# OS-V: 1000 Complex Eigenvalue Analysis of Rotor Bearing System

Rotor Bearing system is an excellent example of rotating machines used in mechanical engineering applications.

Analysis of this system to get unbalanced response, critical speed, resonance
frequency and vibration modes is important to evade the catastrophic failure of
these systems. Here the critical speed of a rotor bearing system using OptiStruct is verified.

^{1}## Model Files

Before you begin, copy the file(s) used in this problem
to your working directory.

## Benchmark Model

The finite element model, as shown in Figure 1 is constrained at all the nodes. Only DOF 1 and 4 are allowed on all the nodes. The model is meshed with beam elements of different sections (Figure 2). Mass is attached at node 5. An isotropic system is assumed.

## Material

The material properties are:

**Property****Value**- Young's modulus
- 207.8 GN/m
^{2} - Density
- 7806 kg/m
^{3}

Bearing (undamped and linear) with the following stiffness matrix are used in this
model.

- k22 = k33
- = 4.378 x e7 N/m
- k23 = k32
- = 0 N/m

Two different approaches are used in OptiStruct to input
the Bearing Stiffness in the model.

- DMIG
- The stiffness matrix of the bearing is defined directly in the model as multiple column entries using K2GG.
- GENEL
- A file (.inc) which contains the details of bearing stiffness is imported in the model.

The problem has been solved for Complex Eigenvalue Analysis (ASYNC).

## Results

The results are plotted over a range of spin speed for 12
different modes. The deformation of the rotor bearing system can be visualized in
HyperView by importing an .h3d
file.

Comparison of results at speed 70k RPM.

Mode | Speed (RPM) | Normalized Value | |
---|---|---|---|

Nelson McVaugh ^{1} |
OS ^{2} |
||

1 | 15470 | 15433.38 | 0.998 |

2 | 17159 | 17069.22 | 0.995 |

3 | 46612 | 46975.50 | 1.008 |

4 | 49983 | 50221.98 | 1.005 |

5 | 64752 | 65122.80 | 1.006 |

6 | 96547 | 92419.20 | 0.957 |

Mode | Speed (RPM) | Normalized Value | |
---|---|---|---|

Nelson McVaugh ^{1} |
OS ^{2} |
||

1 | 4015 | 4002.56 | 0.997 |

2 | 4120.20 | 4102.75 | 0.996 |

3 | 11989.25 | 12063.20 | 1.006 |

4 | 12200 | 12267.90 | 1.006 |

5 | 18184.25 | 18353.40 | 1.009 |

6 | 20162.25 | 20116.80 | 0.998 |

Mode | Speed (RPM) | Normalized Value | |
---|---|---|---|

Nelson McVaugh ^{1} |
OS ^{2} |
||

1 | 15858 | 15810.3 | 0.996992054 |

2 | 16700 | 16626.42 | 0.995594012 |

3 | 47520 | 47853 | 1.007007576 |

4 | 49204 | 49476.78 | 1.005482888 |

5 | 69640 | 70074 | 1.006232051 |

6 | 85552 | 84773.4 | 0.990899102 |

Mode | Speed (RPM) | Normalized Value | |
---|---|---|---|

Nelson McVaugh ^{1} |
OS ^{2} |
||

1 | 14758 | 14731.2 | 0.998184036 |

2 | 18148 | 18027.54 | 0.993362354 |

3 | 44695 | 45108.24 | 1.009245777 |

4 | 51430 | 51627.48 | 1.003839782 |

5 | 58424 | 58599.52 | 1.003004245 |

6 | 111455 | 101374.8 | 0.909558118 |

Here, you have verified that the critical speeds obtained by OptiStruct for various whirl ratios are a close match with
those mentioned in the Nelson McVaugh Paper.

**Nomenclature**- Critical Speed
- The angular speed of a rotor that matches one of its natural frequencies.
- Whirl Ratio
- Ratio of whirl speed to spin speed.
- Campbell Diagram
- The plot of natural frequencies of the system as functions of the spin speeds.

^{1}Nelson,H.D. and McVaugh, J.M. (1976) The Dynamics of Rotor-Bearing Systems Using Finite Elements. ASME Journal of Engineering for Industry, 98,593-600

^{2}Critical speed values calculated by OptiStruct