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.

Fluid-structure interaction can result in dynamic instabilities like flutter. Nonlinear parameters present in the system can stabilize the diverging growth of flutter oscillations to a limit cycle oscillation (LCO). Sustained LCO can lead to fatigue failure of rotating structures such as wind turbine rotors. Hence, it is an important design concern in aeroelastic analysis. Moreover, there is a growing interest in understanding how system uncertainties in structural and aerodynamic parameters and initial conditions affect the characteristics of such dynamical response.

Uncertainty quantification in a stochastic framework with stochastic inputs has traditionally been analyzed with Monte Carlo simulations (MCSs). To apply this procedure one should use the distribution of the input parameters to generate a large number of realizations of the response. Probability density function (PDF) and other required statistics are then approximated from these realizations; however, it is computationally expensive, especially for large complex problems. Hence, there is a need to develop alternate approaches which are computationally cheaper than direct MCS procedure. Perturbation method is a fast tool for obtaining the response statistics in terms of its first and second moments [

Polynomial chaos expansion (PCE) is a more effective approach, pioneered by Ghanem and Spanos [

Galerkin PCE (also called intrusive approach) modifies the governing equations to a coupled form in terms of the chaos coefficients. These equations are usually more complex and arriving at them is quite often a tedious task for some choices of the uncertain parameters. In order to avoid these, several uncoupled alternatives have been proposed. These are collectively called nonintrusive approaches. In a nonintrusive polynomial chaos method a deterministic solver is used repeatedly as in Monte Carlo simulation. The Probabilistic Collocation (PC) method is such a nonintrusive polynomial chaos method in which the problem is collocated at Gauss quadrature points in the probability space [

The intrusive and nonintrusive PCE approaches and their implementation to an aeroelastic model with structural nonlinearity are discussed in detail in the subsequent sections.

Figure

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

Explicitly, the system looks like,

It is increasingly being felt among the aeroelastic community that aeroelastic analysis should include the effect of parametric uncertainties. This can potentially revolutionize the present design concepts with higher rated performance and can also reshape the certification criteria. Nonlinear dynamical systems are known to be sensitive to physical uncertainties, since they often amplify the random variability with time. Hence, quantifying the effect of system uncertainties on the aeroelastic stability boundary is crucial. Flutter, a dynamic aeroelastic instability involves a Hopf bifurcation where a damped (stable response) oscillation changes to a periodic oscillatory response at a critical wind velocity. In a linear system the post flutter response can grow in an unbounded fashion [

The original homogeneous polynomial chaos expansion [

In the classical Galerkin-PCE approach, the polynomial chaos expansion of the system response is substituted into the governing equation and a Galerkin error minimization in the probability space is followed. This results in a set of coupled equations in terms of the polynomial chaos coefficients. The resulting system is deterministic, but it is significantly modified to a higher order and complexity depending on the order of chaos expansion and system nonlinearity. After solving this set of coefficient equations, they are substituted back to get the system response.

As per the Cameron-Martin theorem [

Any stochastic process

Equation (

If the cubic spring constant

Substituting the chaos expansion terms, (

The Galerkin approach is also called the intrusive approach as it modifies the system governing equations in terms of the chaos coefficients. The modification results into a higher order and much more complex form. As a result, this approach may become computationally quite expensive.

A number of nonintrusive variants of PCE have been developed to counter the disadvantages of the classical Galerkin method. Stochastic projection is one of them [

The Hermite polynomials are statistically orthogonal, that is, they satisfy

The main focus of the present study is quantifying the effect of system uncertainties on the bifurcation behavior and the flutter boundary of the nonlinear aeroelastic system. A fourth order variable step Runge-Kutta method is employed for the time integration. The main bifurcation parameter in a flutter system is the nondimensional wind velocity, also called the reduced velocity. In a linear aeroelastic system, the response changes to an exponentially growing solution from a stable damped oscillation at some critical wind velocity, known as the linear flutter speed. Nonlinear aeroelastic system can stabilize the response at the post-flutter regime to limit cycle oscillations [

Deterministic flutter and bifurcation diagram with cubic nonlinearity (supercritical Hopf bifurcation).

We now consider random variations in the system parameters and investigate the influence on the overall dynamics. We consider only single uncertain parametric variation in this paper, that is, a single random variable model. First, the hardening cubic spring constant is considered to be a Gaussian random variable with mean

Uncertain nonlinear stiffness: stochastic bifurcation diagram.

A Galerkin PCE approach is used to quantify the propagation of this uncertainty on the response. The Galerkin approach modifies the

Galerkin-PCE: behavior of the first few random modes.

Galerkin-PCE: PDF comparisons for increasing order of chaos expansions, at

Now the nonintrusive projection approach is followed using a Gauss-Hermite quadrature. Galerkin-PCE and nonintrusive results are compared in terms of their accuracy and simulation time in Figure

Galerkin-PCE: PDF comparisons for intrusive and nonintrusive PCE, at

The response realization time histories for a few samples of random variable

Uncertain nonlinear stiffness: five different realizations time histories at

Uncertain nonlinear stiffness: comparison of a typical time history with

Uncertain nonlinear stiffness: amplitude response PDF as a function of reduced speed.

Next, we consider the viscous damping ratio in pitch (

Uncertain viscous damping: stochastic bifurcation diagram.

The major difference between the uncertain damping and the earlier considered uncertain stiffness is that, variation in damping can show phase shifting behavior in the response realizations. This is presented in Figure

Uncertain viscous damping: five different realizations time history at

Uncertain viscous damping: comparison of the PDFs with increasing order of PCE at nondimensional time 1400 at

Uncertain viscous damping: comparison of the PDFs with increasing order of PCE at

Uncertain viscous damping: a typical time history with (a) The

A typical realization time history with PCE along with its deterministic counterpart are presented in Figure

The amplitude response PDFs for the uncertain damping case is shown in Figure

Uncertain viscous damping: amplitude response PDF as a function of reduced velocities.

For the uncertain damping case, we also see that the critical reduced velocity at which flutter can occur, has come down from its corresponding deterministic value. This value can be read off the bifurcation plot (Figure

Uncertain viscous damping: (a) CDF, (b) PDF of the critical flutter point.

The bifurcation behavior of a nonlinear pitch-plunge flutter problem with uncertain system parameters has been studied. The problem is a simple model problem to understand the mechanism of nonlinear flutter in a stochastic framework. The parameters which have been assumed to be random could attribute their uncertainties to laboratory testing conditions. Moreover, a cubic nonlinear stiffness is used for various sources of analytic nonlinearities; they often represent different control mechanisms and could face modeling uncertainties.

The classical Galerkin Polynomial Chaos method and the nonintrusive Projection method are applied to capture the propagation of uncertainty through the nonlinear aeroelastic system. The focus of this work is to investigate the performance of these techniques and to see how the aeroelastic stability characteristics are altered due to the random effects. The Monte Carlo solution is used as reference solution. The computational cost of the Galerkin Polynomial Chaos method is seen to be very high and subsequently only the Projection method based on Gauss-Hermite quadrature is used for the analysis. The effect of uncertain cubic structural nonlinearity and viscous damping parameter are investigated. Uncertainty in the cubic stiffness does not alter the bifurcation (flutter) point, it only affects the amplitudes of the periodic response in the post flutter stage. The PDF behavior also does not show any qualitative changes. On the other hand, uncertainty in damping affects the bifurcation point. It can lower the onset of flutter; the PDF of the response amplitude also undergoes a qualitative change. In other words, a bifurcation of the response PDF takes place. The results highlight the risk induced by parametric uncertainty and importance of uncertainty quantification in nonlinear aeroelastic systems.

The uncertain damping case by polynomial chaos suffer from long time degeneracy, as is also discussed in the literature. The degeneracy can be controlled by using higher order chaos expansions, though this cannot be a permanent solution. For the uncertain nonlinear stiffness, the problem of time degeneracy is not encountered.

The coefficients introduced in Section