4. B modeling

4.1. Characteristics of modeling

The characteristics of this modeling are identical to the previous one (modeling A). The only difference is that the structure is depreciated. The amortization used is of the proportional type:

\(C=\alpha K+\beta M\) with \(\alpha ={\mathrm{6.5x10}}^{-6}s\) and \(\beta =16.0{s}^{-1}\).

These values correspond to a depreciation reduced by 1% in the first mode of the structure.

4.2. Characteristics of the mesh

The characteristics of the mesh are also identical to those of modeling A (cf. [§ 3.2]).

4.3. Tested features

We test the dynamic calculation functionalities, taking into account depreciation, by substructuring as well as the restoration in physical space.

4.4. Tested sizes and results

The results are reproduced on a skeleton mesh consisting of the two substructures. The initial mesh, which contains 6 nodes, is therefore duplicated to create the substructure on the right. The end node therefore corresponds to node 12.

Identification	Reference (full model)	Sub-Structure	Difference (%)
Node \(12\): move \((m)\)	—9.54882E—7	—9.54882E—7	< 0.1
Node \(12\): speed \(({\mathrm{m.s}}^{-1})\)	1.22190E—3	1.22190E—3	< 0.1
Node \(12\): acceleration \(({\mathrm{m.s}}^{-2})\)	—1.91712E+0	—1.91712E+0	< 0.1

4.5. notes

It is surprising that the reference adopted corresponds to the complete beam modelled by 10 elements and not to the analytical solution. The significant differences between numerical and theoretical solutions are due to the reduced number of elements. The use of 50 elements instead of 10 would have made it possible to approach the theoretical acceleration to within 1%. Apart from that, it can be seen that the use of a substructuring method provides the same results as those of the complete beam.

The calculation by modal superposition is carried out on the complete modal basis of the structure (11 modes). Likewise, the dimension of the projection base used for the calculation by dynamic substructuring is 11 (left substructure: 4 natural modes + 1 constrained mode; right substructure: 5 natural modes + 1 constrained mode). It is therefore normal to obtain an excellent agreement between the modeling of the complete beam and its modeling in two substructures.