The modal deformation
We saw in the previous episode that to determine the modal deformation of a structure in operation, and without imposed excitation, it is necessary to use OperationalModal Analysis (OMA) methods.
We have also seen that for the frequency-domain decomposition method, the modal deformation corresponds to the first eigenvector from the singular-value decomposition of the power spectral density matrix.
This matrix belongs to the complex domain, and therefore the identified modal deformation is also complex. It is important, for a better understanding, to make a transition to the real domain.
How is this transition achieved?
Let the complex deformation U be of size n :
1- The first step is to calculate the amplitude of each component using the following formula:
The phase 𝜭i is necessary to define the sign of the deformation. It is calculated as follows:
3- The sign of the deformation takes the sign of the cosine of the phase 𝜭 :
4- The last step consists in normalizing it with respect to the maximum amplitude
Assuming that all sensors are synchronized, the phases |𝛳𝑖| must be equal as shown in the following figure:
