1. Problème de référence#
1.1. Géométrie#
Un modèle de rotor supporté par 2 paliers (nœuds \(\mathrm{B1}\) et \(\mathrm{B2}\) sur la figure ci-dessous), dont les matrices de raideur et d’amortissement sont non symétriques. Il est composé de 3 disques, et 4 sections d’arbre.
Figure 1.1-a -a: Modèle de rotor avec 3 disques et 2 paliers asymétriques
1.2. Propriétés du matériau#
Les caractéristiques géométriques et matériau sont listées dans le tableau suivant.
Matériau |
\(E={2.10}^{11}N/{m}^{2}\) |
\(\rho =7800\mathrm{kg}/{m}^{3}\) |
\(\nu =0.3\) |
|
Disque |
\(\mathrm{D1}\) |
\(M=20\mathrm{kg}\) |
\({I}_{D}=200.{10}^{-3}\mathrm{kg}{m}^{2}\) |
\({I}_{P}={400.10}^{-3}\mathrm{kg}{m}^{2}\) |
\(\mathrm{D2}\) |
\(M=17\mathrm{kg}\) |
\({I}_{D}=170.{10}^{-3}\mathrm{kg}{m}^{2}\) |
\({I}_{P}={340.10}^{-3}\mathrm{kg}{m}^{2}\) |
|
\(\mathrm{D3}\) |
\(M=10\mathrm{kg}\) |
\({I}_{D}=15.{10}^{-3}\mathrm{kg}{m}^{2}\) |
\({I}_{P}=30.{10}^{-3}\mathrm{kg}{m}^{2}\) |
|
Les caractéristiques des paliers sont renseignés dans les tableaux qui suivent.
\(\Omega =0\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={90.10}^{6}N/m\) |
\({K}_{xx}={50.10}^{7}N/m\) |
\({K}_{yx}=-{8.10}^{4}N/m\) |
\({K}_{xy}={9.10}^{4}N/m\) |
||
\({c}_{yy}=15.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=45.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={60.10}^{6}N/m\) |
\({K}_{xx}={15.10}^{7}N/m\) |
\({K}_{yx}=-{8.10}^{4}N/m\) |
\({K}_{xy}={8.10}^{4}N/m\) |
||
\({c}_{yy}=12.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=19.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
\(\Omega =5000\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={90.10}^{6}N/m\) |
\({K}_{xx}={50.10}^{7}N/m\) |
\({K}_{yx}=-{9.10}^{4}N/m\) |
\({K}_{xy}={9.10}^{4}N/m\) |
||
\({c}_{yy}=15.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=45.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={60.10}^{6}N/m\) |
\({K}_{xx}={15.10}^{7}N/m\) |
\({K}_{yx}=-{8.10}^{4}N/m\) |
\({K}_{xy}={8.10}^{4}N/m\) |
||
\({c}_{yy}=12.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=19.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
\(\Omega =6500\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={100.10}^{6}N/m\) |
\({K}_{xx}={40.10}^{7}N/m\) |
\({K}_{yx}=-{15.10}^{4}N/m\) |
\({K}_{xy}={15.10}^{4}N/m\) |
||
\({c}_{yy}=13.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=33.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={70.10}^{6}N/m\) |
\({K}_{xx}={14.10}^{7}N/m\) |
\({K}_{yx}=-{13.10}^{4}N/m\) |
\({K}_{xy}={13.10}^{4}N/m\) |
||
\({c}_{yy}=10.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=15.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=1.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{1.10}^{2}\mathit{Ns}/m\) |
\(\Omega =8000\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={110.10}^{6}N/m\) |
\({K}_{xx}={35.10}^{7}N/m\) |
\({K}_{yx}=-{20.10}^{4}N/m\) |
\({K}_{xy}={20.10}^{4}N/m\) |
||
\({c}_{yy}=11.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=26.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=2.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{2.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={80.10}^{6}N/m\) |
\({K}_{xx}={14.10}^{7}N/m\) |
\({K}_{yx}=-{20.10}^{4}N/m\) |
\({K}_{xy}={20.10}^{4}N/m\) |
||
\({c}_{yy}=9.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=13.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=2.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{2.10}^{2}\mathit{Ns}/m\) |
\(\Omega =10000\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={115.10}^{6}N/m\) |
\({K}_{xx}={33.10}^{7}N/m\) |
\({K}_{yx}=-{35.10}^{4}N/m\) |
\({K}_{xy}={35.10}^{4}N/m\) |
||
\({c}_{yy}=10.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=20.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=2.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{3.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={90.10}^{6}N/m\) |
\({K}_{xx}={14.10}^{7}N/m\) |
\({K}_{yx}=-{30.10}^{4}N/m\) |
\({K}_{xy}={30.10}^{4}N/m\) |
||
\({c}_{yy}=8.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=10.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=2.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{2.10}^{2}\mathit{Ns}/m\) |
\(\Omega =14000\mathrm{tr}/\min\)
Palier |
\(\mathrm{P1}\) |
\({K}_{yy}={120.10}^{6}N/m\) |
\({K}_{xx}={30.10}^{7}N/m\) |
\({K}_{yx}=-{70.10}^{4}N/m\) |
\({K}_{xy}={70.10}^{4}N/m\) |
||
\({c}_{yy}=7.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=15.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=3.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{4.10}^{2}\mathit{Ns}/m\) |
||
Palier |
\(\mathrm{P2}\) |
\({K}_{yy}={100.10}^{6}N/m\) |
\({K}_{xx}={14.10}^{7}N/m\) |
\({K}_{yx}=-{60.10}^{4}N/m\) |
\({K}_{xy}={60.10}^{4}N/m\) |
||
\({c}_{yy}=6.{10}^{4}\mathrm{Ns}/m\) |
\({c}_{xx}=8.{10}^{4}\mathit{Ns}/m\) |
||
\({c}_{yx}=3.{10}^{2}\mathit{Ns}/m\) |
\({c}_{xy}=-{3.10}^{2}\mathit{Ns}/m\) |
1.3. Conditions aux limites#
Pour bloquer les mouvements de type corps rigide dans la direction \(z\) , on bloque les degrés de liberté \(\mathit{DZ}\) et \(\mathit{DRZ}\) au nœud palier \(\mathrm{B1}\) .