Structural Modeling and Aeroelastic Analysis of High-Aspect-Ratio Composite Wings

ZHAO Yong-hui, HU Hai-yan

(Institute of Vibration Engineering Research, Nanjing University f Aeronautics and Astronautics,

Nanjing 210016, China)

Abstract: A unified structural model for high- aspect- ratio composite wing with arbitrary cross-section is developed. Two types of lay- ups of the composite wing, namely, circumferentially uniform stiffness ( CUS) configuration and circumferentially asymmetric stiffness ( CAS) configuration, are investigated. The present structural modeling method is validated through ANSYS FEM software for the case of a composite box beam. Then, the case of a single- cell composite wing with NACA0012 airfoil shape is considered. To investigate the aeroelastic problem of high- aspect- ratio composite wings, the linear ON ERA aerodynamic model is used to model the unsteady aerodynamic loads under the case of small angle of attack. Finally, flutter speeds of the high aspect- ratio wing with va'ious composite ply angles are determined by using U- g method.

Keywords: structural modeling; aeroelastic analysis; high-aspect-ratio composite wing

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High" aspect"ratio wings have come into prominence recently due to the interest in High Altitude and Long Endurance (HALE) aircrafts for future military as well as civilian missions'1J. Schoor'2J and Tang'3,4] studied the aeroelastic problem of high aspect"ratio wings, but they dealt with the isotropic wing only. Because of the high"performances provided by new composite materials, anistropic composite thirrwalled structures are likely t o play an increasing role in the construction of actual and future generation of high performance flight vehicles, especially for the HALE aircraft s'5J. M any researchers modeled the composite wing as a box beam'6,7J. This excessive simplifica-

tion cannot reflect the real structure of the composite wing.

This paper deals with the high"aspect"ratio composite wing with NACA0012 airfoil shape. The composite wing is modeled as a single-cell, closed cross"sectional shell, and the asymptotically consistent theory for anisotropic thitrwalled beams'8,9 is employed to derive the equations of motion of this system. The structural modeling method for the case of composite box beam is validated by ANSYS FEM software. Then, the present modeling method is straightforwardly extended to the composite wing with NACA0012 airfoil shape. The linear ONERA aerodynamic model'10J is used to

Received date: 2003 09-05; Revision received date: 2004 09 27 Foundation item: National High Tech Projects of China ( 863" 705 2. 3)

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perform aeroelastic analysis of composite wing.

1 Structural Model

1.1 Basic assumptions

The study begins with a single-cell, closed cross-section, fiber-reinforced composite thin-walled shell as shown in Fig. 1 with the following assumptions made for the sake of simplicity.

( 1) All deformations are small enough so that the linear theory of elasticity works well.

(2) The composite shell is a thin-walled structure. That is, its thickness and the cross-section dimension are far smaller than wing semi"span.

(3) It is reasonable to neglect the transverse shear strains in the cross" sectional displacement field.

(4) The out"of-plane cross-sectional warping is incorporated.

Fig. 1 A single cell, closed cross-sectional shell

1.2 Strain energy density

Fig. 2 shows the displacement field of the composite shell, where (x, s, n) denotes the curvilinear coordinate system of a point on the closed contour r in the shell cross"section, s is measured along the tangent to the closed contour r

in a clockwise direction, U1, U2 and U3 denote the displacement components of a point at ( y, z ) = (0, 0), and 'P is the twist angle, positive in the nose-up rotation. The displacement field of the shell can be expressed as u 1 (x, s) = Ui(x ) - y( s) U2(x ) - z (s) U3(x ) +

w(x, s) (1a)

u 2 (x, s) = U2(x )+ z(s) V(x) (1b)

u 3 (x,s) = U3(x)-y(s) f(x) (1c)

where ( ) = d( )/dx ; w(x,s) denotes the out" of-plane warping of the cross"section. The strain energy density of the shell can be written as

O = J (A 11 Y11 + A 22 Y2 + 4 A 66 Y2 +

2A 12 ! 1 Y22 + 4A 16 ! 1 Y12 + 4A 26 Y Y22)

where Yn= £11; Y22= £22; Yj2= ( 1/2) £12.

The relation between the strain and the deformation gives

Y11 = U1(x) - y(s ) U 2( x )- z ( s) U 3 ( x) + w (s, x) (3a)

2!12 = z(s) V(x)

dw(x, s ) ds

Y22 = ( U2(x) + z(s) V(x))

y(s) <P(x) dZ +

d y ( s ) 1 dz ( s )

+ (U 3(x)

y (s) *(x))

d z ( s) 1 dy( s)

3s2 R ds

where R is the radius of curvature of the middle surface. Note that °22 ~ 0. Thus, one obtains

Y22 =- (A 12 Y11 + 2A 26 Y12)/A 22 (4) Substituting Eq. (4) into Eq. (2) gives

O = j(A(s) Y?1 + 2B(s) Y11 Y12 + C(s) Y2)

( 5) A 12 A 26

A(s)= A 11 -

B(s) = 2

A 16 -

C(s) = 4

A ij = ^ '■ Q j ply

(i,j= 1,2,6)

Here, the stiffness A ( s) corresponds to the axial Fig. 2 Displacement field of the composite shell . b( ) h, h ' C( ) i l'

© 1994-2010 China Academic Journal Electronic PubSsSn^^' ( Alto thh e s eai; ( s is a coup ]ngd.net

modulus.

The warping function w ( x, s) can be deter mined from the following conditions'8] 1 I dw (x, s) i a

t # d; d s =0 (6)

Here l is length of the closed contour r. 1.3 Structural modeling of a composite wing

This subrsection focuses on a high"aspect"ratio composite wing with NACA0012 airfoil shape as shown in Fig. 3, where U is the speed of flow, Lc the aerodynamic lift, Mc the aerodynamic mo ment, L the semi-span. The geometry of the cross"section for this composite wing is shown in Fig. 4, where hc is wall thickness, 2b represents for chord length, 'P is pitch angle and positive in the nose-up rotation.

Fig. 3 A higlr aspect- ratio composite wing

Fig. 4 Geometry of the normal cross section

From Eq. (3), Eq. (5) and the warping function obtained from Eq. ( 6) , one has the strain energy of the composite wing as follows

Ui T "Ctt C 12 Cl3 Cl4 Ui

Cl2 C 22 C23 C24 <p

► < „ dx

U3 Ci3 C 23 C33 C 34 U 3

-Cl4 C 24 C34 C44 U 2 V V

u = *JJ

© 1994-2010 China Academic Journal Electr( 7)

where Cj is listed in the appendix for complete^ ness, for more details see Ref. [ 8].

The kinetic energy of the composite wing can be writ ten as

S N. Ui T m c I I I"

I I c - Sz Sy

U3 I - Sz mc I

M - I Sy I mc

Now, the following generalized coordinates are introduced to reduce the dimensions of the sys" t em

U = [U1 ^ U3 U2]T = Oq (9) O = Blockdiag[[ ®U,(x)7 1% n,[ ®9(x)71%o,

[ Ou3(x)] 1%p,[ OU2(x)] 1%¡j (10) Here [ Ou| ( x )] 1% n, [ Op ( x )] 1% o, [ Ou3 (x )] 1%p and [ Ou2 (x )] 1 % q a'e the extensional, torsional, vertical bending, and chordwise bending mode shapes, respectively. Thus, the equations of motion for the composite wing can be derived as

Mq + Kq= Q

mc I I I"

I Ic - Sz Sy

I - Sz mc I

- I Sy I mc

"C11 C12 C13 C14

C 12 C22 C23 C 24

C13 C23 C33 C 34

- C 14 C24 C34 C 44

O* = Block diag^[ Ou{x)] 1% n, [ Ot>(x )] 1%o, [ Ou3(x)] 1%p,[ Ou2(x)] 1%¡j

2 Aerodynamic Loads

The linear ONERA model'10] can be obtained by neglecting the nonlinear part as following

C2 = s2 a+ kvz 0+ C2! ( 12a)

Cz! + A Cz! = X(ao*a+ c '0j + cz( a„za+ c 0') (12b)

-r, .where ( ) = d ( )/ d a= 0- h represents the ef-PublisningHouse.All rights reserved. 1ntfp://www.cnki.net

fective angle of attack; (= Ut/b the dimensionless time; 0 the instantaneous angle of attack; b the semi-span; h the plunge displacement at the 1/ 4 chord; #= h/b the dimensionless plunge displacement at the 1/ 4 chord. Cz represents the lifting coefficient or moment coefficient, a a. the slope of static lifting or moment curve. sz, kVz, A, oz and (% should be identified through wind tunnel tests.

3 Flutter Analysis

To perform the flutter analysis of the composite wing by using v-g method, it is necessary to assume the harmonic motion of aeroelastic system of concern. Combining Eq. ( 11) with Eq. ( 12), one has the following flutter equations

- )2Mq + Kq = pair b3 )2 W(k) q ( 13)

w h ere

2bRM(k) - yacRl(k) R l( k)

R 1(k) G1(k)+ R 2(k) G2(k) ClkXa.

Rz(k) R 1(k) = sz

R2(k) = k Vz — +

C(k) =

kk i cm,

A + %zk2

A2 + k2

. kAz ( (- 1)

i a2+ k2 0]

i k 1 ik 2 1 - b

G,(k) = [ 0 1-

G2(k) = [ 0 ik 0 0] and Rz(k) represents RM(k) or RL(k), k= U is the reduced frequency, Pair the air density, yac= - )b, X a dimensionless parameter reflecting the location of the aerodynamic center, and i = J- 1. Introducing the structural damping ( 1 + ig) into Eq. ( 13) and rearranging the terms, one can arrive at an eigenvalue problem used for flutter analysis.

4 Case Studies

4.1 The case of composite box beam

In order to validate the structural modeling

itebox beam is considered. CU S and CAS configurations of the composite box beam are considered and the material properties listed in Table 1. Table 1 Geometry and material properties of the box beam

Outer width/m 24.21 x 10-3

Outer height/ m 13.46 x 10-3

Length/ m 0. 84455

Ply thickness/ m 127x 10- 6

N umber of plies 6

¿,,/GPa 142

¿22= ¿33/GPa 9. 8

G ,2= G 13/ GPa 6. 0

G23/GPa 4. 83

v , 2= v13 0. 42

v 23 0. 5

P/(kg-m- 3) 1.445 x 103

For CUS configuration of the box beam, the axial, coupling, and shear stiffness A, B and C are constant throughout the cross"section. This type of configuration produces extension-tw ist coupling (ET) , uncoupled vertical bending (VB) and horizontal bending (HB) modes. To validate the present modeling, FEM analysis is performed by using AN SYS. Layered structural shell unit Shell 99 and total 400 elements is used in FEM analysis. The present theoretical results of natural frequencies for composite box beam are compared with a FEM so lution in Table 2. The results show that the present modeling is in good agreement with FEM solu" t ion .

Table 2 Natural frequencies of composite box beam (CUS)

Ply Modes FEM/Hz Present/Hz Error •/%

1VB 28. 30 28. 12 - 0. 64

CLS1: | 151 6: 1HB 45. 03 44. 69 - 0. 76

1ET 499. 88 482. 14 - 3. 55

1VB 33. 91 33. 8 - 0. 21

CLS2: | 0/30| 3 1HB 53. 81 53. 80 - 0. 02

1ET 687. 59 664. 17 - 3. 41

1VB 32. 35 32. 31 - 0. 12

CL S3: | 0/ 4513 1HB 51. 31 51. 37 0. 12

1ET 647. 70 642. 22 - 0. 8 Ui

The CAS configuration of the box beam leads to a set of uncoupled and a set of two coupled equations of motion. The uncoupled equations correspond to the axial extension and the horizontal bending, respectively, while the coupled equations

scribed in previous section, a cantilever compos" describe the bending-twist (BT) coupling modes. © 1994-2010 Chma Academic Journal Electronic Publishing House. AIT rights reserved. nttp:/7www.cnk;i.net

A comparison is made in Table 3 for the predictions from the present model and those in Ref. [ 11] . All the results show that the present modeling is feasible and effective.

Table 3 Natural frequencies of composite box beam ( CAS)

Ply Modes Reference11 '1 / Hz Present/Hz Error/ %

1BT 21. 8 19. 9 - 8.72

CAS1: | 30| 6

2BT 123. 28 122. 13 - 0. 93

CAS2:|45|6 1BT 2BT 15. 04 92 39 14. 39 90. 17 - 4. 32 - 2. 40

4.2 The case of composite wing

Now, a cantilever composite wing with NACA0012 airfoil shape is considered with the same material properties as those listed in Table 1. The chord length is 0. 056m, the semi"span is 1. 68m, ply thickness is 127 % 10 6m, and number of plies is 6. Fig. 5 and Fig. 6 show the results for

'8aö • ° • • a ■ • i e 9 . dS A* J«

° CUS[f>]6 • CAS[0]6

Fig. 5 Bending deflections at the wing tip, Ftip = 1N

"a 7-0

■¡6.0 5.5

5.0 4.5

0 30 60 90

Ply angle/(°)

Fig.6 Twist angles at the wing tip, M tp= 1N'm

the composite wing carrying in turn a Ftip= 1N tip load and a Mtip= 1N *m tip torque. They indicate that there is no significant difference in the tip deflections for CUS[006 and CAS[0]6 configuration, respectively, while CAS[006 configuration has the larger or equal torsional stiffness within 0°~ 90° ply

angles. Therefore, the flutter speed under the CAS €5 1994-20TO China Academic Journal Electronic

[Q]6 configuration is larger than or equal to that under the CUS [ 0]6 configuration (see Fig. 7). The results demonstrate that the laminate layups have the significant influence on the flutter speed. Fig. 8 reveals the influence of aspect"ratio on flutter speed. As expected, the higher aspect"ratio wing has the relatively lower flutter speed.

Ply angle/(n) Fig. 7 Flutter speed vs ply angle

Fig. 8 Flutter speed vs half aspect- ratio

5 Conclusions

A structural model for a high-aspect"ratio composite beam with arbitrary single-cell cross section is developed and validated by using AN SYS software for the case of composite box beam. It is demonstrated that the proposed modeling is in good agreement with the existing methods. Then, the present modeling is extended to the case of the composite wing with NACA0012 airfoil shape. Using the linear ONERA aerodynamic model and the present modeling, the flutter speed of a high"aspect" ratio wing is predicted for different composite ply angles. The results show that the present modeling is applicable to solving the aeroelastic problem

of high-aspect"ratio composite wings. Publishing House. All rights'reserved, http://www.cnki.net

References

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4| Tang D, Dowell E H. Experimental and theoretical study of gust response of high" aspect" ratio wing | J| . AIAA Journal, 2002, 40(3): 419- 429.

51 Patil M J, Hodges D H. Nonlinear aeroelasticity and flight dynam ics of high altitude long- endurance aircraft | J| . Journal of Aircraft, 2001, 38( 1): 88- 94.

6| Patil M J. Aeroelastic tailoring of composite box beams | R | . AIAA Paper 97 0015, 1997.

7| Smith EC, Chopra I. Formulation of an analytical model for composite box-beams | R|. AIAA Paper 90 0962, 1990.

81 Armanios E A, Badir A M. Free vibration analysis of anisotropic thin"walled closed" section beams | J| . AIAA Jour nal, 1995, 33( 10) : 1905- 1910.

9| Berdichevsk V, Armanios E. Theory of anisotropic thin walled closed cross section beams |J|. Composite Engineer

ing, 1992, 2(5): 411- 432.

101 Dunn P, Dungundij J. Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wings | J|. AIAA Journal, 1992, 30( 1) : 153- 162.

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Biographies:

ZHAOYonghui Born in 1969, he receive! his doctoral degree from Harbin Institute of Technology in 1999. From 2002 to 2004, he became a postdoctoral candidate in Nanjing University of Aeronautics and Astronautics (NUAA) . Now he works for Institute of Vibration Engineering Research in NUAA as an associate professor. He has published several scientific papers in different periodicals. Tel: 025- 84891672, E mail: Zhaoyonghui02@163.com

Appendix

= #K ids + #K2ds *

C11 = #K ids + C12 = As #K2ds C i3 = - # K i z ds-C14 = - # K i y d s-

# K 2d s#

# K 2d s #

d sty K 2 z d s

2ds # K 2y ds

C22 = A"s\# K 3d s

C23 =- As#K2zds\#K3ds C24 = - As#K2yds\#K3ds

K 3ds K 3ds

C33 = #K 1 z zds + C34 = #K 1 yz d s +

K 2 z d s

# K 3d s J K 2y d s #K 2 z d s

C44 = # K 1 j d s +

K1 = A(s) - B2(s)/C(s) K 2 = B(s)/C(s) K 3 = 1/C(s)

As = "2/ r"ds =

= #K 1 y 2d s+ #K 2 y d s

K 3d s

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