published in Radfahren 2/1990, pp. 44 - 46
Translated by Damon
Rinard from the original German
language article 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. The forces attacking the bicycle, those the bicycle rider welcomes - strong tail winds or a downhill - as well as ones that make him groan - for instance ascending for hours with heavily laden bicycle in sweltering heat - abstract formulas cover them all.
The force resiting the motion of the bicycle F_{total} consists of the sum of rolling friction F_{roll}, aerodynamic drag F_{wind}, the force needed to accelerate F_{accel}, the upward slope resistance F_{slope}, the bearing friction resistance and the drive efficiency h. Unlike the other quantities, acceleration and upward slope resistance can also be negative - usually a positive thing for the cyclist - thus propelling rather than retarding the bicycle. Naturally the drive losses h apply only if the bicycle rider really pedals the bicycle and does not just let it roll. Bearing friction, such as friction in the hubs, is usually added to rolling friction; accordingly the bearing friction forces in the drive train are added to the drive efficiency h, specifically, pedals, bottom bracket, freewheel and partially also the hubs (the additional forces on the hub due to chain tension). Thus one arrives at the formula:
F_{total} = (F_{roll} + F_{slope} + F_{accel} + F_{wind})/h | |||
where | h : | 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 » 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 [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 strictly applies only with no wind. With any wind the vector sum of wind due to motion of the bicycle plus true wind is to be taken instead of v; however c_{w} and A apply only to incident flow normal to the front. A description of air resistance which is a good aproximation of reality (and only quite the measurement with wind) is not entirely simple (1, 20).
The bicycle's drivetrain efficiency h 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.
To take a simpler view, assume no wind, no upward slope and no acceleration. Figure 1 shows the thrust as a function of velocity for a typical conventional racing bicycle and the effect of the individual retarding forces. It was calculated using:
h = 0.95; m = 80 kg; c_{r} = 0.003; r = 1.2 kg/m^{3}; c_{w}A = 0.39 m^{2}
At approximately 12 km/h rolling and air resistance have equivalent magnitude. At higher velocities air resistance dominates quite strongly.
Figure 2 shows the required power output as a function of velocity for the same bicycle. A normal 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, through the highly variable application of power in traffic conditions in combination with the environment. Pure points race riders can maintain about 500 W for one hour.
Figure 3 shows the power demand for different vehicles with a male rider.
In normal traffic a bicycle cannot proceed at a constant rate. Travel is stopped over and over, or obstructed by crossings and various other obstacles and disturbances. To that extent the acceleration work is not to be neglected, at least in city traffic. Kyle calculated the energy of the individual retarding forces for a trip on a touring bike at a constant 187 W - this corresponds to a speed of 32 km/h under standard conditions (14). It included stops 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 may cause no excessive increase in weight.
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