The singular value decomposition, usually referred to as SVD, is probably one of the most important linear algebra tools used to solve many structural dynamics problems
the mathematical formulation
Let us first present, in the simplest way, the mathematical (matrix) formulation of the SVD and some of its variants, and then describe its use in modal analysis.
Let be a square matrix [A] of size n x n. The basic equation of the SVD is :
The expansion of the formulation gives :
This new formulation implies that the matrix A is composed of a set of vectors and singular values that describe it. It can also be defined as a matrix composed of other matrices that are very simply described by a vector and a corresponding eigenvalue. Finally, the SVD therefore really has the ability to determine the "principal elements" that make up the matrix A. This also implies that the rank of the matrix can be determined.
This mathematical tool is used in the application ofoperational modal analysis (OMA) techniques to determine the modal deformations
Example of the singular value decomposition
In the following example, we will consider the frequency domain decomposition (FDD) method for determining modal deformations: Suppose that we have a structure instrumented with accelerometers (four in this example), distributed in an equidistant manner. Let us also assume that the sensors are oriented in the same way and that each sensor is uniaxial. Let us now imagine that for each sensor, we have the frequency response of the structure (the responses of the sensors are recorded at the same time).
From these data, the power spectral density matrix [Gyy] can be constructed for each frequency index i :
PSD is the power spectral density, CSD is the cross spectral density, indices 1 to 4 denote the sensors and index i denotes the frequency index.
Finally, this matrix becomes particularly interesting when the frequency index i corresponds to a natural frequency of the structure.
The matrix
It can be decomposed into singular values. The number of non-zero elements in the diagonal of the singular matrix S corresponds to the rank of the spectral density matrix. The singular vectors 𝑢𝑖 correspond to an estimate of the unnormalized modal distortions and the corresponding singular values 𝑠𝑖 are the spectral densities of the 1 degree of freedom system.
The singular values are arranged in a decreasing manner along the matrix S. The largest value, i.e. 𝑠1 corresponds to the vector {𝑢1} ({𝑢1}𝑠1{𝑣1}𝑡). Moreover, it highlights the preponderant phenomenon which is the modal deformation in our case when the frequency index corresponds to a natural frequency of the studied system.
The question that can be asked for the singular value decomposition.
All things considered, Gyy is composed of PSD and CSD, the PSD is real and the CSD is complex. The singular value decomposition is about the nature of the data, so the resulting modal deformation is complex. How to interpret a complex deformation?
