. Internally-geared hubs
have efficiencies between 95% in direct drive and 80% in the
worst case. The efficiency of dirty and rusted bicycle chains is
not well-known.

In what follows here, the assumptions are made of no wind, no upward slope and no acceleration. Figure 1 shows the drag as a function of speed for a typical conventional racing bicycle, and the effect of the individual retarding forces. It was calculated using:

*η* = 0.95; *m* = 80 kg; *c _{r}*
= 0.003;

At approximately 12 km/h, rolling and air resistance are equal. At higher speeds, air resistance is strongly dominant.

Figure 2 shows the required power output as a function of speed for the same bicycle. A typical rider can maintain 80 W continuously on an ergometer. Observations in traffic however show that riders there often ride in a range up to 200 W. Probably this is attributable to better cooling by the wind, the highly variable application of power in traffic conditions, and stimulation by the surroundings. Top racers can maintain about 500 W for one hour.

Figure 3 shows the power needed to propel different bicycles with a male rider.

In normal riding, a bicycle cannot proceed at a constant speed. Travel is stopped or slowed over and over due to intersections and various other obstacles and disturbances. To that extent, acceleration work is not to be neglected, at least not in city traffic. Kyle calculated the energy to overcome the individual retarding forces for a trip on a touring bike assuming a constant power input of 187 W - this corresponds to a speed of 32 km/h under ordinary conditions (14). He assumed that there would be a stop every 400 m. Under these conditions 53% of the energy goes into air resistance, 11% into rolling friction and 36% into acceleration work. Thus aerodynamic improvements should not result in any excessive increase in weight.

Contact:

Rainer Pivit

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D-33602 Bielefeld

Germany

Tel.: 0521 / 201 80 81

Fax: 0521 / 201 80 66

© by Rainer Pivit, 08/98