Aeroelastic Stability and Response of Composite

Aeroelastic Stability and Response of Composite Swept Wings in Subsonic Flow Using Indicial Aerodynamics S. A. Sina1 e-mail: [email protected]

T. Farsadi H. Haddadpour Aerospace Engineering Department, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-8639, Tehran, Iran

In this study, the aeroelastic stability and response of an aircraft swept composite wing in subsonic compressible flow are investigated. The composite wing was modeled as an anisotropic thin-walled composite beam with the circumferentially asymmetric stiffness structural configuration to establish proper coupling between bending and torsion. Also, the structural model consists of a number of nonclassical effects, such as transverse shear, material anisotropy, warping inhibition, nonuniform torsional model, and rotary inertia. The finite state form of the unsteady aerodynamic loads have been modeled based on the indicial aerodynamic theory and strip theory in the subsonic compressible flow. Novel Mach dependent exponential approximations of the indicial aerodynamic functions have been developed. The extended Galerkin’s method was used to construct the mass, stiffness, and damping matrices of the nonconservative aeroelastic system. Eigen analysis of the system was performed to obtain the aeroelastic instability (divergence and flutter) boundaries. Also, solving the equations of motion in the time domain leads to the aeroelastic response of wing in different flight speeds. The obtained results are compared with the available results in the literature, which reveals an excellent agreement. The numerical results obtained in this article seek to clarify the effects of geometrical and material couplings and flight Mach number on the aeroelastic instability and response of composite wings in subsonic compressible flow. [DOI: 10.1115/1.4023992] Keywords: aeroelastic stability and response, unsteady subsonic compressible flow, indicial aerodynamic, thin-walled composite beam

1

Introduction

Thin walled beams (TWBs) are widely used in engineering applications with the minimum weight design criteria, ranging from civil to aerospace and many other industrial fields. The weight reduction of the beam leads to the importance of the dynamical and vibrational problems of the structure. The advanced nature of composite TWB is reviewed in a monograph by Librescu and Song [1]. A beneficial behavior obtained from the directionality property of composite materials is structural tailoring, which enables one to achieve desired deformation modes, structural couplings, and response characteristics. A spectacular product of this technology was, among others, the possibility to eliminate, without weight penalties, the occurrence of the chronic aeroelastic instabilities. The tailoring concept applied to composite lifting surfaces, in general, and to forward-swept wings, in particular, has been thoroughly discussed in survey papers by Weisshaar [2] and Shirk et al. [3]. The structural tailoring technique has been widely utilized in the design of composite rotor blades of rotorcraft structures; the pioneering works by Mansfield and Sobey [4,5], Bauchau [6], Rehfield and Atilgan [7], and Ganguli and Chopra [8] should be addressed. The extensive research papers reviewing in depth the problem of rotating composite beams (see, e.g., Hodges [9], Kunz [10], Jung et al. [11], Oh et al. [12], Li et al. [13], Chakravarty [14], and Nicholls-Lee et al. [15]) clearly reveal the extreme importance of this research field. Furthermore, the concept of passive blade twist control was developed through the use of composite materials with proper tension1 Corresponding author. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 2, 2012; final manuscript received February 3, 2013; published online June 18, 2013. Assoc. Editor: Massimo Ruzzene.

Journal of Vibration and Acoustics

torsion elastic coupling, as implemented in the XV-15 tilt rotor aircraft. According to this technology, structural tailoring is utilized in the design of two different blade twist distributions corresponding to two rotor speeds (helicopter and airplane flight modes) to improve the aerodynamic performance [16,17]. The aeroelastic problems play an important role in the safety and air-worthiness of flight vehicle systems constructed from composite materials. Since the aircraft design is primarily based on the principle of TWBs (see, e.g., Bruhn [18]), the aeroelastic instability should be directly investigated within the frame of TWB models. In the context of aeroelastic response and control of advanced composite wings, during the last decade a number of research works have been done by Librescu and his colleagues [19–29]. Qin [19] investigated the dynamics, stability, and control of vibration and aeroelasticity of aircraft wings. He used an advanced structural model, which was developed by Librescu and Song [20]. Subsequently, Qin and Librescu [22,23] presented the aeroelastic analysis of aircraft wings using unsteady aerodynamic theory for different subsonic flight speeds. The obtained results revealed that elastic tailoring and warping restraint play a significant role on the flutter instability and dynamic response of composite wings. Recently, Fazelzadeh et al. [27] studied the aeroelastic instability of advanced swept composite wings subjected to roll angular velocity. Choo et al. [28] studied the robust vibration control and dynamic response of thin walled composite beams featuring bending-torsion elastic coupling in incompressible flow with model uncertainty. Also, Na et al. [29] have investigated the aeroelastic response and active control of aircraft wings in compressible flight regimes. They used indicial aerodynamic theory to represent the unsteady aerodynamic loads in a compressible flight regime. In general, there are three approaches in the modeling of the aerodynamic loads: steady, quasi-steady, and unsteady. The steady and quasi-steady aerodynamic theories contain some errors

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in the prediction of aeroelastic instability and response [30]. The indicial lift response is a useful starting point in the development of a general time-domain unsteady aerodynamic theory. By definition, an indicial function is the response to a disturbance that is applied instantaneously at time zero and held constant thereafter, that is, a disturbance given by a step function. If the indicial response is known, then the unsteady loads to arbitrary changes in angle of attack can be obtained through the superposition of indicial responses using Duhamel’s integral [31–34]. The indicial functions have been used to modify the circulatory part of the lifting force and pitching moment in unsteady compressible aerodynamic models. Mazelski [35] and Mazelski and Drischler [36] used the exponential approximation of the indicial function in compressible flow. The coefficients of the approximation are obtained with an indirect approach by relating numerical results obtained for oscillating airfoil in the frequency domain back into the time domain. The same approach is used by Dowell [37] to obtain such approximations in an incompressible regime using Theodorsen’s exact result and for compressible flows using finitedifference solutions of the unsteady flow problem. Marzocca et al. [38] presented the exponential approximation of the indicial function in the subsonic compressible and supersonic flight speed regimes. Also, the validation of the model, closed-form solutions, and aerodynamic derivatives for different flight regimes are obtained. They applied the obtained indicial aerodynamic functions in the aeroelastic analysis of two-dimensional lifting surfaces in the subsonic compressible, linearized transonic, supersonic, and hypersonic flight speed regimes [39]. Exponential approximations of the compressible indicial functions in the existing research works are available only in limited Mach numbers (M ¼ 0.5, 0.6, 0.7, 0.8). In the present study, a novel exponential approximation is developed, which represent the coefficients of approximations as functions of Mach number. This technique in conjunction with the state-space representation of the aerodynamic loads [19,33] enables one to perform direct stability analysis of aircraft wings in different subsonic Mach numbers. This paper is an extension of the research work of Haddadpour et al. [40] and includes the effects of compressibility in the aeroelastic analysis of advanced aircraft wings based on the indicial aerodynamic theories. Exponential approximations of indicial aerodynamic functions in the subsonic compressible flow regime are utilized, and the related coefficients are represented as functions of Mach number. The structural model incorporates a number of nonclassical effects, such as transverse shear, warping inhibition, nonuniform torsional model, and rotary inertia. The governing differential equations of aeroelastic system are obtained using Hamilton’s principle. Furthermore, the assumptions of small deformations and small strains theory result in a linear relationship between cross-section external loads and the strain measures. In order to study the effects of the fiber orientation and the lay-up configuration on the aeroelastic behavior of composite wing, the circumferentially asymmetric stiffness (CAS) [7] is used as a configuration scheme. Another important design parameter, which is included in this study is the offset between the elastic and the midchord axis of the wing. To the best of the authors’ knowledge, most of the research works in the aeroelastic analysis of the aforementioned TWB structural model have not considered this parameter [19,22–29]. This geometric parameter of the wing plays such an important role that, without it, the obtained aeroelastic behavior via analytical studies is practically in doubt. However, the assessments of the influence of these design parameters and their proper application should constitute an important task in the aeroelastic design of advanced composite wing structures. In summary, the aim of this study is to: (i)

represent a unified model of aeroelastic systems in a subsonic compressible flow regime using aerodynamic states and indicial aerodynamics theory. Mach dependent exponential approximations of the indicial functions have been developed. This feature is the novel aspect of the proposed

051019-2 / Vol. 135, OCTOBER 2013

(ii) (iii) (iv) (v) (vi)

2

model, which enables one to perform direct stability analysis of aircraft wings in different subsonic Mach numbers. represent some numerical results concerning the aeroelastic response of advanced aircraft wings modeled as composite TWBs in a subsonic compressible flight regime do a stability analysis of the aircraft wings in different subsonic Mach numbers ranging from weak subsonic to weak transonic Mach numbers study the effects of fiber angle and sweep angle on the aeroelastic behavior of composite TWBs in subsonic compressible flow study the effects of offset between the elastic and the midchord axis of the wing on the aeroelastic behavior of composite TWBs in subsonic compressible flow study the behavior of the aeroelastic modes in the various subsonic flight Mach numbers

Structural Model

Although the utilized structural model is representative for advanced aircraft wings with general cross section, for the sake of convenience a single-cell, fiber-reinforced composite thin walled box beam with a length of L, width of 2w, height of 2d, thickness of h, wing chord of 2b, and sweep angle of K is considered in this paper (see Fig. 1). The utilized structural model is similar to that developed by Librescu and Song [1]. Two structural reference coordinates are considered: (x, y, z) is a local fixed Cartesian coordinate system with the outward z-direction parallel to the

Fig. 1 Schematic description of the wing structure and its cross section

Fig. 2 Cross section coordinate to define complex cross sections of CAS configuration

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longitudinal axis of the wing, and (s, n, z) is used to define complex cross section profiles (see Fig. 2). Also, the aerodynamic measures are presented in the aerodynamic reference coordinate, which is specified with the subscript of “ae.” The 3-D displacement parameters of (u, v, w) are defined as, uðx; y; z; tÞ ¼ u0 ðz; tÞ  y/ðz; tÞ vðx; y; z; tÞ ¼ v0 ðz; tÞ þ x/ðz; tÞ

    dx dy wðx; y; z; tÞ ¼ w0 ðz; tÞ þ hx ðz; tÞ yðsÞ  n þ hy ðz; tÞ xðsÞ þ n ds ds  /0 ðz; tÞ½Fw ðsÞ þ naðsÞ

(1)

where u0, v0, and w0 are midsurface displacements in the x, y, and z directions, respectively. Also, hx, hy, and / are section normal vector rotations about the x, y, and z directions, respectively. hx ðz; tÞ ¼ cyz ðz; tÞ  v00 ðz; tÞ hy ðz; tÞ ¼ cxz ðz; tÞ  u00 ðz; tÞ

(2)

dy dx aðsÞ ¼ yðsÞ  xðsÞ ds ds

The free stream velocity Un is normal to the leading edge of the wing Un ¼ U1 cos K. So, the downwash can be expressed as     @v0 @/ tan K tan K  x /_ þ Un wa ðx; z; tÞ ¼ v_0  Un / þ Un @z @z

The primary warping function (Fw) can be written as Fw ¼

ðs

phenomena. The noncirculatory part, also called the apparent mass or inertia effect, is generated when the wing motion has a nonzero acceleration. The air surrounding the wing has finite mass, which leads to inertial forces opposing its acceleration. In fluid with a finite speed of sound, noncirculatory flow patterns do not adjust themselves immediately to changing boundary conditions. Contrary to the incompressible case, which the noncirculatory loads become infinite at the start of the impulsive motion, such a singularity is eliminated when compressibility cushions the impact. So in the compressible aerodynamics, any concept of virtual mass is meaningless [32,33]. Downwash velocity about the reference axis placed temporarily through the leading edge is expressed as [32]   @za @za @za @za @za þU1 ¼ þU1 cosK þ sinK wa ðx;z;tÞ ¼ @t @ x @t @x @z     @v0 @/ sinK  x /_ þU1 sinK ¼ v_0 U1 /cosK þU1 @z @z (8)

½rn ðsÞ  Wds

(3)

D ^ ^_ _ a ðz; tÞ  x/ ¼w a ðz; tÞ

0

where the torsional function (W) for a box beam is stated as þ

rn ðsÞ ds 2AC hðsÞ ¼ W ¼ Cþ ds b hðsÞ C

(4)

rn(s) is the perpendicular distance from the section’s centroid to the tangent of the midline beam contour, defined as rn ðsÞ ¼ xðsÞ

dy dx  yðsÞ ds ds

(5)

(9)

^_a ðz; tÞ gives The downwash velocity includes two main parts: w ^_ plunging motion, and /a ðz; tÞ leads to a linear variation of plunging motion along the chord. In the utilized compressible aerodynamic theory, these two parts are treated separately. ^_a ðz; tÞ are The aerodynamic lift and moment produced by w stated as [31] 1 L0T ðz; tÞ ¼ CL/ qUn2 ð2bLÞ 2 " # ðt ^_ a ðz; rÞ ^_ a ðz; 0Þ 1 dw w /c ðt  rÞdr (10) /c ðtÞ þ  Un dr 0 Un

The axial strains associated with the displacement field is

and

ezz ðn; s; z; tÞ ¼ e0zz ðs; z; tÞ þ nenzz ðs; z; tÞ

1 TT0 ðz; tÞ ¼ CL/ qUn2 ð2bÞð2bLÞ 2 " # ðt ^_ a ðz; rÞ ^_ a ðz; 0Þ 1 dw w /cM ðt  rÞdr (11) /cM ðtÞ þ  Un dr 0 Un

e0zz ðs; z; tÞ ¼ w00 ðz; tÞ þ enzz ðs; z; tÞ ¼ h0y ðz; tÞ

h0x ðz; tÞyðsÞ

þ

h0y ðz; tÞxðsÞ

00

 / ðz; tÞFw ðsÞ

dy dx  h0x ðz; tÞ  /00 ðz; tÞaðsÞ ds ds

(6)

where prime denotes derivative with respect to the z coordinate. The tangential shear strain components can be defined as

respectively. Also, the aerodynamic loads due to the angular ve^ locity /_ ðz; tÞ are [18] a

AC csz ðs; z; tÞ ¼ c0sz ðs; z; tÞ þ 2 /0 ðz; tÞ b dx dy þ ½v00 ðz; tÞ þ hx ðz; tÞ ds ds dy dx cnz ðs; z; tÞ ¼ ½u00 ðz; tÞ þ hy ðz; tÞ  ½v00 ðz; tÞ þ hx ðz; tÞ ds ds c0sz ðs; z; tÞ ¼ ½u00 ðz; tÞ þ hy ðz; tÞ

(7)

where e0zz ; c0sz are the normal strain and the shear strain components on the midsurface of the box beam, respectively.

3

Unsteady Aerodynamic Model

An incompressible unsteady aerodynamic theory consists of noncirculatory and circulatory parts from two physically different Journal of Vibration and Acoustics

L0q ðz; tÞ ¼ CL/ qUn2 ð2bLÞ " # ðt ^ ^ b/_ a ðz; 0Þ b d/_ a ðz; rÞ /cq ðt  rÞdr /cq ðtÞ þ  Un dr 0 Un (12) and Tq0 ðz; tÞ ¼ CL/ qUn2 ð2bÞð2bLÞ " # ðt ^ ^ b/_ a ðz; 0Þ b d/_ a ðz; rÞ /cMq ðt  rÞdr /cMq ðtÞ þ  Un dr 0 Un (13) OCTOBER 2013, Vol. 135 / 051019-3

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Total aerodynamic loads about the leading edge are expressed as L0 ðz; tÞ ¼ L0T ðz; tÞ þ L0q ðz; tÞ; T 0 ðz; tÞ ¼ TT0 ðz; tÞ þ Tq0 ðz; tÞ

(14)

In the above equations, q is the air density. To include 3D effects of the finite span wing, lift curve slope CL/ is obtained from the Diederich general formula [41] as 2p cos Ke CL/ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  M2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AR 1  M2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p cos Ke 2p cos Ke 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ AR 1  M 1 þ p pAR 1  M2 (15)

1 Lae ðz; tÞ ¼ CL/ qUn2 ð2bLÞ 2 " ðt ^_a ðz; rÞ ^_a ðz; 0Þ 1 dw w  /c ðt  rÞdr /c ðtÞ þ Un U dr 0 n þ

1 Tae ðz; tÞ ¼ CL/ qUn2 ð2bÞð2bLÞ 2 " ðs ^_a ðz; rÞ ^_a ðz; 0Þ 1 dw w /cM ðt  rÞdr  /cM ðtÞ þ Un U dr 0 n

where M is the Mach number, and AR and Ke are the wings aspect ratio and effective sweep angle, respectively. AR ¼

þ

ðt

^_a ðz; rÞ 1 dw /c ðt  rÞdr ¼ D1 ðz; tÞ U dr 0 n

LT ðz; tÞ ¼ L0T ðz; tÞ

2b Un ðt

ð t ^_ d/a ðz; rÞ  /cq ðt  rÞdr ¼ D2 ðz; tÞ dr 0

^_a ðz; rÞ 1 dw /cM ðt  rÞdr ¼ D3 ðz; tÞ dr 0 Un

2b Un

(20)

ð t ^_ d/a ðz; rÞ  /cMq ðt  rÞdr ¼ D4 ðz; tÞ dr 0

To convert Lae ðz; tÞ and Tae ðz; tÞ into state space form, the exponential approximation of the indicial functions is utilized. These approximations have specific coefficients, which are obtained for each flight Mach number numerically. In the existing research works (see, e.g., [35–39]), these approximations are available only in limited Mach numbers (M ¼ 0.5, 0.6, 0.7, 0.8). In order to represent the indicial response functions as functions of Mach number, new exponential approximations were developed based on the following statements.

TT ðz; tÞ ¼ TT0 ðz; tÞ þ bða þ 1ÞL0T ðz; tÞ 1 bða þ 1Þ Lq ðz; tÞ ¼ L0q ðz; tÞ  CL/ qUn2 ð2bLÞ 2 Un " # ð t ^_ d/a ðz; rÞ ^_ /c ðt  rÞdr  /a ðz; 0Þ/c ðtÞ þ dr 0 1 bða þ 1Þ Tq ðz; tÞ ¼ TT0 ðz; tÞ  CL/ qUn2 ð2bÞð2bLÞ 2 Un " # ð t ^_ d/a ðz; 0Þ ^_ /cM ðt  rÞdr  /a ðz; 0Þ/cM ðsÞ þ dr 0

•

The asymptotic values of the indicial functions are computed by multiplying their counterparts in incompressible flow by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

the Prandtl–Glauert factor 1= 1  M2 . It should be noted that these asymptotic values are equals to the sectional lift and moments coefficients in steady flow [42]. Hence,

(17)

1 1 /c ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; /cM ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  M2 4 1  M2 3 1 /cq ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; /cMq ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 1M 4 1  M2

A new set of aerodynamic indicial functions are defined in the reference coordinate as /c ðtÞ ¼ /c ðtÞ

  a 1 þ /c ðtÞ /cM ðtÞ ¼ /cM ðtÞ þ 2 2   a 1 þ /c ðtÞ /cq ðtÞ ¼ /cq ðtÞ  2 2      a 1  a 1 2 þ /cq ðtÞ  /cM ðtÞ  þ /c ðtÞ /cMq ðtÞ ¼ /cMq ðtÞ þ 2 2 2 2 (18) Unsteady aerodynamic lift Lae ðz; tÞ and pitching moment Tae ðz; tÞ about the reference axis are expressed as 051019-4 / Vol. 135, OCTOBER 2013

# ð t ^_ d/a ðz; rÞ  /cMq ðt  rÞdr dr 0

Let D’s be defined as (16)

The quantities identified as ð/c ; /cM Þ and ð/cq ; /cMq Þ are the compressible lift and moment indicial functions associated with the plunging and the rate of pitch motions, respectively. In the Eqs. (10)–(14), the prime indicates that the aerodynamic loads are calculated in the coordinate passing through the leading edge. Let the aerodynamic loads to be expressed in the so called Theodorsen’s coordinate, which is located at b(a þ 1) behind the leading edge. Hence,

þ Lq ðz; tÞbða þ 1Þ

2b ^_ 2b / ðz; 0Þ/cMq ðtÞ þ Un a Un

(19)

L 2b

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  M2 cos Ke ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos K 1  M2 cos2 K

# ð ^ ^ 2b/_ a ðz; 0Þ  2b t d/_ a ðz; rÞ  /cq ðt  rÞdr /cq ðtÞ þ Un Un 0 dr

•

(21)

The initial values of the indicial functions are the corresponding quantities for simple harmonic motion at infinite frequency [43]. 2 1 ; /cM ð0Þ ¼ pM pM 1 2 /cq ð0Þ ¼ ; /cMq ð0Þ ¼ pM 3pM /c ð0Þ ¼

•

(22)

Lomax et al. [44] have solved the wave equation in the 2D unsteady compressible flow to obtain the chordwise pressure loading on the airfoil in the time range of 0  s  2M=ðM þ 1Þ where s ¼ Ut=b. Hence, Transactions of the ASME

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i 2 h s 1 ð1  MÞ pM 2M   1 s s2 ð1  MÞ þ ðM  2Þ /cM ðsÞ ¼ 1 pM 2M 8M    1 s s2 M 1 ð1  MÞ þ /cq ðsÞ ¼ 1 (23) pM 2M 2 4M  2 3s 3s2 /cMq ðsÞ ¼ 1 ð1  MÞ þ ð1  MÞ2 3pM 4M 32M2   s3 1 3 þ ð 1  M Þ M þ 4 16M3 /c ðsÞ ¼

•

" 3 ^_a ðz; tÞ X a0 w 1 2 Lae ðz; tÞ ¼ CL/ qUn ð2bLÞ c  aic Bic ðz; tÞ 2 Un i¼1 3 ^ 3 2b a0cq /_ a ðz; tÞ X  aicq Bicq ðz; tÞ5 þ Un i¼1 " 3 ^_a ðz; tÞ X 1 a0 w 2 Tae ðz; tÞ ¼ CL/ qUn ð2bÞð2bLÞ cM  aicM BicM ðz; tÞ 2 Un i¼1 3 ^ 3 2b a0cMq /_ a ðz; tÞ X aicMq BicMq ðz; tÞ5  þ Un i¼1

The indicial functions are assumed to be in the form of four term Mach dependent exponentially growing functions. So,

4 /c;cq;cM;cMq ðM;sÞ " ¼ a0c;cq;cM;cMq ðMÞ

3 X

# aic;cq;cM;cMq ðMÞexpðbi sÞ H ðsÞ (24)

i¼1

where HðsÞ is the unit step function. ai are assumed to be Mach dependent, but bi are constants. Let bi be equal to their counterpart in the exponential approximation of /c presented in Ref. [36]. To evaluate four unknowns ai , the collocation method has been used. In the case of higher order exponential approximations, the interpolation techniques can be used to obtain other necessary equations. The present method can easily model as many as necessary aerodynamic lag terms into the finite state space form. In Eq. (24), if three aerodynamic lag terms are used for each indicial function, then 12 aerodynamic lag terms would exist in the description of the 3D unsteady aerodynamic loads in the subsonic compressible flow. D’s are given as D1 ðz; tÞ ¼

3 ^_a ðz; tÞ X a0c w  aic Bic ðz; tÞ Un i¼1

D2 ðz; tÞ ¼

^ 3 2b a0cq /_ a ðz; tÞ X  aicq Bicq ðz; tÞ Un i¼1

3 ^_a ðz; tÞ X a0 w D3 ðz; tÞ ¼ cM  aicM BicM ðz; tÞ Un i¼1

The governing equations of motion are derived using Hamilton’s principle in the absence of body forces, surface shear forces, and thermal loadings. Hamilton’s principle is expressed as follows: ð t2 ðdU þ dW  dT Þdt ¼ 0 (28) t1

where U, T, and W are the potential energy, kinetic energy, and work done by external forces of the system, respectively. t1 and t2 are two arbitrary points of time. The variation of the kinetic energy of the system is ( ð þ "      # ) N ð 1 L X @u 2 @v 2 @w 2 ðkÞ q þ þ dndsdz dT ¼ d 2 0 C k¼1 hðkÞ @t @t @t (29) The variation of the potential energy of the system is  ð 1 dV ¼ d rij eij ds 2 s ( ð þ ) N ð 1 L X ½rzz ezz þ rsz esz þ rnz enx ðkÞ dndsdz ¼d 2 0 C k¼1 hðkÞ (30) The variation of the work of external forces is dW ¼

ðL

½Lae ðz; tÞdv0 ðz; tÞ þ Tae d/ðz; tÞdz

(31)

0

The quantities of B have been defined such that they satisfy the following set of expressions:

(26)

ai and bi are counterparts of ai and bi respectively, corresponding to / via Eq. (18). Assuming the wing motion starts from the rest, the explicit expressions of Lae ðz; tÞ and Tae ðz; tÞ are given as Journal of Vibration and Acoustics

Governing Differential Equations of Motion

(25)

^ 3 2b a0cMq /_ a ðz; tÞ X D4 ðz; tÞ ¼  aicMq BicMq ðz; tÞ Un i¼1

  Un 1 ^ Bic ðz; tÞ ¼ w_ a ðz; tÞ B_ ic ðz; tÞ þ bic Un b   Un 2b ^_ B_ icq ðz; tÞ þ bicq Bicq ðz; tÞ ¼ / ðz; tÞ Un a b   Un 1 ^ B_ icM ðz; tÞ þ bicM BicM ðz; tÞ ¼ w_ a ðz; tÞ Un b   Un 2b ^_ B_ icMq ðz; tÞ þ bicMq BicMq ðz; tÞ ¼ / ðz; tÞ Un a b

(27)

where Lae and Tae are the unsteady aerodynamic lift and pitching moment about the reference axis. Substituting the Eqs. (1)–(7) into Eqs. (29) and (30) and using Hamilton’s principle, the Euler– Lagrange equations are obtained. du0 : Q0x  b1 u€0 ¼ 0

(32)

dv0 : Q0y  b1 v€0 þ Lae ¼ 0

(33)

€0 ¼ 0 dw0 : Tz0  b1 w

(34)

dhy : M y  Qx  ðb5 þ b15 Þh€y ¼ 0

(35)

dhx : M0 x  Qy  ðb4 þ b14 Þh€x ¼ 0

(36)

0

d/ : B00w þ M0 z  ðb4 þ b5 Þ/€ þ ðb10 þ b18 Þ/€00 þ Tae ¼ 0

(37)

Also, the corresponding boundary conditions at two edges of the wing are obtained as OCTOBER 2013, Vol. 135 / 051019-5

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z¼0

Nnz ¼ ½A44  A245 =A55 cnz

z¼L

du0 ¼ 0

Qx ¼ 0

dv0 ¼ 0

Qy ¼ 0

(

dw0 ¼ 0 Tz ¼ 0

(38)

dhx ¼ 0

Mx ¼ 0

dhy ¼ 0

My ¼ 0

d/ ¼ 0

B0w þ Mz þ ðb10 þ b18 Þ/€0 ¼ 0

0

d/ ¼ 0

Bw ¼ 0

The one-dimensional stress measures Tz, Tr, Qx, Qy, My, Mx, Mz, and Bw can be defined in terms of stress resultants and stress couples as þ (39) Tz ðz; tÞ ¼ Nzz ds  dx dy þ Nzn ds ds ds   þ dy dx Qy ðz; tÞ ¼ ds Nsz  Nzn ds ds  þ dx Mx ðz; tÞ ¼ ds yNzz  Lzz ds  þ dy My ðz; tÞ ¼ ds xNzz þ Lzz ds þ Mz ðz; tÞ ¼ ðNsz w þ 2Lsz Þds

Qx ðz; tÞ ¼

þ

Nsz

(40) (41) (42) (43)

Lsz

)

" ¼

k41

k42

k43

k51

k52

k53

k54 > /0 > > > > > > > > : n > ; ezz

(44)

dv0 : a55 ðv000 þ h0x Þ  a56 /000 þ Lae ¼ b1 v€0

¼ ðb4 þ b5 Þ/€  ðb10 þ b18 Þ/€00

ðNzz ; Lzz Þ ¼

rzz ð1; nÞdn

ðNsz ; Lsz Þ ¼

ð h=2

The associated BCs at z ¼ 0 are rsz ð1; nÞdn

(47)

rnz ð1; nÞdn

(48)

h=2

v0 ¼ 0;

C12 C22 C23

C13 C23 C33

0 0 0

0 0 0

0 0 C26

0 0 C36

C44 C45 0

C45 C55 0

8 9 ess > > > > > > > > > ezz > > > > > > > < enn = > czn > > > > > > > > > csn > > > > > : > ; c sz ðk Þ ðk Þ

3 C16 C26 7 7 7 C36 7 7 0 7 7 7 0 5 C66

(49) in which Cij represent stiffness coefficients [1]. Substituting Eq. (49) into the Eqs. (46)–(48) yields 

Nzz Nsz



 ¼

k11 k21

k12 k22

/ ¼ 0;

/0 ¼ 0;

hx ¼ 0

(56)

and at z ¼ L are dv0 : a55 ðv00 þ hx Þ  a56 /00 ¼ 0

and the reduced mass terms bi are defined in the Appendix. Constitutive relations for a general orthotropic material can be written as 8 9 2 C11 rss > > > > > > 6C > > > rzz > 12 > > 6 > > > > 6 < 6 C13 rnn = ¼6 6 0 > > r zn > > 6 > > > > 6 > > > rsn > > 4 0 > > > : ; C16 rsz ðkÞ

(54)

dhx : a33 h00x þ a37 /00  a55 ðv00 þ hx Þ þ a56 /00 ¼ ðb4 þ b14 Þh€x (55)

h=2

Nnz ¼

(53)

(46)

h=2

ð h=2

(52)

00 0000 00 00 d/ : a56 ðv000 0 þ hx Þ  a66 / þ a37 hx þ a77 / þ Tae

(45)

where Nzz, Nsz, and Nzn are stress resultants and Lsz and Lzz are stress couples defined as ð h=2

(51)

where kij are defined in the Appendix. The Navier equations of motion of a composite wing can be expressed in terms of the displacement components by substituting Eqs. (39)–(52) into Eqs. (32)–(37). Rehfield and Atilgan [7] considered two structural configurations for composite TWB, which demonstrate special structural couplings, the first one circumferentially asymmetric stiffness and the other circumferentially uniform stiffness (CUS). They are available using the filament winding technology. For a composite TWB of rectangular cross section in CAS configuration, the ply angle distribution in the flanges and webs follows the relation of hðyÞ ¼ hðyÞ (See Fig. 2), and in CUS configuration, the relation of hðyÞ ¼ hðyÞ. In the case of fixed composite TWB, CAS configuration leads to the decoupling between bending-torsion and extension-transverse shear, while CUS configuration leads to the decoupling between flapping and lagging bending-transverse shear and extension-torsion. In the context of CAS configuration, the equations of motion governing the bending-torsion-vertical transverse shear ðv0 ; /; hx Þ motions in the aeroelastic system can be stated as

þ

Bw ðz; tÞ ¼ ðFw ðsÞNzz þ at ðsÞLzz Þds

Lzz

8 0 9 > > ezz > > > > #> > > > k44 < c0sz =

k13 k23

8 0 9 > ezz > > > > > k14 < c0sz = 0 > k24 > > >/ > > : ; n ezz

051019-6 / Vol. 135, OCTOBER 2013

(50)

d/ : a56 ðv000 þ h0x Þ  a66 /000 þ a37 h0x þ a77 /0 þ ðb10 þ b18 Þ/€0 ¼ 0 dhx : a33 h0x þ a37 /0 ¼ 0 d/0 : a56 ðv00 þ hx Þ  a66 /00 ¼ 0

(57)

where aij are defined in the Appendix. The unsteady aerodynamic lift and pitching moment are expressed in Eq. (27).

5

Solution Methodology

In order to form the mass, stiffness, and damping matrices of the nonconservative aeroelastic system, the extended Galerkin’s method (EGM) and the method of separation of variables have been used. Afterwards, with transforming matrices into the form of state space, the eigenvalue analysis will be performed. So, the flutter and divergence speeds will be obtained. By numerically solving the aforementioned dynamic equations in the time domain, the aeroelastic responses of the composite wing in different flight speeds will be computed. All unknown variables F(z, t) are written as Fðz; tÞ ¼ WF ðzÞqðtÞ

(58)

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where WF ðzÞ are admissible functions required to fulfill the geometric boundary conditions, and q(t) are vectors of generalized coordinates [45]. Substituting Eq. (58) into the weak form of Eqs. (53)–(55) reduces the equations of motion to the following system of equations: €g þ ½Cae fq_ g þ ½Ks þKae fqg ¼ f0g ½Ms þ Mae fq

(59)

where Ms and Mae are the structural and aerodynamic mass matrices, Cae is the aerodynamic damping matrix, and Ks and Kae are the structural and aerodynamic stiffness matrices, respectively. The state vector q is defined as n fqg ¼ qTv qT/ qTx qTB1c qTB2c qTB3c qTB1cq qTB2cq qTB3cq … oT qTB1cM qTB2cM qTB3cM qTB1cMq qTB2cMq qTB3cMq (60)

Table 1 Geometric and material properties of the composite TWB used for validation of structural system Geometric properties Length, L (m) Width, 2w (m) Depth, 2d (m) Wall thickness, h (m) Fiber angle, h (deg)

Table 2

Material properties E1 (GPa) E2 ¼ E3 (GPa) G12 ¼ G13 (GPa) G23 (GPa) Density, q(kg/m3) Poisson’s ratio, 

2.032 0.254 0.068 0.01 45

206.75 5.17 2.55 1.38 1528.15 0.25

Natural frequencies of a clamped-free composite TWB Ref. [46]

Present work

55.31 345.6

56.86 353.1

x1 (rad/s) x2 (rad/s)

X is defined as fXg ¼

fqgT

fq_ gT

T

(61)

Hence, the state space form of the Eq. (59) will become

 X_ ¼ ½AfXg     ½Cae  ½Mae þMs  1 ½Kae þKs  ½0 ½ A ¼  ½I ½0 ½0  ½I

GJ EI Model (Nm2) (Nm2)

(62)

where [I] is the unit matrix and [0] is the zero matrix. In Eqs. (62), the real part of a particular eigenvalue indicates the damping value and the imaginary part represents the frequency. By increasing the flow velocity, the damping value or real part of the eigenvalues of Eq. (62) increases and becomes positive in a certain frequency, which indicates the start of the wing instability. In such case, if the frequency or imaginary part of the eigenvalues of Eqs. (62) becomes zero, this implies that the instability is of divergence type and nonzero frequencies relate to the flutter instability. Furthermore, the various aeroelastic responses of the wing model can be computed by solving Eqs. (62) in the time domain.

6

Validation and Numerical Results

6.1 Validation. For validation of the structural model, the first two natural frequencies of cantilever composite TWB with CAS configuration (Table 1) obtained in the context of this study are compared with the results of Librescu and Na [46] in Table 2, which reveals a good agreement. NACA TN-2121 [47] has been used to evaluate the accuracy of the designed aeroelastic system Table 4

Table 3 Geometric and material properties of the wings used for validation of aeroelastic system

1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 6 7 8 9 10 11 12 13

10.15 10.15 9.55 26.5 15.6 11.7 5.6 13 41.3 13 31.82 26.49 8.056

b (m)

a

0.6299 0.6299 0.6299 0.574 0.6604 0.6604 0.8128 0.8128 0.4216 0.8128 0.4622 0.574 0.553

0.0509 0.0509 0.0509 0.0713 0.0509 0.0509 0.0509 0.0509 0.0978 0.0509 0.0713 0.0713 0.0719

–0.2 –0.2 –0.21 –0.08 –0.12 –0.12 –0.2 –0.08 –0.074 –0.08 –0.08 –0.08 –0.02

0.00024 0.00024 0.00023 0.00054 0.000216 0.00024 0.00013 0.000172 0.0014 0.00017 0.00051 0.00054 0.00038

0.339 0.338 0.322 0.464 0.476 0.524 0.138 0.289 0.637 0.289 0.439 0.461 0.238

0 0 15 45 30 15 60 60 15 60 30 45 45

Note: Ia: wing section mass moment of inertia about elastic axis, m: wing mass per unit length GJ: torsional stiffness (a77), and EI: bending stiffness (a33).

and solution methodology. A number of wings with available flutter speeds are considered (Table 3), and the obtained results have been compared with the results of wind tunnel experiments and available analytical results (Table 4). The compared results reveal that the correlation between the present predictions and the previous results is good in a broad range of geometrical properties and flight speeds.

Flutter Mach numbers and frequencies of different wing models Mach number at flutter (Mf)

Model

15 15 14.5 94.6 16.8 22.4 18.5 54.7 156.8 54.7 152.4 94.62 30.96

m K Ia (Kgm2) (Kg/m) (deg)

L (m)

Frequency at flutter speed (xf)

Experimental [47]

Analytical [47]

Present

Experimental [47]

Analytical [47]

Present

0.5 0.45 0.51 0.81 0.69 0.59 0.79 0.51 0.79 0.62 0.68 0.56 0.54

— — — 0.756 — — 0.848 0.568 0.679 0.685 0.53 0.512 0.447

0.526 0.475 0.482 0.788 0.708 0.61 0.766 0.537 0.812 0.637 0.704 0.577 0.556

61 56 62 37 24 30 29 37 55 36 61 54 49

— — — 53 — — 40 58 69 55 65 61 58

63 59 59 41 27 34 26 41 59 40 63 57 53

Journal of Vibration and Acoustics

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Fig. 3 First four coupled bending-torsion frequencies (rad/s) versus fiber angle (deg) for TWB with CAS configuration

Table 5 TWB

Fig. 5 Mach number at onset of aeroelastic instability versus ply angle (deg) for composite wing with CAS configuration and a 5 20.4, K 5 0 deg

Geometric and material properties of the composite

Geometric properties Length, L (m) Width, 2w (m) Depth, 2d (m) Wall thickness, h (m) Fiber angle, h (deg) 2b (m)

Material properties 14 0.757 0.1 0.03 45 1.6

E1 (GPa) E2 ¼ E3 (GPa) G12 ¼ G13 (GPa) G23 (GPa) Density, q (kg/m3) Poisson’s ratio, 

206.8 5.17 3.1 2.55 1528 0.25

Fig. 6 Frequency (rad/s) of aeroelastic instability versus ply angle (deg) for composite wing with CAS configuration and a 5 20.4, K 5 0 deg

Fig. 4 Variation of bending a 33 (Nm2) and torsion a77 (Nm2) stiffness quantities versus ply angle (deg) for composite wing with CAS configuration

6.2 Numerical Results Effects of the Ply Angle on the Natural Frequencies of the Composite TWB. Inasmuch as the variation of natural frequencies versus ply angle is symmetric to h ¼ 0 deg [48], the results are 051019-8 / Vol. 135, OCTOBER 2013

shown only for h between 0 deg and 90 deg. Figure 3 shows the variation of the first four bending-torsion natural frequencies versus fiber angle for different ply angles of composite TWB characterized in Table 5. At h ¼ 0 deg or 90 deg, where the decoupling between bending and torsion occurs, the modes are either purely bending or torsional, which are denoted by “B” or “T,” respectively, in Fig. 3. The fundamental natural frequency of the composite TWB corresponds to a purely bending dominant mode and increases by increasing ply angle. In general, it should be mentioned that all the natural frequencies of bending dominant modes increase by increasing the ply angle. A similar trend can be seen in torsion dominant modes for ply angles in the range of 0 deg  h  75 deg, but between 75 deg  h  90 deg the natural frequencies will decrease. Changing between bending and torsion dominancy in branches of the frequency spectrum can be explained according to the variation of the bending and torsion stiffness quantities (Fig. 4). Transactions of the ASME

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Fig. 7 Variation of bending-torsion a37 5 a37 =ða37 Þh 5 45 deg and transverse shear-torsion a56 5 a56 =ða56 Þh 5 45 deg stiffness quantities versus ply angle (deg) for composite wing with CAS configuration

Fig. 8 Normalized responses in bending for composite wing with CAS configuration and h 5 230 deg, a 5 20.5, and K 5 0 deg in M 5 0.5 and M 5 0.6

Effects of the Ply Angle on the Aeroelastic Behavior of the Composite Wing. Figures 5 and 6 represent the Mach number and frequency of unstable mode at the onset of aeroelastic instability in different ply angles for a composite wing with CAS configuration and a ¼ 0.4, K ¼ 0 deg. The results are calculated according to both incompressible and compressible aerodynamics. For all positive ply angles, instability is of torsional dominant divergence type and the frequency of unstable mode is zero. Although the geometrical properties of the wing are such that the instability at h ¼ 90 deg, 0 deg, and 90 deg is torsional dominant divergence, which occurred in the same Mach number, utilizing structural tailoring technique would change the characteristics of the instability. In the positive range of fiber orientations, instability occurred in the incompressible regime, and the variation of the critical Journal of Vibration and Acoustics

Fig. 9 Normalized responses in bending for composite wing with CAS configuration and h 5 260 deg, a 5 20.5, and K 5 0 deg in M 5 0.5 and M 5 0.6

Fig. 10 Subcritical (M 5 0.56), flutter (M 5 0.57), and supercritical (M 5 0.58) normalized responses in bending for composite wing with CAS configuration and h 5 220 deg, a 5 20.35, and K 5 0 deg

Mach number is explainable according to the variation of the bending-torsion (a37) and transverse shear-warping (a56) stiffness quantities (Fig. 7). The most beneficial elastic coupling from the aeroelastic standpoints is the bending-torsion elastic coupling. Excepting a37 and a56, variables in governing Eqs. (53)–(55) in the case of straight wing are symmetric about h ¼ 0 deg , and the corresponding critical Mach number and frequency of instability will exhibit symmetry about h ¼ 0 deg when a37, a56 ¼ 0. For a large range of ply angles 0 deg  h  50 deg resulting in positive values of a37, there are significant reductions in the amount of critical Mach number of the divergence instability; for 50 deg < h  90 deg, the opposite trend takes place. These two phenomena are referred to as structural wash-in and wash-out, respectively. Another interesting behavior of this structural bending-torsion coupling has occurred in ply angles with instability of flutter type. The amplitude of bending response of composite wing with OCTOBER 2013, Vol. 135 / 051019-9

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Fig. 11 Subcritical (M 5 0.56), flutter (M 5 0.57), and supercritical (M 5 0.58) normalized responses in torsion for composite wing with CAS configuration and h 5 220 deg, a 5 20.35, and K 5 0 deg

Fig. 13 Modal frequencies and dampings of the aeroelastic modes plotted versus airspeed for composite TWB with CAS configuration and a 5 20.45, K 5 0 deg, and h 5 270 deg

Fig. 12 Normalized responses in bending for composite wing with CAS configuration and a 5 20.45, M 5 0.62, and K 5 0 deg versus a different ply angle

h ¼ 30 deg decreased by increasing Mach number (See Fig. 8) while the opposite trend occurred in ply angle of 60 deg as shown in Fig. 9. Also, a similar behavior is observed in torsion responses. The dominant nature of the aeroelastic instability for 25 deg  h  90 deg is torsional, which gradually change from divergence in the positive ply angles to the flutter in the negative range of ply angles. Variation of the structural bending-torsion coupling in the negative range of ply angles leads to the increasing of the flutter speed from h ¼ 0 deg to h ¼ 25 deg, which shows the wash-out type behavior of a37 in these negative ply angles. Figures 10 and 11 represent the normalized bending and torsional responses of wing tip at onset of flutter and the corresponding sub and supercritical responses for h ¼ 20 deg and a ¼ 0.35, respectively. For 90 deg < h < 25 deg, the instability is bending dominant and the variation of the critical Mach number versus ply angles should be explained as a tradeoff between the effects of 051019-10 / Vol. 135, OCTOBER 2013

different stiffness quantities. The normalized bending response of a composite wing in different ply angles has been shown in Fig. 12. By increasing ply angles from 30 deg to 60 deg, the frequency of response increased while the response amplitude has an increasing trend until 45 deg and will decrease by increasing h to 60 deg. Variation of the torsional stiffness quantities for 85 deg < h < 60 deg (Figs. 4 and 7) leads to local maximum and minimum in the corresponding critical Mach numbers. By increasing flight speeds to the compressible regime, the center of pressure moves toward the trailing edge and the airfoil becomes more stable in pitching motion. As an evidence of this statement, compressible aerodynamic predicts larger Mach numbers for the onset of aeroelastic instabilities in torsion dominant ply angles than those calculated using incompressible aerodynamic theory (See Fig. 5). Also the results displayed in Fig. 6 reveal that compressibility has important effects on the frequency of flutter. Effects of the Flight Speed on the Modal Frequencies and Dampings of Aeroelastic System. A flying wing has an infinite number of aeroelastic modes, which are in one-to-one correspondence with the free vibration modes of the structure. Figure 13 shows the variation of the frequency and damping of the lowest Transactions of the ASME

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Fig. 14 Mach number at onset of the aeroelastic instability versus sweep angle (deg) for composite wing with CAS configuration and a 5 20.4, h 5 220 deg, and 230 deg

Fig. 15 Frequency (rad/s) of aeroelastic instability versus sweep angle (deg) for composite wing with CAS configuration and a 5 20.4, h 5 220 deg, and 230 deg

aeroelastic modes versus flight speed. The frequency branches initiated at the corresponding structural frequencies and continued regularly until aerodynamic damping of the first mode (bending dominant mode) vanished at U ¼ 200 m/s and flutter occurred. The first mode, which was more affected by the airspeed, approaches the second mode at onset of the flutter without frequency coalescence. Although most flutter encountered in practice involves interaction of two or more aeroelastic modes, it has been shown that this is not a necessary condition for aeroelastic instability. Effects of the Sweep Angle on the Aeroelastic Behavior of the Composite Wing. Although the primary motivation for sweeping a wing is to improve aircraft performance by drag reduction, the sweep angle has an important effect on the aeroelastic behavior of the aircraft wing. This topic has been investigated extensively in the literature, and different conclusions have been made (see, e.g., Journal of Vibration and Acoustics

Fig. 16 Subcritical (M 5 0.55) normalized response in torsion for composite wing with CAS configuration and h 5 220 deg, a 5 20.5 in different sweep angles

Fig. 17 Subcritical (M 5 0.49), divergence (M 5 0.5), and supercritical (M 5 0.51) normalized responses in bending for composite wing with CAS configuration and h 5 220 deg, a 5 20.4, and K 5 220 deg

[1,32,47,49,50]). There are two ways in which the sweep influences the aeroelastic behavior. One is the loss of aerodynamic effectiveness (Un ¼ U1 cos K), and the second effect is the influence of bending and torsion slopes on the effective angle of attack and downwash velocity (see Eq. (9)), which leads to an aeroelastic bending-torsion coupling. This coupling has important influence on both divergence and load distribution and makes the forward swept wing more susceptible to divergence. In the design of the X-29 Grumman swept forward aircraft, the bending-torsion elastic coupling has been used to passively stabilize the effects of the bending-torsion aeroelastic coupling. Contrary to the some general conclusions in the literature, which consider an increasing effect for sweep angle on the critical Mach number of aeroelastic instabilities, sweep angle has both increasing and decreasing effects depending on the other material and geometrical properties OCTOBER 2013, Vol. 135 / 051019-11

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Fig. 18 Subcritical (M 5 0.49), divergence (M 5 0.5), and supercritical (M 5 0.51) normalized responses in torsion for composite wing with CAS configuration and h 5 220 deg, a 5 20.4, and K 5 220 deg

Table 6 Variation of Mach number, frequency (rad/s), and type of aeroelastic flutter versus “a” parameter in different ply angles and K 5 0 h ¼ 70 (deg) a –0.3 –0.35 –0.4 –0.45 –0.5

h ¼ 30 (deg)

h ¼ 20 (deg)

Mf

xf

Type

Mf

xf

Type

Mf

xf

Type

0.52 0.529 0.535 0.588 0.6

25.25 25.09 24.19 24.24 24.43

B.D B.D B.D B.D B.D

0.42 0.47 0.72 — —

8.7 9.23 9.48 — —

PT PB PB — —

0.45 0.57 0.85 — —

6.07 6.37 18.28 — —

PT PT PT — —

Note: B.D: bending dominant flutter with frequency coalescence between bending and torsion modes, and PB (PT): bending (torsion) dominant flutter without frequency coalescence between bending and torsion modes.

of the wing. Figures 14 and 15 show the effect of sweep angle on the variation of the critical Mach numbers and corresponding frequencies of the aeroelastic instabilities in two different ply angles of h ¼ 20 deg and  30 deg. As shown in these figures, negative sweep angle has such a strong effect, which changes the type of instability in the case of h ¼ 30 deg, from bending dominant flutter in K ¼ 0 deg (Figs. 5 and 6) to torsion dominant divergence in 30 deg  K < 0 deg. In the latter case, divergence is not predictable by utilizing incompressible aerodynamics. The instability in the positive range of sweep angles is of bending dominant flutter type. Figure 16 shows the effects of sweep angle on the torsional responses of wing tip in subcritical speed (M ¼ 0.55), which demonstrates that backward sweep angle makes the wing more stable in this case. Another interesting phenomenon is the occurrence of torsional dominant flutter in K < 30 deg for both ply angles of h ¼ 20 deg and  30 deg. Figures 17 and 18 display the normalized bending and torsional responses of wing tip at onset of divergence, respectively, and the corresponding sub and supercritical responses for h ¼ 20 deg, a ¼ 0.4, and K ¼ 20 deg. Effects of the Offset Between the Reference Axis and the Midchord Axis on the Aeroelastic Behavior of the Composite Wing. Table 6 shows the effects of offset between the elastic and the midchord axis of the wing (i.e., “a” parameter) on the Mach 051019-12 / Vol. 135, OCTOBER 2013

Fig. 19 Normalized responses in bending for composite wing with CAS configuration and h 5 225 deg, M 5 0.6, and K 5 0 deg in different a parameter

number, frequency (rad/s), and type of aeroelastic flutter in different ply angles and K ¼ 0 deg. As shown, increasing the magnitude of this cross coupling parameter (jaj) makes the wing more stable and postpones the boundaries of the instability. Also, this conclusion was confirmed by the bending response, as depicted in Fig. 19.

7

Conclusion

In this paper, the aeroelastic stability and response of an aircraft swept composite wing in subsonic compressible flow are investigated. The structural model consists of a thin walled composite beam. An unsteady indicial aerodynamic model was developed as a function of Mach number in compressible flight regimes. The extended Galerkin’s method was utilized to construct the proper state space form of the governing equations of motion. Eigen analysis of the system was performed to obtain the aeroelastic instability (divergence and flutter) boundaries. It is shown that material anisotropy, sweep angle, and the offset between the elastic and the midchord axis of the wing lead to elastic, aerodynamic, and cross coupling between bending and torsion, respectively. The effects of these couplings on the aeroelastic instability and response of the wing in subsonic flight speeds were investigated, and a number of conclusions were outlined. The study’s results reinforce the importance of a consistent stiffness model of aeroelastic system in the stability analysis of aircraft wings. The results were validated with the available analytical and experimental results, which reveal an excellent agreement.

Appendix: Reduced Mass Terms and Stiffness ðm0 ; m2 Þ ¼

N ð h ðk Þ X k¼1

hðk1Þ

  qk 1; n2 dn

þ   ðb1 ; b4 ; b5 ; b10 Þ ¼ m0 1; y2 ; x2 ; F2w ds "    # dx 2 dy 2 2 ðb14 ; b15 ; b18 Þ ¼ m2 ; ; at ds ds ds þ

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A212 A11 A12 A16 ¼ A26  ¼ k21 A11   A12 B16 Ac þ 2k12 ¼ k31 ¼ 2 B26  A11 b A12 B12 ¼ B22  ¼ k41 A11 2 A ¼ A66  16 A11   A16 B16 Ac þ 2k22 ¼ k32 ¼ 2 B66  A11 b A16 B12 ¼ B26  ¼ k42 A11   B12 B16 Ac þ 2k24 ¼ k34 ¼ 2 D26  A11 b 2 B ¼ D22  12 A11 B16 A12 ¼ B26  ¼ k15 A11 B16 A16 ¼ B66  ¼ k25 A11   B2 Ac ¼ 2 D66  16 þ 2k25 ¼ k35 A11 b B12 B16 k54 ¼ D26  ¼ k45 A11

k11 ¼ A22  k12 k13 k14 k22 k23 k24 k43 k44 k51 k52 k53

a55 a56 a66

#

 2 dx dx k14 þ k44 ds ds ds  þ dx ¼ yk13  k34 ds ds  2 # þ "  2 dy dx ¼ k22 þ A44 ds ds ds  þ dy dy ¼ Fw k21  at k24 ds ds ds þ   ¼ F2w k11 þ 2Fw at k14 þ a2t k44 ds

a33 ¼ a37

þ"

[8] [9] [10] [11] [12]

[13] [14] [15]

[16]

[17]

[18] [19]

[20]

[21]

[22] [23]

y2 k11  2y

þ a77 ¼ Wk23 ds a33 is bending, a37 is bending-torsion, a55 is transverse shear, a56 is transverse shear-warping, a66 is warping, and a77 is torsion.

[24] [25]

[26] [27]

[28]

[29]

[30]

[31] [32] [33]

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