Experimental Identification of Concentrated Nonlinearity in Aeroelastic System

Identification of concentrated nonlinearity in the torsional spring of an aeroelastic system is performed. This system consists of a rigid airfoil that is supported by a linear spring in the plunge motion and a nonlinear spring in the pitch motion. Quadratic and cubic nonlinearities in the pitch moment are introduced to model the concentrated nonlinearity. The representation of the aerodynamic loads by the Duhamel formulation yielded accurate values for the flutter speed and frequency. The results show that the use of the Duhamel formulation to represent the aerodynamic loads yields excellent agreement between the experimental data and the numerical predictions.


Introduction
Nonlinearities in aeroelastic systems can affect the response of aerospace vehicles and lead to complex phenomena such as flutter, limit-cycle oscillations (LCOs), chaos and bifurcations [1][2][3].Such phenomena are associated with nonlinear aerodynamic or structural conditions that characterize aeroelastic systems.Regarding structural nonlinearities, they can arise from large structural deflections and/or partial loss of structural integrity.Furthermore, the effects of aging, loose attachments, and material features could lead to undesirable and dangerous responses [4].The combination of the structural and aerodynamic nonlinearities can lead to changes in the basic response of the system such as changing it from supercritical to subcritical responses and then catastrophic behavior can occur.As such, assessing the evolution of these inevitable behaviors through modeling and analysis of these nonlinearities is important to avoid or control these dangerous responses.
In this work, we perform a nonlinear analysis to identify the concentrated nonlinearity in the torsional spring of an airfoil section.The objective is to show how the use of a cubic nonlinearity to model the concentrated nonlinearity and the Duhamel formulation to represent the aerodynamic loads can yield relevant characterization of the aeroelastic system.

Experimental setup of the aeroelastic system
The aeroelastic tests were carried out using an open-circuit wind tunnel with 520 mm × 515 mm of test section and its maximum air velocity of approximately 30 m/s.The experimental apparatus has been designed to characterize the aeroelastic response of an airfoil section.Based on the experimental setup, the airfoil is assumed to be two-dimensional a e-mail: mhajj@vt.eduand all parameters are defined per unit span and are assumed to be uniformly distributed.The airfoil is composed of an aluminum rigid wing that is mounted vertically at the 1/4 of the chord from leading edge by using an aluminum shaft.At the edges, the shaft is connected through bearings to the support of the plunge mechanism that includes a bicantilever beam made of two steel leaf-springs.Pictures of the experimental setup and of the pitch spring mechanism are presented in Figs 1 and 2, respectively.Signals of the rotational motion of the elastic axis were collected using an encoder.An accelerometer was used to collect the plunge motion.To induce motions of the wing, an initial displacement in the plunge was applied.This process was repeated starting with low speeds of about 8 m/s.It was observed that all disturbances with the same set of initial displacement would damp at speeds less than 10.9 m/s.Above this speed, limit cycle oscillations (LCO) were observed.

Modeling of the aeroelastic system
We model the rigid airfoil as an aeroelastic system that is allowed to move with two degrees of freedom namely the, plunge (h) and pitch (α) motions as shown in Fig. 3.The equations of motion governing this system are given by [5,6], and are written in the following form In the above equations, m T is the total mass of the wing with its support structure, m w is the wing mass alone, I α is the mass moment of inertia about the elastic axis, b is the semichord length, x α is the nondimensional distance between the center of mass and the elastic axis.Viscous damping forces are described through the coefficients c h and c α for plunge and pitch motions, respectively.Table 1 gives values of the typical section parameters used in the analytical part.In addition, L and M are the aerodynamic lift and moment about the elastic axis.The two spring forces for plunge and pitch motions are represented by k h and k α , respectively.In this setup, the sole source of nonlinearity is the concentrated pitch nonlinearity.For the analysis, this nonlinearity is modeled by quadratic and cubic terms in the torsional spring, i.e, we write

Aerodynamic loads representation
The lift and moment as derived by Theodorsen [7] in incompressible flow are given by and where and k = wb U is the reduced frequency of the harmonic oscillations.Clearly, the aerodynamic loads are dependent on Theodorsen's function C(k).The exact expression of C(k) is given by [5] where F and G are functions of the Hankel and modified Bessel functions.

Duhamel formulation
Considering a simple harmonic assumption in the system's variables [8], the aerodynamic loads can be determined for arbitrary time dependent motion as [8]: and where Using the Wagner function as an inditial admittance and Duhamel's superposition integral, the circulatory term can be written as [8]: (10) where and φ(τ) is Wagner function.The Sears approximation to φ(τ) is given by [8] 03001-p.2Using integration by parts, the Duhamel formulation is rewritten as follows: Using the Pade approximation to transform the integral form into a second order ordinary differential equation [9], we write where x and ẋ are two augmented variables in the state space used to model the wake effects.The lift and moment expressions are then rewritten as: and On the other hand, to consider the effects of the wake dynamics of the airfoil, the augmented state variables are related to the other system parameters through the following differential equation [9] Substituting the expressions of the lift and moment in equation 1, the general form of the equations of motion is then written as: where and Multiplying equation ( 18) from the left with the inverse M −1 1 of M 1 , we obtain where In state space form, we define Using these variables, equation ( 22) is rewritten as: Additionally, we have: Combining these equations, we obtain the following form where N 1q (Z, Z) and N 1c (Z, Z, Z) are the quadratic and cubic vector functions of the state variables which describe the structural nonlinearity and where K * 1 depends on U, U 2 and U 3 and C * 1 depends on U and U 2 .The matrix B 1 (U) has a set of six eigenvalues λ i , i = 1, 2..., 6.The first four eigenvalues are complex conjugates (λ 2 = λ 1 , λ 4 = λ 3 ) which are directly related to the pitch α and plunge h motions.The real parts of these eigenvalues correspond to the damping coefficients and the imaginary parts are the coupled frequencies of the aeroelastic system.The other eigenvalues λ 5 and λ 6 are related to the augmented state variable x which are negative.The solution of the linear part is asymptotically stable if the real parts of the λ i are negative.In addition, if one of the real parts is positive, the linearized system is unstable.The speed at which one or more eigenvalues have zero real parts, corresponds to the onset of the linear instability and is termed as the flutter speed, U f .03001-p.3

Linear analysis: Validation of the aerodynamic model
The variation of the damping term as a function of the freestream velocity when using the Duhamel formulation are presented in Fig. 4.This plot was obtained using the parameters values in Table 1.The results show analytical value of 10.90 m/s for the flutter speed.Clearly, the predicted flutter speed based on the Duhamel formulation is very close to the measured values between 10.89 m/s and 10.92 m/s.Values of the flutter frequency as predicted from the linear analysis is shown in the second column of Table 2.The corresponding experimental values are presented in the third column of Table 2.The results show that the predicted flutter frequency when considering the Duhamel formulation have a very good agreement with the experimental results.

Nonlinear analysis: Validation of the aerodynamic model and identified coefficients
The plotted curves in Figs. 5 and 6 show comparison between the experimental results and numerical integrations of the governing equations for U = 13.68m/s.Clearly, time histories of both the plunge and pitch motions are satisfactorily predicted by the identified parameters which are determined numerically (k α 1 = 3.95 N.m and k α2 = 107 N.m).Figs. 7 and 8 show the bifurcation diagrams for both the plunge and pitch RMS values.In these figures, a comparison is presented between numerical predictions based on Runge-Kutta method and experimental data.We note that there is a very good agreement between the experimental results and the numerical integrations.Finally, we note that the system has a supercritical behavior which is observed in the experiment.

Conclusions
In this work, we have performed a nonlinear analysis to identify the concentrated nonlinearity in the torsional spring of a two-dimensional typical section.The concentrated nonlinearity is modeled by adding quadratic and cubic nonlinear terms in the pitch moment.We use the Duhamel formulation to model the aerodynamic loads.The results 03001-p.4 show that using the Duhamel representation, which takes into consideration the wake effects, yield to an excellent agreement with the experimental values in both linear and nonlinear analyses.

Fig. 4 .
Fig. 4. Variation of the damping term as function of the freestream velocity.

Fig. 8 .
Fig.8.Bifurcation diagram of the pitch RMS amplitude when using numerical integration (dashed lines) and experimental data (red points).

Table 2 .
Comparison between theoretical and experimental results