Cyclist slope physics general help

[keml]

I've volunteered for a project at my uni in which the objective is to develop a "green wave" system to sync traffic signals for cyclists. We have a specific location, but the idea is to build a general case that can be applied to any stretch of road.

The brief is super vague and we didn't manage to get any mechanical engineering students (we're mostly undergrad civils), and the traffic professor that was supposed to be sort of mentoring us is away on maternity leave, so we're feeling a bit out of our depth! But it's still early days, so have plenty of time to do research.

We have figured that it will probably be necessary to model cyclist velocity in relation to slopes. So that the green wave can be slowed down or sped up based on the slope, to match an "average" cyclists speed.

What I wanted to try to do as the first major milestone is to model a cyclist in 2D (in Matlab) in terms of distance and elevation (so no lateral movement/turning etc) for a given stretch of road. So basically produce a graph of one cyclists velocity based on given elevations for a road.

So I'm wondering if anyone could give me some advice about what kind of things we will need to consider in terms of modelling cyclist movement? I guess we want to keep things as simple as possible given that we don't have any expertise in this area and will have to learn things as we go along.

Perhaps what kind of data we need to collect? The point is to have the wave at some speed that most people can keep up with. So we were thinking we might need to collect some data about cyclists weight/power/stamina and things like that?

Cheers!

 

[jack action]

It's a simple $F=ma$ situation, where $F = F_t - \sum R$, i.e. the bicycle traction force minus the sum of the resistances.

The traction force is $F_t = \frac{P}{v}$, where $P$ is the rider's power and $v$ is the bicycle's velocity.

The possible resistances are:

• Aerodynamic drag: $\frac{1}{2}C_dAv^2$;
• Rolling resistance: $f_rmg\cos\alpha$;
• Grade: $mg\sin\alpha$;

Where:

$C_d$ = drag coefficient;
$A$ = frontal area;
$f_r$ = rolling friction coefficient;
$mg$ = bicycle + rider's weight;
$\alpha$ = the slope angle.

With this, you will get a cubic equation of $v$. Simplifying, you can assume $a$ = 0 (constant velocity), you can find the velocity of a rider, given its typical power input $P$, the typical drag $C_dA$ and rolling resistance $f_r$ of a bicycle for any given slope of the road.

Suggested reading:
Accelerating
Power limited
Hill Climbing
Speed, Distance and Time

 

[CWatters]

As it's almost the holiday is there an option to temporarily close the road and time a bunch of cyclists as they travel the route. Nothing like first hand data.

 

[berkeman]

As it's almost the holiday is there an option to temporarily close the road and time a bunch of cyclists as they travel the route. Nothing like first hand data.

loan out a bunch of FitBits...

https://www.fitbit.com/

 

[Rx7man]

It doesn't seem like a very complicated problem... especially considering accuracy isn't of great importance since you want to accommodate many riders, and they're all going to have various capabilities..

There will be an acceleration period.. probably about 10 seconds to a terminal speed, which is determined by the slope..
The terminal speed factor can probably be simplified for most typical grades to Vt = Vf * (1+(S/100))^(factor), where S is slope in %, and factor is an empirically determined number (I found a number between 4 and 8 to provide 'sane' results), Vt is the terminal velocity, and Vf is an empirically determined average speed on flat ground.
Using a factor of 5 and a flat ground speed of 5m/s you get (hope the formatting is legible)
slope speed
-15 = 10.1
-10 = 8.1
-5 = 6.4
0 = 5.0
5 = 3.9
10 = 3.0
15 = 2.2You can probably simplify the acceleration/deceleration portion to a distance (EG, it takes 50 meters to get up to speed, we'll call that Da), so you just deduct half of that from the total distance.
Then the equation for time becomes
T = (D-Da)/Vt
Substitute the formula for Vt and you have T = (D-Da)/(Vf * (1+(S/100))^(factor)).
Is it going to be super accurate? No, but if you're going to have Tour-de-France cyclists on the same path as people trying to stave off heart attacks you're going to have to make concessions one way or another... Perhaps the Tour-de-France cyclists get through it in the time requires, and the unfit people will take 2 light cycles!

 

[keml]

Awesome thanks guys!

I've created a matrix of the road gradients and the corresponding power/velocity based on this article:

https://www.researchgate.net/publication/223922575_Design_speeds_and_acceleration_characteristics_of_bicycle_traffic_for_use_in_planning_design_and_appraisal

Next thing is acceleration. Not sure about just adding distance. The way I've done it so far is calculating the time taken to travel each road segment, but the distance to get up to speed would change depending on the gradient and then if you're accelerating over multiple segments I think that could get a bit tricky?

I was thinking about starting the power at 0 (at rest), and then each second checking if the power is below the "mean power" for that gradient, and if it is increasing the power by some fraction of the mean power. Then I guess decreasing power to the "mean power" can be basically instant, because they're just stopping pedaling?

(these numbers just off the top of my head/not accurate) So for instance, switching from a power of say 100W on flat going at 25kph, to a -2% slope where you only need -50W to go at 25kph can happen instantly?

When a cyclist breaks, I guess that's applying negative power?

I'm assuming that riders will use their breaks when they're going downhill to stay at the mean speed.

Also I guess that braking can be much faster than accelerating, so I could decrease the power by much more each step.

Does that seem like a reasonable way to do it? Or any other ideas?

It would be great if we could close the road and get some uninterrupted data of the route, but I'm not too optimistic because it's in the city and quite a busy road. Our supervisor talked about getting some people to ride the route with pedal power meters fitted on their bikes, but I'm not sure if that will happen because they're pretty expensive. A couple of our group have just ridden the route a few times and gotten data from the Strava app, so we might continue with that. It's a