Lagrangian Dynamics, calculating virtual force

[alanic]

I'm very happy that I found this forum, hello everyone.
I'm studying Lagrangian Dynamics and I can't figure out how to find the generalized forces in a setup like this:

   ____             ____
  |    |    ___    |    |
  | b1 |===|___|===| b2 |
  |____|           |____|

--->x

b1 and b2 are corrected with a prismatic joint that has a motor so it can apply force on the two bodies along the joint. It can also be loose, in which case it won't apply any force and the joint will move in and out freely. b1 and b2 are restricted to move only along the x axis, which is the axis of the joint. Arbitrary external forces can be applied to both bodies. I choose generalized coordinates as the position of b1 (x_b1) and the length of the prismatic joint (l).

Now I would like to find the generalized forces for the generalized coordinates I chose. When doing that, I should take each generalized coordinate one by one and assume there is a small variation on that generalized coordinate. Then I should see how much work is done by the forces in the system. My question is, when finding the generalized force for x_b1, should I assume that the prismatic joint is stiff? It seems like what needs to be done since I should only allow x_b1 to vary. But, at the same time, this would require me to consider a force transfer between the two bodies(wouldn't it?), even though there was no force on the prismatic joint. So, for example, if b1 and b2 had forces that are equal and opposite to each other and if I assume the joint to be stiff, there will be no virtual work done for x_b1. This feels wrong.

So I guess I'm a little confused about the procedure to calculate the generalized forces. Should I allow only one generalized coordinate at a time, force all others to be stiff (which would introduce extra forces that are not there, like the ones that I would need to make the prismatic joint stiff), and consider that setup to find the virtual work for that generalized coordinate? Or should I consider all the forces that are really there and not force the other generalized coordinates to be stiff? Or, does the idea of "not allowing other generalized coordinates to vary" not really make them stiff and do I consider the existing forces like in a normal general body diagram?

Thanks for reading about my confusion. I think I'm missing some fundamental principle here. Any comments would be greatly appreciated.

 

[alanic]

Ok some careful math analysis showed that I should keep the other generalized coordinates absolutely stiff and solve the whole system by varying one generalized coordinate at a time.