Aerospace Science and Technology 8 (2004) 519–532 www.elsevier.com/locate/aescte
Frequency and time domain flutter computations of a wing with oscillating control surface including shock interference effects Hyuk-Jun Kwon a , Dong-Hyun Kim b , In Lee a,∗ a Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu,
Daejeon, 305-701, South Korea b School of Mechanical and Aerospace Engineering (ReCAPT), GyeongSang National University (GSNU), Jinju, Kyungnam, 660-701, South Korea
Received 1 October 2002; received in revised form 8 March 2004; accepted 5 April 2004 Available online 11 May 2004
Abstract In this study, the nonlinear aeroelastic characteristics of a wing with an oscillating control surface have been examined in transonic and supersonic regimes. The various effects of rotational stiffness on flutter have also been observed. A modified transonic small-disturbance (TSD) theory is used to more effectively analyze the unsteady aerodynamics of a wing with an oscillating control surface. In the flutter analysis, a coupled time integration method (CTIM) and a transient pulse method (TPM) were used to examine the effects of rotational stiffness reduction on the control surface. The present study shows that the severe decrease of flutter speed and the flutter mode transition can be induced by the reduction of rotational stiffness. In particular, it is shown that the aerodynamic effects of control surface oscillation play an important role in this flutter speed reduction. 2004 Elsevier SAS. All rights reserved. Keywords: Flutter; Aeroelasticity; Control surface; Rotational stiffness reduction; TSD equation
1. Introduction In the modern aircraft design, most of the control surfaces have a dual actuator system to prepare an emergent flight situation of possible actuator failure. However, when one of them fails, even though it is possible to control the aircraft, the reduction of rotational stiffness is unavoidable on the control surface hinge. This stiffness change can be closely related with the aeroelastic responses of control surface due to unsteady aerodynamic excitations. In particular, when the motion of control surface is interacting with the moving shock in the transonic regime, aeroelastic instabilities can be significantly increased. Hence, it is important to investigate the reduction effects of the flap torsional stiffness on the flutter stability in the transonic and low supersonic regimes. In previous studies, many researchers have studied the effects of oscillating control surfaces on the flutter stability. However, their researches have usually been performed * Corresponding author. Tel.: 82-42-869-3717, Fax: 82-42-869-3710.
E-mail address:
[email protected] (I. Lee). 1270-9638/$ – see front matter 2004 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2004.04.001
in the subsonic or high supersonic flow regions using the linear aerodynamic theories [2,9,12] or focused on the 2-dimensional models [4,6,10,14–17]. Few works have been devoted to the 3-dimensional aeroelastic analyses considering the critical effect of the hinge stiffness reduction of control surface in the transonic and low supersonic regimes. To perform the aeroelastic analysis in the transonic flow region, the computational fluid dynamic (CFD) technology must be used because of the existence of the moving shock. Especially, to capture the full nonlinearity of the flow field such as viscous effects, the Navier–Stokes code with turbulent model should be applied to the aeroelastic analysis. However, it is difficult to use this technology to the 3-dimensional wings with control surface; considering the arbitrary elastic motion with the flap rotation is very hard work on treating the mesh and guaranteeing the numerical stability. Moreover, when the aeroelastic analysis with the local motion of control surface is performed in the transonic flow region, long response time is generally required to clearly determine the aeroelastic stability. At these points, the aeroelastic analysis using an efficient
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unsteady aerodynamic analysis such as transonic small disturbance (TSD) code can have strong computational advantages in various parametric studies. It is well known that the TSD code is one of the most efficient tools among the conventional CFD-based approaches. In addition, for the grid system, a TSD code does not need to have an additional grid remeshing process since it uses the changes of wing airfoil slopes at each wall grid point to simulate the arbitrary wing surface motions. In this paper, nonlinear aeroelastic analyses of a 3dimensional wing with an oscillating control surface have been conducted in the transonic and low supersonic regimes. The effects of the control surface rotational stiffness on the flutter stability are studied by comparing the different rotational stiffness models. An efficient 3-dimensional aerodynamic code based on the transonic small disturbance (TSD) theory is applied to investigate the nonlinear flutter characteristics in transonic and low supersonic regimes. Even though the TSD code cannot consider the viscous effects of the flow field, it shows good correlation with the results of the Navier–Stokes or Euler code when the wing has thin airfoil and its motion has small amplitude. In this study, these basic assumptions are reasonable because we concentrate to show the macroscopic effects of the control surface stiffness changes on the flutter stability and the airfoil of the transonic/supersonic wing commonly has around 4 or 5 percents thickness.
2. Theoretical backgrounds 2.1. Unsteady aerodynamic analysis In the aeroelastic analysis, even though there has been the remarkable growth on the analysis scheme and technique, the panel methods such as Doublet Lattice Method (DLM), piston theory and Doublet Point Method (DPM) are most widely used on the practical applications. The usage of these panel methods, however, has the limitation on transonic and low supersonic regimes because they cannot consider the discontinuity across the shock wave. This is the reason of the development of CFD-based aeroelastic analysis program. Transonic small disturbance theory (TSD) is the most simplified equation among these CFDbased schemes. The transonic small disturbance (TSD) equation can be written in the conservation form as follows [1];
f3 = φz , B = 2M 2 ,
A = M 2,
1 F = − (γ + 1)M 2 , 2
E = 1 − M 2, 1 G = (γ − 3)M 2 , 2
H = −(γ − 1)M 2 , τ = tU∞ /cr , where φ is nondimensional disturbance velocity potential and nondimensionalized by cr U∞ . x, y, z are nondimensional Cartesian coordinate in streamwise, spanwise, and vertical directions, respectively and they are normalized by the root chord length, cr . U∞ is free stream velocity and t is physical time. The imposed flow field boundary conditions are represented in Ref. [1]. The wing flow-tangency boundary condition can be written as φz± = fx± + fτ ,
(2)
where f is defining the position of wing surface and + and − mean the upper and lower surface, respectively. The transonic small disturbance (TSD) equation can be transformed to the computational domain using the modified shearing transformation: −
∂ M2 φτ + 2M 2 φξ ∂τ ξx ∂ + (1 − M 2 )ξx φξ + F ξx2 φξ2 + G(ξy φξ + φη )2 ∂ξ ξy + (ξy φξ + φη ) + H ξy φξ (ξy φξ + φη ) ξx ∂ 1 + (ξy φξ + φη ) + H φξ (ξy φξ + φη ) ∂η ξx ∂ 1 + φς = 0. ∂ς ξx
(3)
In Eq. (3) ξ , η, and ζ represent the axes in the computational domain, which correspond to x-, y- and z-axes in the physical coordinates. ξ(x, y) =
x − xLE (y) , xT E (y) − xLE (y)
η(y) = y,
(4)
ζ(z) = z.
f1 = Eφx + F φx2 + Gφy2 ,
Eq. (3) is solved using a time-accurate approximate factorization (AF) algorithm developed by Batina [1]. The AF algorithm consists of a time linearization procedure coupled with a Newton iteration technique. Thus, the TSD equation is solved using a time-accurate approximate factorization method as briefly described here. First, Eq. (3) can be written in general conservation form at the advanced time level n + 1 as follows:
f2 = φy + H φx φy ,
R(φ n+1 ) = 0,
∂f0 ∂f1 ∂f2 ∂f3 + + + = 0, ∂τ ∂x ∂y ∂z where f0 = −Aφτ − Bφx ,
(1)
(5)
H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
where φ n+1 represents the unknown potentials at time level (n + 1). Next, the solution of Eq. (5) is given by the Newton linearization about an intermediate time level (∼): ∂R ˜ φ = −R(φ), (6) ∂φ ˜ φ=φ
where φ = φ n+1 − φ˜ and if φ approaches to zero then φ˜ may be equal to the next step of φ. The left side of Eq. (6) can be approximately factorized into the following three operators: ˜ Lξ Lη Lζ φ = −R(φ),
(7)
where the operators are defined as 3B ∂ ξx t 4A ∂ξ t 2 ∂ Eξx + 2F ξx φ˜ x + 2Gξx φ˜ y − ξx 2A ∂ξ ξy2 ∂ , + (1 + H φ˜ x ) + H ξx φ˜ y ξx ∂ξ t 2 ∂ 1 ∂ Lη = 1 − ξx , (1 + H φ˜ x ) 2A ∂η ξx ∂η t 2 ∂ 1 ∂ , Lζ = 1 − ξx 2A ∂ζ ξx ∂ζ t 2 A 2φ˜ − 5φ n + 4φ n−1 − φ n−2 ˜ = −ξx R(φ) − 2A ξx t 2 Lξ = 1 +
−B + + + +
(8a)
(8b) (8c)
The aeroelastic equations of motion can be formulated by Hamilton’s Theorem for elastic models and is written in matrix form as follows: ¨ + [Cg ] q(t) ˙ + [Kg ] q(t) = Q(t) , (10) [Mg ] q(t) where {q(t)}T = [q(t)1 , q(t)2 , . . . , q(t)n ] is the generalized displacement vector and [Mg ], [Cg ], and [Kg ] express the generalized mass, damping, and stiffness matrices respectively. {Q(t)} represents the generalized aerodynamic force as follows:
1 (11) Qi (t) = ρU 2 cr2 hi (x, y) Cp (x, y, t) dS ∗ , 2 where subscript ‘i’ indicates the influence mode and S ∗ is 2 c 2 are multhe nondimensional plane area of wing. 1/2ρU∞ r tiplied to make the dimensional force term because the inside of integral are nondimensional. In Eqs. (10) and (11), symbol ‘t’ represents physical time, so we must pay attention to the transition from nondimensional time in the CFD code into physical time in the structural equations of motion. The generalized damping matrix [Cg ] is proportional damping and the structural damping ratio is generally assumed to be 0.005 ∼ 0.02. Though the actual damping ratio is less than 0.1 in real wing structures, a larger value can be assumed for static aeroelastic analysis to get a converged solution quickly. In this study, until the wing is positioned in an equilibrium position, 0.95 is assumed as an artificial damping ratio for a rapid convergence and then it is return to the original damping ratio. The state vector form of Eqs. (10) is introduced for efficient numerical calculations and can be written as x˙ (t) = [A] x(t) + [B] u(t) , (12) where
[I ] , −[Mg ]−1 [Cg ]
[0] −[Mg ]−1 [Kg ] [0] , [B] = [Mg ]−1 {q(t)} x(t) = , {q(t)} ˙
[A] = (8d)
The value of φ is obtained by solving the following three steps: ˜ Lξ φ¯ = −R(φ), ¯ Lη φ¯¯ = φ,
2.2. Time domain flutter analysis
S∗
3φ˜ ξ − 4φξn + φξn−1
2 t ∂ Eξx φ˜ξ + F ξx2 φξ2 + G(ξy φ˜ ξ + φ˜ η )2 ∂ξ ξy (ξy φ˜ ξ + φ˜ η ) + H ξy φ˜ ξ (ξy φ˜ ξ + φ˜η ) ξx ∂ 1 ˜ ˜ ˜ ˜ ˜ (ξy φξ + φη ) + H φξ (ξy φξ + φη ) ∂η ξx ∂ 1 ˜ φζ . ∂ζ ξx
521
(9)
¯¯ Lζ φ = φ. An advanced Engquist–Osher (E-O) type-dependent mixed difference operator has been used in the AF algorithm to achieve numerical stability when the shock waves exist. Nonreflecting far-field boundary conditions were used to achieve more accurate and efficient unsteady calculations under both subsonic and supersonic inflow conditions.
(13)
u(t) =
{0} . {Q(t)}
For the nonlinear structural system, a numerical scheme such as Runge–Kutta method is typically used to solve the equation of motion. However, if the structural system is assumed as linear, an analytical solution can be applied as follows [7,8]: x(t) = e[A]t x(0) +
t
e[A](t −τ )[B] u(τ ) dτ.
(14)
0
In this equation, if the time is advanced from the arbitrary time step, n, to next time step, n + 1, then the initial value,
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{x(0)}, can be changed into the nth time step value, {x}n and {x(t)} becomes next time step value, {x}n+1 . {x}n+1 = e[A] t {x}n +
(n+1) t
e[A]((n+1) t −τ )[B] u(τ ) dτ.
n t
(15)
If we assume the {u(t)} is constant during t, the convolution integral can be simplified as follows: e[A]((n+1) t −τ )[B] u(τ ) dτ
(n+1) t
e[A]((n+1) t −τ ) dτ [B]{u}n
= n t
t =
(16)
0
Eq. (16) can be represented in matrix form: {x}n+1 = [Φ]{x}n + [Θ][B]{u}n ,
(17)
where [Φ] = e
[A] t
t ,
[Θ] =
e[A]t dt.
0
Additionally, to get the more accurate numerical solution, the next input vector, {u}n+1 , is assumed as equal to {u}n + ({u}n − {u}n−1 ) and Eq. (17) can be rewritten as follows: 1 (18) {x}n+1 = Φ{x}n + Θ[B] 3{u}n − {u}n−1 . 2 If the generalized mass, stiffness, and damping matrix are diagonal, then the analytical forms of the matrix [Φ] and [Θ] can be written as follow: φ11 φ12 θ11 θ12 [Φ] = , [Θ] = , (19) φ21 φ22 θ21 θ22 where
ai φ11 = eai t cos(bi t) − sin(bi t) , bi 1 ai t sin(bi t), φ12 = e bi a 2 + bi2 ai t φ21 = − i e sin(bi t), bi ai sin(bi t) + cos(bi t) , φ22 = eai t bi 1 (2ai φ11 − 2ai − φ21 ), θ11 = 2 ai + bi2 1 θ12 = 2 (2ai φ12 − φ22 + 1), ai + bi2
where t is the integration time step and ai and bi are defined as ai = −ζi ωi , bi = ωi 1 − ζi2 . (20) 2.3. Frequency domain flutter analysis
−ω2 [Mg ]{q} ¯ + ω[Cg ]{q} ¯ + [Kg ]{q} ¯ = [Y ]{q}, ¯
e[A]t dt [B]{u}n .
θ22 = φ12 ,
The aeroelastic analysis is also performed in the frequencydomain. The aeroelastic equation of motion, Eq. (10), can be changed into the eigenvalue problem. At this time, the wing motion is assumed as the small harmonic oscillation ({q(t)} = {q}e ¯ iωt ) and Eq. (10) can be rewritten as
(n+1) t
n t
θ21 = φ11 − 1,
(21)
where ω is circular frequency and [Y ] means the generalized aerodynamic influence coefficient (GAIC) matrix. In the subsonic and high supersonic flow region, the GAIC matrix is usually computed using the unsteady aerodynamic panel method, for example, doublet lattice method (DLM), doublet point method (DPM), and doublet hybrid (lattice/point) method (DHM). In the transonic or low supersonic flow region, however, the GAIC matrix must be obtained by the CFD-based unsteady aerodynamic schemes to consider the discontinuity effects of shock wave. Hence, an additional treatment is required to get the GAIC matrix from the aerodynamic responses of CFD-based schemes. In this study, a transient pulse method (TPM) has been applied to calculate the generalized aerodynamic forces, Qij . The GAIC matrix is defined as follows: FFT{Qij (t)} , (22) Yij = FFT{qj (t)} where FFT{ } means the fast Fourier transformation. Qij (t) and qj (t) can be written as follows:
1 2 2 (23) Qij = ρU cr hi (x, y) Cpj (x, y, t) dS, 2 S qj = q0j exp −(t − t0 )2 /4 , (24) where Qij is the GAF matrix in the time domain. It is very similar with Eq. (11) except Cpj (x, y, t) is the unsteady pressure distributions on the wing surface induced by the j th transient pulse modal displacement, qj . In Eq. (24), q0j is the transient pulse magnification factor, and t0 is the peak time of the impulse function. The results indicate that the GAIC matrix Yij is induced by j th mode acting through the displacements of ith mode. In the frequency domain flutter analysis, V-g (Efficient K-method or KE-method) [13] and PK-method [18] are typically used to compute the flutter solution and the V-g method is applied in the present study. To get the aeroelastic solution, the structural damping is assumed as zero and Eq. (21) is changed into the complex eigenvalue problem with multiplying the artificial damping (1 + ig).
H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
kb b
kb =
2
1 [Y ] − λ[K ] ¯ = 0, g {q} 2 U∞ 1 + ig λ= , 2 U∞
[Mg ] + ωb , U∞
523
(25) (26)
where kb is nondimensional reduced frequency and b is half root chord length. In this equation, the artificial damping value, g, can be considered as the required damping to maintain the neutral stability. The procedure to solve this eigenvalue problem is to hold Mach number and altitude (or aerodynamic density) constant but vary kb in order to obtain the V-g plot. The flutter speed and frequency can be obtained when the damping value, g, is equal to zero by the following relations: 1 k b U∞ . (27) , ωf = Uf = √ b Re(λ) In the frequency domain analysis, the system is stable when the structural damping has the negative value. Conversely, the system is unstable when it has the positive value. For this reason, the freestream velocity corresponding to the zero damping point is considered as a flutter speed. Fig. 1. Geometric configuration of the wing model with control surface.
3. Results and discussion To examine the effects of the oscillating control surface, four different models are used, viz. a clean wing (CL) model, a low rotational stiffness (LRS) model, a high rotational stiffness (HRS) model, and a no control surface oscillating (NCO) model. In the clean wing model, the wing has no control surface. The low rotational stiffness (LRS) and high rotational stiffness (HRS) models are classified according to the ratio of the control surface rotational√frequency to the reference frequency, ωβ /ωα . Here ωβ = kθ /Iα and ωα is the reference frequency. The second modal frequency is used for the reference frequency, ωα , because the second mode is the first torsion mode. In the LRS model, ωβ /ωα is 1.72 but in the HRS model, ωβ /ωα is 2.77. The NCO model means that the local rotation of control surface is ignored to eliminate the unsteady aerodynamic effects of control surface. This model is different with the clean wing (CL) model because the root of CL model is clamped entirely but that of NCO model is clamped except the control surface part. It is designed to have a similar structural boundary condition with LRS and HRS model at wing root except the existence of control surface motion. 3.1. Aerodynamic computations The geometric model configuration used in this study is presented in Fig. 1. In this study, every wing model has the same plane form of F-5 wing. This model has a 31.9◦ sweep back angle at the leading edge, the taper ratio is 0.31, and the aspect ratio is 2.98. The control surface hinge is located at the 82% chord and the flap occupies the 60.2% of the wing’s span. The corresponding grid system is shown
Fig. 2. CFD grid for the wing model with control surface (x–y plane).
in Fig. 2. The wing section has the shape of the 64A004.8 airfoil and the number of grid points is 80 × 34 × 40 in x, y, and z direction respectively. The steady aerodynamic analysis results for various Mach numbers are presented in Fig. 3. The upper and lower pressure distributions do not coincide with each other at the zero angle of attack because of the asymmetric airfoil. Fig. 4 shows the steady Cp distributions at various control surface angles and a positive angle indicates downward rotation. This figure informs that the shock position and strength are most sensitive to the angle of control surface around Mach 0.95. Fig. 5 shows the comparison of unsteady pressure distributions at Mach 0.9. The present results show good agreements with the
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H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
12.3% Span 0.4
Cp upper surface (12.3% Span)
46.0% Span 0.4
0.95 1.03 1.2
Cp upper surface (46.0% Span)
-0.4
0
-0.4
0.95 1.03 1.2
0.4
Cp lower surface (12.3% Span)
0
-0.4
Cp lower surface (46.0% Span)
0.95 1.03 1.2 0.7
0.4
Cp lower surface (91.4% Span) 0.95 1.03
0.95 1.03 1.2
1.2 0.7
0
-0.4
Cp
0.7 Cp
Cp
Cp upper surface (91.4% Span)
0.7 Cp
0
0.4
0.4
0.95 1.03 1.2 0.7
Cp
Cp
0.7
91.4% Span
0
-0.4
0
-0.4
Fig. 3. Sectional Cp plots for the various Mach number.
Mach 0.8 0.5
Mach 0.95
Cp upper surface (Mach 0.8)
0.5
Mach 1.2
Cp upper surface (Mach 0.95)
3 2
0.4
Cp upper surface (Mach 1.2)
3
0
0
-2 Cp
0
Cp
Cp
-2 0
3 2
0
0
-2
-0.5
-0.5 -0.4
Cp lower surface (Mach 0.8)
0.5
0
-2
32 Cp
Cp
-2 0
0.4
Cp lower surface (Mach 0.95)
0
Cp lower surface (Mach 1.2)
-2 0
0 Cp
0.5
0
2 3
3
-0.5
-0.5 -0.4
Fig. 4. Sectional Cp plots for the various control surface angles at 12.3% span.
experimental data and the previous Navier–Stokes analysis results [11]. Here, the control surface is oscillating at a frequency of 20 Hz and amplitude of 0.5 deg. As mentioned above, to perform the frequency domain aeroelastic analysis, GAIC matrix must be obtained from the unsteady aerodynamic results by the fast Fourier transform. The procedure to get the GAIC can be affected by the several numerical factors; the time step size, t, total sampling time, tmax , peak position of impulse function, t0 , and the pulse magnification factor q0j . Even though there is no clear guideline for these numerical factors because it can be differ-
ent according to the target model and unsteady aerodynamic solver, we have several recommendations for simple wing model. The time step size is recommended to be smaller than 0.02. If the time step size is too large, a numerical instability is caused during the unsteady aerodynamic calculation and the frequency bandwidth of FFT results is reduced. The magnification factors q0j are usually set to between 0.001 and 0.01% of the amplitude of each nondimensional mode. The larger amplitude requires the smaller time step and sometimes makes the numerical problems on the unsteady aerodynamic solver. The maximum sampling time tmax should
H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
(a)
525
(b)
Fig. 5. Comparisons of upper surface unsteady pressures of the wing model with oscillating control surface (M = 0.9, α = 0◦ , δ = 0.5 sin ωt, ω = 20 Hz). (a) Unsteady pressure distributions at 17% semi-span. (b) Unsteady pressure distributions at 49% semi-span.
0.4
Imaginary
-1.2 -1.6 0.0
Q12
Q11
3
-0.4 -0.8
Real
4
Real 0.0
DLM TSD (Flat, dt=0.02) TSD (Flat, dt=0.01) TSD (Airfoil, dt=0.02) 0.2
0.4
0.6
2 1 Imaginary
0 0.8
-1 0.0
1.0
Reduced Frequency (k b)
0.2
0.4
0.6
0.8
1.0
Reduced Frequency (k b) 6
0.0
Q22
Q21
Real -0.4
-0.8 0.0
Real
4
0.4
0.6
0.8
Reduced Frequency (k b)
0 -2
Imaginary 0.2
2
1.0
-4 0.0
Imaginary 0.2
0.4
0.6
0.8
1.0
Reduced Frequency (k b)
Fig. 6. Comparisons of generalized aerodynamic influence coefficient (GAIC) at M = 0.7.
be selected to have enough frequency resolution because the frequency resolution bandwidth of FFT, f , is the reciprocal value of maximum sampling time. In Fig. 6, the effects of time step size and comparison results with doublet lattice method (DLM) are represented. This figure shows that 0.02 is sufficiently small time step size when the total sampling time is 50 and pulse magnification factor is 0.01. The present GAIC results generally are in good agreement with linear
theory at Mach 0.7 and show little effect of airfoil thickness in the low reduced frequency. The comparison results of various sampling times and pulse magnification factors are not presented in this paper because the selected values are shows good correlation with other results in Fig. 6. To examine the sensitivity of unsteady aerodynamic force for the transonic Mach number, the GAIC are compared in Fig. 7. This figure shows that the variations are mainly ap-
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H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
2
4
Real
0 -1 -2 0.0
0.4
0.6
0.8
2
-2 0.0
1.0
Reduced Frequency (k b)
0.2
0.4
0.6
0.8
1.0
8 Real
4
Q22
Imaginary
Q21 -1
Imaginary
Reduced Frequency (k b)
1
0
Real
0
Imaginary 0.2
Black : Real White : Imaginary
6
Q12
Q11
1
M = 0.8 M = 0.9 M = 0.93 M = 0.97
0
Real -4 Imaginary
-2 0.0
0.2
0.4
0.6
0.8
1.0
Reduced Frequency (k b)
-8 0.0
0.2
0.4
0.6
0.8
1.0
Reduced Frequency (k b)
Fig. 7. Generalized aerodynamic influence coefficient plots for various transonic Mach number.
Table 1 Structural properties of the wing model
Hinge stiffness, Kθ Uncoupled frequency ratio, ωβ /ωα Iα Density, ρ E ν
LRS model
HRS model
73.4 Nmrad−1
212.7 Nmrad−1 2.77
1.72
peared in the low reduced frequency region. This difference in low reduced frequency affects the flutter characteristics because the flutter is generally occurred below the reduced frequency value of 0.5. 3.2. Free vibration analysis and modes spline In this study, a wing model with an oscillating control surface is considered. The various rotational stiffness values are applied to the hinge of the control surface, and for each case we would investigate the changes of aeroelastic characteristics resulting from the oscillating control surfaces. The free vibration analyses are accomplished by MSC/ NASTRAN (Ver.70.5). In the NASTRAN model, RBAR elements, which define a rigid bar with six degree-offreedom (DOF), are used to connect the control surface with the wing structure. In the RBAR elements, the translation
0.000842 kg m2 2770 kg m−3 73.1 GPa 0.33
DOFs at each end are assigned as dependent DOFs, however the rotational DOFs remain as independent DOFs. The CELAS2 elements are used to represent the rotational stiffness of the control surface. The structural parameters are given in Table 1. The fluid-structure coupled analysis usually requires a spline technique to transmit the force and (or) displacement information between the structural mesh and the aerodynamic mesh. In this study, the natural mode data are calculated on the FEM mesh, so they must be transmitted to the aerodynamic mesh. To perform the numerical interpolation, an Infinite Plate Spline (IPS) method is used [5]. In this scheme, a set of discrete structural grid points is distributed within the infinite plate domain and the vertical positions of the deformed plate are defined at each structural grid point only. Next,
H.-J. Kwon et al. / Aerospace Science and Technology 8 (2004) 519–532
527
Mode 1
Mode 2
Mode 3
Mode 4
8.76 Hz
33.96 Hz
45.80 Hz
88.26 Hz
8.31 Hz
28.83 Hz
37.52 Hz
53.64 Hz
8.31 Hz
27.37 Hz
35.39 Hz
49.12 Hz
Clean Model
HRS Model
LRS Model
Fig. 8. Comparisons of natural mode shapes and frequencies.
(a)
(b)
Fig. 9. Flutter boundary comparisons for the clean wing (CL) model. (a) Flutter speed boundary. (b) Flutter frequency boundary.
the IPS code solves the partial differential equation to have the continuous deformed shape for the whole surface. Once the partial differential equation is solved, the deflection at the aerodynamic points can be determined. For the present wing models, the wing structure is divided into two parts: the control surface and the remaining part of the wing. Therefore, each part must be splined independently and then be superposed. The natural mode shapes and frequencies of the first four modes are presented in Fig. 8. These results show that the first mode is simple bending but the higher modes include control surface rotation. In this study, the first six modes were used in the aeroelastic analyses.
3.3. Flutter analysis The aeroelastic analyses are performed for CL, HRS, LRS, and NCO models. The calculated flutter speed and frequency boundaries of the CL model are shown in Fig. 9. In the subsonic region, the results of the Coupled Time Integrated Method (CTIM) are compared with that of the Transient Pulse Method (TPM) and Doublet Lattice Method (DLM). These results are additionally compared with the TPM results of the zero-thickness airfoil model because the airfoil thickness effect is not considered in the DLM and show good agreement. However, in the transonic and low supersonic regimes, there are some discrepancies because
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(b) Fig. 10. V-g and V-f plots for the clean wing (CL) model at Mach 0.7. (a) V-g plot. (b) V-f plot.
the shock interference severely affects the aeroelastic motion. The V-g and V-f plots for the DLM and TPM results are shown in Fig. 10. In these figures, the second mode becomes the flutter mode and the DLM and TPM curves show a similar pattern at Mach 0.7. The previous verifications of the present TSD3KR-AE code were performed in Ref. [15]. Fig. 11 shows the flutter speed and flutter frequency boundaries of the LRS and the HRS models. It also contains the comparison of the no control surface oscillating (NCO) model. In Fig. 11(a), when the oscillating effects are considered, the large decrease of flutter speed appears near Mach 1.0. Moreover, unlike the NCO models, the control surface oscillating models have higher flutter frequencies than the third modal frequency. Therefore, we can find that nonlinear aerodynamic effects such as shock interference can cause severe decreases in flutter speed. Also, these phenomena are closely related to the transition of the dominant flutter mode. Additionally, at Mach 1.2, the flutter speed of the HRS model is close to that of the NCO model, even though their flutter frequencies exist in different regions. The rise of the flutter speed following Mach number is delayed in low supersonic regime when the control surface rotational stiffness has decreased. Therefore, comparing the HRS model, the transonic dip of LRS model is extended to higher Mach
(b) Fig. 11. Comparisons of computed flutter characteristics for different rotational stiffness models. (a) Flutter speed. (b) Flutter frequency.
number. Fig. 11 also shows that the large increments of flutter speed can be accompanied by the transition of the flutter mode to higher modes. This trend can be observed at Mach 1.1. Detailed aeroelastic responses are presented in Figs. 12– 16. In these responses, static aeroelastic analyses are performed to find the structural equilibrium under the aerodynamic loading before the initial disturbances. The flutter responses are obtained after initial disturbances that have been applied to the first bending and the first torsion modes. The frequency and damping ratio of flutter response are calculated using the moving block method (MBM) [3]. Even though only the physical responses of the wing tip and the control surface are presented in Figs. 12–16, the entire modal responses were observed to find the modal damping ratio. Figs. 12 and 13 show the aeroelastic responses and phase plots at Mach 0.8 and 0.95 respectively. In these responses, the control surfaces show more unstable responses than the wing responses because a flutter occurs in higher mode including the control surface rotation. With the exception of amplitude, the motion pattern is maintained despite the increase of ωβ /ωα from 1.72 to 2.77. Figs. 14 and 15
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Fig. 12. Aeroelastic responses and phase plots of the wing model with control surface at M = 0.8.
Fig. 14. Aeroelastic responses and phase plots of the wing model with control surface at M = 1.03.
Fig. 13. Aeroelastic responses and phase plots of the wing model with control surface at M = 0.95.
Fig. 15. Aeroelastic responses and phase plots of the wing model with control surface at M = 1.1.
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(b) Fig. 16. Aeroelastic responses and phase plots of the wing model with control surface at M = 1.2.
show the flutter responses at Mach 1.03 and Mach 1.1, respectively. As shown in Fig. 14, when ωβ /ωα is 1.72 (LRS model), the response of the control surface shows a simple harmonic motion. This means that a single mode, which is third mode, is dominant in the unstable response. In the high rotational stiffness (HRS) model, however, the fourth mode participates in the unstable motion and the phase plot represents the multi-periodic pattern. Fig. 16 shows the responses near the flutter speed at Mach 1.2. In this figure, the LRS model shows unstable response in the control surface only. However, in the HRS model, the wing tip response is also unstable. Fig. 17 shows the spectral density function of the physical responses at Mach 1.03. It is found that the higher frequency mode is included in the responses of the HRS model. The spectral density plots shown in Fig. 17(a) is similar to that of simple harmonic motion and the corresponding flutter frequency is close to the third natural frequency. However, as shown in Fig. 17(b), the spectral density plot for the HRS model indicates that the control surface response has two modal frequencies and each of them is close to the third and fourth natural frequency. Moreover, for the responses of the HRS model, the MBM results represent that only the fourth modal response is unstable. This is also found in the case of Mach 1.1. In Fig. 15 the pattern of the control surface response is changed simultaneously with the increase of the uncoupled frequency ratio, ωβ /ωα . Also, the phase plot of control surface motion indicates multi-periodic
Fig. 17. Auto spectral density plots for the wing and control surface responses at M = 1.03. (a) ωβ /ωα = 1.72 and U ∗ = 0.87. (b) ωβ /ωα = 2.77 and U ∗ = 1.13.
characteristics. The auto spectral density plots for Fig. 15 are presented in Fig. 18. In Fig. 18(a), the spectrums of the responses of the LRS model are similar to that of a simple harmonic response with the normalized frequency of 1.39. However, when the rotational stiffness is increased, the auto-spectral density plot shows that the response has two modal frequencies and, then, the lower mode of them disappears as shown in Fig. 18(b). Therefore, the autospectral density function represents the simple harmonic motion with a normalized frequency of 2.01. At Mach 1.1, in the LRS model, the second and third modes are unstable modes, however in HRS model, only the fourth mode is unstable. The wing and control surface responses of the HRS model at Mach 1.2 are not correlated with each other because they have different modal frequencies as shown in Fig. 16. This uncorrelated response pattern is more clearly observed in auto-spectral density plots shown in Fig. 19. In this figure, when ωβ /ωα = 2.77, the frequency of wing response is different from that of the control surface response. This uncorrelated motion between the wing and the control surface is distinct from the motion pattern in the lower Mach numbers. Fig. 20 shows the drawings of flutter mode shapes at various Mach numbers. As mentioned above, the difference of oscillating shape between the LRS model and HRS model is observed at Mach 1.1 and Mach 1.2. In the LRS model,
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Fig. 18. Auto spectral density plots for the wing and control surface responses at M = 1.1. (a) ωβ /ωα = 1.72 and U ∗ = 1.09. (b) ωβ /ωα = 2.77 and U ∗ = 1.55.
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Fig. 19. Auto spectral density plots for the wing and control surface responses at M = 1.2. (a) ωβ /ωα = 1.72 and U ∗ = 1.96. (b) ωβ /ωα = 2.77 and U ∗ = 2.00.
Fig. 20. Selected drawings of flutter mode shapes.
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the wing motion near the flutter speed is mainly affected by the control surface rotation. In the HRS model, however, bending or another motions as well as the control surface rotation affects the flutter response. 4. Conclusions In this study, the transonic and supersonic flutter characteristics of wings with oscillating control surfaces were investigated. In the frequency domain analysis, TPM and DLM were applied to calculate the generalized aerodynamic influence coefficients and the V-g method was used to determine the aeroelastic stability. The time domain flutter analyses were performed using a coupled time integration method (CTIM). The aeroelastic characteristics of the control surface models were compared to those of the NCO model to investigate the unsteady aerodynamic effects of control surface oscillation. These results showed that the unsteady aerodynamic interference cause severe decrease of the flutter speed. To examine the effects of rotational stiffness, aeroelastic analyses were performed for two different rotational stiffness models. In addition, the large change on the flutter boundary was induced when the dominant mode on unstable motion was altered to another mode and the predicted dominant mode usually contained the control surface motion. Acknowledgements This work was sponsored by the Korea Ministry of Science and Technology. The authors are grateful for the support as the National Laboratory Program. References [1] J.T. Batina, Efficient algorithm for solution of the unsteady transonic small-disturbance equation, J. Aircraft 25 (10) (1988) 962–968.
[2] D. Borglund, J. Kuttenkeuler, Active wing flutter suppression using a trailing edge flap, J. Fluids and Structures 16 (3) (2002) 271–294. [3] W.G. Bousman, D.J. Winkler, Application of the moving block analysis, in: 22nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Atlanta, Georgia, 1981, AIAA81-0653-CP. [4] W.J. Chyu, L.B. Schiff, Nonlinear aerodynamic modeling of flap oscillations in transonic flow: a numerical validation, AIAA 21 (1) (1983) 106–113. [5] R.L. Harder, R.N. Desmarais, Interpolation using surface splines, J. Aircraft 9 (2) (1972) 189–191. [6] K. Isogai, K. Suetsugu, Numerical calculation of unsteady transonic potential flow over three-dimensional wings with oscillating control surfaces, AIAA 22 (4) (1984) 478–485. [7] D.H. Kim, I. Lee, CFD-based matched-point transonic and supersonic flutter computations using a modified TSD equation, Comput. Fluid Dynam. J. 11 (1) (2002) 44–54. [8] D.H. Kim, Y.M. Park, I. Lee, O.J. Kwon, Nonlinear aeroelastic computation of a wing with a finned-store using a parallel unstructured Euler solver, in: 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf. & Exhibit, Denver, 2002, AIAA Paper 2002-1289. [9] I. Lee, S.H. Kim, Aeroelastic analysis of a flexible control surface with structural nonlinearity, J. Aircraft 32 (4) (1995) 868–874. [10] T.J. Leger, J.M. Wolff, Improved determination of aeroelastic stability properties using a direct method, Math. Comput. Modeling 30 (1999) 95–110. [11] S. Obayashi, G.P. Guruswamy, Navier–Stokes computations for oscillating control surfaces, J. Aircraft 31 (3) (1994) 631–635. [12] S.K. Paek, I. Lee, Flutter analysis for control surface of launch vehicle with dynamic stiffness, Computers and Structures 60 (4) (1996) 593– 599. [13] W.P. Rodden, E.H. Johnson, MSC/NASTRAN Aeroelastic Analysis USER’S GUIDE, Version 68, MSC. [14] J.L. Steger, H.E. Bailey, Calculation of transonic aileron buzz, AIAA 18 (3) (1980) 249–255. [15] S. Weber, K.D. Jones, J.A. Ekaterinaris, M.F. Platzer, Transonic flutter computations for the NLR 7301 supercritical airfoil, Aerospace Science and Technology 5 (2001) 293–304. [16] S. Yang, I. Lee, Aeroelastic analysis for flap of airfoil in transonic flow, Computers and Structures 61 (3) (1996) 421–430. [17] T.Y. Yang, C.H. Chen, Transonic flutter and response analyses of two 3-degree-of-freedom airfoils, AIAA 19 (10) (1982) 795–884. [18] ZAERO Theoretical Manual Version 5.1, Zona Technology.
