### B 4. Rolling resistance

Figure 1 From pedaling power to cycling speed pedaling power (Ppe), intrinsic resistance (Rb), rolling resistance (Rr),
air resistance (Rd), slope resistance (Rsl) and velocity (v)

The rolling resistance (Rr) in cycling depends upon the road surface (asphalt, cobblestones, sand), the mass (m) of bicycle plus cyclist, gravity (g), the rolling resistance coefficient (Cr) and the speed (v). The pedaling power needed to overcome the rolling resistance is: Pr = m × g × Cr × v (1).
The mass (m) is measured in kg, the gravity g = 9.81 m/s², the speed (v) is expressed in m/s.

Mass and rolling resistance coefficient (Cr)
The mass is the ready-to-ride bicycle weight (including a filled water bottle, spare tires, pump and bicycle computer) plus the rider’s ready-to-start weight (with helmet, shoes, contents of his pockets). The regular and recumbent racing bicycle that we are comparing both have a ready-to-ride weight of 10 kg. The cyclist’s ready-to-start weight is 75 kg. So the mass is 85 kg.
The size of the wheels is significant to the rolling resistance coefficient: larger wheels run lighter (3). In addition, the thickness and the profile of the tires play a role, as well as the tire pressure: because the greater the distortion under strain, the higher the Cr-value (2). Which is why in road races the tires (or tubes) are as light as possible, having little or no profile and being inflated to their maximum. Punctured tires are annoying for every cyclist. And touring cyclists don’t have the assistance of a support car with spare wheels. An anti-puncture layer (made of kevlar) renders the tire virtually non-pierceable for nails, glass splinters and the like. But the extra layer somewhat increases the Cr-value of the tire (and therefore the rolling resistance) (1). Our regular racing bicycle and the high racer have the same 28 inch wheels and the same tires. As a result, the Cr-value of both bicycles is equal (an estimated 0.006).

Regular racing bicycle
A 75-kg male is riding a 10-kg regular racing bicycle on a level and smooth asphalt road with his hands in the curves of the handlebars. The weather is calm and his speed is 30.0 km/hour (8.333 m/s). In this situation, the pedaling power needed to overcome the rolling resistance is: Pr = m × g × Cr × v
Pr = (75 + 10) × 9.81 × 0.006 × 8.333 = 41.7 watt
Moreover, 128.1 watt of pedaling power is required to overcome the air resistance (next chapter). So Pd = 128.1 watt. As a result
Pr + Pd = 169.8 watt.
The intrinsic resistance of the racing bicycle accounts for a loss of 5% of the pedaling power (previous chapter). This brings the total pedaling power to:
Ppe = 169.8/0.95 = 178.7 watt
Pb = 0.05 × 178.7 = 8.9 watt (table 1).

Table 1 Pedaling power at 30 km/hour on a level road
for the regular racing bicycle and the recumbent high racer

 racing bicycle vkm/h Pbwatt Prwatt Pdwatt Ppewatt regular 30 8.9 41.7 128.1 178.7 recumbent 30 8.0 41.7 64.0 113.7 difference (%) -50.0 -36.3

Ppe = Pb + Pr + Pd

Recumbent high racer
The recumbent high racer has the same rolling resistance as the racing bicycle. And so, with a speed of 30.0 km/h on the same level road, it applies again:
Pr = (75 + 10) × 9.81 × 0.006 × 8.333 = 41.7 watt
Overcoming air resistance requires Pd = 64.0 watt (next chapter). So Pr + Pd = 105.7 watt.
The intrinsic resistance of the high racer costs 7% of the pedaling power. That total power is therefore:
Ppe = 105.7/0.93 = 113.7 watt
Pb = 0.07 × 113.7 = 8.0 watt
So the high racer requires 36.4% less pedaling power on a level road compared to the regular racing bicycle (113.7/178.6) when cycling 30 km/hour (table 1). Figure 2 Recumbent high racer with 28-inch wheels
(photo M5 Recumbent Bicycles)

Larger wheels run lighter
The recumbent high racer (figure 2) has two 28 inch wheels. Another type of recumbent bicycle is the low racer (figure 3). It also weighs 10 kg (ready-to-ride), but it has one 26-inch and one 20-inch wheel. Consequently, the Cr-value (rolling resistance coefficient) of the low racer is estimated to be 0.007, slightly higher than the Cr-value of the high racer (0.006). The tire profile, tire thickness and tire pressure are the same for both recumbent bicycles.
What is the effect of the different wheel size on speed?

Recumbent low racer
On the low racer, with the same cyclist and riding with the same 30 km/h (8.333 m/s), overcoming the rolling resistance needs:
Pr = (75 + 10) × 9.81 × 0.007 × 8.333 = 48.6 watt
To overcome the air resistance requires 64.0 watt. Which is exactly the same as on the high racer. Thus Pr + Pd = 112.6 watt.
Both recumbent bicycles have the same long chain, so that the intrinsic resistance of the low racer also requires 7% of the pedaling power. Which makes the total pedaling power
Ppe = 112.6/0.93 = 121.1 watt
Pb = 0.07 × 121.1 = 8.5 watt
To maintain an average speed of 30 km/h on the recumbent low racer 6.5% more pedaling power is needed than on the high racer (121.1/113.7). Because the rolling resistance is 16.5% greater on the low racer (48.6/41.7) (table 2). So larger wheels run lighter.

Table 2 Pedaling power at 30 km/hour on a level road
for the recumbent high racer and the recumbent low racer

 racing bicycle vkm/h Pbwatt Prwatt Pdwatt Ppewatt rec. high racer 30.0 8.0 41.7 64.0 113.7 rec. low racer 30.0 8.5 48.6 64.0 121.1 difference (%) +16.5 +6.5

Ppe = Pb + Pr + Pd Figure 3 Recumbent low racer with a 26 and a 20-inch wheel
(photo Optima)

Conclusions
1. The rolling resistance depends upon the road surface, the weight of the bicycle plus the rider, rolling resistance coefficient and speed.
2. The rolling resistance coefficient is determined by the size of the wheels and by the type and pressure of the tires.
3. Larger wheels run lighter.

Sources
1. Wiel van den Broek: Technische artikelen over de fiets: Vermogen en krachten. juni 2013
2.Jan Heynen: Snelheid en weerstand
3.Wikipedia: Rolweerstand