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Research Papers

Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

[+] Author and Article Information
Y. F. Chen

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600 
e-mail: yfcheng@uw.edu

I. Y. Shen

Professor
Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600
e-mail: ishen@uw.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 9, 2013; final manuscript received February 24, 2015; published online March 23, 2015. Assoc. Editor: Yukio Ishida.

J. Vib. Acoust 137(4), 041007 (Aug 01, 2015) (13 pages) Paper No: VIB-13-1236; doi: 10.1115/1.4029945 History: Received July 09, 2013; Revised February 24, 2015; Online March 23, 2015

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.

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References

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Figures

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Fig. 1

Finite element model of a cyclic symmetric rotor

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Fig. 2

Discrete eigenvalue (natural frequency) loci

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Fig. 3

Family map of the cyclic symmetric system

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Fig. 4

Contour plot of mode shapes with 1 nodal diameter

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Fig. 5

Mistune patterns with standard deviation 5%

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Fig. 6

Visual method for frequency regions (a)–(d), respectively

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Fig. 7

Mode shape of mistuned mode 87 in numerical analysis set 1

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Fig. 8

Mode shape of mistuned mode 137 in numerical analysis set 1

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Fig. 9

Visual method for numerical analysis set 2: L∧ in the regions (c) and (d), respectively

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Fig. 10

Mode shape of mistuned mode 94 in numerical analysis set 2

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Fig. 11

Mode shape of mistuned mode 140 in numerical analysis set 2

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Fig. 12

Visual method for numerical analysis set 3: L∧ in the regions (c) and (d), respectively

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Fig. 13

Visual method for numerical analysis set 4: L∧ in the regions (c) and (d), respectively

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Fig. 14

Mode shape of mistuned mode 90 in numerical analysis set 4

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Fig. 15

Mode shape of mistuned mode 137 in numerical analysis set 4

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