# Modal Analysis

The modal dynamic analysis is concerned with the calculation of the natural frequencies and the mode shapes of the structure. As in the static analysis a stiffness matrix can be constructed, but in a modal dynamic analysis a mass matrix is also constructed. The free vibration of the model is then given by

$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{Ku} = 0$

The natural frequencies are then given when

$\left| \mathbf{K} - \lambda\mathbf{M} \right| = 0$

The eigenvalue problem is then

$\mathbf{K}\boldsymbol{\phi} - \lambda\mathbf{M}\boldsymbol{\phi} = 0$

where $\{ \lambda, \phi \}$ are the eigenpairs. The eigenvalue, $\lambda$, gives the frequency $f = \frac{\sqrt{\lambda}}{2 \pi}$ and the eigenvector, $\phi$, is the mode shape.

## Finding specific eigenpairs​

When running modal analysis, we find eigenpairs with an iterative eigensolver. These work in various ways, but there are three main ways we can choose the eigenpairs the solver will find. These three different methods can be used for various aspects of dynamic analysis. The first method of finding the lowest $n$ frequencies is useful for getting general insight into structural dynamics, the frequency interval method is more useful for footfall analysis, the effective mass method is typically used for seismic analysis.

### Find the $n$ eigenvalues corresponding to the lowest frequencies​

There are options within this to find the $n$ eigenvalues closest to a given frequency. This is done by applying a shift, $\sigma$, to the above eigenvalue problem to get

$\mathbf{K}\boldsymbol{\phi} - (\lambda-\sigma)\mathbf{M}\boldsymbol{\phi} = 0$

making the solver now find the $n$ eigenvalues corresponding to the lowest values of $(\lambda-\sigma)$, i.e. the values that are the closest to $\sigma$.

### Find all the eigenvalues in a frequency interval​

This is done by first counting the eigenvalues in the interval using a Sylvester inertia check, then running the above solver to find that many modes closest to the midpoint of the frequency interval.

### Find enough eigenvalues to provide a given effective mass​

The effective mass is a measure of the mass of the structure participating in this mode. The summation of effective mass over all the modes in the model would sum to the unrestrained mass of the structure, but usually a target such as 90% is sufficient.

This can be achieved in two ways:

• Using the iterative solver, which finds more and more eigenvalues until the target effective mass is achieved. This option returns all the modes starting from the mode with lowest frequency until what is needed to achieve the target mass participation.
• Using the MASIL (mass accumulating shift-and-invert lanczos algorithm [1]) solver, which calculates a collection of frequency intervals that contain the effective mass and then runs the frequency interval solver to recover the necessary modes, before purging the results and returning a near-minimal set of eigenpairs covering the required effective mass. This option returns smallest amount of modes needed to achieve the target mass participation (and skips the modes with low contribution to the effective mass).

[1] M. Zemaityte, F. Tisseur, R. Kannan. "Filtering frequencies in a shift-and-invert Lanczos algorithm for the dynamic analysis of structures." SIAM Journal on Scientific Computing 41.3 (2019): B601-B624.