The forces generated by the pedaling action are, obviously, the most rigorous that the bicycle frame and components have to handle. Knowing how these forces change with the time is really very important to design an optimal component, specially for fatigue considerations. As many of you know, the pedaling motion and the forces induced by it aren't continuous, the dead spot position and chainring shape affect in some degree the force exerted to the pedals. Knowing that the expression of this function could be slightly affected by some biomechanical and equipment limitations, here I will take a more general approach to this topic.
First, let's take a look at the pedaling system from a mechanical standpoint, as we can see, it has 5 different elements: hip, thigh, foot, crankarm and the frame (assumed fixed). Knowing that there are also 5 joints and apliying Grubler's formula, we conclude that the pedaling system has 2 DOF. Here we find the first problem as is complicated to replicate the muscular system of a leg to know the relation between these 2 independent movements (Andy Ruina have published a paper about this e. g.) so let's try to find experimental relations between the movement of different elements. As many will agree, the simplest movement of the whole system is done by the crankarm, it's a simple rotation around the BB and can be easily caracterized only with its angular velocity and acceleration as there isn't any change in direction during the pedaling cycle. Ok, now we've got 1 DOF covered, how to define the other? Let's relationate the movement of the crankarm with any of the other 3 non-fixed elements.
Personally, I have chosen foot's movement because it's the closest element to the crankarm. To find the relation between these two movements, I needed a side view of a cyclist pedaling. After looking for a video like this for a few days, I finally found a nice video to do this study, HD quality and even recorded at 60 fps. That's what I needed. The video tries to show the diferences between the most effective ankling type of pedaling and the standard one. Obviously, I choose the ankling fragment to determine the relation.
It's important to note that the ankling way of pedaling is only feasible up to a determinated cadences. After measuring both angles in 17 different options, I made this table (angles measured counterclockwise):
The next step is trying to fit the data, for this I first used the program CurveFitter to view the dispersion graphic and try different models. The cosinus/sinus expression adjust was very good so I tried a custom expression (y=a*sin(x+b)+c) but despite adjusting the coefficents manually until the fit was nearly perfect, the final fit given by this program wasn't good (R=0.37). Then I decided to use Mathematica's BestFit function but it gave me an unspecified error. As the last option, I tried the Curve Fitter toolbox of MATLAB and this worked very well. I tried both y=a*sin(x+b)+c and y=a*sin(bx+c)+d but I chose the former as the difference of the parameter R was negligible. Here it's the fit that I obtained:
As you can see, the fitting is very good despite the rudimentary way of determining the angles. Now the 2 DOF pedaling system is reduced to a 1 DOF linkage where the only parameter is the position, angular velocity and angular acceleration of the crankarm, very easy to determine. I'd also like to point out that while searching papers about this topic, I have noticed that the sinusoidal approximation of this relation had been already proposed by Redfield and Hull
That's all for today. Thanks for reading!