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Research Papers

Uncertainty Quantification of a Nonlinear Aeroelastic System Using Polynomial Chaos Expansion With Constant Phase Interpolation

[+] Author and Article Information
Ajit Desai

Department of Aerospace Engineering,
IIT Madras,
Chennai 600036, India

Jeroen A. S. Witteveen

Center for Turbulence Research,
Stanford University,
Stanford, CA 94305

Sunetra Sarkar

Associate Professor
Department of Aerospace Engineering,
IIT Madras,
Chennai 600036, India
e-mail: sunetra.sarkar@gmail.com

1Present address: Department of Civil and Environmental Engineering, Carleton University, Ottawa ON K1S 5B6, Canada.

2Present address: Scientific Staff Member, Center for Mathematics and Computer Science, Amsterdam, The Netherlands.

3Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 16, 2011; final manuscript received June 3, 2013; published online July 9, 2013. Assoc. Editor: Bogdan Epureanu.

J. Vib. Acoust 135(5), 051034 (Jul 09, 2013) (13 pages) Paper No: VIB-11-1048; doi: 10.1115/1.4024794 History: Received March 16, 2011; Revised June 03, 2013

The present study focuses on the uncertainty quantification of an aeroelastic instability system. This is a classical dynamical system often used to model the flow induced oscillation of flexible structures such as turbine blades. It is relevant as a preliminary fluid-structure interaction model, successfully demonstrating the oscillation modes in blade rotor structures in attached flow conditions. The potential flow model used here is also significant because the modern turbine rotors are, in general, regulated in stall and pitch in order to avoid dynamic stall induced vibrations. Geometric nonlinearities are added to this model in order to consider the possibilities of large twisting of the blades. The resulting system shows Hopf and period-doubling bifurcations. Parametric uncertainties have been taken into account in order to consider modeling and measurement inaccuracies. A quadrature based spectral uncertainty tool called polynomial chaos expansion is used to quantify the propagation of uncertainty through the dynamical system of concern. The method is able to capture the bifurcations in the stochastic system with multiple uncertainties quite successfully. However, the periodic response realizations are prone to time degeneracy due to an increasing phase shifting between the realizations. In order to tackle the issue of degeneracy, a corrective algorithm using constant phase interpolation, which was developed earlier by one of the authors, is applied to the present aeroelastic problem. An interpolation of the oscillatory response is done at constant phases instead of constant time and that results in time independent accuracy levels.

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Figures

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Fig. 1

The schematic of a symmetric airfoil with pitch and plunge degrees-of-freedom

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Fig. 2

Uncertain βα and ω¯: stochastic bifurcation plot

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Fig. 3

Uncertain βα and ω¯: convergence behavior of the PCE at U = 6.8 at nondimensional time t = 2000

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Fig. 4

Uncertain βα and ω¯: a typical time history with a 25th order PCE at U = 6.8

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Fig. 5

Uncertain βα and ω¯: five different realizations of the time history with a 25th order PCE at U = 6.8

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Fig. 6

Uncertain βα and ω¯: comparison of the PDFs obtained by the PCE (25th order) and the MCS with U = 6.8 at the nondimensional time (a) t = 2000, (b) t = 4000, (c) t = 6000, and (d) t = 8000

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Fig. 7

Uncertain βα and ω¯: PCE coefficients at U = 6.8

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Fig. 8

Uncertain βα and ω¯: PCE coefficients at U = 6.8

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Fig. 9

Uncertain βα and ω¯: PCE coefficients at U = 6.8

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Fig. 10

Uncertain βα and ω¯: PCE coefficients at U = 6.8

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Fig. 11

Uncertain βα and ω¯: LCO amplitude response PDF as a function of reduced velocity (a),(b) around the first deterministic bifurcation point, and (c),(d) around the second deterministic bifurcation point

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Fig. 12

Uncertain βα and ζα: stochastic bifurcation plot

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Fig. 13

Uncertain βα and ζα: a typical time history with the 25th PCE at U = 6.8

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Fig. 14

Uncertain βα and ζα: five different realizations of the time history with a 25th order PCE at U = 6.8

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Fig. 15

Uncertain βα and ζα: comparison of the PDFs obtained by the PCE (25th order) and the MCS with U = 6.8 at the nondimensional time (a) t = 4000, (b) t = 6000, and (c) t = 8000

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Fig. 16

Uncertain βα and ζα: LCO amplitude response PDF as a function of the reduced velocity about the second bifurcation point (a) 3D, and (b) 2D

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Fig. 17

Oscillatory samples as a function of time and phase (a) samples vk(t), and (b) samples v∧k(φ)

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Fig. 18

Uncertain βα and ω¯: comparison of the time histories of the mean of the pitch angle (α) by the MCS, PCE, and constant phase at U = 6.8

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Fig. 19

Uncertain βα and ω¯: comparison of the time histories of the standard deviation of the pitch angle (α) by the MCS, PCE, and constant phase at U = 6.8

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Fig. 20

Uncertain βα and ω¯: comparison of the response PDF by the MCS, PCE, and constant phase for U = 6.8 at time level (a) 6000, and (b) 8000

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Fig. 21

Uncertain βα and ω¯: Comparison of the time histories of the pitch angle (α) by the MCS, PCE (25th order), and constant phase at U = 6.8

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Fig. 22

Uncertain βα and ζα: comparison of the PDFs obtained by the MCS, PCE, and constant phase with five samples, with U = 6.8 at the nondimensional time (a) t = 6000, and (b) t = 8000

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