# A Novel Approach of Modelling and Predicting Track Cycling
## 2 Materials and Methods
### Physical Model  
The physical model should be capable of specifying, at every instance, the counter-balance between power input by the cyclist and the main power loss factors in cycling, i.e., aerodynamic drag and tire friction. Since a well-trained athlete will keep the bike close to the 250 m measuring line in time-trial runs, changes in potential energy can be neglected. The effects of dynamic load that occur during cornering are considered in the model. Perhaps the most elaborate physical description of track cycling to date was published by [PERSON] et al. in [8]. These authors addressed all contributing factors, including bank angle of the track, cycling up and downhill and the increase in rolling resistance due to a steering angle. Judging from practical experience, we are convinced that these additional features require numerous additional coefficients to be established while delivering only a minor gain in information. A suitable physical description was published by [PERSON] et al. in [12], which we will adopt, but reformulate it in terms of force, not power, for convenience in implementation. Furthermore, their consideration of potential energy will be omitted. The system is comprised of a cyclist and bicycle that is considered as a single mass point (mass \(m\)), which moves along a straight trajectory with constant height level. In this situation, the propulsive force \(F_{P}\) acts collinear to the resistance forces \(F_{RR}\) for rolling resistance, and \(F_{D}\) for aerodynamic drag. The dynamic force balance for this single mass point system reads as:  
\[m\cdot a=F_{P}-F_{RR}-F_{D} \tag{6}\]  
where a surplus of available propulsive force over the force required to sustain the current speed results in an acceleration of the system, while a lack of force input leads to a decrease in velocity. The rolling resistance is modelled as [PERSON]'s friction:  
\[F_{RR}=m\cdot g\cdot\mu \tag{7}\]  
Ambient conditions, such as air temperature, humidity and barometric pressure, needs to be established to calculate the air density \(\varrho\) accordingly. With the drag area (\(c_{D}\cdot A\)) being known from, e.g., wind-tunnel measurements, the aerodynamic drag can be estimated by:  
\[F_{D}=\varrho/2\cdot v^{2}\cdot(c_{D}\cdot A) \tag{8}\]  
The propulsive force \(F_{P}\) can be deducted from \(F_{T}\) the tangential force at the pedals averaged per revolution, the crank arm length \(l\), the rear wheel radius \(r\) and the gear ratio \(i\) as:  
\[F_{P}=\frac{F_{T}\cdot l}{r\cdot i} \tag{9}\]  
Assuming that a non-slip condition between the rear tire and the track surface exists, the pedalling rate is connected to the velocity by:  
\[PR=\frac{v}{2\pi\cdot i\cdot r} \tag{10}\]  
Cycling in a velodrome is characterised by a periodic sequence of straights and curves, with lengths and corner radii \(R\) specific to the particular track. In the curves, the rider has to lean inwards to compensate for the centrifugal forces acting. The lean angle \(\varphi\) for cycling free of transverse forces is given by:  
\[\tan\varphi=\frac{v^{2}}{g\cdot R} \tag{11}\]  
and leaning inwards creates an additional dynamic load factor \(n\):  
\[n=\frac{1}{\cos\varphi} \tag{12}\]  
which increases the downward force on the wheels, resulting in the rolling resistance becoming a function of the velocity and the corner radius. Moreover, with the contact point of the wheels travelling at track radius \(R\), we assume in contrast to [12], that the centre of gravity moves with approximately constant, not reduced, tangential speed along a concentric trajectory smaller in radius by \(\Delta R\), depending on its height \(H\) above ground:  
\[\Delta R=\sin\varphi\cdot H \tag{13}\]  
The bicycle's velocity is usually captured by a frequency counter at the wheel or at the crank arm.