Unconstrained motion in hamiltonian mechanics

[dvisacker]

Hello everyone,

I am currently working on a bar-linkage simulation and decided to use multibody dynamics to try to perform this analysis. So basically everything seems simple and clear for examples such as double-pendulums or 4-bar linkages. And i obtain the lagrangian equations or hamiltonian without too much problems.

However, I would like to model bar-linkage (actually for a start just a bar chain without any ramifications) that is not constrained. By this, I mean that the bar is not attached to any support. For example a bar is in free-fall. I think in this case the system of equations would be singular.

I don't really see how to handle this problem. I thought of constraining the center of mass to an imaginary frame but the center of mass is actually always moving in regard to the whole system.

If anyone got any thoughts about this problem, your help would be greatly appreciated .
Best regards and thanks in advance for your help.

 

[Dougt115]

As complex as it may be I don't see a better option than the Center of Mass as a reference. While it is true that it is not connected to any single point on the system it is the only point that is stable. The bars in the linkage will gyrate and move in complex patterns but the center of mass will move smoothly in space, linearly or parabolic, in a predictable manner.

 

[3DDave]

If the acceleration is uniform there isn't much to model. All parts are accelerating at the same time at the same rate. As far as they are concerned there is no acceleration force on them at all.

 

[dvisacker]

The bars would be in free-fall but still with some different forces applying on different bars and different initial velocities. But yes, I the case of a uniform acceleration and no other forces applying on the system, indeed there is not much to model.

Thanks for the suggestion Dougt
Anyhow, can't really find a way to do it properly, the constraints between the bars seem to mess it up. Maybe hamiltonian/lagrangian mechanics is not the way to do it.

 

[jlnsol]

For bar-linkage simulation I recommend SAM mechanism designer

http://www.artas.nl/en

If you upload your model sketch I can try to help you.

 

[dvisacker]

Thanks for the suggestion jlnsol.

I tried making the model on SAM, but as for previous things I tried, it says my model is underdetermined which comes from the fact that in my case there is no support.
Anyways, looks like a good piece of software, I will keep it for later, thanks!

I attached a quick sketch (made in SAM) of what I would like to model. As you see there is no support for the mechanism, which is odd but there should be some method able to solve the system, right ? I started with only 2 bars which can rotate relative to each other. There is gravity and we apply a force on the tip of one bar for example. The details like where the force is applied, if there is gravity or not, etc. are actually not really important. I'm more looking for a method or an idea to solve this. The final system would contain more than 2 linkages. All the theories seem to be fit for models with support/constrained models.

Best regards

 

[prex]

Lagrangian mechanics is perfectly capable of solving unconstrained motions, so I don't see where is your problem (but never tried myself a similar one).
As you cerytainly know, the critical choice in lagrangian mechanics is about the coordinates. Personally I would start, for two bars, with the x,y of the pivot point, plus the angle of each bar with respect to the reference axis. The position of the CoG, that of course is in the equations, would be expressed as a function of those other coordinates.

prex
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[jlnsol]

Wikipedia:
a uniform gravitational field does not by itself cause stress or strain, and a body in free fall in such an environment experiences no g-force acceleration and feels weightless.

So, how can you apply a force on a part of a mechanism in free fall?
Assumed there is no air resistance.

 

[dvisacker]

jlinsol : Of course i agree with that wikipedia statement. Actually, the gravitational field does not even matter. Let's just assume a bar-linkage floating around in space.
If we apply a force on one part of the mechanism (via let's say an external operator or some kind of engine fixed on one of the parts) and that the parts can move relatively to each other (in my case, there is a rotational degree of freedom between the 2-bars), the part will experience different accelerations and undergo complex movement. Do i got anything wrong ?

prex : Thanks I tried with your system of coordinates. Looks like Mathematica managed to find and solve the equations . Will see if it works as wanted soon.