1. Reference problem

1.1. Geometry

Beam \(\mathit{PA}\):

Section: \(0.1\mathrm{mm}\times 0.1\mathrm{mm}\)

Length: \(3.0m\)

Beam \(\mathit{PB}\):

Section: \(0.001\mathrm{mm}\times 0.001\mathrm{mm}\)

Length: \(2.0m\)

Mass ratio between \(\mathrm{PB}\) and \(\mathrm{PA}\): \(0.33E–04\)

1.2. Material properties

Young’s module beams \(A\) and \(B\):	\(E=2.1E+11N/{m}^{2}\)
Poisson ratio beams \(A\) and \(B\):	\(\nu =0.3\)
Density beam \(\mathrm{PA}\):	\({\rho }_{A}=2000\mathrm{kg}/{m}^{3}\)
Density beam \(\mathit{PB}\):	\({\rho }_{B}=1000\mathrm{kg}/{m}^{3}\)

1.3. Boundary conditions and loads

The movement is authorized in plan \((\mathrm{DX},\mathrm{DY})\).

Beam \(\mathrm{PA}\) is embedded in the support.

Beam \(\mathrm{PB}\) is connected to Beam \(\mathrm{PA}\) by three points. In each, the movements in the \(\mathrm{DX}\) and \(\mathrm{DY}\) directions of the \(\mathrm{PA}\) node and the \(\mathrm{PB}\) node are identical. The rotations are not linked.

The interspectral matrix that transmits the movements from structure \(\mathrm{PA}\) to structure \(\mathrm{PB}\) in the chained calculation is of dimension 6 (6 degrees of freedom of transmission).