@dc42 said in Scientific take on pressure advance:

@h975 ... There is a further complication: I don't believe the compression of filament within the Bowden tube behaves like a linear spring. So there are two non-linear effects to compensate for.

I think the solution will be to use a non-linear PA algorithm, which is more or less what you are suggesting. However, before we can say what sort of nonlinear PA curve is needed, we need to do some more measurements on systems with and without long Bowden tubes, and work out a good method of calibration. Your suggestions will be welcome!

I agree that the filament in the Bowden tube will not behave as a linear spring.

The reason I say that is because I am proposing that as a system with certain parallels, the Bowden tube could be viewed as a loaded structural column with fixed-free end conditions and lateral support. The buckling of such a column will produce a buckled shape which is a string of half-waves in the form of a sin wave. The number of half waves produced is associated with defined critical load limits being reached in the column. Each successive critical load limit that is reached will produce another half wave. The relationship between the number of half waves and the critical load limit is non-linear. For that reason I would say that the filament will not load up linearly.

The equation describing the critical load points is : F(cr) = n^2Pi^2EI/4L^2 where F is the load, n is the number of half waves produced, L is the length of the 'column' and EI is the product of the Youngs modulus and the second moment of area of the filament.

As the above formula is derived for statically loaded columns and the filament is in reality moving in a tube thereby adding friction effects to the driven load, this will not allow an exact application of this formula to predict all variables, but the point is that the same principles of column loading still apply and therefore an analysis of it can, I believe provide valid insight into the system behaviour.

As far as the friction effects are concerned it is my view that these will also not behave linearly. Each half wave will produce another contact point with the Bowden tube which will have the effect of increasing the frictional resistance to movement in the same non linear way as for the spring effect. The friction effects will stack up as you go up the length of the tube thereby adding to the load carried by the filament column.

Note that the critical loads are discrete quantities meaning that the filament would load up in a stepwise fashion as each new critical load level is reached.

I have an idea for a test that can be done to characterize the dynamics in the Bowden tube in order to gain greater insight into this part of the system. I will post this later in a response to your proposal on using a force transducer to test the response of the system.

I agree that a non linear PA algorithm would be the correct way to go. Hopefully the way forward will become clearer once the above hypothesis has been tested.