1. Reference problem

1.1. Geometry

Beam length	\(L=20.0m\)
Inner diameter	\(d=0.388m\)
Outside diameter	\(D=0.400m\)
Mass concentrated at the top	\(M=300.\mathrm{kg}\)
Mass moment of inertia	\(J=200.\mathrm{kg}{m}^{2}\)

1.2. Material properties

Tube density	\(\rho =7850{\mathrm{kg.m}}^{-3}\)
Young’s module	\(E=210.E+9{\mathrm{N.m}}^{-2}\)
Poisson’s Ratio	\(\nu =0.\)

1.3. Boundary conditions and loading

The tube is embedded at the base. The mass is free. Movement is allowed in a \((\mathrm{DX},\mathrm{DRZ})\) vertical plane.

A random effort \(F(t)\), applied to the concentrated mass, is assimilated to a stationary random Gaussian, centered process, of the white noise type with a limited band from \(\mathrm{1.Hz}\) to \(\mathrm{101.Hz}\). It is characterized by a standard deviation \({\sigma }_{F}=1\mathrm{kN}\), and a one-sided spectral density in frequency \({S}_{F}(f)\) such that:

\(\begin{array}{}\forall f\in [1\mathrm{Hz},101z]\\ {S}_{F}(f)=\frac{{\sigma }_{F}^{2}}{100}={10}^{4}{N}^{2}s\end{array}\)