Modeling and identification of freeplay nonlinearity

Journal of Sound and Vibration 331 (2012) 1898–1907

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Modeling and identification of freeplay nonlinearity A. Abdelkefi a,n, R. Vasconcellos b, F.D. Marques b, M.R. Hajj a a b

Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA ~ Paulo, Sao ~ Carlos, Brazil Laboratory of Aeroelasticity, University of Sao

a r t i c l e in f o

abstract

Article history: Received 24 March 2011 Received in revised form 26 September 2011 Accepted 17 December 2011 Handling Editor: A.V. Metrikine Available online 11 January 2012

A nonlinear analysis is performed for the purpose of identification of the pitch freeplay nonlinearity and its effect on the type of bifurcation of a two degree-of-freedom aeroelastic system. The databases for the identification are generated from experimental investigations of a pitch-plunge rigid airfoil supported by a nonlinear torsional spring. Experimental data and linear analysis are performed to validate the parameters of the linearized equations. Based on the periodic responses of the experimental data which included the flutter frequency and its third harmonics, the freeplay nonlinearity is approximated by a polynomial expansion up to the third order. This representation allows us to use the normal form of the Hopf bifurcation to characterize the type of instability. Based on numerical integrations, the coefficients of the polynomial expansion representing the freeplay nonlinearity are identified. Published by Elsevier Ltd.

1. Introduction Nonlinear sources can affect the aeroelastic response of aerospace vehicles and lead to limit cycle oscillations (LCO), chaotic motions, coexisting stable solutions and bifurcations [1–4]. The nonlinearities can arise from unsteady aerodynamic sources, large structural deflections, and/or partial loss of structural or control integrity [5]. Furthermore, nonlinearities associated with moving surfaces or external stores are inevitable. In these cases, freeplay is one of the most common nonlinearity. Freeplay can arise from worn hinges and loose attachments which are generally related with aircraft aging. The combination of freeplay nonlinearities with the aerodynamic and structural nonlinearities can vary the system’s response and change its instability from a supercritical to a subcritical one. These changes can result in catastrophic damages. To that end, many researchers have investigated numerically and experimentally the effects of freeplay nonlinearity on aeroelastic systems [6–11]. Because of the discontinuous representation of freeplay nonlinearity, it is quite difficult to analyse it mathematically and include it in numerical simulations. Consequently, there is a need to develop mathematical representations or models that can be used to determine the impact of such nonlinearities on the system’s response. These models can be helpful in the design stages of varying configurations, development of control strategies and uncertainty quantification of response characteristics. The objective of this work is to determine whether the representation of a freeplay nonlinearity by a polynomial approximation is valid for certain responses. This is achieved by performing system identification in a pitch–plunge airfoil using data from experiments where freeplay nonlinearity is introduced in the torsional spring. The system identification

n

Corresponding author. Tel.: þ 1 5405773982; fax: þ 1 5402314574. E-mail address: [email protected] (A. Abdelkefi).

0022-460X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.jsv.2011.12.021

A. Abdelkefi et al. / Journal of Sound and Vibration 331 (2012) 1898–1907

1899

makes use of pitch–plunge governing equations with quasi-steady approximation of the aerodynamic loads. The quasisteady approximation has been shown to be valid over the range of motions considered in the experiments. Once modeled, the normal form that represents the observed LCO is derived from the governing equations. This form is then used to identify the type of bifurcation associated with the freeplay nonlinearity of these experiments. 2. Experimental setup

Torque

The experimental apparatus has been designed to simulate a typical wing section in a two-dimensional incompressible flow. As shown in Fig. 1, a freeplay nonlinearity (d) that is equal to 2.251 is introduced in the pitch motion. Since the typical section is two-dimensional, all parameters of the system are defined per unit span and are assumed to be uniformly distributed. The airfoil is composed of a foam-wood made NACA0012 rigid wing that is mounted vertically at the 1/4 of the chord from leading edge by using an aluminium shaft having a 15 mm diameter. At the edges, the shaft is connected through bearings to the support of the plunge mechanism that is a bi-cantilever beam made of two steel leaf-springs which are of 28 cm long, 2 cm wide, and 0.12 cm thick. The distance between the two cantilever beams is 18.2 cm. The chordwise center of gravity is adjusted by adding a balance weight to a tube placed inside the wing section at the spanwise position of 19 cm from the leading edge. The torsional pitch stiffness and freeplay mechanism comprises a steel leaf spring inserted tightly into a slot in the main shaft at the top of the wing section. The free end of the leaf spring is placed into a support that allows for freeplay variations. A picture of the experimental setup and a schematic of the freeplay mechanism are presented in Fig. 2(a) and (b), respectively. The aeroelastic tests were carried out using an open-circuit blower-like wind tunnel with 500 mm  500 mm of test section and its maximum air velocity of approximately 17.0 m/s. Signals of the rotational motion of the elastic axis were collected using a USdigital HEDS-9000-T00 encoder. To induce motions of the wing, an initial displacement in the plunge was applied. This process was repeated starting with low speeds of about 2 m/s. It was observed that all disturbances with the same set of initial displacement would damp at speeds less than 8.4 m/s. Above this speed, limit cycle oscillations (LCOs) were observed as shown in Fig. 3. In addition, we observed that LCO could be initiated at lower speeds than 8.4 m/s if larger initial displacements are introduced. This indicates that the system has a subcritical behavior. We have also

2δ

−8

−6

−4

−2

0 α (deg)

2

4

6

Fig. 1. Freeplay nonlinearity representation.

Fig. 2. (a) Experimental setup. (b) Pitch freeplay mechanism.

8

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α (deg)

5 0 −5 0

1

2

3

4

5

3

4

5

3

4

5

3

4

5

3

4

5

time (s)

α (deg)

10 0 −10 0

1

2 time (s)

α (deg)

10 0

α (deg)

−10 0

1

2

0

1

2

10 0 −10

α (deg)

time (s)

10 0 −10 0

1

2 time (s)

Fig. 3. Experimental time histories for different freestream velocities (a) U ¼8.44 m/s, (b) U ¼8.62 m/s, (c) U¼ 8.94 m/s, (d) U¼ 9.10 m/s and (e) U ¼9.42 m/s.

observed that the pitch angle increases dramatically when the freestream velocity is larger than 9 m/s. In this region, the experimental results show limit cycle oscillations that have larger amplitudes than the ones observed at freestream velocities that are less than 9 m/s. Additionally, the pitch angle became larger than 151 as shown in Fig. 3(d) and (e). Clearly, aerodynamic stall became relatively important. Because the effort here is to determine how freeplay nonlinearity can be identified. The identification is performed over the region up to a freestream velocity of 9 m/s so that aerodynamic nonlinearities are excluded. 3. Governing equations We model the wing section as an aeroelastic system that is allowed to move with two degrees of freedom namely the plunge (w) and pitch (a) motions as shown in Fig. 4. The equations of motion governing this system have been derived in many references [12,13], and are written in the following form: " #  " #  " #    _ € kw 0 cw 0 mw xa b w mT w L w þ þ ¼ (1) 0 ka ðaÞ 0 ca Ia mw xa b a_ a M a€

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Fig. 4. Schematic of an aeroelastic system under uniform airflow.

Table 1 Parameters of the aeroelastic wing. Wing geometry and air density Span b a xa

0.5 0.125  0.5 0.1776

rp

Wing span (m) Wing semi-chord (m) Position of elastic axis relative to the semi-chord Nondimensional distance between center of gravity and elastic axis Air density (kg/m3)

Section parameters of the aeroelastic wing per unit span (m) mw mT Ia ca cw ka0 kw

Mass of the wing (kg) Mass of wing and supports (kg) Moment of inertia about the elastic axis (kg m2) Pitch damping coefficient (kg m2/s) Plunge damping coefficient (kg/s) Stiffness in pitch (N m) Stiffness in plunge (N/m)

1.314 4.46 0.004836 0.0184 2.6 1.4228 4200

1.1

In the above equations, mT is the total mass of the wing with its support structure, mw is the wing mass alone, Ia is the mass moment of inertia about the elastic axis, b is the semi-chord length, xa is the nondimensional distance between the center of mass and the elastic axis. Viscous damping forces are described through the coefficients cw and ca 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 and are assumed to be given by the quasi-steady representation: L ¼ rU 2 bcla aeff 2

M ¼ rU 2 b cma aeff

(2) (3)

where U is the freestream velocity and cla and cma are the aerodynamic lift and moment coefficients. The effective angle of attack due to the instantaneous motion of the airfoil is given by [14].   h_ 1 a_ (4) aeff ¼ a þ þ a b U 2 U The two spring forces for plunge and pitch motions are represented by kw and ka , respectively. In this experimental setup, the sole source of nonlinearity is the freeplay pitch nonlinearity. For the analysis, we approximate the freeplay nonlinearity by a polynomial expansion, i.e, we write ka ðaÞ ¼ ka0 þ ka1 a þ ka2 a2 þ    To express the equations of motion in state space form, we define the following state of variables: 2 3 2 3 w X1 6 _ 7 6 7 6w 7 6 X2 7 6 7 7 6 7 X¼6 6 X3 7 ¼ 6 a 7 4 5 6 _ 7 4a5 X4

(5)

(6)

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A. Abdelkefi et al. / Journal of Sound and Vibration 331 (2012) 1898–1907

The equations of motion are then rewritten as X_ 1 ¼ X 2

(7)

  Ia kw mw xa bka0 X 1 c1 X 2  k1 U 2  X 3 c2 X 4 N1a ðXÞ X_ 2 ¼  d d X_ 3 ¼ X 4   mw xa bkw ka0 mT X_ 4 ¼ X 1 c3 X 2  k2 U 2 þ X 3 c4 X 4 N2a ðXÞ d d where d ¼ mT Ia ðmw xa bÞ2 3

c1 ¼ ½Ia ðcw þ rUbcla Þ þ rUb mw xa cma =d 2

4

c2 ¼ ½Ia rUb cla ð12 aÞmw xa bca þ mw xa b rUcma ð12aÞ=d 2

c3 ¼ ½mw xa bðcw þ rUbcla ÞmT cma rUb =d 3

3

c4 ¼ ½mT ðca b rUcma ð12 aÞÞmw xa b rUcla ð12aÞ=d 3

k1 ¼ ½Ia rbcla þ mw xa b rcma =d 2

2

k2 ¼ ½rb cla mw xa þmT rb cma =d N1a ¼ mw xa b½ka1 X 23 þka2 X 33 =d N2a ¼ mT ½ka1 X 23 þ ka2 X 33 =d These equations have the following form: _ ¼ BðUÞX þQ ðX,XÞ þ CðX,X,XÞ X

(8)

where Q ðX,XÞ and CðX,X,XÞ are, respectively, the quadratic and cubic vector functions of the state variables which describe the freeplay pitch nonlinearity, and 2 3 0 1 0 0 6  Ia kw c 2 mw xa bka0 ðk1 U  Þ c2 7 6 7 1 d d 7 BðUÞ ¼ 6 6 0 0 0 1 7 4 5 2 ka0 mT mw xa bkw c3 ðk2 U þ d Þ c4 d The matrix BðUÞ has a set of four eigenvalues li , i ¼ 1; 2, . . . ,4. These eigenvalues are complex conjugates (l2 ¼ l1 and l4 ¼ l3 ). The real parts of these eigenvalues correspond to the damping coefficients and the positive imaginary parts are the coupled frequencies of the aeroelastic system. The solution of the linear part is asymptotically stable if the real parts of the li are negative. In addition, if one of the real parts is positive, the linear system is unstable. The speed, U f , for which one eigenvalue has zero real part, corresponds to the onset of the linear instability and is termed as the flutter speed. 4. Linear model vs experimental results In this section, we validate the coupled natural frequencies and the flutter speed as determined from the analysis against the experimental values. The plotted curves in Fig. 5 show the variation of the real parts and the imaginary parts of the eigenvalues as function of the freestream velocity. These plots were obtained using the parameters values in Table 1. Using these curves, we determine an analytical value of 8.42 m/s for the flutter speed which is close to the measured values between 8.35 m/s and 8.43 m/s. Values of the coupled natural frequencies and the flutter frequency as predicted from the linear analysis of Eq. (8) are shown in the second column of Table 2. Fig. 6 shows plots of the magnitudes of the frequency response function when the freestream velocity was set to zero and to the flutter speed of 8.4 m/s. Experimental values of the coupled frequencies as well as flutter frequency as determined from these plots and are presented in the third column of Table 2. A comparison of the values in the second and third columns of Table 2 shows that the predicted coupled and flutter frequencies are within an acceptable margin of the experimental values. This shows that the values of the parameters of Table 1 are valid for the identification.

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Re (λi)

2 0 -2 -4

0

2

4

6

8 U (m/s)

10

12

14

0

2

4

6

8 U (m/s)

10

12

14

4

Im(λi)

2 0 -2 -4

Fig. 5. Variation of (a) the real part of the eigenvalues, (b) imaginary part of the eigenvalues as function of the freestream velocity. Table 2 Comparison between theoretical and experimental results. Parameter

Theoretical results

Experimental results

Pitch coupled natural frequency (Hz) at U¼ 0 m/s Plunge coupled natural frequency (Hz) at U¼ 0 m/s Flutter frequency (Hz) U f (m/s)

2.69

2.80

5.02

5.10

4.73 8.42

4.98 8:35 o U f o 8:43

5. Identification of the nonlinear torsional spring coefficients As discussed in Section 3, the freeplay nonlinearity is approximated by a polynomial expansion that includes quadratic, cubic and higher order terms. To identify the terms in this approximation that are mostly associated with the freeplay nonlinearity, we present in Fig. 7 the power spectrum of the measured pitch motion when 8.62 m/s. The spectrum shows peaks at the response frequency of f¼5.85 Hz and its third superharmonic and subharmonic. The presence of these peaks at 3f and 13f and the absence of strong peaks at 2f and 12f indicate that of all terms in the expansion the cubic term is the most relevant. As such, only cubic nonlinearity, ka2 , is considered to model the freeplay nonlinearity and determine the associated type of bifurcation using the normal form. Fig. 8 shows the variation of the pitch angle as a function of the cubic nonlinear term ka2 when the freestream velocity U is set to 8.62 m/s. We note that increasing the coefficient of the cubic nonlinearity results in a decrease of the amplitude of the pitch angle. Therefore, we can easily identify the value of this coefficient to obtain the experimentally measured value of the pitch amplitude. Using this figure, it is concluded that this coefficient should be equal to 380 N m to obtain pitch angle of 8.41. Fig. 9(a) shows the variation of the pitch angle as a function of the freestream velocity when ka2 ¼ 380 N m. The solid curve presents the analytical results while the red points show the measured values. Fig. 9(b) and (c) shows the time histories of the pitch angle for 1 s and 0.3 s, respectively, when U ¼8.62 m/s. We note that there is a slight shift in the frequency between the experimental and theoretical results. This behavior is associated with a change in the pitch stiffness that is caused by higher angles of attack which stressed the spring wire as discussed by Conner et al. [9].

1904

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Magnitude (dB)

Fig. 6. Power spectra as determined from the experiments when the freestream velocity is set to (a) zero and (b) the flutter speed of 8.4 m/s.

20 0 −20 −40 0

5

10

15

20 25 30 frequency (Hz)

35

40

45

50

Fig. 7. Experimental power spectrum when U ¼8.62 m/s.

30 25

α (deg)

20 15 10 5 0 0

100

200

300 400 kα2 (N m)

500

600

700

Fig. 8. Variation of the pitch angle as function of the cubic nonlinear coefficient ka2 when U ¼8.62 m/s.

6. Nonlinear normal form: subcritical behavior In this section, we determine the type of instability that is associated with the flutter of the wing. For this, we compute the normal form of the Hopf bifurcation of the aeroelastic system (Eq. (8)) near the flutter speed U f . To derive the nonlinear normal form, we first add a perturbation term, E2 sU U f , to the flutter speed ðU ¼ U f þ sU E2 U f Þ which leads to the appearance of the secular terms at the third order. Taking account of the perturbation term of the flutter speed, the matrix

A. Abdelkefi et al. / Journal of Sound and Vibration 331 (2012) 1898–1907

1905

20

α (deg)

15 10 5 0 8.2

8.4

8.6 U (m/s)

8.8

10

α (deg)

5 0 -5 -10 0.0

0.2

0.4

0.6

0.8

1.0

t (s) 10

α (deg)

5 0 -5 -10 0.00

0.05

0.10

0.15 t (s)

0.20

0.25

0.30

Fig. 9. Comparison of experimental (red dots) and numerical (solid line) of (a) the bifurcation diagram of the pitch degree of freedom when ka2 ¼ 380 N m (b,c) the time histories of the pitch angle for ka2 ¼ 380 N m and U ¼8.62 m/s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

BðUÞ is written as the sum BðU f Þ þ E2 B1 ðU f Þ where 2

0

60 6 B1 ðU f Þ ¼ 6 60 4 0

0

0

d1 U f

2k1 U 2f

0

0

d3 U f

2k2 U 2f

0

3

d2 U f 7 7 7 0 7 5 d4 U f

Eq. (8) is then written as _ ¼ BðU f ÞX þ E2 sU B1 ðU f ÞX þCðX,X,XÞ X where

2

0

(9)

3

6 7 6 C2 7 7 C¼6 6 0 7 4 5 C4 Letting P be the matrix whose columns are the eigenvectors of the matrix corresponding to the eigenvalues 7 j o1 m1 and 7jo2 of BðU f Þ, we define a new vector Y such that X ¼ PY and rewrite Eq. (9) as PY_ ¼ BðU f ÞPY þ E2 sU B1 ðU f ÞPY þ CðPYÞ

(10)

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Multiplying Eq. (10) from the left with the inverse P1 of P, we obtain Y_ ¼ JY þ E2 sU KY þP1 CðYÞ

(11)

where J ¼ P BðU f ÞP is a diagonal matrix whose elements are the eigenvalues 7 j o1 m1 and 7 j o2 of BðU f Þ and K ¼ P1 B1 ðU f ÞP. We note that Y 2 ¼ Y 1 , Y 4 ¼ Y 3 , and hence Eq. (11) can be written in component form as 1

Y_ 1 ¼ jo1 Y 1 m1 Y 1 þ E2 sU

5 X

K 1i Y i þN 1 ðY,Y,YÞ

(12)

1

Y_ 3 ¼ jo2 Y 3 þ E2 sU

5 X

K 3i Y i þ N3 ðY,Y,YÞ

(13)

1

where the Ni ðY,Y,YÞ are tri-linear functions of the Y. Because we considered only cubic nonlinearities, the solution of Y1 decays to zero. Consequently, we retain only the non-decaying solution (Y3). Moreover, to compute the normal form of the Hopf bifurcation of Eqs. (12) and (13) near U ¼ U f , we follow Nayfeh and Balachandran [15] and search for a third-order approximate solution of Eq. (13) in the following form: Y 3 ¼ EY 31 ðT 0 ,T 2 Þ þ E2 Y 32 ðT 0 ,T 2 Þ þ E3 Y 33 ðT 0 ,T 2 Þ þ OðE4 Þ

(14)

where T n ¼ En t. In terms of the Ti, the time derivative can be expressed as d q q ¼ þ E2 ¼ D0 þ E2 D2 dt qT 0 qT 2

(15)

Substituting Eqs. (14) and (15) into Eq. (13). These new equations must hold for all values of E, therefore the coefficients of like powers of E must satisfy these new equations. This leads to two different set of relations corresponding to E and E3 as follows: Order (E) D0 Y 31 jo2 Y 31 ¼ 0

(16)

D0 Y 33 jo2 Y 33 ¼ D2 Y 31 þ sU ðK 33 Y 31 Þ þ NðY 31 ,Y 31 ,Y 31 Þ þ cc þNST

(17)

3

Order (E )

where NST stands for terms that do not produce secular terms and cc stands for the complex conjugate of the preceding terms. The solution of Eq. (16) can be expressed as Y 31 ¼ AðT 2 Þejo2 T 0

(18)

Substituting Eq. (18) into Eq. (17), eliminating the terms that lead to secular terms, we obtain the modulation equation D2 A ¼ sU K 33 A þ ae A2 A

(19)

The effects of the cubic nonlinearity (ka2 ) on the system are expressed through the expression of ae . For convenience, we write Eq. (19) as D2 A ¼ bA þ ae A2 A

(20)

where b ¼ sU K 33 . Letting AðT 2 Þ ¼ 12a e igðT 2 Þ and separating the real and imaginary parts in Eq. (20), we obtain the following nonlinear normal form of Hopf bifurcation: 1 aer a3 4

(21)

1 4

(22)

a 0 ¼ br a þ

g0 ¼ bi þ aei a2

where a is the amplitude and g is the shifting angle of the periodic solution. Eq. (21) has generally three equilibrium solutions which are sffiffiffiffiffiffiffiffiffiffiffiffi 4br a ¼ 0, a ¼ 7

aer

a ¼0 which corresponds to the fixed points (0,0). The other two solutions are the nontrivial ones. The origin is asymptotically stable for br o 0 or br ¼ 0 and aer o 0 , unstable for br 40 or br ¼ 0 and aer 4 0. For the nontrivial solutions, they exist when br aer o0. Furthermore, they are stable (supercritical Hopf bifurcation) for br 40 and aer o 0 and unstable (subcritical Hopf bifurcation) for br o 0 and aer 4 0. Upon substituting the physical parameters, the expressions of the linear and nonlinear coefficients of the normal form are given by

br ¼ 1:013sU

aer ¼ 0:00166ka2

(23)

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Based on the numerical integrations, the cubic nonlinear coefficient ka2 is set equal to 380 N m. Consequently, because (aer 4 0), this system has a subcritical instability. This result confirms the experimental observation of the existence of (LCO) for larger initial displacements at freestream velocities that are smaller than the flutter speed. The ability to model and identify the type of bifurcation associated with the freeplay nonlinearity provides a capability that can be used to avoid subcritical responses such as the one observed in these experiments. 7. Conclusion In this work, we have performed identification of freeplay nonlinearity in a two degree-of-freedom aeroelastic system. The system is composed of a rigid airfoil supported by a nonlinear torsional spring. The databases for the identification are generated from experimental data of periodic responses that included the flutter frequency and its third harmonics. Based on this characterization of the responses, linear and nonlinear analyses that represent the freeplay nonlinearity of the torsional spring with a polynomial expansion up to the third order were performed. This representation allowed the use of the normal form of the Hopf bifurcation to characterize the type of instability of this system. It was determined that the ensuing Hopf bifurcation is of the subcritical type. The representative model and identified parameters were validated by integrating the governing equations and comparing response values from this integration with experimental responses.

Acknowledgment The authors acknowledge the financial support of the State of Sa~ o Paulo Research Agency (FAPESP), Brazil (grant 2007/ 08459-1) and the Coordination for the Improvement of Higher Education Personnel (CAPES), Brazil (grant 0205109). References [1] E.H. Dowell, D. Tang, Nonlinear aeroelasticity and unsteady aerodynamics, AIAA Journal 40 (2002) 1697–1707. [2] H.C. Gilliat, T.W. Strganac, A.J. Kurdila, An investigation of internal resonance in aeroelastic systems, Nonlinear Dynamics 31 (2003) 1–22. [3] A. Raghothama, S. Narayanan, Non-linear dynamics of a two-dimensional air foil by incremental harmonic balance method, Journal of Sound and Vibration 226 (1999) 493–517. [4] L. Liu, Y.S. Wong, B.H.K. Lee, Application of the centre manifold theory in non-linear aeroelasticity, Journal of Sound and Vibration 234 (2000) 641–659. [5] T. O’Neil, Nonlinear aeroelastic response—analyses and experiments, 34th AIAA, Aerospace Sciences Meeting and Exhibit, 1996. [6] M.D. Conner, Nonlinear Aeroelasticity of an Airfoil Section with Control Surface Freeplay, PhD Thesis, Department of Mechanical Engineering and Materials Science, Duke University, 1996. [7] D.S. Woolston, An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter, Journal of the Aeronautical Sciences (1957) 9936–9956. [8] B.H.K. Lee, S. Price, E.H. Dowell, Y. Wong, Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos, Progress in Aerospace Sciences 35 (1999) 205–334. [9] M.D. Conner, D.M. Tang, E.H. Dowell, L.N. Virgin, Nonlinear behavior of a typical airfoil section with control surface freeplay: a numerical and experimental study, Journal of Fluids and Structures 11 (1997) 89–109. [10] R.M.G. Vasconcellos, F.D. Marques, M.H. Hajj, Time series analysis and identification of a nonlinear aeroelastic section, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Orlando, FL, April 12–15, 2010. [11] L. Daochun, G. Shijun, X. Jinwu, Aeroelastic dynamic response and control of an airfoil section with control surface nonlinearities, Journal of Sound and Vibration 239 (2010) 4756–4771. [12] Y.C. Fung, An Introduction to the Theory of Aeroelasticity, Wiley, NY, 1955. [13] E.H. Dowell, A Modern Course in Aeroelasticity, Kluwer, Dordrecht, 1995. [14] T.W. Strganac, J. Ko, D.E. Thompson, A.J. Kurdila, Identification and control of limit cycle oscillations in aeroelastic systems, Proceedings of the 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, AIAA Paper No. 99-1463, 1999. [15] A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley Series in Nonlinear Science, NY, 1994.