2. Benchmark solution

2.1. Calculation method

The aim here is to couple an experimental model representing the real structure and a numerical model of the modification that is desired to be made to the initial structure according to the technique proposed by Mr. Corus [1]. This coupling is done via a « support » model making it possible to condense the measurement at the interfaces between the real structure and the modification. The two models are then assembled using the substructuring technique. A more detailed description of the modification procedure used is shown in [U2.07.03]. The extension of this technique on damping structures was studied by B. Groult [2].

The input data for the calculation are: the identified eigenmodes of the initial structure, the numerical model of the « support » and the numerical model of the « modification ».

It is proposed to calculate the variations of the first two natural frequencies of the free-embedded beam, following the modification made on a portion of the beam.

The « support » model chosen for this test case is a numerical finite element model of the free-embedded beam described in the preceding paragraph. The first two natural frequencies of this beam are: \(9.31\mathrm{Hz}\) and \(39.32\mathrm{Hz}\).

The condensation of the information measured at the interfaces is obtained by carrying out an expansion of the measurement by means of a previously chosen projection base, defined on this « support » model. The quality of the result depends on the choice of this projection base.

The measurement was simulated numerically based on a calculation from the support model.

The modification was modelled numerically using finite elements.

2.2. Reference quantities and results

The first two natural frequencies of the modified structure are compared to the natural frequencies obtained by a direct calculation on the complete model. The first two natural frequencies of the modified structure are: \(7.78\mathrm{Hz}\) and \(32.85\mathrm{Hz}\).

The correct progress of the structural modification procedure is also verified by comparing the field obtained at the interfaces of the modified structure in two different ways.

The first calculation corresponds to the calculation of the field at the interfaces on the coupled model.

The second calculation corresponds to the calculation of the field at the interfaces by static expansion of the field obtained at the measurement points of the modified structure.

The difference between these two fields can be evaluated by calculating the sum of the terms of the MAC matrix (Modal Assurance Criterion) between these two fields. A MAC matrix that is close to the identity matrix indicates that the two vectors are nearly parallel. Criterion IERI (Energy Indicator of Interface Regularity) is also evaluated. This energy criterion tends to 0 if the two fields are very close.

2.3. Uncertainties

The reference solution on the natural frequencies of the modified structure is obtained by direct calculation on the modified structure.

We consider that the discretization chosen leads to results that are very close to the analytical solution.