Rigid body motion (RBM) transformation

  • #1
Mechanics_student
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0
TL;DR Summary
Rigid body motion (RBM) transformation of interface points between a rigid and an elastic bodies in contact
68maW.png


ABfVc.png

7FCA8.png


Above are pictures of a problem in mechanics. An elastic body occupying domain ##\Omega## is supported below with a fixed support and from above with a rigid body. The following calculations aim to express the movements of the points located at the interface ##\Gamma_F## between the rigid and elastic bodies in terms of a reference point ##B##:
I don't understand them well, so I will explain my understanding (not sure it's correct), and confusion.
$$\vec{y}(x) = \vec{x} + \vec{u}(x)$$
The new position ##\vec{y}## of a point is equal to its original position ##\vec{x}## plus its displacement ##\vec{u}##.

Rigid body motion transformation:
$$\vec{y}(\vec{x}) = \vec{y}^B + \mathcal{R}(\beta) (\vec{x} - \vec{x}^B)$, where $\vec{x} \in \Gamma_F$$
where ##\mathcal{R}## is the rotation matrix as the rigid body rotates at an angle ##\beta##.
$$\mathcal{R} = \begin{bmatrix}
cos(\beta) & sin(\beta)\\
-sin(\beta) & cos(\beta)
\end{bmatrix}$$
Linearization for small deformation:
$$\vec{u}(x) - \vec{u}^B = \vec{y} - \vec{y}^B - (\vec{x} - \vec{x}^B)\\
= (\mathcal{R}(\beta) - I)(\vec{x}-\vec{x}^B)$$
where ##I## is the identity matrix.
For linearization we take,
$$\mathcal{R} = \begin{bmatrix}
0 & \beta\\
-\beta & 0
\end{bmatrix}$$
$$\Rightarrow \vec{u}(x)\mid_{\Gamma_F} = \vec{u}^B + (\mathcal{R}(\beta) - I)(\vec{x}-\vec{x}^B)$$
I don't understand the displacement of the points on the interface are parametrize by the displacement of a reference point that is also located on the interface undergoing transformation as well.

Isn't a reference point supposed to be fixed?
 
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  • #2
Welcome, @Mechanics_student ! :cool:

I can't help with that complicated math, but I can tell you that from the point of reference of point B, the only movements that an observer sees are the deformation of the elastic body occupying domain Ω, and the rotation of the fixed support located below it.

You could also assume that two springs located where the green lines are replacing the elastic body.
For ideal conditions, the left side spring will compress the same distance that the left side spring is stretched, and in the same period of time.

I would assume that a point moment applied at a midpoint of the top interface replaces the actual moment created by the force F.
The vertical displacement of that midpoint would be zero, constantly increasing with the distance at which a point of that interface is located at.
 
  • #3
Lnewqban said:
Welcome, @Mechanics_student ! :cool:

I can't help with that complicated math, but I can tell you that from the point of reference of point B, the only movements that an observer sees are the deformation of the elastic body occupying domain Ω, and the rotation of the fixed support located below it.

You could also assume that two springs located where the green lines are replacing the elastic body.
For ideal conditions, the left side spring will compress the same distance that the left side spring is stretched, and in the same period of time.

I would assume that a point moment applied at a midpoint of the top interface replaces the actual moment created by the force F.
The vertical displacement of that midpoint would be zero, constantly increasing with the distance at which a point of that interface is located at.
Hi Lnewqban, thanks I'm glad you to be here. The thing is, point B is part of the rigid body, and so when the rigid body rotates, I assume that it rotates as well. That's why I don't understand why its taken as a reference. Another thing is that, the fixed support doesn't rotate, because it is fixed. So the upper interface of the elastic body is in contact and rotates exactly like the rigid body, and the lower interface of the elastic body in contact with the fixed support, doesn't undergo any displacements.
 
  • #4
Mechanics_student said:
Hi Lnewqban, thanks I'm glad you to be here. The thing is, point B is part of the rigid body, and so when the rigid body rotates, I assume that it rotates as well....
That could be, but in that case, force F should have been represented as being perpendicular to the top interface for any angle.
It has been represented as not solidly linked to point B, rather than rotating respect to it (case of F being applied on a shaft that rotates inside bushing B).

The other option could be that reference point B is part of the shaft rather than a fixed point of the top interface or rigid body.

Again, this is over my head, and I am only responding because nobody else with much better understanding of Math has.

I have found this article, which you may find helpful:
https://ucb-ee106.github.io/ee106a_jupyterbook/ForwardKinematics.html

revoluteJoint.png
 
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