Dynamical model. Wheel loads

As I've already commented, I think that slip force and rolling resistance moment are the two main loads that could play a role efficiency-wise. Consequently, I have made an effort to complexify their models as much as possible. The following 4 sheets explain the contact model and the loads acting on the wheel (chain, reactions in the axles, normal force, tangential force and rolling resistance moment).

After reading these sheets, you can imagine why the solver has problems under certain conditions. Complexity is very high.

That's all for today

Dynamical model. Modelling bike compliance

How to modelize the compliance of a bike in a dynamic model? That's a very good question. I have chosen the simplest type of model used in elastodynamics, the area that analyzes the deformation of elements in dynamic conditions. There are more complex models based in a FEA formulation of the deformable components and various choices of kinematical coordinates that can be found in some commercial codes like ADAMS Flex or Altair Hyperworks. For the moment, I didn't want to go as far.

This model is based on connecting certain elements with springs whose stiffness is derived from statical tests. It's based on linearity so the amount of deformation is proportional to the force between them. There are some features that the model isn't able to capture like the inertia associated to the deformation of the components but we will consider that it's a second order effect.

I've used this model to take into account the connection between the wheels and the frame to analyze the effect of bike compliance on performance. A simple diagram is shown below.

The wheels are free to move with respect to the frame and they are connected to the undeformed configuration of the chassis using springs. Obviously, the wheel axle and the dropouts of the bike in the deformed configuration should be coaxial so the stiffness of the springs is equivalent to the stiffness of the frame in the defined directions. Now the question that arises is: what's the stiffness of those springs and what's the relation between them and the statical tests?

Correlation between the results of static test benches and those measured in real world is a difficult issue. I recommend you to take a look at  two very interesting articles (here and here) that Damon Rinard wrote about how to improve correlation. The process that I've followed to obtain the stiffness of the vertical springs is explained in the following sheet:

As a ROT, we can say that the stiffness of these springs is half of the BB stiffness of the bike so significantly lower than the stiffness of a road tire. Some typical values are shown below:

Tour Magazin data

Once we have calculated the stiffness of the vertical springs, it's time for the horizontal ones. We can modelize the rear end as a structure clamped in the ST-Seatstay junction and in the BB and with a symmetry plane. Similarly, the fork has a symmetry plane and it's clamped in the HT. As there isn't data available under this type of loads, I've done some tests using ANSYS.

Rear end test. Steel. BEAM 188. 201 nodes. Krx=71000 N/mm for the whole rear end

A similar test was done for a steel fork with straight legs and 30mm of spacement between the crown and the dropouts. Using the same tubing than in the previous case, Kfx equals 127N/mm.

Greetings

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!