FS 69er Concept

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!

Pedaling Model III

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!

Pedaling Model II

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!

Pedaling Model I

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.

With this high-quality video, I can determine the relation between the crankarm angle and the foot angle to restrict the other DOF. I did this taking screen captures of the video at different crankarm angles and measuring with the Measure tool of GIMP both the crankarm and the foot angles for each position. Here it's an example:

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!

First

Let's see what we can gather