### 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 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, for example, water bottle, spare tires, pump and bicycle computer) plus the rider’s ready-to-start weight (with helmet, shoes and 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: large 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 professional cyclists use tubes: narrow tires with no profile, that have been inflated to their maximum. But, on the other hand, that means frequent flat tires, which is inconvenient for non-professionals who cannot dispose of a following car with spare wheels and bicycles. 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 wheels and tires. As a result, the Cr-value of both bicycles is an estimated 0.006

Calculations
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 (table 1). Moreover, 128.1 watt of pedaling power is required to overcome the air resistance (next chapter). So Pd = 128.1 watt (table 1). 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 (Pb) (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

 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

The recumbent high racer has the same rolling resistance as the racing bicycle. And so 41.7 watt is required to overcome this resistance at a speed of 30.00 km/hour on the same road (table 1). 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 (previous chapter). That total power is therefore: Ppe = 105.7/0.93 = 113.7 watt and Pb = 0.07 × 113.7 = 8.0 watt (table 1).

Large wheels run lighter
The high racer (figure 2) is one of the few recumbent bicycles with two large 28-inch wheels, the same as on the regular racing bicycle. Figure 2 Recumbent high racer with 28-inch wheels (photo M5 Ligfietsen) Figure 3 Recumbent low racer with a 26 and a 20-inch wheel (Optima baron x-low)

Another type of recumbent bicycle is the low racer (figure 3). It also weighs 10 kg, 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 that of the high racer (0.006). The tire profile, tire thickness and tire pressure are the same for both recumbent bicycles. What then is the effect of the different wheel size on the speed?

On the high racer, overcoming the rolling resistance on a level road at a speed of 30.00 km/hour (8.333 m/s) requires 41.7 watt. The total pedaling power is 113.7 watt. Pb = 0.07 × 113.7 = 8.0 watt (table 2).

Table 2 Difference in pedaling power between the recumbent high and low racer
at 30 km/hour on a level road

 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

On the low racer, overcoming the rolling resistance needs: Pr = (75 + 10) × 9.81 × 0.007 × 8.333 = 48.6 watt. This is 16.5% more than on the high racer (48.6/41.7), due to the smaller size of the wheels. 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. The extra guidance of the long chain is the same for both recumbent bicycles. So 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. That is 6.5% more than on the high racer (121.1/113.7). Pb = 0.07 × 121.1 = 8.5 watt (table 2).
Thus large wheels run lighter.

Conclusions
1. The rolling resistance depends upon the road surface, the weight of bicycle plus 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. Large wheels run lighter.

Sources
1. Wiel van den Broek (2017): Technische artikelen over de fiets: Vermogen en krachten
2.Jan Heynen (2017): Snelheid en weerstand
3.Wikipedia.nl (2017): Rolweerstand The hardcover edition can be ordered here price € 29.95