Limit cycle oscillations of two-dimensional panels in low subsonic flow

International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

Limit cycle oscillations of two-dimensional panels in low subsonic #ow Deman Tang ∗ , Earl H. Dowell1 Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA

Abstract Limit cycle oscillations of a two-dimensional panel in low subsonic #ow have been studied theoretically and experimentally. The panel is clamped at its leading edge and free at its trailing edge. A structural non-linearity arises in both the bending sti.ness and the mass inertia. Two-dimensional incompressible (linear) vortex lattice aerodynamic theory and a corresponding reduced order aerodynamic model were used to calculate the linear #utter boundary and also the limit cycle oscillations (that occur beyond the linear #utter boundary). ? 2002 Published by Elsevier Science Ltd. Keywords: Reduced order aerodynamic model; Flutter; Limit cycle oscillation

1. Introduction Flutter and limit cycle oscillations of plates with free edges presentative of some low aspect ratio wings have been studied and these results [1– 4] have provided a good physical understanding of the physical phenomena for such plates in a high Mach number supersonic #ow. In particular, it has been demonstrated that even with only a single streamwise edge of a plate restrained, bending tension or geometrical non-linearities can produce limit cycle oscillation amplitudes of the order of the plate thickness. More recently, comparable studies for low subsonic #ow speeds [5 –7] have used Von Karman plate equations and a three-dimensional time domain vortex lattice aerodynamic model and reduced order aerodynamic technique to investigate the #utter and limit ∗

Corresponding author. Tel.: +1-919-660-5306. E-mail address: [email protected] (D. Tang). 1 Director of the Center for Non-linear and Complex Systems. Dean Emeritus, Pratt School of Engineering.

cycle oscillation characteristics of a cantilevered low aspect ratio, rectangular or delta wing-panel structure. Again limit cycle oscillations were found and correlated with experiment. As distinct from the above conAgurations, a two-dimensional cantilevered panel clamped at its leading edge and free at its trailing edge in uniform incompressible #ow is considered in this paper. For this model, early studies have been made by Kornecki et al. [8] and also by Huang [9] and Shayo [10]. The results of these investigators were concerned with linear aeroelastic instability. In the present paper, the structural non-linearities arising in the bending sti.ness and mass inertia are included. A two-dimensional incompressible (linear) vortex lattice aerodynamic theory and a corresponding reduced order aerodynamic model are used in the present analysis. 2. State-space equations A schematic of the airfoil=panel geometry with a two-dimensional vortex lattice model of the unsteady

0020-7462/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 1 4 0 - 8

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D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

Nomenclature c

total length of airfoil and panel in the chordwise direction, c = L0 + L D ≡ Eh3 =12(1 − 2 ), panel bending sti.ness E modulus of elasticity h panel thickness L length of elastic panel L0 length of rigid airfoil in chordwise direction km numbers of vortex elements on both the rigid airfoil and panel kmm total number of vortices on the rigid airfoil, the panel and the wake in the x-direction mp mass=area of elastic panel, mp = h m nx number of structural modal functions deAning w qi generalized coordinate in the z-direction Qi generalized aerodynamic force size of the reduced order aerodynamic Ra model t time u in-plane displacements

#ow is shown in Fig. 1. The aeroelastic structure=#uid state-space equations are described as follows. 2.1. Aerodynamic model To model the above aeroelastic structural=#uid system, we consider a #at elastic panel of inAnite width and Anite length, L. It is clamped at its leading edge to a inAnite rigid airfoil with Anite length L0 . The total length of the rigid airfoil plus elastic panel system is L + L0 . The #ow about the rigid airfoil=cantilevered panel is assumed to be incompressible, inviscid and irrotational. Here, we use an unsteady (linear) vortex lattice method to model this #ow [5 –7]. The rigid airfoil=cantilevered panel and wake are divided into a number of elements. In the wake and on the rigid airfoil=cantilevered panel all the elements are of equal size, d x, in the streamwise direction. Point vortices are placed on the rigid airfoil=cantilever panel and in

U Uf w

airspeed linear #utter airspeed elastic panel de#ection in the transverse or z-direction x; z streamwise and normal coordinates [X ]; [Y ] right and left eigenvector matrices of vortex lattice aerodynamic eigenvalue model [Z] eigenvalue matrix
of vortex lattice model  time parameter, mp L4 =D Gp aerodynamic pressure loading on panel Gp non-dimensional aerodynamic pressure, Gp=( ∞ U 2 ) dt the time step, d x=U dx panel element length in the streamwise direction  the vortex strength

∞ ; m air and panel material densities  Poisson’s ratio !f ; ! #utter frequency and oscillation frequency !m natural frequency (˙) d()=dt () d()=d x

the wake at the quarter chord of the elements. At the three-quarter chord of each panel element, a collocation point is placed for the downwash, i.e. we require the velocity induced by the discrete vortices to equal the downwash arising from the unsteady motion of the cantilever panel. Thus, we have the relationship wit+1 =

kmm 

Kij jt+1 ;

i = 1; : : : km;

(1)

j

where wit+1 is the downwash at the ith collocation point at time step t + 1, j is the strength of the jth vortex, and Kij is an aerodynamic kernel function. For the two-dimensional incompressible #ow, the kernel function is given by Kij = 1=[2 (xi − !j )];

(2)

where xi is the location of the ith collocation point, and !j is the location of the jth vortex.

D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

Top of

Rigid Airfoil

1201

Flat Elastic Panel

Wind Tunnel

Horseshoe Vortex

Air Flow U

X kmm Wake Elements

Z

km

Bottom of Wind Tunnel Lo

dx

L

Fig. 1. Schematic of the airfoil=panel geometry with a two-dimensional vortex lattice model of the unsteady #ow.

The aerodynamic matrix equation is given by [A]{}t+1 + [B]{}t = {w}t+1 ;

(3)

where [A] and [B] are aerodynamic coeJcient matrices. The non-dimensional pressure distribution on the rigid airfoil=#at panel is given by   j  c (jt+1 + jt )=2 + Gpj = (it+1 − it ) (4) dx i and the aerodynamic generalized force is calculated from  c 2 Qi = ∞ U Gp$i d x; (5) 0

where $i is ith downwash mode function  0 for x 6 L0 ; $i = %i for L + L0 ¿ x ¿ L0

and %i is ith structural mode function of twodimensional cantilevered #at panel. We now proceed to calculate the kinetic and potential energy of the two-dimensional cantilevered #at elastic panel. (Note that the energies of the rigid airfoil are neglected.) Thus, the equations of motion may be derived via Hamilton’s principle. 2.2. Structural equations of motion For the present structural model, an axially inextensible assumption is used here for the in-plane motion of the elastic panel. Following the analysis of Semler and Paidoussis [11], the inextensibility condition may be expressed as  2  2 @u @w 1+ + = 1: (6) @x @x We use the classical expressions for kinetic and potential energy of an elastic body, but including

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D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

non-linear sti.ness and inertia e.ects. Thus, the expressions for the energies are as follows. 2.2.1. Kinetic energy  1 L mp (u˙ 2 + w˙ 2 ) d x: T= 2 0

t1

(9)

where  is the non-linear curvature of the two-dimensional panel [11]:   2 1=2 @w @2 w  1− = : @x2 @x Thus for (@w=@x) 1;  2   2 2  @w @w  2 1+ : ( ) ≈ 2 @x @x Substituting (10) into (9) gives  2   2 2   @w @ w 1 L 1+ d x: D V= 2 2 0 @x @x

(10)

(11)

2.2.3. Hamilton’s principle The energy method is based on Hamilton’s principle which may be written as  t2  t2 + L dt + +W dt = 0; (12) t1

D + 2

t2



L

0

t1



@w @x

1+

 +

@2 w @x2

2

2  dt d x:

(13)

Integrating by parts, one obtains  t2  L  t2 + T dt = −mp [u+u L + w+w] L dt d x; t1

0

t1

(14)

where  x

1 uL = − 2  +

t2

t1

0

V dt = D  +

t1

@wL @x 

@2 w @x2

2 d x; 

t2

t2

t1

L



0

t1

3

 +W dt =

@4 w + 4 @x

 0

L

@4 w @w @2 w @3 w + 4 @x4 @x @x2 @x3



@w @x

2  +w dt d x

Gp+w d x dt:

(15)

(16)

A relationship between the virtual displacements +u and +w may be obtained using the inextensibility condition. It is   2  1 @w @w @u @w 1+ + : + =− (17) @x @x 2 @x @x Integrating Eq. (17) and applying the boundary conditions, +w = 0 at x = 0, one obtains

 3  @w 1 @w + +w +u = − @x 2 @x

t1

where L is the Lagrangian of the system (L = T − V ) and +W is the virtual work due to aerodynamic force.





and  t2

2

0

t1

(7)

If the panel transverse de#ection in the z-direction (w) is considered to be small relative to the panel length, then the term, (@u=@x)2 , in Eq. (6) may be neglected compared to @u=@x. Thus, for u = 0 at x = 0 and from Eq. (6) we have   2 1 x @w u(x) = − d x: (8) 2 0 @x 2.2.2. Potential energy  1 L D(  )2 d x; V= 2 0

The variational operations on L lead to  t2  L  t2 L dt = mp (u+u ˙ + w+w) ˙ dt d x +

+

 x

0

@2 w 3 + @x2 2



@w @x

2

@2 w @x2

 +w d x:

(18)

D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

Substituting (18) into (14) and using (12), (15), (16) and the following integration formula,  x   L g(x) f(x)+w d x d x 0

0

L 

 =

0

L

x

−

@w @2 w @3 w + @x @x2 @x3

@2 w @x2

@w + @x



L



s



x

0

x

0

 mp

@2 w @x2

 mp @w˙ @x

FM =

 n



g(x) d x f(x)+w d x

Mii ≡

one obtains the Anal structural equation of motion as

  2  @w @4 w mp wL + D 1+ @x4 @x +4

and r

s

Minrs qn q˙r q˙s ;

where





1203

@w˙ @x

2

3 

2 +

@w @wL + @x @x

mp %2i d x;

0

 Kinrs =

1

0

 Minrs =

@w @wL @x @x

1

        D%i [% n %r %s + 4%n %r %s + %n %r %s ]d x;

1

mp %i %n

0



 dx dx

 d x = Gp: (19)

We now expand the transverse or out-of-plane displacement, w, as follows:  w= qm (t)%m (x); (20)

−

0

1



mp %i %n

0

x

%r %s

 x

1



 0

dx x

 %r %s

dx dx

2.3. Aeroelastic state-space equations Consider a discrete time history of the panel, q(t), with a constant sampling time step, Gt. The structural dynamic equations, Eq. (21), can be reconstituted as a state-space equation in discrete time form, i.e. [D2 + dM (qnt ; qst )]{3}t+1 + [D1 ]{3}t + [C2 ]{}t+1 + [C1 ]{}t = −{FN }t+1=2 ;

m

where the transverse natural mode function, %m (x) is that of a cantilevered beam. These functions satisfy the boundary conditions of the cantilevered panel. Substituting (20) into (19), multiplying by %i (x) and integrating from 0 to 1, gives the (non-linear) equations of motion.  Mii qLi + Minrs qn qr qLs n

r

+!i2 Mii qi

s

+ F K + FM = Qi ;

(21)

where qi is normalized by L. The non-linear force FK is induced by the non-linear of the re curvature strained panel and FM plus n r s Minrs qn qr qLs are the non-linear inertia terms. They are, respectively,  FK = Kinrs qn qr qs n

r

s

d x:

(22)

where the vector {3} is the state of the panel, {3} = {q; ˙ q} and [D1 ]; [D2 ] are matrices describing the panel structural behavior. [dM (qnt ; qst )] is an additional non-linear mass matrix which depends on structural response itself. [C1 ]; [C2 ] are matrices describing the vortex element forces on the airfoil=panel. The non-dimensional non-linear force FN is given by FN = FK =2 + FM =(mp L2 ): There is a linear relationship between the downwash w at the collocation points and panel response, {3}. It is deAned by {w} = [E]{3}:

(23)

Thus, combining Eqs. (3), (22) and (23), we obtain a complete aeroelastic state-space equation in matrix

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D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

form:  A

3.1. Non-linear response due to forced excitation 

−E

C2 D2 +

dM (qnt ; qst )





×



t

3

=

0



t+1

 +

3

B 0



C 1 D1

t+1=2

−FN

:

(24)

Following a similar treatment as described in Refs. [5 –7], a reduced order aerodynamic model with static correction is constructed and the Anal aeroelastic state space model is given by  

C2 XRa  +

  t+1  6d     3 D2 + dM (qnt ; qst ) + C2 (A + B)−1 E T −YRa [I − A(A + B)−1 ]E

I

t+1=2   t   6d   0   = ;     3 −FN D1 + C1 (A + B)−1 E (25)

T B(A + B)−1 ]E −ZRa −YRa

C1 XRa

where 6 is the vector of the aerodynamic modal coordinates and  = XRa 6.

3. Numerical studies For the present numerical studies, we consider an aluminum panel (7075 material) ( m = 2:84 × 103 kg=m3 , E = 7:2 × 109 kg=m2 ,  = 0:3, L = 0:27 m, L0 =0:072 m and h=0:36 mm). The structural critical damping ratio is taken as 0.005. The Arst three structural modes were included and the natural frequencies were computed to be 4.13, 25.85 and 71:67 Hz. For the aerodynamic parameters, the airfoil=panel was modeled using 60 vortex elements, i.e. km = 60. The wake was also modeled using 60 vortex elements. The total number of vortex elements (or aerodynamic degrees of freedom) was thus 120, i.e. kmm = 120. The vortex relaxation factor was taken to be 8 = 0:992. A convergence study was done to determine the number of structural modes and vortex elements needed in the analysis. The numerical study includes determination of the forced structural non-linear response and also the aeroelastic response (linear #utter boundary and limit cycle oscillations).

In this case, the aerodynamic force is removed. A single harmonic force f0 %i (1)cos(!t) is placed at the trailing edge of the panel with constant force amplitude (f0 = 0:05). One hundred (1 0 0) excitation frequencies ! from 2 to 7 Hz (near the Arst natural frequency) and ! = 23 to 28 Hz (near the second natural frequency) with G! = 0:1 Hz are considered. At each frequency, the transient time history is computed until the system achieves a steady-state response. In 1 general, it takes about 20 s (a time step of dt = 2048 s is used). The time history for the last one second is used to calculate the response amplitude. For the next frequency (increasing G!), we use initial conditions that are provided by the previous state. This process is continuous in time until the frequency increases to ! = 7 and 28 Hz, respectively. The non-dimensional trailing edge response amplitude (w=L) vs. excitation frequency (!) is shown in Fig. 2(a) (the range of Arst natural frequency) and Fig. 2(b) (the range of second natural frequency) as indicated by the solid line. The Arst two peak frequencies are 4.2 and 25:3 Hz. One is slightly higher than the Arst natural frequency for the linear system and other is smaller than the second natural frequency. A jump response is found near ! = 25:3 Hz. In order to consider the mass and sti.ness e.ects of the structural non-linearity on the response, several di.erent cases were considered as shown in the Agures. The broken line shows the linear results, i.e. all non-linear e.ects are omitted. These have the largest amplitudes and the corresponding frequencies are equal to the Arst two natural frequencies. The dashed line shows the results with the sti.ness non-linearity only (dM = FM = 0). The Arst two peak frequencies are 4.3 and 26:4 Hz. These are higher than those for the linear system. A jump response is found near ! = 26:4 Hz but the response is less than for the linear case (note that there is no jump response near the Arst natural frequency). The dash–dot line shows the results for mass inertia non-linearity only (FK = 0). The Arst two peak frequencies are 4.0 and 25:1 Hz and these are smaller than the natural frequencies. Also a jump is found near ! = 25:1 Hz and there is a net smaller amplitude in the range of the second peak frequency. As shown in Fig. 2(a), the e.ects of the mass and sti.ness non-linearities are proportionally smaller for the

D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209 0.3 0.25

w/L (tip)

w/L (tip)

0.2 0.15 0.1 0.05 0 3

4 5 (a) frequency,Hz

6

0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0

5 10 15 20 25 30 35 40 45 50 (c) time (sec.)

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

7

23 23.5 24 24.5 25 25.5 26 26.5 27 27.5 28 (b) frequency,Hz

w/L (tip)

w/L (tip)

2

1205

0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

5 10 15 20 25 30 35 40 45 50 (d) time (sec.)

Fig. 2. (a) and (b) frequency response curve at trailing edge; (c) and (d) frequency response “time history” for increasing frequency.

frequency range near the Arst natural frequency, but larger for the frequency range near the second natural frequency as shown in Fig. 2(b). A typical “time history” of the steady-state response for the last one second at each excitation frequency is shown in Fig. 2(c) and (d). The time axis (from 0 to 50 s) in the Agure corresponds to the frequency ranging from 2 to 7 Hz and 23 to 28 Hz. A very clear amplitude jump at ! = 25:3 Hz was found. 3.2. Aeroelastic response (self-excitation) 3.2.1. Flutter boundary of the linear aeroelastic model When the structural non-linear forces FN and dM in Eq. (24), or Eq. (25), are set to zero, a linear aeroelastic model is obtained. The aeroelastic eigenvalues obtained from solving these equations determine the stability of the system. When the real part of any one

eigenvalue becomes positive, the entire system becomes unstable. Fig. 3(a) and (b) show a typical graphical representation of the eigenanalysis in the form of real eigenvalues (damping) vs. the #ow velocity and also a root-locus plot for the nominal linear system using all aerodynamic eigenmodes and a reduced order aerodynamic model for the airfoil=panel system. There is an intersection of damping with the velocity axis at Uf = 12:5 m=s, the critical #utter velocity using all the aerodynamic modes (Ra = 120), with a corresponding #utter oscillatory frequency, !f = 22:37 Hz, as indicated by the symbol . Using a reduced order aerodynamic model with a static correction and only 12 aerodynamic eigenmodes (Ra = 12), i.e. the Arst 12 eigenmodes in the Arst branch of aerodynamic eigenvalues, the corresponding values are indicated by the symbol ◦. The #utter velocity and frequency using the reduced order aerodynamic model are virtually

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D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

80

10

70

5

60 frequency,Hz

damping

0 -5 -10

50 40 30 20

-15

10

-20 0

5

10

15

20

(a) flow velocity,m/s

0 -20

-15

-10

-5

0

5

10

(b) damping

Fig. 3. Eigenvalue solution of the linear aeroelastic system (a) for real part (damping) and (b) for root-locus.

Fig. 4. LCO response for U = 13:5 m=s, (a) for time history and (b) for FFT analysis.

identical to those from the all eigenmodes aerodynamic model. 3.2.2. Limit cycle oscillations We have used a standard discrete time algorithm to calculate the non-linear response of this aeroelastic system using the full aerodynamic model, Eq. (24), and also the reduced order aerodynamic model, Eq. (25). It is found that it is almost impossible to calculate the response using the full aerodynamic model because of long CPU times even when using a supercomputer, T916, in the North Carolina Supercomputing Center. Due to the mass inertia non-linearity, the matrix with t + 1 time in Eq. (25) is no longer constant (independent of time) and thus one must invert this matrix each time step. However, the reduced

order aerodynamic model makes these calculations possible. Another beneAt from the reduced order aerodynamic model is that we avoid a numerical divergence in the higher velocity range using a small dt. This is because the time step, dt, can be selected to small enough to insure numerical stability whatever the d x of the original aerodynamic model (unlike the original vortex lattice model). A typical non-dimensional transverse displacement time history and corresponding FFT analysis for U = 13:5 m=s ¿ Uf (12:5 m=s) are shown in Fig. 4(a) and (b) which uses the reduced order aerodynamic model with 12 aerodynamic eigenmodes (Ra = 12). There is a steady-state limit cycle oscillation when the time is larger than about 15 s. The dominant LCO frequency is ! = 22:0 Hz as shown in Fig. 4(b) by

0.06

0.15

0.05

0.1

0.04

0.05 w/L (tip)

RMS of w/L

D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209

0.03

0

0.02

-0.05

0.01

-0.1

0

1207

-0.15 0

0.2

0.4

0.6

0.8

1

0

x/L

the solid line. Note that the LCO frequency is lower than the #utter frequency (!f = 22:37 Hz) which is indicated by a dash–dot line. This is because the effect of the mass inertia non-linear terms on the LCO behavior is signiAcant. This result is not surprising based upon the results for forced excitation. Recall Fig. 2(b). The non-dimensional rms amplitude (w=L) is 0.054. The results corresponding to the dashed line are discussed below. Corresponding to Fig. 4, the LCO (rms) vibration mode shape for U = 13:5 m=s is shown in Fig. 5. Recalling Fig. 3(b), it was found that the #utter mode is dominated by the second chordwise bending mode. The LCO mode shape is similar to the #utter mode with a node line near x=L ≈ 0:8− . To gain further insight into the LCO, the mass inertia non-linear terms were removed and only the structural sti.ness non-linearity was retained, i.e. dM = FM = 0 in Eq. (25). The results are shown in Fig. 6. It is found the system still has a steady-state limit cycle oscillation for the same #ow velocity as in Fig. 4. The non-dimensional rms amplitude (w=L) is 0.093 or nearly double the value for the case with both structural and inertia non-linearities. The FFT analysis of the steady-state LCO for this case is shown in Fig. 4(b) by a broken line. The LCO frequency is 22:55 Hz and is slightly higher than the linear #utter frequency. It is evident that a structural sti.ness non-linearity alone leads to a higher LCO amplitude than when both mass and sti.ness non-linearities are included. Similar results were also observed in Fig. 2 for the forced excitation case. The mass or inertia non-linear terms

10

15

20

25

30

(a) time (sec.) Fig. 6. Time history for U = 13:5 m=s and only sti.ness non-linearity is included.

1 0.8 0.6 0.4 w/L (tip)

Fig. 5. LCO vibration mode shape for U = 13:5 m=s.

5

0.2 0 -0.2 -0.4 -0.6 -0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec.) Fig. 7. LCO response for U = 31:0 m=s.

not only provide an additional inertia force but also equivalent cubic spring forces which render the LCO amplitude smaller. When the #ow velocity increases, the response has a higher amplitude with a single oscillatory frequency. As a typical result, Fig. 7 shows the time history for U = 31 m=s. The peak frequency is 16:65 Hz which is much lower than the linear #utter frequency. Such large amplitudes, however, may be beyond the physical validity of the mathematical model. Fig. 8 shows the rms non-dimensional transverse amplitude (a) and corresponding frequency (b) of the limit cycle oscillation vs. the #ow velocity using the reduced order aerodynamic model with Ra = 12. As seen in this Agure, as #ow velocity increases, the

D. Tang, E.H. Dowell / International Journal of Non-Linear Mechanics 37 (2002) 1199 – 1209 0.6

23

0.5

22 LCO frequency,Hz

RMS nondimensional amplitude at tip

1208

0.4 0.3 0.2 0.1

21 20 19 18 17

0

16 12 14 16 18 20 22 24 26 28 30 32

12 14 16 18 20 22 24 26 28 30 32

(a) flow velocity, m/s

(b) flow velocity, m/s

Fig. 8. Non-linear aeroelastic response vs. #ow velocity (a) for rms response amplitude at the tip or trailing edge and (b) for frequency.

response amplitude increases and LCO frequency decreases. This suggests the LCO response is dominated by the non-linear inertial terms. 4. Concluding remarks Non-linear equations of motion of two-dimensional panels in low subsonic #ow are derived by an energy method. Non-linear sti.ness and inertia e.ects are included in the governing di.erential equation of motion. The #utter stability of the linear system was investigated Arst. It was shown that the #utter mode is dominated by the second panel bending mode which conArms the previous work [8–10] using linear theory. A limit cycle oscillation is found when the #ow velocity is beyond the linear #utter boundary. The LCO amplitude increases and LCO frequency decreases as the #ow velocity increases. The dynamic response of the LCO is a single harmonic oscillation. For the present model, both the non-linear structural sti.ness and inertia lead to LCO. However, the inertia non-linearity is relatively more signiAcant. A more reAned model would include aerodynamic non-linear e.ects. Here the work of Mook and colleagues [12,13] could be used as a basis for such a model. Acknowledgements This work was supported by DARPA through AFOSR Grant F49620-99-1-00253, “Aeroelastic

Leveraging and Control Through Adaptive Structures” under the direction of Dr. Ephrahim Garcia and Dr. Dan Segalman. Dr. Robert Clark is the Duke University Principal Investigator. All numerical calculations were done on a supercomputer, T916, in the North Carolina Supercomputing Center (NCSC).

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