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Research Papers

Flutter Amplitude Saturation by Nonlinear Friction Forces: Reduced Model Verification

[+] Author and Article Information
Carlos Martel

E.T.S.I. Aeronáuticos,
Universidad Politécnica de Madrid,
Madrid 28040, Spain
e-mail: Carlos.Martel@upm.es

Roque Corral

Technology and Methods Department,
Industria de TurboPropulsores S.A.,
Madrid 28108, Spain
e-mail: Roque.Corral@itp.es

Rahul Ivaturi

E.T.S.I. Aeronáuticos,
Universidad Politécnica de Madrid,
Madrid 28040, Spain
e-mail: Rahul.Ivaturi@upm.es

1Corresponding author.

2Present address: Associate Professor at Department of Propulsion and Themofluid Dynamics at the School of Aeronautics, UPM, Madrid 28040, Spain.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received July 25, 2014; final manuscript received August 25, 2014; published online October 28, 2014. Editor: Ronald Bunker.

J. Turbomach 137(4), 041004 (Oct 28, 2014) (8 pages) Paper No: TURBO-14-1178; doi: 10.1115/1.4028443 History: Received July 25, 2014; Revised August 25, 2014

The computation of the final, friction saturated limit cycle oscillation amplitude of an aerodynamically unstable bladed-disk in a realistic configuration is a formidable numerical task. In spite of the large numerical cost and complexity of the simulations, the output of the system is not that complex: it typically consists of an aeroelastically unstable traveling wave (TW), which oscillates at the elastic modal frequency and exhibits a modulation in a much longer time scale. This slow time modulation over the purely elastic oscillation is due to both the small aerodynamic effects and the small nonlinear friction forces. The correct computation of these two small effects is crucial to determine the final amplitude of the flutter vibration, which basically results from its balance. In this work, we apply asymptotic techniques to consistently derive, from a bladed-disk model, a reduced order model that gives only the time evolution on the slow modulation, filtering out the fast elastic oscillation. This reduced model is numerically integrated with very low computational cost, and we quantitatively compare its results with those from the bladed-disk model. The analysis of the friction saturation of the flutter instability also allows us to conclude that: (i) the final states are always nonlinearly saturated TW; (ii) depending on the initial conditions, there are several different nonlinear TWs that can end up being a final state; and (iii) the possible final TWs are only the more flutter prone ones.

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References

Corral, R., and Gallardo, J. M., 2008, “Verification of the Vibration Amplitude Prediction of Self-Excited LPT Rotor Blades Using a Fully Coupled Time-Domain Non-Linear Method and Experimental Data,” ASME Paper No. GT2008-51416. [CrossRef]
Corral, R., and Gallardo, J. M., 2014, “Nonlinear Dynamics of Bladed Disks With Multiple Unstable Modes,” AIAA J., 52(6), pp. 1124–1132. [CrossRef]
Schwingshackl, C. W., Petrov, E. P., and Ewins, D. J., 2012, “Effects of Contact Interface Parameters on Vibration of Turbine Bladed Disks With Underplatform Dampers,” ASME J. Eng. Gas Turbines Power, 134(3), p. 032507. [CrossRef]
Petrov, E., Zachariadis, Z., Beretta, A., and Elliot, R., 2012, “A Study of Nonlinear Vibration in a Frictionally-Damped Turbine Bladed Disk With Comprehensive Modelling of Aerodynamics Effects,” ASME Paper No. GT2012-69052. [CrossRef]
Sinha, A., and Griffin, J. H., 1985, “Effects of Friction Dampers on Aerodynamically Unstable Rotor Stages,” AIAA J., 23(2), pp. 262–270. [CrossRef]
Martel, C., and Corral, R., 2013, “Flutter Amplitude Saturation by Nonlinear Friction Forces: An Asymptotic Approach,” ASME Paper No. GT2013-94068. [CrossRef]
Corral, R., Gallardo, J. M., and Ivaturi, R., 2013, “Conceptual Analysis of the Non-Linear Forced Response of Aerodynamically Unstable Bladed Disks,” ASME Paper No. GT2013-94851. [CrossRef]
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Figures

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Fig. 1

Typical time evolution of the vibration amplitude (from Ref. [2]), the thick line approximately marks the slow time envelope over the fast elastic cycle

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Fig. 2

Simplified mass-spring model of a bladed-disk

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Fig. 3

Sketch of the friction load versus displacement hysteresis loop of the Olofsson model (P: preloading, L: loading, U: unloading)

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Fig. 4

Nondimensional aerodynamic frequency correction (top) and critical damping ratio (bottom)

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Fig. 5

Time evolution of the TW amplitudes of the solution of Eq. (4) starting from a random initial condition. The final surviving TW has ND = 7.

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Fig. 6

Time evolution of the TW amplitudes of the solution of Eq. (4) starting from a random initial condition. The final surviving TW has ND = 6.

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Fig. 7

Time evolution of the TW amplitudes of the solution of Eq. (4) starting from a pure TW with ND = 12 initial condition. The final surviving TW has ND = 7.

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Fig. 8

Time evolution of the TW amplitudes of the solution of Eq. (4) starting from a pure TW with ND = 3 initial condition. The final surviving TW has ND = 9.

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Fig. 9

Complex friction coefficient for the Olofsson microslip model (circles: real part, squares: imaginary part)

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Fig. 10

Time evolution of the TW amplitudes of the solution of Eq. (30). The initial condition corresponds to a random distribution, and the final surviving tw has ND = 6.

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Fig. 11

Time evolution of the TW amplitudes of the solution of Eq. (30). The initial condition corresponds to a random distribution, and the final surviving TW has ND = 8.

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Fig. 12

Limit cycle oscillation of blade #1 for the TW with ND = 8. Full model: results from the bladed-disk model given by Eq. (4) with θ = 0.1. ROM: results from the reduced order model given by Eq. (30).

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Fig. 13

Limit cycle oscillation of blade #1 for the TW with ND = 7. Full model: results from the bladed-disk model given by Eq. (4) with various values of θ = 0.1, 0.01, and 0.001. ROM: results from the reduced order model given by Eq. (30).

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