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Research Papers: Gas Turbines: Structures and Dynamics

# Nonparametric Stochastic Modeling of Uncertainty in Rotordynamics—Part II: Applications

[+] Author and Article Information
Raghavendra Murthy, Marc P. Mignolet

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106

Aly El-Shafei

Department of Mechanical Design and Production, Cairo University, Giza 12316, Egypt

J. Eng. Gas Turbines Power 132(9), 092502 (Jun 07, 2010) (11 pages) doi:10.1115/1.3204650 History: Received March 24, 2009; Revised March 25, 2009; Published June 07, 2010; Online June 07, 2010

## Abstract

In the first part of this series, a comprehensive methodology was proposed for the consideration of uncertainty in rotordynamic systems. This second part focuses on the application of this approach to a simple, yet representative, symmetric rotor supported by two journal bearings exhibiting linear, asymmetric properties. The effects of uncertainty in rotor properties (i.e., mass, gyroscopic, and stiffness matrices) that maintain the symmetry of the rotor are first considered. The parameter $λ$ that specifies the level of uncertainty in the simulation of stiffness and mass uncertain properties (the latter with algorithm I) is obtained by imposing a standard deviation of the first nonzero natural frequency of the free nonrotating rotor. Then, the effects of these uncertainties on the Campbell diagram, eigenvalues and eigenvectors of the rotating rotor on its bearings, forced unbalance response, and oil whip instability threshold are predicted and discussed. A similar effort is also carried out for uncertainties in the bearing stiffness and damping matrices. Next, uncertainties that violate the asymmetry of the present rotor are considered to exemplify the simulation of uncertain asymmetric rotors. A comparison of the effects of symmetric and asymmetric uncertainties on the eigenvalues and eigenvectors of the rotating rotor on symmetric bearings is finally performed to provide a first perspective on the importance of uncertainty-born asymmetry in the response of rotordynamic systems.

Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

Sample rotor

Figure 2

Campbell diagram of the rotor of Fig. 1. (a) Imaginary part of eigenvalues. (b) Real part of eigenvalues.

Figure 3

Relative errors in (a) imaginary and (b) real parts of the eigenvalues corresponding to the first forward and first backward modes

Figure 4

Probability density function of the normalized rotor alone natural frequencies (first four nonzero and nonrepeated); uncertainty in stiffness

Figure 5

Campbell diagrams corresponding to 1000 stiffness matrices. (a) Imaginary part of eigenvalues. (b) Real part of eigenvalues.

Figure 6

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, second bearing mode, and uncertainty in stiffness

Figure 7

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, second backward mode, and uncertainty in stiffness

Figure 8

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, second forward mode, and uncertainty in stiffness

Figure 9

Probability density function of the normalized eigenvalue real part for different rotor speeds, first forward mode, and uncertainty in stiffness

Figure 10

Probability density function of the eigenvector deviation Δj for different rotor speeds, first backward mode, and uncertainty in stiffness

Figure 11

Probability density function of the eigenvector deviation Δj for different rotor speeds, second backward mode, and uncertainty in stiffness

Figure 12

Probability density function of the whip instability speed, uncertainty in stiffness (“Stiffness”), in mass/gyroscopic (“Mass”), and bearing (“Bearing”) properties

Figure 13

Forced unbalance response at the last overhung disk, uncertainty in stiffness

Figure 14

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, second bearing mode, and uncertainty in mass/gyroscopic matrices

Figure 15

Probability density function of the normalized eigenvalue real part for different rotor speeds, first forward mode, and uncertainty in mass/gyroscopic matrices

Figure 16

Probability density function of the eigenvector deviation Δj for different rotor speeds, first backward mode, and uncertainty in mass/gyroscopic matrices

Figure 17

Probability density function of the eigenvector deviation Δj at 800 and 3000 rpm, second forward mode, uncertainty in stiffness (K), and uncertainty in mass/gyroscopic matrices (MGi,λG, i is the simulation algorithm number, 1 or 2)

Figure 18

Probability density function of the eigenvector deviation Δj at 800 rpm, first backward mode, uncertainty in stiffness (K), and uncertainty in mass/gyroscopic matrices (MGi,λG, i is the simulation algorithm number, 1 or 2)

Figure 19

Probability density function of the eigenvector deviation Δj at 3000 rpm, first backward mode, uncertainty in stiffness (K), and uncertainty in mass/gyroscopic matrices (MGi,λG, i is the simulation algorithm number, 1 or 2)

Figure 20

Evolution versus rotor speed of the mean model values of KB,xz and of the mean and mean ±1 standard deviation of the simulated values of this coefficient. (a) Bearing 1. (b) Bearing 2.

Figure 21

Campbell diagrams corresponding to 1000 bearing stiffness and damping matrices. (a) Imaginary part of eigenvalues. (b) Real part of eigenvalues.

Figure 22

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, first bearing mode, and uncertainty in bearing properties.

Figure 23

Probability density function of the eigenvector deviation Δj for different rotor speeds, first bearing mode, and uncertainty in bearing properties

Figure 24

Probability density function of the normalized eigenvalue magnitude for different rotor speeds, first backward mode, and uncertainty in bearing properties

Figure 25

Probability density function of the eigenvector deviation Δj for different rotor speeds, first forward mode, and uncertainty in bearing properties

Figure 26

Isotropic bearing stiffness (a) and damping (b) coefficients versus rotor speed

Figure 27

Campbell diagrams of the rotor on isotropic bearings. (a) Imaginary part in the fixed frame. (b) Imaginary part in the rotating frame. (c) Common real part.

Figure 28

Probability density functions of the normalized deviations of the average of the first two rotor alone natural frequencies; symmetric and asymmetric uncertainties in stiffness

Figure 29

Campbell diagrams corresponding to 1000 asymmetric stiffness matrices. (a) Imaginary part of eigenvalues. (b) Real part of eigenvalues.

Figure 30

Probability density function of the absolute deviation in eigenvalue imaginary part at 2200 rpm for mode I, asymmetric and symmetric uncertainties in stiffness

Figure 31

Probability density function of the absolute deviation in eigenvalue imaginary part at 2590 rpm for mode I, asymmetric and symmetric uncertainties in stiffness

Figure 32

Probability density function of the absolute deviation in eigenvalue imaginary part at 3260 rpm for mode III, asymmetric and symmetric uncertainties in stiffness

Figure 33

Probability density function of the absolute deviation in eigenvalue real part at 2590 rpm for mode I, asymmetric and symmetric uncertainties in stiffness

Figure 34

Probability density function of the eigenvector deviation at 2590 rpm for mode I, asymmetric and symmetric uncertainties in stiffness

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