2. Modal adjustment of a conservative system#
Two modal adjustment techniques are presented. The first uses the sensitivity of natural modes and the second uses the diagonality of generalized matrices.
2.1. Exploitation of the difference between modal deformations and the difference between natural frequencies#
An attempt is made to find the parameters of the numerical model such that the calculated modal deformation restricted to the observation points is collinear to the modal deformation obtained experimentally and that the associated natural frequency is equal to the natural frequency identified experimentally.
To do this, the normalized dot product between the measured natural deformation and the calculated natural deformation restricted to the ddls associated with the measurement points is calculated. This calculation corresponds to the calculation of MAC (Modal Assurance Criterion).
If we designate respectively by \(({y}_{{i}_{\mathit{num}}},{f}_{{i}_{\mathit{num}}})\) and \(({y}_{{i}_{\mathit{mes}}},{f}_{{i}_{\mathit{mes}}})\) the calculated eigenmodes and the identified eigenmodes, we have:
\(\mathit{MAC}({y}_{{i}_{\mathit{num}}},{y}_{{i}_{\mathit{mes}}})=\frac{{({y}_{{i}_{\mathit{num}}}^{T}{y}_{{i}_{\mathit{mes}}})}^{2}}{({y}_{{i}_{\mathit{num}}}^{T}{y}_{{i}_{\mathit{num}}})({y}_{{i}_{\mathit{mes}}}^{T}{y}_{{i}_{\mathit{mes}}})}\)
Vectors \(V\) and \({F}_{r}\) are defined as:
\(V=\left(\begin{array}{c}⋮\\ \mathit{MAC}({y}_{{i}_{\mathit{num}}},{y}_{{i}_{\mathit{mes}}})-1\\ ⋮\end{array}\right)\)
\({F}_{r}=\left(\begin{array}{c}⋮\\ {f}_{{i}_{\mathit{num}}}-{f}_{{i}_{\mathit{mes}}}\\ ⋮\end{array}\right)\)
The functional to be minimized is formulated as follows:
\(ϵ={V}^{T}{W}_{\mathit{MAC}}V+{F}_{r}^{T}{W}_{\mathit{freq}}{F}_{r}\)
Of course, it is necessary to assess the difference between two analogous methods. Thus, this technique is not suitable for a structure where modal density is important.
This procedure is used by MACR_RECAL (option DYNAMIQUE) in the sdls121a [V2.03.121] test case.
2.2. Exploitation of the orthogonality of the measured eigenmodes#
On the basis of the modal deformations noted at the observation points, an expansion is carried out on the support numerical model. Next, we try to find the parameters of the numerical model so that the generalized mass and stiffness matrices relating to the modes identified experimentally are diagonal and that the difference between the measured natural pulsation and the calculated natural pulsation is minimal.
The natural pulsation (or more exactly the square of the pulsation) is estimated by calculating the ratio between the generalized stiffness and the generalized mass of the identified mode.
This technique requires neither a pairing between the experimental mode and the numerical mode, nor a modal calculation. It is therefore suitable for structures where modal density is high. However, it requires the expansion of the modes identified on the numerical model.
The expansion of the i-th mode identified on the numerical model can be done in the following way:
We choose an expansion base composed of modal deformations calculated with the support numerical model:
\(Y=[{y}_{1}\dots {y}_{n}]\)
The coordinates \({\eta }_{i}\) of the identified modal deformation \({{\Phi }_{i}}^{\mathit{mes}}\) are then calculated on the basis \(Y\), restricted to observation points. These coordinates can be obtained by minimization of the least squares type.
\(\epsilon ={\left({{\Phi }_{i}}^{\mathit{mes}}-Y{\eta }_{i}\right)}^{T}{W}_{i}\left({{\Phi }_{i}}^{\mathit{mes}}-Y{\eta }_{i}\right)\)
We then perform an expansion of the deformation identified on the ddls of the numerical model: \({\Phi }_{i}=Y{\eta }_{i}\)
The next step is to calculate the standardized generalized matrices:
\({\mathit{MAC}}_{W}(i,j)=\frac{{({{\Phi }_{i}}^{T}W{\Phi }_{j})}^{2}}{({{\Phi }_{i}}^{T}W{\Phi }_{i})({{\Phi }_{j}}^{T}W{\Phi }_{j})}\)
If the weighting matrix \(W\) is equal to the mass matrix \(M\) or the stiffness matrix \(K\), \({\mathit{MAC}}_{W}\) becomes a diagonal matrix.
It is then a question of finding the terms of the matrices \(K\) and \(M\) that minimize at the same time:
\({\mathit{MAC}}_{K}(i,j)\) for \(i\ne j\)
\({\mathit{MAC}}_{M}(i,j)\) for \(i\ne j\)
Difference between the i-th natural pulsation identified \({\widehat{{\omega }_{i}}}^{2}\) and \({{\omega }_{i}}^{2}=\frac{{\Phi }_{i}^{T}K{\Phi }_{i}}{{\Phi }_{i}^{T}M{\Phi }_{i}}\)
The matrix \({\mathit{MAC}}_{W}\) is symmetric, its lower triangular part can be stored in a vector called \({\mathit{MAC}}_{W(i<j)}\). The functional to be minimized can then be formulated as follows:
\(\epsilon ={\mathit{MAC}}_{K(i<j)}^{T}{W}_{K}{\mathit{MAC}}_{K(i<j)}+{\mathit{MAC}}_{M(i<j)}^{T}{W}_{M}{\mathit{MAC}}_{M(i<j)}+\sum _{i}{\left({\widehat{{\omega }_{i}}}^{2}-\frac{{\Phi }_{i}^{T}K{\Phi }_{i}}{{\Phi }_{i}^{T}M{\Phi }_{i}}\right)}^{T}{W}_{i}\left({\widehat{{\omega }_{i}}}^{2}-\frac{{\Phi }_{i}^{T}K{\Phi }_{i}}{{\Phi }_{i}^{T}M{\Phi }_{i}}\right)\)
The following weighting matrices can be chosen:
\({W}_{K}\) = nb_modes_identified* \({I}_{d}\)
\({W}_{M}\) = nb_modes_identified* \({I}_{d}\)
\({W}_{i}\) = 0.5 nb_modes_identified (nb_modes_identified_— 1) * \({I}_{d}\)
Where \({I}_{d}\) is the identity matrix
This choice of weighting makes it possible to assign the same weight to the equations on the frequencies and to the equations on the extra-diagonal terms of the generalized matrices.
The implementation of this modal adjustment approach is illustrated in the modeling d of the sdls121 test case [V2.03.121].