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Research Papers

Enhanced Campbell Diagram With the Concept of $H∞$ in Rotating Machinery: Lee Diagram

[+] Author and Article Information
Chong-Won Lee1

Department of Mechanical Engineering, Center for Noise and Vibration Control (NOVIC), KAIST, Daejeon 305-701, Koreacwlee@kaist.ac.kr

Yun-Ho Seo

Department of Mechanical Engineering, Center for Noise and Vibration Control (NOVIC), KAIST, Daejeon 305-701, Koreayunho@kaist.ac.kr

1

Corresponding author.

J. Appl. Mech 77(2), 021012 (Dec 14, 2009) (12 pages) doi:10.1115/1.3173610 History: Received March 24, 2008; Revised April 19, 2009; Published December 14, 2009; Online December 14, 2009

Abstract

The Campbell diagram, a frequency-speed diagram, has been widely used for prediction of possible occurrence of resonances in the phase of design and operation of rotating machinery since its advent in 1920s. In this paper, a set of new frequency-speed diagrams, which is referred to as the Lee diagram, is newly proposed, where the conventional Campbell diagram is incorporated with the concept of the infinity norm of directional frequency response matrix (dFRM) associated with a rotor with rotating and stationary asymmetry in general. The dFRM is constructed based on complete modal analysis of a linear periodically time-varying rotor model formulated in the complex coordinates. It is shown that the Lee diagram is powerful in that it can identify the modes of symmetry, rotating and stationary asymmetry, and extract only a few critical resonances out of the, otherwise, overcrowded ones without a measure of priority as in the Campbell diagram. In order to demonstrate the power of the Lee diagram in design and operation of rotating machines, three examples are treated: a typical anisotropic rigid rotor, a simple general rotor, and a two-pole generator.

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Figures

Figure 1

An anisotropic rigid rotor

Figure 2

Campbell diagram for the anisotropic rotor: unbalance force (----); F(B): forward (backward) mode; under-bar: complex conjugate mode

Figure 3

(a) Campbell diagram incorporated with modal strength, (b) complex, and (c) real modal vector norm for the anisotropic rigid rotor

Figure 4

(a) Normal and (b) reverse dFRFs, and (c) FRFs for the anisotropic rigid rotor: —dFRFs or FRFs; ∙∙∙∙∙∙∙Lee diagram; —Upper bound to Lee diagram

Figure 5

Lee diagram for the anisotropic rigid rotor (a) L11(ω,Ω) and (b) L12(ω,Ω)

Figure 6

Conventional Campbell diagram for the simple general rotor

Figure 7

Lee diagram for the simple general rotor (a) L11(ω,Ω), (b) L12(ω,Ω), (c) L21(ω,Ω), (d) L22(ω,Ω), and (e) three-dimensional L11(ω,Ω)

Figure 8

Commercial two-pole generator

Figure 9

Conventional Campbell diagram for two-pole generator

Figure 10

Campbell diagram incorporated with modal strength for two-pole generator

Figure 11

Lee diagram (upper bound) for two-pole generator (a) L̂11(ω,Ω), (b) L̂12(ω,Ω), (c) L̂21(ω,Ω), and (d) L̂22(ω,Ω)

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