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Journal Article•DOI•

Analysis of a Nonlinear Aeroelastic System with Parametric Uncertainties Using Polynomial Chaos Expansion

18 Jul 2010-Mathematical Problems in Engineering (Hindawi)-Vol. 2010, pp 1-21
TL;DR: In this article, a nonlinear aeroelastic system with parametric uncertainties is considered, and a projection-based nonintrusive polynomial chaos approach is shown to be much faster than its classical Galerkin method based counterpart.
Abstract: Aeroelastic stability remains an important concern for the design of modern structures such as wind turbine rotors, more so with the use of increasingly flexible blades. A nonlinear aeroelastic system has been considered in the present study with parametric uncertainties. Uncertainties can occur due to any inherent randomness in the system or modeling limitations, and so forth. Uncertainties can play a significant role in the aeroelastic stability predictions in a nonlinear system. The analysis has been put in a stochastic framework, and the propagation of system uncertainties has been quantified in the aeroelastic response. A spectral uncertainty quantification tool called Polynomial Chaos Expansion has been used. A projection-based nonintrusive Polynomial Chaos approach is shown to be much faster than its classical Galerkin method based counterpart. Traditional Monte Carlo Simulation is used as a reference solution. Effect of system randomness on the bifurcation behavior and the flutter boundary has been presented. Stochastic bifurcation results and bifurcation of probability density functions are also discussed.

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Citations
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01 Jan 2007
TL;DR: Two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weightfunction are known or can be calculated.
Abstract: Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

1,007 citations

01 Jan 1994
TL;DR: In this article, a two-dimensional airfoil with a free-play nonlinearity in pitch subject to incompressibl e flow was analyzed and the aerodynamic forces were evaluated using Wagner's function and the resulting equations integrated numerically to give time histories of the air-foil motion.
Abstract: A two-dimensiona l airfoil with a free-play nonlinearity in pitch subject to incompressibl e flow has been analyzed. The aerodynamic forces on the airfoil were evaluated using Wagner's function and the resulting equations integrated numerically to give time histories of the airfoil motion. Regions of limit cycle oscillation are detected for velocities well below the linear flutter boundary, and the existence of these regions is strongly dependent on the initial conditions and properties of the airfoil. Furthermore, for small structural preloads, narrow regions of chaotic motion are obtained, as suggested by power spectral densities, phase-plane plots, and Poincare sections of the airfoil time histories. The existence of this chaotic motion is strongly dependent on a number of airfoil parameters, including, mass, frequency ratio, structural damping, and preload.

110 citations

Journal Article•DOI•
TL;DR: In this article, the transition in aeroelastic response from an initial state characterised by low-amplitude aperiodic fluctuations to aero-elastic flutter when the system exhibits limit cycle oscillations was studied.

37 citations

Journal Article•DOI•
TL;DR: In this article, the quantification of uncertainty effects on the dynamic responses and vibration properties of a rotor system was studied, and the authors quantified the effect of uncertainty on the response prediction of rotor systems.
Abstract: Parametric uncertainties play a significant role in the response predictions of a rotor system. In this paper, the quantification of uncertainty effects on the dynamic responses and vibration chara...

22 citations


Cites methods from "Analysis of a Nonlinear Aeroelastic..."

  • ...To deal with the uncertainties, a non-intrusive PCE, which is proposed by Walters (2003), is employed to predict mean responses and corresponding probability density functions of the vibration amplitude in this work, and its availability and convergence have been verified in some literature (Desai and Sarkar, 2010; Hosder et al., 2006)....

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  • ...…PCE, which is proposed by Walters (2003), is employed to predict mean responses and corresponding probability density functions of the vibration amplitude in this work, and its availability and convergence have been verified in some literature (Desai and Sarkar, 2010; Hosder et al., 2006)....

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Journal Article•DOI•
TL;DR: In this article, the uncertainty quantification of an aeroelastic instability system with multiple uncertainties has been studied and a quadrature based spectral uncertainty tool called polynomial chaos expansion is used to quantify the propagation of uncertainty through the dynamical system of concern.
Abstract: 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.

20 citations


Cites background or methods or result from "Analysis of a Nonlinear Aeroelastic..."

  • ...In our earlier work on an uncertain aeroelastic system, we have used a Gauss–Hermite quadrature scheme to evaluate the chaos coefficient in a one-dimensional approach [19,20]....

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  • ...Convergence of the PCE with an increasing order of PC expansion is given in our earlier work [19,47]....

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  • ...With an increasing order of expansion, it is required to solve higher order inner products which increases the computational time required to obtain the chaos coefficients and, in some cases, it takes almost the same computational time as the MCS [19]....

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References
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Book•
20 Dec 1990
TL;DR: In this article, a representation of stochastic processes and response statistics are represented by finite element method and response representation, respectively, and numerical examples are provided for each of them.
Abstract: Representation of stochastic processes stochastic finite element method - response representation stochastic finite element method - response statistics numerical examples.

5,495 citations

Journal Article•DOI•
TL;DR: This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error.
Abstract: We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs

4,473 citations

Journal Article•DOI•

3,788 citations

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01 Jan 1964
TL;DR: The general nature of Monte Carlo methods can be found in this paper, where a short resume of statistical terms is given, including random, pseudorandom, and quasirandom numbers.
Abstract: 1 The general nature of Monte Carlo methods.- 2 Short resume of statistical terms.- 3 Random, pseudorandom, and quasirandom numbers.- 4 Direct simulation.- 5 General principles of the Monte Carlo method.- 6 Conditional Monte Carlo.- 7 Solution of linear operator equations.- 8 Radiation shielding and reactor criticality.- 9 Problems in statistical mechanics.- 10 Long polymer molecules.- 11 Percolation processes.- 12 Multivariable problems.- References.

3,226 citations

Journal Article•DOI•

2,718 citations