Cycling Power
The physics behind cycling
To move forward with constant speed $V$ you have to provide energy (power) to overcome total resistive force:
$$ P = F_r \cdot V = ( F_{downhill} + F_{rolling} + F_{drag} ) \cdot V $$
Gravity
Cycling uphill or downhill force:
$$ F_{downhill} = m \cdot g \cdot sin(\theta) $$
where:
- $m$ - weight of cyclist and bike;
- $g = 9.80665~m/s^2$ - earth-surface gravitational acceleration.
Rolling resistance
$$ F_{rolling} = C_{rr} \cdot m \cdot g \cdot cos(\theta) $$
where:
- $C_{rr}$ - coefficient of rolling resistance.
The coefficient of rolling resistance of the air filled tires on dry road:
$$ C_{rr} = 0.005 + \frac 1 p \left( 0.01 + 0.0095 \left(\frac V {100}\right)^2 \right) $$
where:
- $p$ - the wheel pressure (Bar);
- $V$ - the velocity (km/h).
The angle $\theta$ can be calculated using elevation gain and total distance:
$$ tan(\theta) = \frac H L \Rightarrow \theta = arctan\left(\frac H L\right) $$
where:
- $H$ - height (opposite side);
- $L$ - length (adjacent side).
Aerodynamic Drag
Drag force:
$$ F_{drag} = \frac 1 2 \cdot \rho \cdot (V - V_w)^2 \cdot C_d \cdot A $$
where:
- $\rho$ - the density of the air;
- $V$ - the speed of the bike;
- $V_W$ - the speed of the wind;
- $A$ - the projected frontal area of the cyclist and bike;
- $C_d$ - the drag coefficient.
Approximated body surface area can be estimated from the measurement of the body height and body mass (Du Bois & Du Bois, 1916; Shuter & Aslani, 2000):
$$ A = 0.00949 \cdot (H/100)^{0.655} \cdot m^{0.441} $$
where:
- $H$ - the body height in $m$;
- $m$ - the body mass in $kg$.
Drag coefficient in cycling can be related to the body mass also and depends on cyclist position.
Density
The density of the air is its mass per unit volume:
$$\rho = \frac m V$$
where:
- $m$ - the mass;
- $V$ - the volume.
It decreases with increasing altitude and changes with variation in temperature or humidity.
The density of dry air:
$$ \rho = \frac {p_0 M} {R T_0} \left(1 - \frac {Lh}{T_0}\right)^{gM/RL-1} $$
where air specific constants:
- $p_0 = 101325~Pa$ - sea level standard pressure;
- $T_0 = 288.15~K$ - sea level standard temperature;
- $M = 0.0289654~kg/mol$ - molar mass of dry air;
- $R = 8.31447~J/(mol \cdot K)$ - ideal gas constant;
- $g = 9.80665~m/s^2$ - earth-surface gravitational acceleration;
- $L = 0.0065~K/m$ - temperature lapse rate.
Density close to the ground is:
$$ \rho_0 = \frac {p_0 M} {R T_0} $$
At sea level and at 15℃, air has $1.225~kg/m^3$.
Using exponential approximation:
$$ \rho = \rho_0 e^{(\frac {gM}{RL} - 1) \cdot ln(1 - \frac {Lh}{T_0})} \approx \rho_0 e^{-(\frac {gMh} {R T_0} - \frac {Lh} {T_0})} $$
Thus:
$$ \rho \approx \rho_0 e^{-h / H_n} $$
where:
$$ \frac 1 H_n = \frac {gM} {R T_0} - \frac L T_0 $$
So $H_n = 10.4~km$.
Coefficients Table
Rolling resistance coefficient:
Tire type | $C_{rr}$ |
---|---|
Bicycle | 0.006 |
Road bike | 0.004 |
Surface area and drag coefficient of cyclist:
Position | $A~m^2$ | $C_d$ | $C_d A$ |
---|---|---|---|
Back Up | 0.423 | 0.655 | 0.277 |
Back Horizontal | 0.370 | 0.638 | 0.236 |
Back Down 1 | 0.339 | 0.655 | 0.222 |
Back Down 2 | 0.334 | 0.641 | 0.214 |
Elbows | 0.381 | 0.677 | 0.258 |
Froome | 0.344 | 0.677 | 0.233 |
Top Tube 1 | 0.371 | 0.644 | 0.239 |
Top tube 2 | 0.355 | 0.611 | 0.217 |
Top Tube 3 | 0.345 | 0.588 | 0.203 |
Top Tube 4 | 0.333 | 0.604 | 0.201 |
Pantani | 0.343 | 0.618 | 0.212 |
TT* & TT Helmet | 0.370 | 0.641 | 0.237 |
TT Top Tube | 0.331 | 0.568 | 0.188 |
TT & Helmet | 0.374 | 0.679 | 0.254 |
Superman | 0.244 | 0.615 | 0.150 |
* Time trial
Conclusion
Based on all this formulas we are able to calculate power effort, burned calories and fat loss of bike ride activity.
Links
- http://bikecalculator.com
- http://thecraftycanvas.com/library/online-learning-tools/physics-homework-helpers/incline-force-calculator-problem-solver/
- https://www.gribble.org/cycling/power_v_speed.html
- https://www.researchgate.net/publication/51660070_Aerodynamic_drag_in_cycling_Methods_of_assessment
- https://www.sciencedirect.com/science/article/pii/S0167610518305762