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.

 

[charng]

I can understand your confusion about finding the generalized forces in a setup like this. It is important to note that in Lagrangian Dynamics, we use generalized coordinates to describe the motion of a system, and these coordinates can be any set of variables that uniquely define the configuration of the system. In your case, you have chosen the position of b1 and the length of the prismatic joint as your generalized coordinates.

When finding the generalized forces, it is important to consider all the forces that are acting on the system, including external forces and forces between the bodies. In your example, the forces between b1 and b2 can affect the motion of b1, even if the prismatic joint is not stiff. Therefore, it is not necessary to assume the joint to be stiff when finding the generalized force for x_b1.

To find the generalized force for x_b1, you should consider the virtual work done by all the forces acting on the system when x_b1 is varied. This includes the forces between b1 and b2, as well as any external forces applied to the system. You should not force the other generalized coordinates to be stiff, as this would introduce extra forces that are not actually present in the system.

In summary, when finding the generalized forces in Lagrangian Dynamics, it is important to consider all the forces that are acting on the system and not force any of the generalized coordinates to be stiff. I hope this helps clarify your confusion. Keep up the good work in your studies of Lagrangian Dynamics!