# OS-V: 1010 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

## Benchmark Model

The finite element model, as shown in 図 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 (図 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 following stiffness matrix are used in this model.
k22 = k33
= 3.503 x e7 N/m
k23 = k32
= -8.756 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).

Compare the whirl speeds at spin speed being 100,000 RPM.

## 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 100,000 RPM.

Mode Whirl Speed (RPM) Normalized Value
Nelson McVaugh 1 OS 2
1 (BW) 10747 10764 1.002
2 (FW) 19665 19518 0.993
3 (BW) 39077 39300 1.006
4 (FW) 47549 47964 1.009
5 (BW) 55079 55338 1.005
6 (FW) 94510 91680 0.970
Here, you have verified that the whirl speeds obtained by OptiStruct for various modes are a close match with those mentioned in the Nelson McVaugh Paper.
Nomenclature
Whirl Speed
The damped natural frequency of the rotor.
Backward Whirl (BW) and Forward Whirl (FW)
At zero shaft speed, the forward and backward frequencies are identical (repeated eigenvalues). As speed increases, each vibration mode is split into two modes, known as forward and backward precision modes, due to gyroscopic effect.

## Model Files

The model file used in this problem includes:

Rotor_Bearing_2.fem

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