Consistent and Lumped Mass Matrices in Modal Analysis
There are a number of options for constructing the mass matrix for a beam, but these fall into two groupings: consistent and lumped mass.
The consistent mass matrix uses the same displacement shape functions as the element stiffness matrix.
The consistent mass matrices will be exact if the actual deformed shape (under dynamic conditions) is contained in the displacement shape functions. Since this shape is unknown, the static displacement distribution is typically used, making the mass distribution approximate but generally accurate for most practical purposes. The consistent mass matrix includes coupling between the ends of the beam through off-diagonal terms in the mass matrix.
In the normal derivation of the consistent mass matrix the mass is assumed to be concentrated along a line between the nodes. For beams with significant depth relative to length, additional rotary inertia terms - associated with the offset of the mass from the neutral axis - can be included. GSA excludes these rotary inertia terms.
In the lumped mass approach, the mass (and inertia if included) is assumed to be concentrated at the ends with no coupling effects between the ends. This leads to a much simpler, diagonal matrix. This gives reduced fidelity in modelling the mass distribution in an individual element, but it has benefits when looking at the overall structural response, since it is less sensitive to modes within individual elements.
With the lumped mass approach, inertia terms can be included or excluded. For real structures the omission of the inertia terms is usually not significant.
The lumped mass matrix is usually derived from the consistent mass matrix. There are two main methods for achieving this: row summation and the HRZ method.
The row summation method makes use of the fact that the mass matrix is symmetric, and sets the diagonal terms to the sum of the mass or inertia terms in the row. This ensures that the total mass of the element is properly accounted for.
The HRZ method described in E. Hinton, T. Rock, O. Zienkiewicz, A note on mass lumping and related processes in the finite element method, Earthquake Engineering & Structural Dynamics 4 (3) (1976) 245–249 is:
- For each coordinate direction, select the DOFs that contribute to motion in that direction and separate into translational DOF and rotational DOF subsets.
- Sum the diagonal entries of the translational DOF subset only.
- Apportion mass to the diagonal entries of both translational and rotational subsets by dividing these entries by the sum.
- Repeat for all coordinate directions.
There is very little different between consistent and lumped mass models for axial and torsional modes. Most differences are in the flexural response.
Example
Consider a simple cantilever beam, fixed at the left hand end and modelled with a single element (and a reference model where the same beam is modelled with 40 elements).
The first two flexural modes flexural modes are a shown below
The table below shows how, with the reference model, there is little difference in the modal frequencies
| mode | consistent (1 element) | lumped (1 element) | consistent | lumped |
|---|---|---|---|---|
| Hz | Hz | Hz | Hz | |
| minor axis - first | 1.488 | 1.032 | 1.481 | 1.481 |
| major axis - first | 5.393 | 3.756 | 5.366 | 5.365 |
| minor axis - second | 14.66 | - | 9.246 | 9.236 |
| major axis - second | 52.74 | - | 31.27 | 31.24 |
The single element model, with consistent mass matrix, gives a good prediction of the first fexural modes, but does not reliably predict the second flexural modes, under-estimating the effect of the inertia terms. This is as expected as the shape functions do not properly model the mass distribution for the second mode shape. With the consistent mass matrix, the effective mass summed over the modes is 150% in the transverse directions, indicating that the model is not sufficiently refined.
With lumped mass the frequencies are underestimated for the first modes and are not available for the second mode. In this case it is clear that the model is not sufficiently refined. However it is worth nothing that effective masses sum to 100%.
So in conclusion to capture the dynamic behaviour properly it is important to ensure that there are sufficient elements to properly model the problem under consideration.
To see the effect of increaing the number of elements (and consequently the number of degrees of freedom the consistent mass and lumped mass models were refined to give results for 1, 2, 4 and 8 elements models.
For the consistent mass the frequencies are
| mode | 1 element | 2 elements | 4 elements | 8 elements | reference |
|---|---|---|---|---|---|
| Hz | Hz | Hz | Hz | Hz | |
| minor axis - first | 1.488 | 1.482 | 1.481 | 1.481 | 1.481 |
| major axis - first | 5.393 | 5.371 | 5.367 | 5.367 | 5.366 |
| minor axis - second | 14.66 | 9.324 | 9.258 | 9.247 | 9.246 |
| major axis - second | 52.74 | 31.54 | 31.39 | 31.30 | 31.27 |
and for the lumped mass matrix
| mode | 1 element | 2 elements | 4 elements | 8 elements | reference |
|---|---|---|---|---|---|
| Hz | Hz | Hz | Hz | Hz | |
| minor axis - first | 1.032 | 1.330 | 1.440 | 1.471 | 1.481 |
| major axis - first | 3.756 | 4.828 | 5.221 | 5.329 | 5.365 |
| minor axis - second | - | 6.824 | 8.432 | 9.023 | 9.236 |
| major axis - second | - | 23.20 | 28.64 | 30.56 | 31.24 |
Increasing the number of elements shows a rapid improvement in the natural frequencies, particularly for the consistent mass model. However, even with four elements modelling the cantilever, the effective mass was overpredicted with the consistent mass, albeit by a small amount. For the lumped mass matrix, the solution converges on the reference solution and for most engineering purposes the eight element model is close enough.
Real models have much greater complexity so these results do not transfer directly to a real structural model. In general it is better to use the lumped mass matrix, though it may lose accuracy if the mass distribution is too coarse.