1. Reference problem#

1.1. Geometry#

The structure is modelled by a set of 3 springs and 2 point masses.

Link stiffness: \({k}_{1}={k}_{3}=k=\mathrm{100000 }N/m\); \({k}_{2}=2k=200000N/m\)
Point mass: \({m}_{2}={m}_{3}=m=2533\mathrm{kg}\).

The only authorized movements are translations according to axis \(x\).

Points \(\mathrm{NO1}\) and \(\mathrm{NO4}\) are embedded:

\(\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0\).

The other points are free to translate in the direction \(x\):

\(\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0\).

Modeling A: the structure is subjected to a decorrelated multiple spectral seismic excitation.

The pseudo-acceleration oscillator response spectra are defined by:

at node \(\mathrm{NO1}\): \({\mathrm{SRO}}_{\mathrm{NO1}}=\frac{{a}_{1}{\omega }^{2}}{∣{\omega }_{1}^{2}-{\omega }^{2}∣}\)
at node \(\mathrm{NO4}\): \({\mathrm{SRO}}_{\mathrm{NO4}}=\frac{{a}_{2}{\omega }^{2}}{∣{\omega }_{2}^{2}-{\omega }^{2}∣}\)

with \({\omega }_{1}=2\pi {f}_{1}\) \({\omega }_{2}=2\pi {f}_{2}\)

\({f}_{1}=1.5\mathrm{Hz}\), \({f}_{2}=2.\mathrm{Hz}\), \({a}_{1}={a}_{2}=0.5{\mathrm{ms}}^{\text{-2}}\)

They do not depend on depreciation.

B modeling: the structure is subjected to seismic excitation identical to both supports.

The pseudo-acceleration oscillator response spectrum is defined by:

at node \(\mathrm{NO1}\) and at node \(\mathrm{NO4}\): \(\mathrm{SRO}=\frac{{a}_{1}{\omega }^{2}}{∣{\omega }_{1}^{2}-{\omega }^{2}∣}\)

with \({\omega }_{1}=2\pi {f}_{1}\)

\({f}_{1}=1.5\mathrm{Hz}\), \({a}_{1}=0.5{\mathrm{ms}}^{\text{-2}}\)

It does not depend on depreciation.

The system is initially at rest