Calculating power and speed on a bike
In order to move the cyclist, we want to take into account the effects of the wind, the slope and the athlete's health. We also want to be able to measure the physiological effort made by the athlete. To do this, we need to calculate the forces that have to be overcome to move. There are at least three forces to take into account: the force of the air, the force of the wind and the force of friction. So we're going to talk about speed and power. Speed refers to displacement in time and power to energy expended in time.
We started this work with realistic calculations. The websites of Jacques Jacques Fine and Bernard Mischler were a great help in developing the calculations. The general principle is to be able to switch from a speed (km/h) to a power (W) and vice versa. So, for a given power and depending on the conditions, we can find out how fast a cyclist is travelling. We have called these conditions "performance factors".
var performance_factors_values = {
health: 0, // float
athlete_position: 0.4, // float
athlete_weight: 75, // kg
bike_condition: 0.01, // float
bike_weight: 10, // kg
road_condition: 0.004, // float
slope_gradient: 0, // %
g_force: 9.81, // m/s2
real_wind_speed: 0, // km/h
air_temperature: 20, // °C
air_pressure: 101325, // Pa
air_molar_mass: 0.028965, // kg/mol
gas_constant: 8.3144621, // J·K-1·mol-1
}
The object above lists a set of performance factors and values that are useful for starting the calculations. For example, we've chosen a bike weighing 10kg, a wind speed of 0km/h and a temperature of 20°C. These performance factors are variable: the friction of the road, the condition of the bike and the position of the athlete can all change and therefore affect performance differently. Even the force of gravity can vary if we choose to send our cyclist to the Moon...
So we've created a set of JavaScript functions that take this object as an argument. Here's the function that calculates the influence of the slope:
function calc_slope_power(values) {
var v = values;
var slope_power = v.speed * (v.athlete_weight + v.bike_weight) * v.g_force * v.slope_gradient / 100;
return slope_power;
}
By adding a series of functions that calculate power, we then obtain the power required to move the cyclist. Calculating power as a function of speed is easier than calculating speed as a function of power. The latter requires us to solve a third-degree polynomial, which we have done by adopting a numerical solution method.
Once all these calculations were in place, we developed a tool to visualise the results. It takes the form of two graphs showing speeds as a function of power output. The first graph shows the speed in relation to the slope and the second the speed in relation to the wind.
We have also added inputs so that we can vary the performance factors. For example, we can see the effect of the athlete's weight on the speed achieved for a given power output.
We then added an input to manage inertia. The previous speed is taken into account to improve or reduce the new speed. If the cyclist was stationary, he will have a penalty when he tries to move. On the other hand, if their previous speed was high, they will have a bonus. We have also taken into account the inertia involved in a slope: it's better to arrive on a hill in a hurry.
We think this approach is an interesting way of estimating cyclist effort. It will be possible to calculate training loads and deduce risks of injury, fatigue or physical fitness.