Thursday, September 19, 2013

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 componentand 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 ishown 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.


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