Friday, November 25, 2011
I have found an interesting article with some torque profiles of a track cyclist during a standing start measured at 200Hz using a SRM Torque Box. I have found this very interesting because nearly all of the studies about pedaling in cycling are conducted at constant cadence and power outpout because they want to study stationary state efficiency but this one shows how the torque is affected by a variable cadence and power. As you can see the shape of the torque profiles are pretty much the same independently of the cadence so the steady state pedaling model is a good aproximation for acceleration, even though, a cadence-torque plot could be useful to obtain an even better approximation. I will consider these torque profiles for a standing start load case, the severer that a bicycle frame should withstand but first I need to develop a model to calculate total pedaling forces from propulsive forces. Here you can see the article:
I don't know who is the rider that generated those torque profiles but his peak torque is about 450Nm. It's a really high value so I will scale later those plots depending on the load case. I have read that Sir Chris Hoy can generate up to 600Nm of peak torque during an standing start, something really impresive. You can see him making his chain suffer here:
That's all for today. Greetings
at 3:20 AM
Saturday, November 5, 2011
In this post, I want to take a critical approach to the method that I have utilized to obtain the force plots that I have shown in the previous posts. To do this, I will list the process that I have followed and the assumptions that I've done to obtain these plots:
- First, I obtained a mathematical relation between the foot angle and the crank arm angle based on a 60fps video of a random cyclist pedaling round at an aproximated cadence of 60rpm. Both the method used to calculate the angles and the fitting routine induced errors.
- Next, I used previously published data of tangential and normal forces at pedals of a cyclist pedaling at 84rpm in order to obtain a mathematical expression of these two components of the force.
- Following, I calculated the x and y components of the force at pedals using the mathematical expression obtained previously and the relation between the foot angle and the crank arm angle. Doing this, I assumed that: 1) the pedaling style of both cyclists is the same and that 2) in the 60-84rpm range, the relation between angles is constant. This two considerations are questionable and they will modify to some extent the final result.
- Finally, I used vectorial decomposition to obtain all the other plots.
In the two following plots you can see some small errors induced by this method:
The correlation between the plots that I have posted and previously published data of pedaling forces at lower wattages is good so I will consider my results as valid. If I don't found pedal force profiles for higher wattages I will scale these ones to replicate higher load cases. Considering the variety of different pedaling styles and some of the testing methods utilized in the industry, I think that this approach is solid. Obviously, putting a professional sprinter in a ergometer pedaling at different power oupouts and positions will be a better way to obtain these forces profiles but that isn't an option for me. Finally, a last graphic of the pedal force components that a crank arm must resist for the indicated power outpout (radial force sign defined by the theory of elasticity):
That's all for today. Greetings
at 1:22 AM
Thursday, October 27, 2011
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
Wednesday, October 5, 2011
Sunday, September 25, 2011
Just a small pause in the serie of posts dedicated to the pedaling motion to show this bike concept. I designed it during the year 2010 and it was finished by the end of August, before the Eurobike 2010 took place in Friedrichshafen where some of the "innovations" of this concept were presented (Hope's cassette with a 9t cog e.g.). The aim was to design a really light FS XC bike, as simple as possible, inspired by Cannondale's Zero Pivot system. As many of you know, a softail system isn't the best on from a suspension's performance standpoint but its simplicity allows very low frame weights. I really think that this system combined with a well studied pivot placement for a concrete drivetrain configuration and a platform shock could be the definitive weapon for XC racing. Some of the highlights of this concept are:
- Full carbon softail frame designed for 29er front wheels. BB30. 29er front wheel gives the main benefit of 29er bikes without many of their problems. 70.8º head tube angle and 73.8º seat tube angle. 1.125-1.75" tapered headtube. Vertical seatpost tube that allows shorter chainstays, a lower standover height and good mud clearance without the need of a ST-TT reinforcement. ISP. Syntace X12-142 axle system make the rear end stiffer and allow the chainstays to be wider in order to increase the lateral stiffness of the rear end. Shock mounted perpendicular to the rear axle to reduce the lateral loads that it has to handle. Integrated front end to reduce as much as possible the stack height. PostMount caliper fixing system.
- Stem with adjustable angle to allow any stack/reach value.
- Single sided 29er fork. 1.125"-1.75" tapered head tube. 480mm A-C length. PostMount caliper fixing system. 51.2mm fork offset to reduce trail as much as possible. Result: 73.8mm of trail, the same as many of the current 26" XC racing bikes.
- 32t chainring combined with a 9-34 cassette. Just a gear shifter.
- Tubular rear wheel to increase rear wheel traction, one of the weaknesses of the suspension system chosen.
And finally, some pics of the concept. Comments are welcomed
More soon. Thanks for reading!
at 6:13 PM
Thursday, September 22, 2011
This update arrives a bit late because I was having some major problems with the MATLAB code that I've created. As I have already commented, I solved the non-linear using the Newton-Raphson method with success but I got some big errors after doing some tests changing mechanism's movement and cinematical parameters. First, I saw some strange values in both the thigh and shank angles caused by the periodicity of the trigonometrical functions. That was simple to solve but, just after that, I noticed that the mechanism was doing some strange things during the ascending portion of the crankarm motion but applying some trigonometrical relations that was also easy to solve. The big problem of that program was some strange things that happened with the residuals of the restriction equations after applying my Newton-Raphson algorithm. The minimal residual that I can achieve was dependent of the crankarm position, in certain positions the residuals went down to nanometers but in the "problematic positions", I can't achieve residuals lower than 0.035m. That clearly wasn't acceptable so I decided to try another option.
The option chosen was to use the matricial method for solving both the velocity and acceleration problems and use MATLAB's function fsolve in order to solve the position problem. This function uses a modified Powell's method to solve non-linear systems of equations. It worked flawlessly right from the first test and I even can change the maximal desired residuals for maximal accuracy. As I had got the position problem solved, I adapted the rest of the program to the new algorithm and I did some tests. It worked perfectly. Here you can see the parameters of a first example (COG distances measured from the lower joint: crankarm from the BB, shank from the ankle, e.g.):
And some plots of the variation of the position and cinematical parameters during a crankarm rotation of some natural coordinates chosen randomly:
And finally, a small study about the evolution of the position of the COG during the pedaling motion.I've to note that I have done this for the whole propulsive mechanism (both legs). As you can see, both x and y coordinates of the COG have a sinusoidal relation with the time:
As I have already commented, I will do some dynamical studies of bicycles using the program JBike6. For that reason, I have added the average position of the COG of the mechanism during the pedaling motion. Knowing this and the COG and weight of the upper body, I can determine the position of the COG of the whole body. I will try to do this later because I have to determine the position of the COG and also its inertia through 3D modelling.
That's all for today. Thanks for reading!
at 9:31 PM
Wednesday, September 14, 2011
In order to solve the mechanism's movement, the next step is to define the restriction equations of the system. I chose the COG of each moving element and the angles of the thigh, shank and foot as natural coordinates, total 11 coordinates. These equations restrict 10 of them through geometrical relations and the last equation to completely restrict the mechanism was obtained in the previous post. The next step is to solve the matricial problem of the mechanism using the Newton-Raphson method. The MATLAB program created can handle any logical value of the dimensions of the elements and even a simulation with the crankarm angularly accelerated. Here you can see the parameters used for this first simulation:
And a small video of the leg movement defined by these parameters. As my 1.6Ghz laptop can't candle the MATLAB code fast enough, the crank velocity is slower than the 90rpm defined for the simulation and the movement isn't very smooth. As you can see, the ankling movement is noticeable:
Logically, all the MATLAB code hasn't been done just for a nice animation of a 5-bars system. After solving the mechanism, I've got matrices of the position, velocity and aceleration of the natural coordinates chosen every few miliseconds. These will allow me to know the inertial forces and moments acting on the mechanism and, hopefully, the forces in the pedals for a given power outpout. Moreover, knowing the weight of every element of the mechanism, I can track its COG and determine the average position during the pedaling motion of the COG of the whole mechanism that which is interesting for future dynamical studies that I will do using the program JBike6.
More soon. Thanks for reading!
at 3:33 PM
Friday, September 9, 2011
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!
at 10:58 PM