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Research Papers

A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part I: Theoretical Basis

[+] Author and Article Information
O. G. McGee, III

Professor
Howard University,
Washington, D. C. 20059

C. Fang

Postdoctoral Research Associate
Howard University,
Washington, D. C. 20059

Y. El-Aini

Senior Fellow
Pratt and Whitney Rocketdyne,
West Palm Beach, FL 33401

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received December 16, 2005; final manuscript received January 1, 2011; published online September 13, 2013. Assoc. Editor: Matthew Montgomery.

J. Turbomach 135(6), 061001 (Sep 13, 2013) (20 pages) Paper No: TURBO-05-1153; doi: 10.1115/1.4004445 History: Received December 16, 2005; Revised January 01, 2011

In this paper, a reduced order model for the vibrations of bladed disk assemblies was achieved. The system studied was a 3D annulus of shroudless, “custom-tailored,” mistuned blades attached to a flexible disk. Specifically, the annulus was modeled as a spectral-based “meshless” continuum structure utilizing only nodal data to describe the arbitrary volume in which the system's dynamical energy was minimized. An extended Ritz variational procedure was used to minimize this energy, subjected to constraints imposed by an assumed 3D displacement field of mathematically complete, orthonormal “blade-disk” polynomials multiplied by generalized coefficients. The coefficients were determined by constraining the polynomial series to satisfy the extended Ritz stationary equations and essential boundary conditions of the bladed disk. From this, the governing equations of motion were generated into their usual dynamical forms to calculate upper-bounds on the actual free and forced responses of bladed disks. No conventional finite elements and element connectivity or component substructuring data were needed. This paper, Part I, outlines the theoretical foundation of the present model, and through extensive Monte Carlo simulations, establishes the analytical basis, predictive accuracy, and re-analysis efficiency of the present technology in the prediction of 3D maximum response amplitude of mistuned bladed disks having increasing numbers of nodal diameter excitations. Further applications validating the 3D approach against conventional finite element procedures of free and forced response prediction of a mistuned Integrally-Bladed Rotor used in practice is presented in a companion paper, Part II (Fang, McGee, and El-Aini, 2013, “A Reduced-Order Meshless Energy Model for the Vibrations of Mistuned Bladed Disks—Part II: Finite Element Benchmark Comparisons, ASME J. Turbomach., to be published.

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Topics: Disks , Blades , Polynomials
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Figures

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Fig. 1

Structural model of a mistuned bladed disk

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Fig. 2

Gauss quadrature points for numerical volumetric integration

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Fig. 6

Representative Campbell diagram [83] of a typical IBR in modern aeroengines showing definitions of flutter frequencies, integral order forced response (F.R.) resonant frequencies, separated flow vibration (SFV) frequencies, and nonsynchronous vibration (NSV) (or nonintegral order response) frequencies (see Refs. [4,5,8-5,8].)

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Fig. 3

Idealized 20-bladed disk

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Fig. 4

Modern aeroengine featuring fan integrally bladed rotors (IBRs) (Pratt and Whitney, private communication of third author, Y. El-Aini)

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Fig. 5

Integrally bladed rotor (IBR)

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Fig. 7

Campbell diagram [83] of a 3D ROME idealized 20-bladed disk (shown in Fig. 3), tuned resonant speeds (left), mistuned at σ = 5% resonant speeds (right) (Ω = 10,700 rpm)

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Fig. 8

Campbell diagram [83] of a I-DEAS FEM idealized 20-bladed disk (shown in Fig. 3), tuned resonant speeds (left), mistuned at σ = 5% resonant speeds (right) (Ω = 10,700 rpm)

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Fig. 9

Campbell diagram [83] of a 3D ROME model of an industry IBR (shown in Fig. 5), tuned resonant speeds (left), mistuned at σ = 5% resonant speeds (right) (Ω = 10,700 rpm)

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Fig. 10

Statistical results and parameters

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Fig. 11

Maximum response amplitude magnification with increasing IBR solidity (see Kenyon and Griffin [35]; Whitehead [37])

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