Representation and analysis of control surface freeplay nonlinearity

Journal of Fluids and Structures 31 (2012) 79–91

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Representation and analysis of control surface freeplay nonlinearity R. Vasconcellos a, A. Abdelkefi b, F.D. Marques a, M.R. Hajj b,n a b

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

a r t i c l e i n f o

abstract

Article history: Received 20 April 2011 Accepted 6 February 2012 Available online 22 March 2012

Different representations for a control surface freeplay nonlinearity in a three degree of freedom aeroelastic system are assessed. These are the discontinuous, polynomial and hyperbolic tangent representations. The Duhamel formulation is used to model the aerodynamic loads. Assessment of the validity of these representations is performed through comparison with previous experimental observations. The results show that the instability and nonlinear response characteristics are accurately predicted when using the discontinuous and hyperbolic tangent representations. On the other hand, the polynomial representation fails to predict chaotic motions observed in the experiments. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Aeroelasticity Control surface Freeplay nonlinearity Experimental identification Duhamel formulation Poincare´ section

1. Introduction Aeroelastic responses including instabilities, flutter and limit cycle oscillations (LCO) limit the operation regime of aircraft and may lead to structural problems and material fatigue (Abdelkefi et al., 2011d, 2012; Conner et al., 1997; Fung, 1993; Eftekhari et al., 2011; Lee et al., 1997; Li et al., 2010; Song et al., 2012). Particularly, nonlinearities could lead to bifurcations and abrupt response changes from damped motions to LCO, from LCO to chaos, or from any of these motions to catastrophic responses. As such assessing the evolution of these behaviors through modeling and analysis of nonlinearities can help in exploiting specific physical aspects for system identification (Chabalko et al., 2008), control of undesirable responses or avoidance of dangerous ones. More recently, aeroelastic response of beams have been proposed as means for energy harvesting to power microsensors or actuators (Abdelkefi et al., 2011a–c; Dunnmon et al., 2011). Nonlinearities can arise from unsteady aerodynamic sources, large structural deflections, and/or partial loss of structural integrity (O’Neil, 1996; Shen and Hsu, 1958; Woolston et al., 1955; Woolston, 1957). Of the latter, nonlinearities associated with moving surfaces or external stores are inevitable. In these cases, freeplay, which arises from worn hinges and loosening of attachments, is the most common nonlinearity. Lee and Tron (1989) showed that freeplay nonlinearities can lead to chaotic motions. Modeling a freeplay nonlinearity in the control surface of a pitch–plunge airfoil, and adding a cubic nonlinearity in the pitch stiffness, Li et al. (2010) showed that the pitch spring induces LCOs at speeds higher than the flutter speed. Virgin et al. (1999), Conner et al. (1997), and Trickey et al. (2002) associated freeplay nonlinearity with transitions from damped to periodic LCOs to quasi-periodic responses and then to chaotic motions in an experimental setup of a similar system. These transitions were observed at speeds lower than the linear flutter speed. Using a discontinuous representation of the freeplay nonlinearity and following Henon’s (1982) method, Conner et al. (1997) identified the transition points. This method can be applied to any form of discrete nonlinearity. However, to perform a

n

Corresponding author. Tel.: þ1 5402314190; fax: þ1 5402314574. E-mail address: [email protected] (M.R. Hajj).

0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2012.02.003

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R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

complete analysis, this method requires multiple time-integrations. Other techniques have also been developed to locate the discontinuity in freeplay nonlinearity. Such techniques can generate precise results but are time consuming (Jones et al., 2007; Roberts et al., 2002). The objective of this work is to assess the validity of three different representations of control surface freeplay nonlinearity. These representations include the discontinuous representation, hyperbolic tangent approximation and polynomial expansion. The validity of each of these representations is determined through comparison of instability and nonlinear response characteristics with the experiments of Conner et al. (1997). The aerodynamic loads are approximated using the Duhamel formulation. This analysis aims at determining the goodness of these representations in the modeling and analysis of complex behaviors associated with freeplay nonlinearities.

2. Nonlinear aeroelastic model The aeroelastic system considered here and shown in Fig. 1 consists of a two-dimensional airfoil that has three degrees of freedom including pitch, plunge and control surface motions. The plunge and pitch motions are denoted by w and a, respectively, and the control surface motion is denoted by b. The plunge and the pitch are measured at the elastic axis and the b angle of control surface is measured at the hinge line. The distance from the elastic axis to midchord is represented by ab where a is a constant and b is the semichord length of the entire airfoil section. The distance between the midchord and the hinge line of control surface is represented by cb. The mass center of the entire airfoil is located at a distance xa from the elastic axis and the mass center of the control surface is located at a distance xb from the hinge line, kw and ka are used to represent the plunge and pitch stiffnesses, respectively and kb is used to represent the stiffness of the control surface hinge. Finally, U is used to denote the freestream velocity. Using Lagrange’s equation, the equations of motion of the typical airfoil section as considered above are written as (Fung, 1993) 2

mT 6 6 mW bxa 6 4 mW bxb

mW bxa 2 2

mW b r a 2

3 2 3 2 € dw 7 w 2 2 2 7 6 7 mW b r b þ mW b xb ðcaÞ 76 4 a€ 5 þ 4 0 5 € 2 0 b mW b r 2b mW bxb

2

mW b r 2b þ mW b xb ðcaÞ

0 da 0

3 2 _ kw w 6 6 7 0 54 a_ 7 5þ4 0 db 0 b_ 0

32

0 ka 0

3 2 3 L w 6 6 7 7 7 0 54 a 5 ¼ 4 Ma 5, M kb FðbÞ=b b b 0

32

ð1Þ where xa ¼

Sa , mW b

xb ¼

Sb , mW b

r 2a ¼

Ia 2

mW b

,

r 2b ¼

Ib mW b

2

,

and m W is the mass of the wing, mT is the mass of the entire system (wing þ support blocks). d w, da and db are damping coefficients for the plunge, pitch, flap motions, respectively. Ia is the airfoil mass moment of inertia about the elastic axis and Ib is the control surface mass moment of inertia about the elastic axis. L and M a are the aerodynamic lift and moment measured about the elastic axis and M b is the aerodynamic moment on the flap about the flap hinge; Sa and Sb are the static moments of the wing mass. FðbÞ is used to represent the control surface freeplay nonlinearity.

Fig. 1. Structural representation of the aeroelastic model.

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

81

4 3

increasing ε

2

F(β) (N.m)

1 0 −1 2δ

−2 increasing ε

−3 −4 −4

−3

−2

−1

0 β (deg)

1

2

3

4

Fig. 2. Flap displacement b versus torque obtained with Eq. (3), e increasing from 0 to 100, bl ¼ 2:121 , bu ¼ 2:121 and P ¼0.

3. Control surface freeplay representations Three representations are considered for the freeplay nonlinearity. In the discontinuous representation, FðbÞ is given by ( 0, 9b9 r d, ð2Þ FðbÞ ¼ bd, 9b9 4 d: To integrate the equations of motion with discontinuous freeplay representation, the method described by Henon (1982) is used to locate and integrate at the discontinuity. This method is usually known as the technique of inverse interpolation and is well described by Conner et al. (1997). The hyperbolic tangent representation of the freeplay nonlinearity is given by FðbÞ ¼ 12 ½1tanhðeðbbl ÞÞðbbl Þ þ 12½1 þtanhðeðbbu ÞÞðbbu Þ þ P,

ð3Þ

where bl and bu are the lower and upper boundaries of the freeplay region respectively, P represents a preload upon the system and e is a variable which determines the accuracy of the approximation. Fig. 2 shows the flap displacement b versus torque as obtained using Eq. (3) for bl ¼ bu ¼ 2:121, P ¼0, and for various values of e. Clearly, as e goes to infinity, the hyperbolic tangent representation approaches that of a real discontinuity. In a third representation, the freeplay nonlinearity is modeled by polynomial expansion. To that end, FðbÞ is approximated by 3

FðbÞ ¼ kb0 b þkb2 b þ    :

ð4Þ

4. Aerodynamic loads The Theodorsen approach is used to model the aerodynamic lift and moments L, Ma and M b . In this approach, the unsteady aerodynamic forces and moments are calculated using the linearized thin airfoil theory and written as (Theodorsen, 1935)   U b 2 € þU a_ baa€  T 4 b_  T 1 b€ 2prUbQCðkÞ, L ¼ prb w ð5Þ

p

p

"

      1 U2 Ub 1 2 1 a a_ b þa2 a€  ðT 4 þT 10 Þb þ T 1 þ T 8 þ ðcaÞT 4  T 11 b_ 2 8 2 p p #   2 b  1 2 QCðkÞ, T þ ðcaÞT 1 b€ þ 2prb a þ þ 2 p 7 2

M a ¼ prb

€ bawUb

" 2

M b ¼ prb

b

p

€þ T1w

ð6Þ

#      2  2 2 1 2b U Ub _ þ b T b€ rUb2 T QCðkÞ, T 4 a_  2T 9 þ T 1  a T 13 a€  ðT 5 T 4 T 10 Þb þ T T b 4 11 3 12 2 p p p p 2p2

Ub

ð7Þ

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R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

where

  1 U b _ þ a_ b a þ T 10 b þ Q ¼ Ua þ w T 11 b_ 2 2p p

ð8Þ

and the T functions are defined in Appendix A. The aerodynamic loads are dependent on Theodorsen’s function C(k), where k is the reduced frequency of harmonic oscillation. To simulate the arbitrary motion of the system, the loads associated with Theodorsen’s function are replaced by the Duhamel formulation in the time domain and written as (Li et al., 2010) Z t @f ðsÞ Lc ¼ CðkÞf ðtÞ ¼ f ð0ÞfðtÞ þ fðtsÞ ds, ð9Þ @s 0 where

    1 1 _ þb a a_ þ 1=pT 10 U b þb T 11 b_ f ðtÞ ¼ U a þ w 2 2p

ð10Þ

and fðtÞ is Wagner function. Sears approximated fðtÞ by

fðtÞ  c0 c1 ec2 t c3 ec4 t ,

ð11Þ

where c1 ¼0.165, c2 ¼0.0455, c3 ¼ 0.335 and c4 ¼ 0.3. Using integration by parts and following the state space method proposed by Lee et al. (1997, 2005) and Li et al. (2010), Eq. (9) is rewritten as ! Z t @fðtsÞ U2 Lc ¼ f ðtÞfð0Þ þ f ðsÞ ds ¼ ðc0 c1 c3 Þf ðtÞ þc2 c4 ðc1 þ c3 Þ x þ ðc1 c2 þ c3 c4 ÞU x_ : ð12Þ @s b 0 With the introduction of two augmented variables, xa1 ¼ x and xa2 ¼ x_ , Eq. (1) is rewritten as     1 1 x_ þ Ks KNC  xRS3 xa ¼ 0, ðMs MNC Þx€ þ Bs BNC  2RS2 2RS1

ð13Þ

where x ¼ ½a b w=bT and xa ¼ ½xa1 xa2 . In the state space form, this equation is written as _ ¼ AðXÞX: X

ð14Þ

Details of the matrices in Eqs. (13) and (14) are presented in Appendix B. 5. Prediction of flutter speed The above approach is used to predict the flutter speed and validated through comparison with the experiments of Conner et al. (1997). All parameters used in the mathematical models can be found in the same reference. The variations of the real parts of the eigenvalues for the three degrees of freedom and which correspond to their damping coefficients are shown in Fig. 3(a). The flutter speed, Uf, at which one or more eigenvalues have zero real parts and which corresponds to the onset of linear instability is determined to be 24.4 m/s. The variations of the imaginary parts which corresponds to the coupled frequencies of the aeroelastic system are shown in Fig. 3(b). The variations of the ratios of these frequencies as a function of the wind speed are shown in Fig. 3(c). This figure is used to determine the possibility of internal resonances that may lead to unstable responses before the onset of flutter. It is noted that 2:1 and 4:1 frequency ratios for the flap and pitch and flap and plunge motions exist for specific wind speeds. However, such internal resonances can only arise if the nonlinearities include quadratic or fourth order terms which is not the case of freeplay nonlinearity. As for the possibility of a 3:1 internal resonance, we note that it can take place only at speeds above the flutter speed where the flap frequency is almost equal to three times the flutter frequency (Abdelkefi et al., 2011d). Table 1 compares the flutter speeds (Uf) and coupled natural frequencies ðoc Þ of the aeroelastic system for U¼0 and U ¼ U f obtained using the Duhamel formulation with the experimental and numerical results of Conner et al. (1997). A comparison of the values in Table 1 shows that the predicted flutter speed with the current model is close to that predicted numerically by Conner et al. (1997). Both of these values are slightly larger than the experimental value as reported by the same reference. 6. Nonlinear analysis Tang et al. (1998) observed that a system such as the one considered here can include many bifurcations or regions. In region I, the response is asymptotically stable around an equilibrium position for the flap. In region II, the response is complex and consists of distinct subregions; an initial aperiodic one that is followed by two bi-stable LCOs subregion. Because the governing dynamics are nonlinear, one should expect that the exhibition of either or both of these specific responses is dependent on the system’s parameters and initial conditions. The experiments used in this work for the assessment and validation of different models for the freeplay nonlinearity reported nonperiodic responses only. Region III includes only unique and stable LCOs. Region IV is a part of the flutter region and is characterized by the coexistence of a

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

30

flap pitch plunge

20

0 −10

flap pitch plunge

30 f(Hz)

Re

10

83

20 10

−20 −30

0

0.25

0.5

0.75

1

0

1.25

0

0.25

0.5

U/Uf

5 Freqn. ratio

0.75

1

1.25

U/Uf

pitch/plunge flap/pitch flap/plunge

4 3 2 1

0

0.25

0.5

0.75

1

1.25

U/Uf

Fig. 3. Variations of (a) the damping terms, (b) the coupled frequencies and (c) the frequencies ratio as function of the freestream velocity for the Duhamel formulation. Table 1 Comparison between different methods and Conner et al. (1997) results. Parameter

Duhamel

Numerical [1]

Experiment [1]

owc (Hz) oac (Hz) obc (Hz)

4.38 9.15 18.60 24.4 6.16

4.45 9.218 19.44 23.9 6.112

4.37 9.125 18.625 20.6 5.47

Uf (m/s)

of (Hz)

stable and unstable LCOs. Beyond this region, the response is characterized by a stable flap equilibrium. The experiments considered here did not report on regions IV and V and as such could not be used to validate the models used for representing the freeplay nonlinearity. Instead the validation will be based on characterizing the complex response behaviors in regions II and III that are clearly associated with the freeplay nonlinearity. Fig. 4 shows the variation of the amplitude of the flap motion with the freestream velocity when using the discontinuous, hyperbolic tangent and polynomial representations of the freeplay nonlinearity at speeds lower than the linear flutter speed. The experimental results of Conner et al. (1997) are also shown in the same figure. The results show three distinctive regions for the system’s response. In region I, the flap motion is damped. Region II is characterized by high response amplitudes and nonperiodic motions as reported by the experiments of Conner et al. (1997) and Trickey et al. (2002). Both the hyperbolic tangent approximation and the discontinuous representation of the control surface freeplay nonlinearity show agreement with the experimental results in this region. The polynomial representation yielded smaller flap deviations. In region III, both Conner et al. (1997) and Trickey et al. (2002) reported periodic motions with amplitudes higher than those of region II. In the polynomial approximation, the flap spring coefficients (kb0 ¼ 0:05 and kb2 ¼ 80) were identified to yield amplitudes that are equal to those of the experimental values in regions I and III. Clearly, the response amplitudes of region II when using these coefficients are underpredicted. Noting, the differences and agreements between the different representations of the freeplay nonlinearity and the experimental measurements, a characterization of the different response regions is performed next. Fig. 5(a)–(c) shows the bifurcation diagrams of the numerical results for the hyperbolic tangent, discontinuous and polynomial representations, respectively. In these figures, response amplitude variations with both forward and backward sweeps of the freestream velocity are investigated. Clearly, a hysteretic response is identified in the region (II). This hysteresis is a characteristic of a subcritical instability. Figs. 6 and 7 show time histories and power spectra of the responses in the subregion where only one response is obtained, represented by U=U f ¼ 0:33. The results show that the polynomial representation yielded periodic responses only and, as such, failed to predict the aperiodic/chaotic responses as observed in the experiments of

84

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

4.5

Flap max. ampitude (deg)

4 3.5 3 2.5 2 1.5 1 Polynomial Hyperbolic Tangent Discontinuous Conner

0.5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

U/Uf Fig. 4. Variation of the flap maximum amplitude with the freestream velocity forward sweep of the discontinuous freeplay representation, hyperbolic tangent approximation, polynomial approximation and Conner et al. (1997).

4.5 Flap max amplitude (deg)

Flap max amplitude (deg)

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4 3.5

3 2.5

2 1.5

1 0.5 0

1

0

0.1

0.2

0.3

0.4

U/Uf

0.5

0.6

0.7

0.8

0.9

1

U/Uf

Flap max amplitude (deg)

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

U/Uf

Fig. 5. Variation of the flap maximum amplitude with increasing and decreasing the freestream velocity for (a) hyperbolic tangent, (b) discontinuous and (c) polynomial representations of the freeplay.

Conner et al. (1997). On the other hand, nonperiodic/chaotic responses for both the hyperbolic tangent and discontinuous representations of the freeplay nonlinearity were observed. As the freestream velocity is increased (forward sweeps), the time series and power spectra presented in Figs. 8(a), (b) and 9(a), (b) show that the response is always aperiodic when using either the discontinuous or the hyperbolic tangent representations. In contrast, as the freestream velocity is decreased (backward sweeps), periodic responses are observed in the hysteresis region for both the discontinuous and hyperbolic tangent representations. Clearly, the initial conditions determined the response characteristics in this region. Using the forward and backward sweeps and resulting bifurcation diagrams as presented in Fig. 5, it is noted that both discontinuous and hyperbolic tangent representations predict the aperiodic and periodic subregions of region II. This observation is in agreement with the characterization given by Tang et al. (1998) who noted the existence of these two

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

3.4

3

2.2

2

1.1 β (deg)

β (deg)

1 0

0

−1

−1.1

−2

−2.2

−3

85

0

0.5

1

1.5

2

2.5

−3.4

3

0

0.5

1

1.5

time (s)

2

2.5

3

time (s) 2

1.5

β (deg)

1 0.5 0 −0.5 −1 −1.5 −2 0

0.5

1

1.5

2

2.5

3

time (s)

0

0

−10

−10 Magnitude (dB)

Magnitude (dB)

Fig. 6. Flap time series at 0:33U f (region II) using (a) hyperbolic tangent representation, (b) discontinuous representation and (c) polynomial representation.

−20 −30 −40 −50

−20 −30 −40 −50

−60

−60 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

frequency (Hz)

40

50

60

70

80

90

100

frequency (Hz)

−30 −40 Magnitude (dB)

−50 −60 −70 −80 −90 −100 −110 −120 0

10

20

30

40

50

60

70

80

90

100

frequency (Hz)

Fig. 7. Power spectra at 0:33U f (region II) using (a) hyperbolic tangent representation, (b) discontinuous representation and (c) polynomial representation.

86

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

4

4

3

3

2

2

1

1

0 0

−1

−1

−2

−2

−3 −4

0

0.5

1

1.5

2

2.5

−3

3

0

0.5

1

1.5

2

2.5

3

2

2.5

3

time (s)

time (s) 4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4

−4 0

0.5

1

1.5

2

2.5

0

3

0.5

1

1.5 time (s)

time (s)

Fig. 8. Flap time series at 0:52U f (region II) for the increasing case (a) hyperbolic tangent representation and (b) discontinuous representation, and for the decreasing case (c) hyperbolic tangent representation and (d) discontinuous representation.

10

Magnitude (dB)

Magnitude (dB)

0 −10 −20 −30 −40 −50

0

10

20

30

40

50

60

70

80

90

100

10 0 −10 −20 −30 −40 −50 −60 −70 −80

0

10

20

30

40

frequency (Hz)

60

70

80

90

100

70

80

90

100

10

10

0 Magnitude (dB)

0 Magnitude (dB)

50

frequency (Hz)

−10 −20 −30 −40 −50

−10 −20 −30 −40 −50 −60

−60

−70 0

10

20

30

40

50

60

frequency (Hz)

70

80

90

100

10

20

30

40

50

60

frequency (Hz)

Fig. 9. Power spectra at 0:52U f (region II) for the increasing case (a) hyperbolic tangent representation and (b) discontinuous representation, and for the decreasing case (c) hyperbolic tangent representation and (d) discontinuous representation.

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

87

subregions. Figs. 10 and 11 show the time histories and the power spectra, respectively, at U=U f ¼0.93 as a representation of the response in region III. Inspecting these figures, it is noted that the system’s response in this region is periodic when using any of the three representations. 4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4

0

0.5

1

1.5

2

2.5

−4

3

0

0.5

1

1.5

time (s)

2

2.5

3

time (s)

4 3 2 1 0 −1 −2 −3 −4

0

0.5

1

1.5

2

2.5

3

time (s)

Fig. 10. Flap time series at 0:93U f (region III) using (a) hyperbolic tangent representation, (b) discontinuous representation and (c) polynomial representation.

10

10

0

−10

Magnitude (dB)

Magnitude (dB)

0 −20 −30 −40 −50

−20 −30 −40 −50

−60 −70

−10

0

10

20

30

40

50

60

70

80

90

−60

100

0

10

20

30

frequency (Hz)

40

50

60

70

80

90

100

frequency (Hz)

10

Magnitude (dB)

0 −10 −20 −30 −40 −50 −60

0

10

20

30

40

50

60

70

80

90

100

frequency (Hz)

Fig. 11. Power spectra at 0:93U f (region III) using (a) hyperbolic tangent representation, (b) discontinuous representation and (c) polynomial representation.

88

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

The plotted curves in Fig. 12(a)–(c) show the evolution of the phase portraits with the freestream velocity for different freeplay representations. As predicted, there is a transition from nonperiodic/chaotic motions to periodic motions when using the discontinuous and hyperbolic tangent representations. In contrast, the polynomial approximation does not show this transition as it always yields periodic responses. To better characterize these responses, we show in Figs. 13–15 the

Fig. 12. Evolution of the phase portraits with increasing speed for the (a) hyperbolic tangent, (b) discontinuous and (c) polynomial representations.

Fig. 13. Poincare´ sections for the hyperbolic tangent representation (a) in region II (U ¼ 0:33U f ) and (b) in region III ðU ¼ 0:93U f Þ.

Fig. 14. Poincare´ sections for the discontinuous freeplay representation (a) in region II ðU ¼ 0:33U f Þ and (b) in region III ðU ¼ 0:93U f Þ.

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

89

Fig. 15. Poincare´ sections for the polynomial representation (a) in region II ðU ¼ 0:33U f Þ and (b) in region III ðU ¼ 0:93U f Þ.

Poincare´ map of the motions obtained from the three different representation of the control surface freeplay nonlinearity based on the Duhamel formulation for the aerodynamic loads. Fig. 13(a) and (b) shows the Poincare´ map for the hyperbolic tangent representation for two different freestream velocities. The first considered speed, U ¼ 0:33U f , is in region II and the second one, U ¼ 0:93U f , is in region III. The strange pattern of the crossing points in Fig. 13(a) implies that the response in this region is chaotic. On the other hand, Fig. 13(b) which shows only two points in the Poincare´ section, stresses that the response in region III is periodic. Poincare´ sections of the responses obtained when using the discontinuous representation are presented in Fig. 14(a) and (b). Similar characteristics to those of hyperbolic tangent representation are obtained. Fig. 15(a) and (b) shows the Poincare´ map for the flap motions when the freeplay nonlinearity is approximated by a polynomial expansion for the same freestream velocities. It is noted that while similar results are observed in region III, only a periodic response is obtained in region II as presented in Fig. 15(a). Through comparison with the experimental observations of Conner et al. (1997) that the response in region II is nonperiodic/chaotic and in region III is periodic, the above results show that representing the freeplay nonlinearity by a polynomial expansion fails to predict the complex responses. On the other hand, both the discontinuous and hyperbolic tangent representations are suitable for characterizing the response of the flap motion at speeds lower than the linear flutter speed. 7. Conclusion The modeling of a control surface freeplay nonlinearity by discontinuous, hyperbolic tangent and polynomial representations has been assessed. The validity of these representations in characterizing the effects of control surface freeplay nonlinearity on the flap motion at speeds below the flutter speed is determined through comparison with results from the experiments of Conner et al. (1997). The results show that the discontinuous and hyperbolic tangent representations of the control surface freeplay nonlinearity can predict complex aperiodic responses as observed in the experiments. On the other hand, the polynomial representation fails to predict these responses. The ability to represent freeplay nonlinearities with a hyperbolic tangent representation as presented here has a significant impact in that it can be used effectively in analytical modeling and analysis of these nonlinearities. Acknowledgment The authors acknowledge the financial support of the Sa~ o Paulo State Research Agency, FAPESP, Brazil (Grant 2007/ 08459-1) and the Coordination for the Improvement of Higher Education Personnel (CAPES), Brazil (Grant 0205109). Appendix A. Theodorsen constants

T1 ¼ 

2 þc2 pffiffiffiffiffiffiffiffiffiffiffi2ffi 1c þ c cos1 c, 3

T3 ¼ 

  pffiffiffiffiffiffiffiffiffiffiffiffi 1c2 1 1 þc2 ðcos1 cÞ2 , ð5c2 þ 4Þ þ cð7 þ 2c2 Þ 1c2 cos1 c 4 8 8

T4 ¼ c

pffiffiffiffiffiffiffiffiffiffiffiffi 1c2 cos1 c,

T 5 ¼ ð1c2 Þðcos1 cÞ2 þ 2c T7 ¼ c

pffiffiffiffiffiffiffiffiffiffiffiffi 1c2 cos1 c,

  7 þ 2c2 pffiffiffiffiffiffiffiffiffiffiffi2ffi 1 þc2 cos1 c, 1c  8 8

90

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

T 8 ¼ 13ð1 þ2c2 Þ

pffiffiffiffiffiffiffiffiffiffiffiffi 1c2 þc cos1 c,

"pffiffiffiffiffiffiffiffiffiffiffiffi # 1c2 ð1c2 Þ þaT 4 , 3

1 T9 ¼ 2

pffiffiffiffiffiffiffiffiffiffiffiffi 1c2 þ cos1 c,

T 10 ¼

pffiffiffiffiffiffiffiffiffiffiffiffi T 11 ¼ ð2cÞ 1c2 ð12cÞ cos1 c, pffiffiffiffiffiffiffiffiffiffiffiffi T 12 ¼ ð2 þcÞ 1c2 ð1 þ 2cÞ cos1 c, T 13 ¼ 12½T 7 þ ðcaÞT 1 :

Appendix B. Matrices of Eqs. (13) and (14) 2

r 2a 6 2 6 Ms ¼ 4 r b þ ðcaÞxb xa 2 6 Ks ¼ 6 4

xa

r 2b

xb

xb

M T =mW

r 2a o2a

0

0

r 2b o2b FðbÞ=b

0

0 2

6 Bs ¼ ðKT Þ1 4

3

r 2b þ ðcaÞxb

7 7, 5

3

0

7 0 7, 5

o2w

2ma oa xa

0

0

3

0

2mb ob xb

0

0

0

2mw ow xw

7 1 5K ,

where K is the eigenvector matrix from Ms x€ ¼ Ks x and ma , mb , mw are modal masses. 2 2 3 pb 18 þa2 ðT 7 þ ðcaÞT 1 Þb2 pab2 7 r 6 2 2 6 2T 13 b2 MNC ¼  T 3 b =p T 1 b 7 5, mW 4 2 2 2 pab T 1 b pb 2

3

pð12aÞUb T 1 T 8 ðcaÞT 4 þT 11 =2ÞUb 0 r 6 7 BNC ¼  T 4 T 11 Ub=ð2pÞ 0 5, 4 ð2T 9 T 1 þT 4 ða12ÞÞUb mW pUb UT 4 b 0 2

0

ðT þT ÞU 2

0

0

0

0

3

4 10 r 6 7 KNC ¼  4 0 ðT 5 T 4 T 10 ÞU 2 =p 0 5,

mW

    1 2prU T =mW rUT 12 =mW  , R ¼ 2prU a þ 2 mW S1 ¼ ½U T 10 U=p 0, S2 ¼ ½bð12aÞ bT 11 =2p b, S3 ¼ ½c2 c4 ðc1 þc3 ÞU 2 =b ðc1 c2 þ c3 c4 ÞU, 2

0 1 6 A ¼ 4 Mt Kt E1

I33

0

3

M1 t Bt

M1D t

E2

F

7 5

being, Mt ¼ Ms MNC ,

Bt ¼ Bs BNC 1=2RS2 ,

Kt ¼ Ks KNC 1=2RS1 ,

D ¼ RS3 ,

R. Vasconcellos et al. / Journal of Fluids and Structures 31 (2012) 79–91

" E1 ¼ " E2 ¼ " F¼

0

0

0

U=b

UT 10 =ðpbÞ

0

# ,

0

0

0

ð1=2aÞ

T 11 =ð2pÞ

1

0

# ,

1 2

2

c2 c4 U =b

91

ðc2 þ c4 ÞU=b

# :

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