1. Reference problem#
1.1. Geometry#
For purely demonstrative purposes, we study in this test case a « non-real » shaft line with a total length of 0.4 m and a full circular cross section with an external diameter of 0.02 m, supported by three levels. This shaft line has a disc positioned 130 mm from the left end.
Figure 1.1-1
The coordinates of the points shown in the figure above are:
Node |
X (m) |
Y (m) |
Z (m) |
|
N1Z1 |
0.000000E+00 |
0.000000E+00 |
0.000000E+00 |
0.000000E+00 |
N2Z1 |
0.000000E+00 |
0.000000E+00 |
1.333340E-01 |
|
N3Z1 |
0.000000E+00 |
0.000000E+00 |
2.666670E-01 |
|
N4Z1 |
0.000000E+00 |
0.000000E+00 |
4.000000E+00 |
4.000000E-01 |
AP1 |
1.000000E-05 |
0.000000E+00 |
0.000000E+00 |
|
P1 |
1.000000E-05 |
0.000000E+00 |
2.666670E-01 |
|
AP2 |
1.000000E-05 |
0.000000E+00 |
4.000000E-01 |
Table 1.1-1
1.2. Material properties#
The material made of the shaft line has the following elastic characteristics:
density: = 7800kg/m3
Poisson’s ratio: n = 0.3
Young module: E = 200 GPa
Link stiffness modeling the bearing at the N1Z1 node: \({k}_{x}={k}_{Y}={10}^{12}N/m\);
Link stiffness modeling the bearing at the N3Z1 node: \({k}_{x}=2.0\times {10}^{5}N/m\) and \({k}_{Y}=5.0\times {10}^{5}N/m\);
Link stiffness modeling the bearing at the N4Z1 node: \({k}_{x}={k}_{Y}={10}^{12}N/m\);
Rotor point mass: \(m=16.47\mathit{kg}\)
The inertia tensor of the rotor with respect to the N2Z1 node are: \({I}_{\mathit{XX}}=0.1861{\mathit{kgm}}^{2}\); \({I}_{\mathit{YY}}={I}_{\mathit{ZZ}}=0.09427{\mathit{kgm}}^{2}\) and the coupling terms and the eccentricity terms are zero.
1.3. Boundary conditions and loads#
Boundary conditions:
The nodes belonging to the tree line are blocked in terms of vertical movement as well as rotation around this axis: \(\mathit{dz}=\mathit{drz}=0\)
Nodes \(\mathit{AP}1\), \(P1\), and \(\mathit{AP}2\) are embedded: \(\mathit{dx}=\mathit{dy}=\mathit{dz}=\mathit{drx}=\mathit{dry}=\mathit{drz}=0\).
Loading:
The structure is subjected to an unbalance at point N2Z1 with the amplitude \(A=15\) g.mm with a zero phase. The rotation speed is 2500 rpm.
1.4. Initial conditions#
The system is at rest.