Mathematical models of cycling locomotion: crash course

Arguably, one of the most famous research work on the topic is the one presented by Martin and colleagues (⊕Martin et al. 1998Martin, James C, Douglas L Milliken, John E Cobb, Kevin L McFadden, and Andrew R Coggan. 1998. “Validation of a Mathematical Model for Road Cycling Power.” Journal of Applied Biomechanics 14 (3): 276–91.). A model can be formulated by means of the equations of motion, which describe the balance between the forces acting on the athlete. With due approximations and assumptions, the propulsive force delivered by the cyclist is balanced by three main resistive forces: aerodynamic resistance, gravity, and rolling resistance.

Aerodynamic resistance: the force that the athlete must provide against the air can be estimated as:

\[ F_D=\frac{1}{2}C_DA(v-v_w)^2\\ P_D=F_D\cdot v \]

Where \(\rho\) is the air density, \(C_D\) is the drag coefficient, \(A\) is the frontal area, \(v\) is the cycling locomotion speed and \(v_w\) is the wind speed. Some of these parameters depend on athlete’s position on the bike and athlete’s anthropometry. If these parameters are considered as constant numbers, it means that assumptions about their 3D-areo effects have been neglected. On the other hand, air density and wind speed are “environmental” variables, that might vary with race location, geography and weather conditions.

With the goal of reducing both the frontal area and the coefficient of drag, athletes use to pedal in an “areo-position.” However, areo-positions are usually not convenient from the biomechanical standpoint, as muscles need to operate outside the optimal contraction length ranges and speeds. Prolonged training sessions are required to get used to these extreme positions, where also vaso-restriction considerations must be considered. The athlete should therefore find the best trade-off between power savings and power delivery ability.

The rolling friction force \(F_{rr}\) is generated by the wheels on the concrete. This can be formulated as:

\[ F_{rr}=C_{rr}\cdot(m+m_B)\cdot g \cdot \cos(\beta) \]

Where \(m\) and \(m_B\) are the masses of the rider and the bike, and \(\beta\) is the slope of the course, g=9.81 \(m/s^2\), and \(C_{rr}\) is a rolling friction coefficient (~0.0035-0.004 on asphalt). What kind of difference can a super smooth racetrack road surface have, as opposed to an ‘average’ road surface? It can be estimated with modeling. It is usually assumed that the friction rolling resistance coefficient can decrease with smoother surfaces. This is not a conservative force, but the power required to overcome the rolling resistance goes linearly with the speed. E.g. if you are averaging 280 W and going at 44.52 kph with a Crr of 0.004, you go at 45.11 kph with Crr of 0.003.

The contribution of the gravity force \(F_{g}\) can be formulated as:

\[ F_{g}=(m+m_B)\cdot g \cdot \sin(\beta) \]

The only propulsive force is provided by the cyclist, and this can be measured with power meters, but then it must be reduced because of the drive-train efficiency (~97-98%).

One of the best readings on the topic is the paper by (⊕Jeukendrup and Martin 2001Jeukendrup, Asker E, and James Martin. 2001. “Improving Cycling Performance.” Sports Medicine 31 (7): 559–69.), which discuss how improvements in cycling performance can be obtained by tweaking many different aspects of aerodynamics, physiology, and nutrition. Highly recommended! You might also find the paper by (⊕Olds 2001Olds, Tim. 2001. “Modelling Human Locomotion.” Sports Medicine 31 (7): 497–509.) equally interesting and, if you want to go classic and classy, read the paper by (⊕Prampero et al. 1979Prampero, Pietro Enrico di, G Cortili, P Mognoni, and F Saibene. 1979. “Equation of Motion of a Cyclist.” Journal of Applied Physiology 47 (1): 201–6.).

Air density

Air density (often indicated with \(\rho\)) measures the mass of air per unit volume (e.g. \(kg/m^3\)). In a given location, air density depends on many factors, such as: temperature, relative humidity, and pressure. For dry air, the density at sea level 15 °C and 1013 hPa is approximately 1.23 \(kg/m^3\). With altitude, these factors can change dramatically, hence altitude is often regarded as the main player in air density determination. As a rule of thumb, you can expect a change of ~0.12 \(kg/m^3\) per ~1000 meters of altitude change. However, when location and altitude are pre-established, then time of the year and of the day play the major role in air density variations.

Computing air density from weather data is challenging, and requires some assumptions. We estimated air density with reasonable accuracy with the following methodology. First, we calculate the saturation vapor pressure (i.e., the vapor pressure at 100% relative humidity) at given temperature \(T\) using the formula:

\[ p_1 = 6.1078 \cdot 10^{7.5 \cdot t/T} \]

where t is temperature in degrees Celsius (C°) and T is temperature in degrees Kelvin (K). Second, we find the actual vapor pressure \(pv\), multiplying the saturation vapor pressure by the relative humidity (\(RH\)):

\[ pv = p_1 \cdot RH \]

To find the pressure of dry air (in Pascal), subtract the vapor pressure (in Pascal) from the total air pressure

\[ pd = p - pv \]

Finally, \(\rho\) is computed as:

\[ \rho = \frac{pd}{Rd \cdot T} + \frac{pv}{Rv \cdot T}\\ \] with \(Rd=287.058~J/(kg\cdot K)\) and \(Rv=461.495~J/(kg\cdot K)\) are the specific gas constant for dry air and water vapor.

Weather data from Lausitzring Weather Station

We all joke about statistics from time to time… “Statistician can have their heads in the oven and their feet in the freezer, and they will say that on average they feel fine.” However, there is no better way to try to estimate environmental race conditions, than relying on historical data and make predictions. With historical data, we can try to make predictions about future events, hoping that the future will not be much different than the past. For this particular case, weather data have been retrieved from the Lausitzring Weather Station and consist in detailed information about air temperature, humidity, pressure and wind direction and speed. Air density has been computed as detailed in the previous section.

Let’s go through all these weather variables from the past two years, comparing the months of May and June to have a sense of how much conditions can change.

What is the best starting time?

To answer to this question, we can do some simulations of the bike course. However, we need to use a more sophisticated set of modelling techniques here. A modified version of the 3D model detailed in (⊕Zignoli and Biral 2020Zignoli, Andrea, and Francesco Biral. 2020. “Prediction of Pacing and Cornering Strategies During Cycling Individual Time Trials with Optimal Control.” Sports Engineering 23 (1): 1–12.) was adopted. It is important to use a multi-dimensional model of cycling and take into consideration the relative angle of the wind with the riders. The bike course was manually tracked and created with Google Maps and Google Earth services. The course geography plays a key role in determining the orientation (i.e. the heading) and the slope of the course. The drag coefficient was adjusted with a virtual athlete’s anthropometric characteristics (e.g. h=1.83 m and m=70 kg). The drag coefficient \(C_D\) and the frontal area \(A\) were estimated as:

\[ A=0.0293 \cdot h^{0.725} \cdot m^{0.425} + 0.0604\\ C_D=4.45 \cdot m^{-0.45} \cdot 0.5 \] The drag contribution was adjusted for team time trials (⊕Blocken et al. 2018Blocken, Bert, Yasin Toparlar, Thijs van Druenen, and Thomas Andrianne. 2018. “Aerodynamic Drag in Cycling Team Time Trials.” Journal of Wind Engineering and Industrial Aerodynamics 182: 128–45.) assuming a number of 4-5 riders (the individual athlete should position himself as second-last rider to optimize for aero effects and benefit from a ~50% reduction in drag force).

Different starting times were simulated, from 5am to 6pm with 1-h interval. With optimization, the mathematical model was used to estimate the final time and, most importantly, the time benefit over the bike section of different weather conditions.

Results suggest that starting at 1 am might be the best option in terms of final performance time. However, improvements in the performance due to a reduction in air density in the afternoon, seem to be well mitigated by the action of the wind. However, it must be considered that the average wind speed was included in the calculations. We already discussed how wind can be detrimental to performance, so to avoid wind gusts (wind data is more variable in the afternoon) riders would better start biking in the morning.

This data confirm, once more, how difficult it might be to take into account all the possible variables to make conclusions about an athletic performance lasting for hours.