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Research Paper: Gas Turbines: Structures and Dynamics

Nonparametric Stochastic Modeling of Structural Uncertainty in Rotordynamics: Unbalance and Balancing Aspects

[+] Author and Article Information
Raghavendra Murthy

Mem. ASME
SEMTE,
Faculties of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287
e-mail: rnmurthy@asu.edu

Joseph C. Tomei

SEMTE,
Faculties of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287
e-mail: jctomei@asu.edu

X. Q. Wang

Mem. ASME
SEMTE,
Faculties of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287
e-mail: xiaoquan.wang.1@asu.edu

Marc P. Mignolet

Fellow ASME
SEMTE,
Faculties of Mechanical and Aerospace Engineering,
Arizona State University,
Tempe, AZ 85287
e-mail: marc.mignolet@asu.edu

Aly El-Shafei

Mem. ASME
Department of Mechanical
Design and Production,
Cairo University,
Giza 12316, Egypt
e-mail: elshafei@gmail.com

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received November 3, 2013; final manuscript received December 3, 2013; published online February 11, 2014. Editor: David Wisler.

J. Eng. Gas Turbines Power 136(6), 062506 (Feb 11, 2014) (12 pages) Paper No: GTP-13-1394; doi: 10.1115/1.4026166 History: Received November 03, 2013; Revised December 03, 2013

This paper focuses on extending an earlier investigation on the systematic and rational consideration of uncertainty in reduced order models of rotordynamics systems. The current effort concentrates on the consistent introduction of uncertainty in mass properties on the modal mass and gyroscopic matrices and on the unbalance force vector. The uncertainty in mass is separated into uncertainty that maintains the rotor symmetry and the one which disrupts it. Both types of uncertainties lead to variations in the system modal matrices but only the latter induces an unbalance. Accordingly, the approach permits the selection of separate levels on the uncertainty on the system properties (e.g., natural frequencies) and on the unbalance. It was first found that the unbalanced response is increased by considering the uncertainty in the rotor modal mass matrices. It was next noted that the approach presented not only permits the analysis of uncertain rotors but it also provides a computational framework for the assessment of various balancing strategies. To demonstrate this unique feature, a numerical experiment was conducted in which a population of rotors were balanced at low speed and their responses were predicted at their first critical speed. These response predictions were carried with the uncertainty in the system modal mass matrices but with or without the balancing weights effects on these matrices. It was found that the balancing at low speed may, in fact, lead to an increase in both the mean and 95th percentile of the response at critical speed.

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References

Figures

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Fig. 1

Structure of the random H¯¯ matrices (figures for n = 8, i = 2, and λ = 1 and 10)

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Fig. 2

Example rotor: (1) electric motor, (2) flexible coupling, (3) bearing housing, (4) steel disks, and (5) steel shaft

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Fig. 3

Campbell diagrams of the rotor on isotropic bearings: (a) imaginary part in the fixed frame, (b) imaginary part in the rotating frame, and (c) common real part

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Fig. 4

Campbell diagram corresponding to both symmetric and asymmetric uncertainty in the rotor mass (blue), and the mean rotor (red): (a) imaginary part of eigenvalues, and (b) real part of eigenvalues

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Fig. 5

Campbell diagram corresponding to symmetric uncertainty only in the rotor mass (blue), and the mean rotor (red): (a) imaginary part of eigenvalues, and (b) real part of eigenvalues

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Fig. 6

Campbell diagram corresponding to uncertainty in the rotor stiffness (blue), and the mean rotor (red): (a) imaginary part of eigenvalues, and (b) real part of eigenvalues

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Fig. 7

Probability density functions of the normalized average of the first two rotor-alone natural frequencies; uncertainties in rotor stiffness (symmetric and asymmetric, from Ref. [3]) and uncertainty in mass

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Fig. 8

Standard deviations of the average of the natural frequencies of the first two flexible modes and unbalance level me1 for combinations of δS and δNS

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Fig. 9

Forced unbalanced response at the last overhung disk shown for mean and uncertain rotors

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Fig. 10

The 95th percentile of the forced unbalanced response at the last overhung disk shown for mean rotors (“Mean”) and uncertain rotors with both asymmetric and symmetric uncertainty (“Both Unc.”), symmetric uncertainty only (same λS, “Symm. Unc.”), and increased symmetric uncertainty to match variability on the natural frequencies (lower λS, “Symm. Eq. Var.”)

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Fig. 11

Forced unbalanced response at the last overhung disk; uncertain rotors before and after low speed balancing

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Fig. 12

Forced unbalanced response at the last overhung disk; mean rotors before and after low speed balancing

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Fig. 13

Left eigenvector; first critical speed

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Fig. 14

Forced unbalanced response at the last overhung disk; balanced uncertain rotors with and without updating the modal matrices with the balancing masses effects

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Fig. 15

Left eigenvector; second critical speed. Stiffer bearings.

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Fig. 16

Forced unbalanced response at the last overhung disk shown for mean and uncertain rotors. Second critical speed; stiffer bearings.

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Fig. 17

The 95th percentile of the forced unbalanced response at the last overhung disk shown for mean rotors (“Mean”) and uncertain rotors with both asymmetric and symmetric uncertainty (“Both Unc.”), symmetric uncertainty only (same λS, “Symm. Unc.”), and increased symmetric uncertainty to match variability on natural frequencies (lower λS, “Symm. Eq. Var.”). Second critical speed; stiffer bearings.

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Fig. 18

Forced unbalanced response at the last overhung disk; uncertain rotors before and after low speed balancing. Second critical speed, stiffer bearings.

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