Thoughts about chassis stiffness and efficiency

As an introduction to the upcoming post about how I modelized frame compliance in the dynamical model, I would like to give a general overview about some possible links between frame compliance and efficiency.  

First of all, we shouldn't forget that the bike is controlled by a rider whose input could be modified by the mechanical properties of the frame and the components. For example, a rear end too stiff could cause excessive bouncing of the rider and have a negative effect on his power output. For this reason, we should always consider the negative effects that could have a particular design feature on the vibration in the contact points, intersegmental loads, joint torques and steering inputs and not treat the bike as an isolated system.

Leaving this influence on the rider input aside, I have identified some possible efficiency loss mechanisms due to the deformation of the chassis:

- Rolling resistance. Although the rolling resistance coefficient (Crr) has been always defined as a constant value, there is a relation between it and sinkage depth. The explanation for this is pretty simple: Rolling resistance moment is caused by the difference in normal pressure  between the leading and the trailing edge of the contact patch due to the hysteretic cycle of the material. If sinkage depth is increased, the tire deforms more, the severity of the hysteretic cycle is maximized and the losses increase. There is also second order effects like the radius of the contact point between the ground and the tire.

- Drivetrain misalignement. Both the torques perpendicular to the BB axle caused by the pedalling forces and the combination of asymmetric chain loads and symmetric rear ends produce misalignements between the BB axle and the rear wheel axle. Those misalignements could cause torsional loads on the chain and increase friction due to the contact between the side plates and the sprockets/chainrings.

- Sideslip and camber of the rear wheel. Once again, the combination of asymmetric chain loads (both in the longitudinal and horizontal planes) and symmetric rear ends produce two effects: 1) a misalignement between the bike speed and the speed of the contact point with respect to the bike and 2) a small camber angle. Dissipation increases due to the presence of a yawing moment that tends to align those two speeds and the effect of camber on rolling resistance.

- Wheel slip. Traction is a function of wheel pressure and frame/fork stiffness so an optimization of these parameters could minimize slip and, consequently, power losses.

- Losses in the frame. The harmonic excitation of the bike causes losses in the structure due to both hysteresis and viscoelaticity that can affect negatively the performance of the bike. Alternatively, a well tuned placement of materials with these characteristics can increase damping and, consequently, comfort.

As you can imagine, the analysis of such complex interactions would need a very complete system. A deformable 3D bike model controlled by a virtual rider capable of balancing the bike in a similar way an human would do would be needed. Additionally, both the effect of drivetrain misalignement and losses in the frame have to be quantified using experimental methods or complex FEA models.

In my case, I haven't gone so far. I have modelled just two of these mechanisms: rolling resistance and wheel slip. I think that these are the ones that could play a major role on efficiency.

That's all for today. Greetings

Dynamical model. Introduction

Those of you that follow this blog from some time ago know that the development of an accurate bike model was one of my main objectives. From the first model that I presented here (very similar to the one that is used by analyticcycling.com), passing by a second one that was richer, I have tried to increase complexity progressively to give answer to some questions that, to the best of my knowledge, nobody has answered. Those questions appear regularly in bike forums without a clear answer, showing that there is still job to be done to explain properly the link between the traditional methods for quantifying bike performance and how a bike behaves in real world. Bike design and construction have improved enormously in the last few years and I think that the effort done to explain scientifically how those improvements are beneficial to the consumer isn't enough.

After those two simpler models, I decided to devote more time to the development of the dynamical model. It started with a tire model able to handle discontinuous contact. Later, I tried to build the whole system around this tire model without success so I decided to step back and add elements to the model progressively.

A virtual ergometer was the first sub-model built. Everything worked flawlessly until I decided to include the elastic couplings between the torso and the bike (arms and saddle) so that feature was removed. Next, the bike-wheels sub-model was built with better results so I only needed to merge those two models. Once again there were problems so I have to calculate saddle loads using the virtual ergometer and use those forces as input in the full model.

I have to say that this trial and error procedure is really frustrating. When you spend some days/weeks gathering all the data needed to add a certain element to the model and integrating it and finally the software fails to solve and it doesn't give a meaningful explanation to the error, you want to give it up. I could have used other solving procedure and maybe this would have speed up the process but I also value all the things that I have learned finding the limits of the solver. For example, a dynamics program (Adams, Working Model or SIMPACK) could have done the job but I doubt I could have integrated the most complex and interesting features.

It's isn't perfect yet but I think that it is accurate enough to give some interesting answers for the future direction of bike design.

A quick animation. Take a look at how the BB and the chain moves during the downstroke ;)

More soon. Thanks for reading!

Yaw calculator v.1.1

A new version of the yaw calculator is available. Nothing revolutionary, some statistical outputs have been added and now there is a proper Readme file with a clear explanation of the calculation procedure and the signs convention. The next version will bring some interesting updates in the equipment selection module. This is the full list of updates:

- The calculation of statistical parameters (Mean, Unbiased standard deviation, Kurtosis and Skewness) of the yaw probability distributions has been added

- The plot of the cumulative distribution functions (CDF) of the module of yaw has been added. This plot can be useful for symmetrical components (wheels,handlebars,etc).

- Normal distribution fitting capability has been removed

- Reports output has been modified

- Plot saving capability of the CDF of yaw module has been added

It can be downloaded from the following link

Yaw calculator v. 1.1

As always, don't forget to run MCRInstaller before running the app

Greetings

A virtual ergometer

Long time without posting because I was very busy the last few months. Now I've got some free time and I will try to complete the dynamical model.

Initially, I modified the tire model that I presented some time ago to take into account the whole system but I was finding a "small" error that I couldn't solve. After some days trying to solve it, I decided to step back and build the model step by step to monitor better possible errors. I decided to start with a static bike-rider (something similar to an ergometer) and add the kinematical DOFs of the bike and the wheel later.

Right now, I am developping accurate pedal force profiles for variable cadence and/or power and a tuneable automatic drivetrain (something similar to the SRAM Automix IGH). The upper body and the elastic couplings between the rider and the bike aren't modelised yet but I will add them later.

Assuming RWS, reduced inertia of the wheel and equivalent moments are defined to modelise properly angular velocity fluctuations due to the variations of rider torque. The model involves a 2nd order DAE system of 43 equations that is solved using MAPLE numerical DAE solvers to avoid using index reduction techniques. An animation:

Some possible applications:

- The effect of gear ratio selection on inter-segmental forces, joint moments and muscular fatigue while maintining power

- An accurater loading spectra of drivetrain components and BB bearings

- Analysis of transient behaviour during gear changes

More soon. Thanks for reading!

Tire model

Just a short post to show the tire-road contact model that I will use for the dynamical model of a bicycle that I'm developping. The model is fully "analytical", no multi-body software used. There are some secundary features lacking but the core is already done.

The GIF shows a 2kg wheel (R=0.34m) falling from 0.5m to a irregular road (maximum amplitude of irregularities=0.07m). Tire stiffness and damping are taken from previous posts of this blog. IC have been chosen randomly.

As you can see just after the impact, this road profile-wheel combination is near the limit of the single contact point condition that the majority of tire load models use. Some plots:

That's all for today. Greetings

Yaw calculator

To sum up all the posts about the topic and following advice by ST member Aralo, I have written a program that some of you will find very helpful. This program is divided in two modules, the objective of the first one (product development) is to compute the yaw-independent CdA given probability distributions or general data of bike and wind speed and relative orientation. This parameter allows you to compare different iterations of a product and make a reasonable choice based on the real world conditions that the component will face. The second module is designed to help with equipment selection for a given course and bike and wind speed. It allows to import bike position and speed data obtained from GPS tracking or any other source and compare the average power consumption needed for each setup.

The program can be downloaded from the following link. I have enclosed some data samples in the file to show the data structure that the program handles.

Yaw calculator v.1.0.

Don't forget to run MCRInstaller before running the app. Have fun and please report any bugs

Greetings

About yaw II

I wasn't going to write more posts about this subject but I've got some results to share. If you remember my previous post about yaw, I calculated the relation between the standard deviation of the Maxwellian distribution and the average of the Rayleigh distribution for wind speeds. One of the main problems of the Rayleigh distribution is that there is only one tuning parameter. That means that average value and variability can't be chosen indepently or, alternatively, that a high average wind speed means high variability.

For that reason, I have tried to modify Dan Connelly's yaw model for a more evolved wind speed model. The Weibull distribution is a two-parameter distribution that is used to model wind speed in the energy industry but its complexity means that no analytical expression can be obtained for the yaw probability model for the fixed bike speed case. If we consider that bike speed is variable, it becomes too much computing intensive and impractical.

As I wanted to take into account variable bike speeds, the Rayleigh model was the only option for wind speeds. Assuming a Weibull distribution of bike speeds, the yaw probability function for variable bike and wind speed and bike and wind direction is:

κ, shape factor of the Weibull distribution. λ, scale parameter of the Weibull distribution. va, average wind speed

The main problem of this distribution is that it isn't normalized and, as no analytical solution exists, the normalization constant can't be computed. The normalization constant could be calculated numerically but this would be too computing intensive so I have used an alternative method. If you calculate the probability for a given yaw, the real probability would be that value divided by the normalization constant. So, if you calculate the probability for different yaws and you fit that data to a null-average gaussian distribution (as Mavic's data) multiplied by that unkown constant, the standard deviation and the normalization constant can be determined.

An example. Weibull distribution for bike speeds:

Rayleigh distribution for wind speeds:

And the resulting probability function and CDF of yaw:

More soon. Thanks for reading!