1. Reference problem

1.1. Geometry

The structure is composed of a rotor of length \(L\) with a variable circular section with two infinitely rigid bearings at the level of the supports. The diameters of the rotor section vary linearly depending on the length.

1.2. Material properties

The geometric and material characteristics are listed in the following table.

Material		\(E=2{10}^{11}N/{m}^{2}\)	\(\rho =7800\mathrm{kg}/{m}^{3}\)	\(\nu =0.3\)
Rotor length	\(L=1m\)
Section diameter	original outer diameter: \(\mathrm{De1}=\mathrm{0,2}m\)
	final outer diameter: \(\mathrm{De2}=\mathrm{0,1}m\)
Thickness	at any point on the rotor: \(\mathrm{De}–\mathrm{Di}=\mathrm{0,04}m\)

Table 1.2-1

The translation coefficients of the stages are:

\(\mathrm{Kxx}=\mathrm{Kyy}=1.0E+12{\mathrm{kg.s}}^{-2}\)

\(\mathrm{Kxy}=\mathrm{Kyx}=0.0\mathrm{kg.s}-2\)

\(\mathrm{Cxx}=\mathrm{Cyy}=\mathrm{Cxy}=\mathrm{Cyx}=0.0{\mathrm{kg.s}}^{-1}\)

The torsional coefficients of the bearings are:

\(\mathrm{Krz}=\mathrm{Kry}=1.0E+12{\mathrm{kg.s}}^{-2}\) \(\mathrm{Kxy}=\mathrm{Kyx}=0.0\mathrm{kg.s}-2\) \(\mathrm{Crx}=\mathrm{Cry}=\mathrm{Cxy}=\mathrm{Cyx}=0.0{\mathrm{kg.s}}^{-1}\)

1.3. Boundary conditions

The rotor is supported on two rigid bearings at both ends.