# 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/m2
Density
7806 kg/m3
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.
Table 1. Whirl/Critical Speeds Comparison for Whirl Ratio 1 (Engine Order-1X)
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
Table 2. Critical Speeds Comparison for Whirl Ratio 0.25 (Engine Order-4X)
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
Table 3. Critical Speeds Comparison for Whirl Ratio 0.5 (Engine Order-2X)
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
Table 4. Critical Speeds Comparison for Whirl Ratio 2 (Engine Order-0.5X)
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.
Used for estimating the critical speed and resonance frequencies.
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