Model Reduction and Substructuring in Dynamics, CMS

[Zirkus]

Dear engineers and physicists,
I would like to ask you a question about Component Mode Synthesis (CMS), which is the topic of my bachelor thesis. My primary resource is the classical AIAA article "Coupling of Substructures for Dynamic Analyses: an Overview" by Prof. Craig.
For now I am considering a simple mechanical system consisting of two connected bodies (arms) according to the first picture. The individual bodies have been modeled in ANSYS and their mass, stiffness, nodal coordinates and adjacency matrices were imported to Matlab, where reduction and interconnection themselves are carried out.
First the FEM solution without reduction (i.e. in physical coordinates) is calculated for reference and then two CMS methods are implemented, namely the Craig-Bampton's (fixed-interface normal modes & constraint modes) and Rubin's (free-interface normal modes, rigid-body modes & residual flexibility modes).
Now to my question, which might be a bit funny, bacause I am complaining about the methods being too precise. :) It seems that the natural frequencies of both methods correspond exactly (!) to the FEM solution, errors being of the order 0.01% which can be accounted to numerical round-offs. If I keep 30 natural modes for each body corresponding to 30 lowest frequencies, the first 50 natural frequencies of the connected system are exact! Then the error increases and the last calculated frequency (60th) has an error of 20% (talking about the Craig-Bampton now). I was really surprised by this, do you somebody have an insight where this "exact" corresponence comes from? If I compare the normal modes of the assembled reduced model with the reference solution by the Modal Assurance Criterion (MAC), I get the expected inaccuracies (please see 2nd included picture for the 20 lowest freq. normal shapes).

Thank you very much for any answers, tips, further questions or any other form of reactions. All the best from the Czech Republic.

 

[AlephZero]

CMS works well when the interface between the components is "simple" compared with the dynamics of each component.

if you want a problem that doesn't work so well, try something like a square plate cut into two parts.

Or try a thin cylinder with diameter = length, fixed at one end. You could cut it into two cylinders (with different lengths, of you want), or two semicircular shells.

 

[Zirkus]

I see, so this very good result is in fact an exception caused by the problem being too simple. I was planning to consider a truss system as a second example, but in that case the interface would again consist of only several DOFs, so I will rather attempt one of your tips.
Could I ask you another question about the atachment methods? In many papers the so-called "inertia-relief attachment modes" are defined for components with rigid body freedom. But I haven't really seen them being used. The "residual flexibility attachment modes" are typically included in the transformation matrix instead. I am a little bit confused about the physical interpretation of these modes, because I thought attachment modes represented a static deformation due to a unit force applied to one interface DOF (after solving the problem with rigid body motion by self-balancing the load by d'Alembert forces), which should be unique, so how come there are more of them? If I calculate this deformation using the full stiffness matrix, I don't need any modal residuum, do I? So far I was only so to say "re-typing the formulas to Matlab" without too much thinking, but I would like to understand what's going on more thouroughly...
The following plot shows one of the two residual modes in green and one of the two inertia-relief modes in yellow and by just looking they seem pretty much the same. As I understand it both of the shapes represent a deformation by a vertical force in the left "hole" after including the inertia effects of acceleration, so what is their relationship?
And maybe one last thing—I hope it's not too many...—does it matter how the assumed modes (e.g. the columns of the transformation matrix between physical and reduced coordinates) are scaled? The attachment modes have much smaller amplitudes, but that should'nt make any difference, right? For example in the posted graph I rescaled the two modes and changed the sign of one of them. The different sign of the two shapes is also puzzling, if they both are generated by the same force!
If you find time to answer any part of this long post, I will be very happy and thankful. Jan

 

[AlephZero]

I see, so this very good result is in fact an exception caused by the problem being too simple.

It's not usually a problem when a method works very well! A real-world example like that is the dynamics of rocket launches for satellites etc. The payload is attached to the rocket at only a few points which always have the same locations. So you can make one model of the rocket and tweak it to match the dynamics of the real structure as accurately as possible, and then couple it to models of many different payloads.

In many papers the so-called "inertia-relief attachment modes" are defined for components with rigid body freedom. But I haven't really seen them being used. The "residual flexibility attachment modes" are typically included in the transformation matrix instead.

I'm not familiar with those descriptions, but the modal model of a component may be measured experimentally, not created from a FE model. In that case the measurements might be done on a structure that is restrained in some way (for practical reasons) and the rigid body components of the motion added in separately.

If you want to create a measured stiffness matrix, it is much easier to do it with a structure that it restrained. As a simple example, you can easily measure the axial stiffness of a rod by fixing one end, and applying a force and measuring the displacement at the other end. After measuring the stiffness ##k## of a 1-DOF freedom model of the rod, you can then create a 2-DOF free stiffness ##\begin{bmatrix} k & -k \\ -k & k \end{bmatrix}## using the theoretical rigid body motion of the structure.

If I calculate this deformation using the full stiffness matrix, I don't need any modal residuum, do I?

Yes, you do. Otherwise, you are not modelling the inertia effects of the mass of the component correctly. If most of the mass of the component is at the interface boundary, of course that may not be very important, but then the internal modes will have high frequencies so you would not include many of them anyway.

And maybe one last thing—I hope it's not too many...—does it matter how the assumed modes (e.g. the columns of the transformation matrix between physical and reduced coordinates) are scaled?

Provided you keep the math consistent, it doesn't matter. if you change the scale of the mode shape, the modal mass and modal stiffness will also change. The "sign" of a mode shape is arbitrary, so it doesn't matter if a mode shape changes sign for no obvious reason.

 

[Zirkus]

(...)the modal model of a component may be measured experimentally, not created from a FE model.

I see, that is an important aspect when accounting for the residual flexibility as I probably can't measure natural shapes corresponding to high frequencies. But if I only work with a FE model, I usually compute all of the eigenvalues and eigenshapes of the components. For example in Matlab I would write

[Phi, Omega_square] = eig(K, M);

and get all the free-interface eigenvalues (K and M include all coordinates). Then I would divide the [itex]\Phi[/itex] matrix with eigenvectors as columns into
[tex]\Phi=\Phi_k+\Phi_d[/tex]
where k stands for kept and d for deleted. If I use the matrix [itex]\Phi_k[/itex] as it is as a transformation matrix between physical and reduced (modal) coordinates as
[tex]\vec{x} = \Phi_k \vec{q}[/tex]
the results will be poor, as I checked myself. In the literature I found that this is because the free-interface modes only do not span the whole space of deformations on the interface, whatever that means (I think I have an idea what it wants to say actually). In order to improve my new basis (basically a Ritz-method basis), I add some static shapes to describe the interface displacements more accurately. The already mentioned article "Craig: Coupling of substructures..." defines the residual flexibility matrix in the following way:

The portion of the flexibility matrix contributed by modes [itex]\Phi_d[/itex] is called the residual flexibility matrix. It is given by
[tex]G_d = \Phi_d (\Omega^2_{dd})^{-1}\Phi_d^T = G - \Phi_k (\Omega^2_{kk})^{-1}\Phi_k^T\quad (36)[/tex]
where G is the total flexibility matrix. Since it is not usually feasible to compute or measure the [itex]\Phi_d[/itex] modes, Eq. (36) is only useful because Eg. (32) exists as an alternative to Eg. (34) for determining the elastic flexibility matrix G.

The Eq. (32) calculates G by inverting K and the Eq. (34) reads [itex]\displaystyle G = \Phi (\Omega^2)^{-1} \Phi^T = \sum_{j=1}^n \frac{\phi_j\phi_j^T}{\omega_j^2}[/itex], where n is the number of natural modes not including rigid-body modes.

The previous quotation lead me to two questions:
1) I know all the modes, including the ones I truncated, so why can't I use the left part of Eq. (36) as it is?
2) The columns of [itex]G_d[/itex] do NOT represent any static deformation shapes, but only their contributions from the deleted natural modes? To sum up: the columns of [itex]G[/itex] represent deformation shapes caused by unit forces at the corresponding DOF and can be written as [itex]G = G_k + G_d[/itex]. The residual modes that I now include to my transformaion matrix to complement the free-interface body modes are the columns of [itex]G_d[/itex] corresponding to the interface DOFs. Is that correct?

I wish a nice day to all readers of this thread.

 

[Kun]

Can you please put code here because after doing all these analysis, I am getting large deviation in mode frequency.

 

[Kun]

I am seeking help for free interface cmc coupling. I did after all the coding in MATLAB by taking the effect of residual attachment modes and residual inertia attachment mode for rigid dof. After doing all this I am getting approx 10% error in modes frequency. I might be doing some mistake in MATLAB . Can anyone help.