3. Modal realignment of a dissipative system#
In the case of a dissipative system, the norm relationship of complex modal deformations is exploited.
The modal adjustment used here consists in finding the parameters of the model in such a way that the experimentally identified eigenmodes verify the norm relationships associated with the numerical model.
The dissipative structure is modelled as follows:
\(M\ddot{y}+B\dot{y}+\mathit{Ky}=0\)
Where: \(M\) refers to the mass matrix
\(B\) refers to the amortization matrix
\(K\) refers to the stiffness matrix
\(y\) refers to displacement
It is assumed that the matrices of the system are symmetric.
In the majority of cases, the modal matrix of the conservative system associated with this dissipative system does not simultaneously diagonalize the three matrices \(M\), \(B\) and \(K\). We then come back to a first-order differential system in 2N dimensional space.
We introduce the state vector: \(x=\left[\begin{array}{c}y\\ \dot{y}\end{array}\right]\)
We transform the N equations of the second order into 2N equations of the first order in the following way:
\(\left[\begin{array}{cc}B& M\\ M& 0\end{array}\right]\left[\begin{array}{c}\dot{y}\\ \ddot{y}\end{array}\right]-\left[\begin{array}{cc}-K& 0\\ 0& M\end{array}\right]\left[\begin{array}{c}y\\ \dot{y}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\)
That can be written as: \(U\dot{x}-\mathit{Ax}=0\)
With this transformation, the clean solutions are in the form: \({x}_{\nu }=\left[\begin{array}{c}{y}_{\nu }\\ {s}_{\nu }{y}_{\nu }\end{array}\right]\)
Where \({s}_{\nu }\) is the eigenvalue associated with \({x}_{\nu }\).
Eigenvalues can be real or complex. If the structure is weakly damped, all the eigenvalues are conjugated complex.
The spectral matrix is then in the following form: \(\underset{(\mathrm{2N}\mathrm{,2}N)}{S}=\left[\begin{array}{cc}{S}_{2}& 0\\ 0& \stackrel{̄}{{S}_{2}}\end{array}\right]\)
And the associated modal matrix: \(\underset{(\mathrm{2N}\mathrm{,2}N)}{X}=\left[\begin{array}{cc}Y& \stackrel{̄}{Y}\\ {\mathit{YS}}_{2}& \stackrel{̄}{{\mathit{YS}}_{2}}\end{array}\right]\)
With: \(\underset{(N,N)}{Y}=\left[{y}_{\nu }\right]\)
The modal matrix \(X\) verifies the following orthonormality relationships: \(\{\begin{array}{c}{X}^{T}\mathit{UX}={N}_{0}\\ {X}^{T}\mathit{AX}={N}_{0}S\end{array}\)
Where \({N}_{0}\) is a diagonal matrix that defines the norm of \(X\): \({N}_{0}=\left[\begin{array}{cc}{N}_{2}& 0\\ 0& \stackrel{̄}{{N}_{2}}\end{array}\right]\)
Taking into account the sub-matrix divisions, the development of the first line of orthonormality relationships is written as follows:
\(\{\begin{array}{c}{Y}^{T}\mathit{BY}+{S}_{2}{Y}^{T}\mathit{MY}+{Y}^{T}{\mathit{MYS}}_{2}={N}_{2}\\ {Y}^{T}B\stackrel{̄}{Y}+{S}_{2}{Y}^{T}M\stackrel{̄}{Y}+{Y}^{T}M\stackrel{̄}{{\mathit{YS}}_{2}}=0\end{array}\) (1)
and: \(\{\begin{array}{c}{S}_{2}{Y}^{T}{\mathit{MYS}}_{2}-{Y}^{T}\mathit{KY}={N}_{2}{S}_{2}\\ {S}_{2}{Y}^{T}M\stackrel{̄}{{\mathit{YS}}_{2}}-{Y}^{T}K\stackrel{̄}{Y}=0\end{array}\) (2)
The diagonal terms in the first line of the systems of equations (1) and (2) lead to the following relationships:
\(\{\begin{array}{c}{y}_{\nu }^{T}B{y}_{\nu }+2{s}_{\nu }{y}_{\nu }^{T}M{y}_{\nu }={n}_{\nu }\\ {s}_{\nu }^{2}{y}_{\nu }^{T}M{y}_{\nu }-{y}_{\nu }^{T}K{y}_{\nu }={s}_{\nu }{n}_{\nu }\end{array}\)
The combination of these two equations leads to:
\({s}_{\nu }^{2}{y}_{\nu }^{T}{\mathit{My}}_{\nu }+{s}_{\nu }{y}_{\nu }^{T}{\mathit{By}}_{\nu }+{y}_{\nu }^{T}{\mathit{Ky}}_{\nu }=0\) (3)
Likewise, the diagonal terms in the second line of the two systems of equations (1) and (2) lead to the following equations:
\({y}_{\nu }^{T}B\stackrel{̄}{{y}_{\nu }}+2\Re ({s}_{\nu }){y}_{\nu }^{T}M\stackrel{̄}{{y}_{\nu }}=0\) (4)
\({s}_{\nu }\stackrel{̄}{{s}_{\nu }}{y}_{\nu }^{T}M\stackrel{̄}{{y}_{\nu }}-{y}_{\nu }^{T}K\stackrel{̄}{{y}_{\nu }}=0\) (5)
These three equations (3) (4) and (5) must also be verified for all the eigenmodes identified on the real structure. The calibration technique presented here consists in finding the parameters of the numerical model that make it possible to verify the three equations.
Experimentally, the modal deformation is measured only on the directions of observation (sensitive directions of the sensors). An expansion of this deformation on the numerical model is necessary in order to obtain \({y}_{\nu }\).
It is appropriate to make the various equations associated with each mode to the same dimension. The following system of equations is then obtained:
\(\frac{{y}_{\nu }^{T}B\stackrel{̄}{{y}_{\nu }}+2\Re ({s}_{\nu }){y}_{\nu }^{T}M\stackrel{̄}{{y}_{\nu }}}{\mid {n}_{\nu }\mid }={z}_{1\nu }\)
\(\frac{{s}_{\nu }\stackrel{̄}{{s}_{\nu }}{y}_{\nu }^{T}M\stackrel{̄}{{y}_{\nu }}-{y}_{\nu }^{T}K\stackrel{̄}{{y}_{\nu }}}{\mid {n}_{\nu }{s}_{\nu }\mid }={z}_{2\nu }\)
\(\frac{{s}_{\nu }^{2}{y}_{\nu }^{T}{\mathit{My}}_{\nu }+{s}_{\nu }{y}_{\nu }^{T}{\mathit{By}}_{\nu }+{y}_{\nu }^{T}{\mathit{Ky}}_{\nu }}{{n}_{\nu }{s}_{\nu }}={z}_{3\nu }\)
For convenience, we choose \({n}_{\nu }\) equal to the Euclidean norm of \({y}_{\nu }\).
These quantities are then deposited in the vectors \({Z}_{1}\), \({Z}_{2}\) and \({Z}_{3}\) in order to be able to define the general form of the functional \(\epsilon\) to be minimized.
\({Z}_{1}=\left[\begin{array}{c}⋮\\ {z}_{1\nu }\\ ⋮\end{array}\right]={Z}_{\mathrm{1r}}\)
\({Z}_{2}=\left[\begin{array}{c}⋮\\ {z}_{2\nu }\\ ⋮\end{array}\right]={Z}_{\mathrm{2r}}\)
\({Z}_{3}=\left[\begin{array}{c}⋮\\ {z}_{3\nu }\\ ⋮\end{array}\right]={Z}_{\mathrm{3r}}+j{Z}_{\mathrm{3i}}\)
\(\epsilon =\left({Z}_{\mathrm{1r}}^{T}{W}_{1}{Z}_{\mathrm{1r}}\right)+\left({Z}_{\mathrm{1i}}^{T}{W}_{1}{Z}_{\mathrm{1i}}\right)+\left({Z}_{\mathrm{2r}}^{T}{W}_{2}{Z}_{\mathrm{2r}}\right)+\left({Z}_{\mathrm{3r}}^{T}{W}_{3}{Z}_{\mathrm{3r}}\right)\)
Where \({W}_{1}\), \({W}_{1}\), and \({W}_{1}\) are the weights associated with the various blocks of equations.
However, it should be noted that the modal deformation must be expressed on the numerical model. It is obtained by expanding the measurement on the numerical model. Numerical model modes are generally used as a basis for expansion.
An illustration of this technique can be found in sdld21st.