Home  Analysis of Lean in Flying 200m 

Including lean in the analysis of a general model of a rider on a surface leads to several important conclusions:
 The faster the rider, the shorter the distance. As a rider leans in the turns, the rider's center of mass travels a shorter path than the rider's wheel.
 The taller a rider's center of mass, the shorter the distance. The length of the path of a rider's center of mass in a turn depends on the height of the rider's center of mass. The greater the height, the shorter the path in the turn.
 The shorter distance contributes significantly to faster times. It's not just that powerful riders go faster, they also travel a shorter distance through the turns.
 Times depend on the configuration of the velodrome. Velodromes with longer straights will have shorter distances in the turns (for a given length). This gives different radii and different speeds for nominally the same path.
 Speeds in the turns are faster because potential energy is converted to speed as the rider leans. This is offset by increased drag from the faster speed. The rider looses this speed when potential energy increases in the straights.


One sets up the model by first defining a velodrome. The velodrome can be any of several predefined velodromes or some custom configuration. Next one defines a path for study. This is the path on the surface of the velodrome which will be ridden by the rider. Next define the parameters that describe a rider such as weight, drag, and power. The model computes a time for the Flying 200m by writing the differential equations of motion for the rider and solving them based on the chosen path and velodrome configuration. 
A rendering of the Dunc Gray velodrome, the one chosen for analysis.
(The velodrome that will be used for the 2000 Olympics.)
 Length of straight = 20,
 Radius of turn = 85/Pi,
 Track Width = 7,
 Distance from center of straight to start line = 10,
 Slope at center of straight = 12.5,
 Slope in turn = 42,



Here is a path of a rider (path of front wheel) on the above velodrome. The rider jumps hard from an initial speed of 13 m/s for 130 m along the fence staying within a third of a meter of the fence, starts diving at the start of the turn, crosses the start line at the measurement line, and rides the measurement line to the finish. This is an arbitrary path, not necessarily an optimal one. 


The model used a frontal area of 0.6 m^2, Coefficient of Drag of 0.5, Air Density of 1.226 kg/m^2, a Coefficient of Rolling Resistance of 0.003, and weight of 100 kg for bike and rider. 
riderData={A>.6, Cw>.5, Rho>1.226, Crr>.003, Wkg>100} 

The path of the rider on the velodrome is the path of the rider's front wheel. The rider's center of mass leans into the turns by a distance that depends on the speed of a rider, the radius of curvature of a rider's path, and the height of the center of mass of he rider.
In this first case the model is run with the rider's center of mass at zero distance from the path of the rider's wheel (h = 0). This establishes a base case for comparison. The path of the wheel for the final 200 is set to be on the measurement line.
We see that the length of the path of the wheel and the length of the path of the center of mass is the same. The distance covered in the final 200 m as calculated by the model is very close to 200 meters as it should be. The time for the Flying 200 is estimated to be 14.455 s. 
Length of Path relative to Measurement Line 
370.00 
m 
Length of Path of wheel 
398.35 
m 
Length of Path of center of mass 
398.35 
m 
Fly 200 time 
14.455 
s 
Total time to cover path of center of mass 
30.515 
s 
Distance cover by center of mass in final 200m 
199.99 
m 
Average speed for center of mass in final 200m 
13.84 
m/s 
Average speed for nominal measurement line distance in final 200m 
13.84 
m/s 
Speed at 200mtogo line 
15.17 
m/s 
Total work from start of effort 
9155. 
N m 
Total work for Fly 200 
4818. 
N m 


This is a path with lean. We see that the time for the Flying 200 is less. The distance covered by the center of mass is less by about 3 meters. The shorter distance covered by the center of mass accounts for 0.3 s faster time. 
Length of Path relative to Measurement Line 
370.00 
m 
Length of Path of wheel 
398.35 
m 
Length of Path of center of mass 
392.67 
m 
Fly 200 time 
14.166 
s 
Total time to cover path of center of mass 
30.066 
s 
Distance cover by center of mass in final 200m 
196.70 
m 
Average speed for center of mass in final 200m 
13.89 
m/s 
Average speed for nominal measurement line distance in final 200m 
14.12 
m/s 
Speed at 200mtogo line 
14.78 
m/s 
Total work from start of effort 
9020. 
N m 
Total work for Fly 200 
4770. 
N m 

This shows a plot of the center of mass of the rider. Notice that the path of the center of mass is inside of the measurement line where it should be. 

Plot of lean angle from vertical (degrees) verses position relative to measurement line. (Lean is zero in straights.) The sloping section corresponds to the dive down the bank where the rider's radius of curvature, speed and lean angle is constantly changing. (The zero point for the plot is at the center of a straight.) 


Let's try a case where the rider goes faster. Make the power 450 w instead of 300 w. We notice that the length of the path of the center of mass is shorter. As riders go faster, their distance decreases. 
Length of Path relative to Measurement Line 
370.00 
m 
Length of Path of wheel 
398.35 
m 
Length of Path of center of mass 
392.19 
m 
Fly 200 time 
13.049 
s 
Total time to cover path of center of mass 
28.150 
s 
Distance cover by center of mass in final 200m 
196.42 
m 
Average speed for center of mass in final 200m 
15.05 
m/s 
Average speed for nominal measurement line distance in final 200m 
15.33 
m/s 
Speed at 200mtogo line 
15.66 
m/s 
Total work from start of effort 
12668. 
N m 
Total work for Fly 200 
6796. 
N m 


We measure the time for another rider the same in all respects to our previous rider except the new rider has longer legs and sits higher by 0.2 m. The new tall rider goes a shorter distance and at a faster speed. Of course sitting higher could have an adverse effect on wind resistance. This rider has a faster time despite having a lower speed at the start of the final 200m. 
Length of Path relative to Measurement Line 
370.00 
m 
Length of Path of wheel 
398.35 
m 
Length of Path of center of mass 
391.35 
m 
Fly 200 time 
13.012 
s 
Total time to cover path of center of mass 
28.090 
s 
Distance cover by center of mass in final 200m 
195.94 
m 
Average speed for center of mass in final 200m 
15.06 
m/s 
Average speed for nominal measurement line distance in final 200m 
15.37 
m/s 
Speed at 200mtogo line 
15.60 
m/s 
Total work from start of effort 
12640. 
N m 
Total work for Fly 200 
6785. 
N m 


What if the shape of the velodrome is different? Let's try a different velodrome, one with longer straights and tighter turns, but still 250 m.
 Length of straight = 31.084,
 Radius of turn = 20,
 Track Width = 7,
 Distance from center of straight to start line = 20,
 Slope at center of straight = 28,
 Slope in turn = 42,


Keep the path about nominally the same. 

Going back to the original rider at 300w but using a new velodrome, one with longer straights and tighter turns gives a slower time for the Flying 200m by about 0.2 s. This rider leans over more in the turns because of the shorter radius. This gives a shorter path of the center of mass in the final 200 m. 
Length of Path relative to Measurement Line 
370.00 
m 
Length of Path of wheel 
398.70 
m 
Length of Path of center of mass 
391.94 
m 
Fly 200 time 
13.951 
s 
Total time to cover path of center of mass 
29.328 
s 
Distance cover by center of mass in final 200m 
196.46 
m 
Average speed for center of mass in final 200m 
14.08 
m/s 
Average speed for nominal measurement line distance in final 200m 
14.34 
m/s 
Speed at 200mtogo line 
14.92 
m/s 
Total work from start of effort 
8799. 
N m 
Total work for Fly 200 
4613. 
N m 

Plot the lean for this case. (The zero point for the plot is at the center of a straight.) 

Here are the lean angles for the two velodrome configurations plotted on the same plot. From the plots of lean angle, one can see that the leans are significantly different. (The zero point for the plot is at the center of a straight.) 



Copyright © 2000 Tom Compton All rights reserved. 