2. Force modeling and calculation

2.1. Modal-based modeling, observability and controllability

The assumed linearity of the structure’s behavior allows the decomposition of the model on a modal basis:

\(\underline{y}\mathrm{=}\underline{\underline{C\Phi }}\text{.}\underline{\underline{{Z}^{\mathrm{-}1}}}\text{.}\underline{\underline{{\Phi }^{T}B}}\text{.}\underline{f}\)

Using the concept of inter-spectrum, preferred in macro [1] _ :

\(\underline{\underline{{S}_{\text{yy}}}}\mathrm{=}\underline{\underline{C\Phi }}\text{.}\underline{\underline{{Z}^{\mathrm{-}1}}}\text{.}\underline{\underline{{\Phi }^{T}B}}\text{.}\underline{\underline{{S}_{\text{ff}}}}\text{.}{\underline{\underline{{\Phi }^{T}B}}}^{H}\text{.}\underline{\underline{{Z}^{\mathrm{-}1}}}\text{.}{\underline{\underline{C\Phi }}}^{H}\) (1)

\(\underline{\underline{C}}\) and \(\underline{\underline{B}}\) are the observability and control matrices. \(\underline{\underline{C}}\) allows you to project a field defined on a model, on a more restricted field of sensors. \(\underline{\underline{B}}\) projects this same field onto the points of application of the efforts. The matrices \(\underline{\underline{C\Phi }}\) and \(\underline{\underline{{\Phi }^{T}B}}\) are the modal deformation matrices defined on a sensor mesh and on a control mesh. They can be obtained with the operator PROJ_CHAMP or with the operator OBSERVATION, which makes it possible, in addition to the previous one, to define unmeasured directions for each node or group of nodes, or to define local references.

\(\underline{\underline{Z}}\) is the \(\text{diag}{(\mathrm{-}{\omega }^{2}+\mathrm{2j}{\xi }_{i}{\omega }_{i}\omega +{\omega }_{i}^{2})}_{1\mathrm{\le }i\mathrm{\le }N}\) impedance matrix associated with the base of the modal deformations \(\underline{\underline{\Phi }}\). This base is generally based on a good quality model. A good quality base must in fact have modal parameters that are close to reality. The deformations must also be close to reality, and in addition be sufficiently regular. Thus, an experimental deformation base is defined on a reduced number of sensors, and is therefore not very regular (there is a risk of solving the problematic inverse problem). We will prefer the deformed base of a corrected numerical model, or, better still, an extended experimental base on a numerical model. This point is specified below.

Note on modal expansion

We have an experimental \(({f}_{i},{\xi }_{i},{m}_{i},{\underline{{\varphi }_{\text{exp}}}}_{i})\) modal base. The modal parameters are supposed to be identified with a good approximation, but the deformed ones, of good quality, are only defined on a limited number of sensors. An attempt is therefore made to extend this deformation base on a numerical model that is fairly representative of the structure (but not necessarily wrong). The expansion base can be the basis for the modes of the numerical model (obtained with CALC_MODES). Achieving a modal expansion therefore means finding the generalized parameter vector \(\eta\) that minimizes:

\({\mathrm{\parallel }{\underline{{\varphi }_{\text{exp}}}}_{i}\mathrm{-}\underline{\underline{{\Phi }_{\text{num}}}}\text{.}\underline{\eta }\mathrm{\parallel }}^{2}\)

The « correlation » tab in CALC_ESSAI allows this expansion to be achieved, thanks to the use of PROJ_MESU_MODAL. For more details on the principle of expansion, refer to the documentation in PROJ_MESU_MODAL (U4.73.01).

2.1.1. Solving the opposite problem

Under certain conditions, equation (1) of the direct problem can be reversed:

\(\underline{\underline{{S}_{\text{ff}}}}\mathrm{=}{\underline{\underline{{\Phi }^{T}B}}}^{\mathrm{\oplus }}\text{.}\underline{\underline{Z}}\text{.}{\underline{\underline{C\Phi }}}^{\mathrm{\oplus }}\text{.}\underline{\underline{{S}_{\text{yy}}}}\text{.}{\underline{\underline{C\Phi }}}^{{H}^{\mathrm{\oplus }}}\text{.}\underline{\underline{Z}}\text{.}{\underline{\underline{{\Phi }^{T}B}}}^{{H}^{\mathrm{\oplus }}}\)

Since the matrices \(\underline{\underline{C\Phi }}\) and \(\underline{\underline{{\Phi }^{T}B}}\) are rectangular, the sign \(\mathrm{\oplus }\) designates their pseudo-inverse of Moore Penrose. This pseudo-inverse can be obtained by using a SVD algorithm (Singular Value Decomposition, cf doc R6.03.01). In CALC_ESSAI, two regularization options are available:

1. truncation of small singular values: we note \({\sigma }_{\text{max}}\), the largest singular value. SVD consists in setting the value 0 for all singular values less than \(\varepsilon {\sigma }_{\text{max}}\), for any given \(\varepsilon\). They are therefore not taken into account in the inverse calculation. Truncation eliminates information about matrices to be inverted, but improves conditioning;

2. Tikhonov regularization: the inverse of the matrix of singular values is not worth \(\text{diag}{(\frac{1}{{s}_{i}})}_{1\le i\le N}\) but \(\text{diag}{(\frac{{s}_{i}}{{s}_{{i}^{2}}+\alpha })}_{1\mathrm{\le }i\mathrm{\le }N}\). Parameter \(\alpha\) is called Tikhonov parameter, and makes it possible to limit the divergence of the inverse solution.

In CALC_ESSAI, the inter-spectra are calculated in succession:

• generalized movements (reversal of \(\underline{\underline{C\Phi }}\)),

• reconstructed physical movements (to verify the quality of the reversal),

• generalized efforts (multiplication by the impedance matrix),

• physical efforts (reversal of \(\underline{\underline{{\Phi }^{T}B}}\)),

• generalized efforts reconstituted (for verification)

• physical movements by synthesis on a modal basis.

A good criterion for the quality of the results of the inversion can be the comparison between the movements measured, and those reconstituted on the same modal basis that was used for the inversion (impedance matrix, and modal deformation matrices).

[1]

The concept of inter-spectrum, or DSP, makes it possible to describe random phenomena. For a deterministic phenomenon, \(\underline{\underline{{S}_{\mathit{yy}}}}\mathrm{=}\underline{y}\mathrm{.}\underline{{y}^{H}}\), where \(H\) refers to the Hermitian operator. For a random signal, \(\underline{\underline{{S}_{\mathit{yy}}}}\) contains, on its diagonal, the spectral powers (auto-spectra) and on its extra-diagonal terms, the correlations between the signals. For example, in the case of a fluid force applied to a wire structure, it is easy to imagine that the vortices created at the base tend to propagate along the structure. Extra-diagonal terms therefore describe this propagation with the phase (see R4.07.02 for more details).