# Experimental modal analysis – Algorithm

The algorithm used to calculate modes in DIRAC is based on the Principal Response Function (PRF). The PRF helps to reduce the effect of noise and to fit modes on a large number of FRFs quickly. The LSCF algorithm (Guillaume et al., 2003, link) is used to find the poles of the system.

The algorithm implemented in DIRAC consists of 6 steps:

1. PRF application.
2. Frequency band selection.
3. Least-squares complex frequency (LSCF) pole estimation.
4. Stabilization diagram visualization.
5. Residue fit.
6. Globalization.

## Step 1: PRF application

The first operation that is applied to the input FRF matrix is the principal response function (PRF). This operation can be viewed as a pre-processing step and the goal is to reduce the system size and to reject measurement noise. The input-output FRF matrix is converted to a flat format containing each individual FRF. From a singular value decomposition of this matrix, the most dominant singular vectors are used as a set of pseudo-measurements, called the Principal Response Functions. These singular vectors are selected based on a predefined threshold parameter.

## Step 2: Frequency band selection

The second step consists of reducing the frequency band where the modes will be calculated. For example, you may want to exclude the range of frequencies in which your measurement has a poor signal-to-noise ratio. The PRF will be cut to only include the data in the defined frequency range.

## Step 3: LSCF pole estimation

The PRF matrix is used as the input to the Least-squares complex frequency (LSCF) pole estimation algorithm. This algorithm solves a linear least-squares problem to find the numerator and denominator polynomials of a transfer function, best describing the observed measurement for a given number of poles. The power of the method lies in the fact that this computation is repeated for many different numbers of poles. The true number of poles is then determined by finding those poles which appear in many of these iterations.

## Step 4: Stabilization diagram visualization

The stabilization diagram visualizes how the algorithm determines the true number of poles in the system. Similar poles from different iterations are grouped together. The stabilization of a pole is then determined by comparing the poles in a group to each other. A pole is marked as stable in frequency if its eigenfrequency is within the specified tolerance of the eigenfrequencies of the other poles in the group. A similar comparison is performed for the damping value. Only poles that are found to be both stable in frequency and damping are considered in the subsequent steps of the algorithm. From each group with a minimum number of stable poles, one pole is selected. You can manually select other poles or deselect poles by clicking on them in the stabilization diagram. The final set of poles is considered to be the set of true poles of the system, marked in the stabilization diagram with .

## Step 5: Residues fit

The poles obtained from the LSCF algorithm are taken to the next step, where the residues and residuals are computed based on the measured FRF data. The residues and residuals describe the contribution of each mode to a particular FRF. They are found by constructing the denominator polynomial for each complex pair of poles and minimizing the squared error of the sum of all mode contributions and the measured FRFs. Note that the residues are found by fitting the measured FRFs, the PRF is only used for the pole estimation part of the algorithm.

$$\boldsymbol{\mathbf{Y}}_i (\omega ) = \sum_{j=1}^{N}d_j(\omega )\mathbf{R}_{i,j} + \mathbf{R}_{low} – \omega ^2 \mathbf{R}_{high}$$

$$d_j(\omega )= \frac{-\omega ^2}{-\omega ^2 + 2 i \zeta_j \omega_{n,j} \omega + \omega^2_{n,j}}$$

## Step 6: Globalization

The final step to obtain a set of modes is to convert the residues for each individual FRF into mode shapes of the system. In this stage of the algorithm, we make use of the input-output relation of the measured FRF matrix. The residues for each mode are converted to a matrix of the same dimension as the input FRF matrix. In a modal model, each input can excite a mode to a greater or lesser extent and the outputs are then activated according to the mode shape of that mode. The mode shape is independent of the input. This condition is enforced by taking the first left singular vector of each residue matrix as the mode shape. The first right singular vector supplies the participation factor of each input to that mode. The input FRF matrix is then reconstructed using these vectors.