A quick update with some additional plots. The following graphics could also be very interesting in order to dimensionate the crank arms properly:
Saturday, October 22, 2011
It's been long time since my last post about this topic but finally I've got an update. The biomechanical model of the leg that I solved in the last post of the serie has 17 dynamic unknowns (14 joint reactions including pedal's ones and 3 joint moments) but I can only write 15 l.i. equations. This means that I have to input the evolution with the time of 2 parameters in order to solve all the dynamic unknowns for any position of the mechanism. The common practice in nearly all the papers that I have read is to measure experimentally both pedal reactions and then solve the system of equations. This is done in order to know the moment functions and determine how is muscular fatigue influenced by chainring shape, cadence... The problem is that I want to know the forces at the pedals, not the joint moments.
First, as I had got some pedal force profiles at low wattages (200W) and the parameters of the leg that generated them, I solved the inverse problem and I obtained all the reactions and joint moments. Next, I wanted to scale the joint moments to higher loads because 200W generated by a 70kg cyclist isn't enough to dimensionate correctly a bicycle frame but many questions arised: How do joint moments functions change when you increase the wattage? Can I use those moment functions to determine heavier cyclists' force profiles? Some difficult questions to answer because everybody hasn't got the same pedaling style, so I decided to take a different approach.
I decided to search pedal force profiles of the heaviest rider that I could found putting as much power as possible (something respresentative of what could be a very severe fatigue test) and I found them in this paper: "The effects of rider weight on rider-induced loads during common cycling situations" by C. Stone and M.L Hull. These force profiles were generated by a 778N cyclist riding at 7.2m/s at a cadence of 84rpm on a 6% grade treadmill, that's 480W in a real situation, a value high enough to test long-term fatigue of a bicycle. There was a small problem with this data because the forces were measured in the normal and tangential directions of the pedals and I hadn't the exact force values so I had to obtain them from the graphics. I used GIMP to obtain a dispersion graphic of the values and, after that, I fitted an 8 degree Fourier polynomial to the normal and tangential force profiles. Finally, using vector decomposition and the relation between crankarm angle and ankle-pedal axle angle that I obtained in the first post of the serie about the Pedaling Model, I generated the following graphics of the evolution of X and Y forces at the pedals for this cyclist (crankarm's angle measured clockwise from the vertical position)
From now on, I will use these pedal force profiles as representative of the average forces that a cyclist generates during a race, considering that Armstrong averaged 497W during the 39 minutes 2004 ITT to Alpe d'Huez this level of continuous frame load is very severe.
More soon. Thanks for reading!
at 12:30 AM