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Research Papers

Uncertainty Quantification Using Generalized Polynomial Chaos Expansion for Nonlinear Dynamical Systems With Mixed State and Parameter Uncertainties

[+] Author and Article Information
Rajnish Bhusal

Department of Mechanical and
Aerospace Engineering,
The University of Texas at Arlington,
Arlington, TX 76019
e-mail: rajnish.bhusal@mavs.uta.edu

Kamesh Subbarao

Department of Mechanical and
Aerospace Engineering,
The University of Texas at Arlington,
Arlington, TX 76019
e-mail: subbarao@uta.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 31, 2018; final manuscript received September 10, 2018; published online January 7, 2019. Assoc. Editor: Yan Wang.

J. Comput. Nonlinear Dynam 14(2), 021011 (Jan 07, 2019) (14 pages) Paper No: CND-18-1245; doi: 10.1115/1.4041473 History: Received May 31, 2018; Revised September 10, 2018

This paper develops a framework for propagation of uncertainties, governed by different probability distribution functions in a stochastic dynamical system. More specifically, it deals with nonlinear dynamical systems, wherein both the initial state and parametric uncertainty have been taken into consideration and their effects studied in the model response. A sampling-based nonintrusive approach using pseudospectral stochastic collocation is employed to obtain the coefficients required for the generalized polynomial chaos (gPC) expansion in this framework. The samples are generated based on the distribution of the uncertainties, which are basically the cubature nodes to solve expectation integrals. A mixture of one-dimensional Gaussian quadrature techniques in a sparse grid framework is used to produce the required samples to obtain the integrals. The familiar problem of degeneracy with high-order gPC expansions is illustrated and insights into mitigation of such behavior are presented. To illustrate the efficacy of the proposed approach, numerical examples of dynamic systems with state and parametric uncertainties are considered which include the simple linear harmonic oscillator system and a two-degree-of-freedom nonlinear aeroelastic system.

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Figures

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Fig. 1

Mixed sparse grid nodes of k=4 for d=3 of Z=[Z1,Z2,Z3], where Z1∼U[−1,  1], Z2∼U[−1,  1], and Z3∼N(0,12): (a) nodes on [Z1, Z2], (b) nodes on [Z1, Z3], (c) nodes on [Z2, Z3], and (d) sparse grid nodes on Z = [Z1, Z2, Z3]

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Fig. 2

Process flowchart of MSG-based gPC expansion

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Fig. 3

Probability density estimate of position of the system at various time instants: (a) probability density estimate of x1 at t = 10 s, (b) probability density estimate of x1 at t = 15 s, and (c) probability density estimate of x1 at t = 25 s

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Fig. 4

Probability density estimate of velocity of the system at various time instants: (a) probability density estimate of x2 at t = 10 s, (b) probability density estimate of x2 at t = 15 s, and (c) probability density estimate of x2 at t = 25 s

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Fig. 5

Schematic representation of 2DOF airfoil

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Fig. 6

Pitch response of the aeroelastic system with deterministic initial conditions and parameters: (a) pitch response for Vr = 6.2, βα = 3, (b) pitch response for Vr = 6.3, βα = 3, and (c) pitch response for Vr = 6.4, βα = 3

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Fig. 7

Probability density estimate of pitch angle α at various nondimensional time (supercritical regime): (a) probability density estimate of α at τ = 2000, (b) probability density estimate of α at τ = 5000, and (c) probability density estimate of α at τ = 8000

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Fig. 8

Probability density estimate of nondimensional plunge ξ at various nondimensional time (supercritical regime): (a) probability density estimate of ξ at τ = 2000, (b) probability density estimate of ξ at τ = 5000, and (c) probability density estimate of ξ at τ = 8000

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Fig. 9

Probability density estimate of α at τ=1000 (subcritical regime)

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Fig. 10

Estimated response PDF of pitch angle when different accuracy levels of sparse grid are used with different orders of gPC expansion: (a) probability density estimate of α at τ = 8000 and (b) probability density estimate of α at τ = 8000

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