# Lumped Mass vs Consistent Mass?

1. Oct 4, 2012

### Ronankeating

Hi All,

Basically I 've 2 question in mind.

1. If I use always consistent masses in my calculations, does that mean that I'm always doing more exact analysis and obtaining the more exact results than lumped mass techniques, with sacrifacing the computer storage?
2. If I choose lumped masses, and if I do some reductions due to unwanted degrees of freedom. Then what like should that mass look after the reduction with lumped masses?

To elaborate the 2nd question:
The 2nd question above is usually named Floor by Floor Assembly(FFA) in structural eng. Where floors are assumed as rigid diphragm and whole floor weight is lumped at the mass or rigidity center of floor. Each floor will have only $\delta_x$, $\delta_y$ and $\theta_z$ two translational and one rotational degree of freedom(as per the code request) and will be treated as master, other joints as slave. FEM books suggest to transfer slave joint mass and force stiffness to master joint, but in my case slave joints are massless and whole floor weight will be at the mass center. So basically what those mx, my and mz should look like in reduced mass matrix.

Master Mass Matrix
\begin{bmatrix}m_x &0&0\\0&m_y&0\\0&0&m_z \end{bmatrix}

Can I assume ??
mx = (Total weight of Floor) /3
mY = (Total weight of Floor )/3
mz = (Total weight of Floor) /3

Best Regards,

2. Oct 4, 2012

### AlephZero

The "nice" thing about consistent mass matrices is that you don't have to think about how to formulate them, you just calculate. In some situations they are no more accurate than a diagonal mass matrix, but it's not obvious what is the "best" diagonal mass matrix except for simple elements. For example the best diagonal mass matrix for an 8-noded plate element has different mass terms on the corner and midside nodes.

Often a diagonal mass matrix gives eigenvalues which converge from below as you refine the mesh, while consistent mass matrices converge from above. That suggests there might a better formulation than either of the above, and some commercial FE packages have more options than "diagonal" or "consistent".

Most reduction methods will tend to convert the diagonal mass matrix into a full one.

It's quite common to include additional "non-structural" mass terms in a FE model. In your application, if you want to use finite elements just to formulate the stiffness and create the complete mass matrix some other way, there's no reason not to do that.

You can eliminiate massless DOFs by static condensation (Guyan). The mass matrix doesn't change, except for deleting the massless rows and columns.

If you condense out a DOF with a diagonal mass term, usually you create off-diagonal terms, so you lose any speed advantage from having a diagonal matrix.

But eliminating massless DOFs also reduces the sparsity of the stiffness matrix, and there are good iterative eigensolution algorithms to find some of the modes with the lowest frequencies which work fine with massless DOFs in the model (e.g. Wilson subspace interation and Lanczos), so reducing out the massless DOFs may not give the fastest solution.

That looks wrong. It should be
mx = (Total weight of Floor)
mY = (Total weight of Floor)
mz = (Total moment of inertia about z for the Floor)

You want the whole mass included in "force = mass x acceleration" or "torque = MI x angular acceleration", for any direction of the acceleration vector, not 1/3 of it.

3. Oct 15, 2012

### afreiden

Ronankeating, I wanted to comment on your first question regarding whether to use a consistent mass matrix or a lumped mass matrix. I think you are saying that you sacrifice speed when using a consistent mass matrix, but gain accuracy. If I misunderstand, forgive me. However, it's not clear to me that you are sacrificing speed in your situation.

It's true that when you use a lumped mass matrix, you can do an "explicit" analysis (e.x. Central Difference Method), where you don't have to invert the stiffness matrix when solving the equation of motion. The computational time to perform one time step of analysis is reduced, no doubt. However, explicit analyses are not always stable from my understanding... your time-step will therefore need to be very small -- in other words, you'll need many more time-steps compared to a traditional implicit (e.x. Newmark) analysis with consistent mass matrix.

Since you are talking about "rigid diaphragms" and whatnot, I assume that you are doing some sort of time history analysis of a decent-sized building and your analysis will have a time range of at least a few seconds. I was under the impression that the benefit of doing an explicit analysis with lumped mass matrix only shows up in analyses with a time range on the order of milliseconds. Furthermore, I was under the impression that there's no reason to use a lumped mass matrix when you are doing an implicit analysis like what you are likely trying to do...

However, I see that one of the best structural engineering software (OpenSEES) does in fact do implicit with lumped mass for analyses of buildings like what you appear to be trying to do... can anyone explain why they use a lumped mass in this case?

4. Oct 16, 2012

### Ronankeating

You got it correctly. I was trying to get info from those who are familiar with that procedure beforehand. I thought that using consistent mass is more accurate than lumped, but that is not completely valid for structures because you have something moving as rigid-body(storey slabs) at storey level, so you actually doesn't diverge from actual result for eigen problems.This is the special case only for buildings. Probably this can't be assumed for car engines or plane wings attached to the aircraft to behave like a rigid-body (lumped masses) unless you want to do very very rough analyse, instead you should use consisten masses. Also AlephZero stated that there are other mass matrices other than those mentioned above
"some commercial FE packages have more options than "diagonal" or "consistent masses"

Although I've asked it in my post, I'm completely uninterested with benefiting from storage and speed, that will be in 2nd stage task(optimization) of my problem. For now, I'm focusing simply on composing the global stiffness matrix from local matrices which takes into account the rigid diaphragm.

I stuck at the phase that I thought that I can use the same techniques similar to direct stiffness method to compose the global stiffness matrix, where we map each local matrix into global stiffness with help of steering vector. And also I thought that with reordering of columns and rows in global stiffness matrix I can be able to write it in that form ready to apply the Guyan reduction

\begin{bmatrix}K_(mm) & K_(ms)\\K_(sm)&K_(ss)\end{bmatrix}
\begin{bmatrix}X_(mm) \\X_(ss)\end{bmatrix}
+w''*\begin{bmatrix}M_(mm) & 0_(ms)\\0_(sm)&0_(ss)\end{bmatrix}=0
where M_mm is lumped masses of each storey.
But it seems that it doesn't turn out to be easy like that. One way or another local matrices should be composed in way that they eliminate unwanted DOFs somehow.

Again this is the special case for buildings under seismic loading in my idead (I'm unsure). It's been said that for "5ms time interval seismic excitation can be assumed linear"(reference Edward Wilson book Structural Dynamics) IMHO we don't deviate from actual solution here also.

Ditto #1 paragraph in my post.