1. Reference problem

1.1. Geometry

The system studied is composed of 3 masses (\(m\)) and 4 springs (\(k\)). The set is embedded at its ends.

Image 1.1-a : Geometry of the system studied

1.2. Material properties

Spring stiffness: \(k=\mathrm{1 }N/m\).

Point masses: \(m=\mathrm{1 }\mathrm{kg}\).

1.3. Boundary conditions

Points \(A\) and \(B\) embedded.

1.4. Initial conditions

Structure initially at rest.

1.5. Observation

It is considered that the two translations x1 and x2 are perfectly observed.

It is considered that the first two natural modes are observed:

• For the first eigenmode, we assume the two translations x1 and x2 as well as its natural pulsation perfectly observed (\(\widehat{{\omega }_{1}}={\omega }_{1}\)).

• For the second eigenmode, we consider the two translations x1 and x2 perfectly observed while the information on the natural pulsation is assumed to be subject to an error of 25% (\(\widehat{{\omega }_{2}}=\mathrm{1,25}\ast {\omega }_{2}\)).

1.6. Error functional

We choose to minimize the error of the energy functional error of the error type in the following behavioral relationship:

\({e}_{\omega }^{2}(u,v,w)=\frac{\gamma }{2}{(u-v)}^{T}[K](u-v)+\frac{1-\gamma }{2}{(u-w)}^{T}{\omega }^{2}[M](u-w)+\frac{1-\alpha }{\alpha }{(\mathit{Hu}-\widehat{u})}^{T}\text{}[\mathit{Gr}](\mathit{Hu}-\widehat{u})\)