[american institute of aeronautics and astronautics 47th aiaa/asme/asce/ahs/asc structures,...

Transonic Flutter Prediction of Supersonic Jet Trainer with Various External Store Configurations

In Lee*

Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

Hyuk-Jun Kwon†

Agency for Defense Development, Daejeon, P.O. Box35, Korea

Jong-Yun Kim‡ and Jae-Han Yoo‡

Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

Seung-Kil Paek§ and Young-Ik Kim**

Korea Aerospace Industries, Ltd., Sacheon, Gyeongnam, 664-802, Korea

and

Jae-Sung Bae††

Hankuk Aviation University, Goyang, Gyeonggi, 412-791, Korea

In this study, the aeroelastic analysis is performed to investigate the flutter characteristics of the aircraft model with external stores in the transonic and low supersonic regimes. The nonlinear aeroelastic analysis for the aircraft model is used using the TSD (Transonic Small Disturbance) theory in the transonic regime. A time-accurate AF (Approximate Factorization) algorithm is applied to solve the TSD equation. The aeroelastic analysis system is verified with the flight test data and the DLM results of the wing model. It is confirmed that the system is quite accurate. The mode shapes and natural frequencies of the aircraft model are obtained by the structural analysis with MSC/NASTRANTM. This study shows that a flutter safety margin depends on store configurations in the transonic and low supersonic regimes.

Nomenclature φ = disturbance velocity potential M = free stream Mach number [Mg] = generalized mass matrix [Cg] = generalized damping matrix [Kg] = generalized stiffness matrix {Q(t)} = generalized aerodynamic forces vector {q(t)} = generalized displacement vector

* Professor, Aerospace Engineering, 373-1 Guseong-dong Yuseong-gu Daejeon 305-701 Korea, AIAA Associate Fellow. † Senior Researcher, Rotorcraft System Division/MADC, P.O. Box35 Yuseong Daejeon Korea. ‡ Graduate Research assistant, Aerospace Engineering, 373-1 Guseong-dong Yuseong-gu Daejeon 305-701 Korea. § Senior Researcher, Research & Development Division, 55 Yongdang-Ri Sacheon-Up Sacheon Gyeongnam 664-802 Korea. ** Principal Researcher, Research & Development Division, 55 Yongdang-Ri Sacheon-Up Sacheon Gyeongnam 664-802 Korea. †† Assistant Professor, School of Aerospace and Mechanical Engineering, 200-1 Hwajeong-dong Deogyang-gu Goyang Gyeonggi 412-791 Korea.

American Institute of Aeronautics and Astronautics

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47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

AIAA 2006-2079

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction LUTTER is a very dangerous phenomenon in aircraft wings, tails, and control surfaces, because it can result in structural failure at a certain critical speed. Hence, it is important to satisfy a safety margin against the flutter

instability in whole flight envelope. The anomalous aerodynamic changes around the transonic flow region aggravate the flutter stability. The aeroelastic instability can be predicted using experimental and numerical methods. In the numerical approaches, the panel methods such as Doublet Lattice Method (DLM) and Doublet Point Method (DPM) are most widely applied for the flutter prediction. The use of these panel methods, however, has limitation on the transonic and low supersonic regimes because they cannot consider the aerodynamic nonlinearity induced by shock waves. Therefore, it is necessary to apply approaches of Computational Fluid Dynamics (CFD) in these regimes. In particular, to capture the full nonlinearity of the flow field, the Navier-Stokes equations with turbulent model should be applied to the aeroelastic analysis.1 However, it is difficult to apply this technology for three dimensional wings with a control surface.2 The Transonic Small Disturbance (TSD) theory is not only very efficient but also appropriate to consider the complex motions of aircraft, such as the rotation of control surfaces.3

F

The aim of this study is to investigate the transonic flutter characteristics of the T-50 supersonic jet trainer with external store configuration. Because the aircraft can speed up to Mach number of 1.5, which means over the transonic regime, it must have an enough safety margin at the transonic regime in which there is the flutter dip. In this study, to consider the aerodynamic nonlinearity induced by shock waves in the transonic and low supersonic regimes, the flutter analysis in a time-domain is performed using the TSD equation. The accuracy of the present program had been verified several times. Especially, the present program of the flutter analysis was compared with the public flight test data of the F-16 wing model which has external stores and wing tip launchers.4 To investigate the transonic flutter characteristics of the T-50 aircraft model with external store configurations, the flutter analysis is implemented for two external store configurations. The mode shapes and natural frequencies of the aircraft model have been obtained via free vibration analysis using MSC/NASTRANTM, and 40 elastic modes are applied to the flutter analysis. In this study, the mode shapes on the structural mesh are interpolated to the aerodynamic grid using the infinite plate spline method. In flutter analysis, the flutter boundary is predicted by an interpolation of damping ratios between the two dynamic pressures corresponding to convergent and divergent responses. From the flutter analysis, this study shows a relationship between a flutter safety margin and store configuration in the transonic and low supersonic regimes.

II. Governing Equations

A. TSD (Transonic Small Disturbance) Equation Although the Navier-Stokes equations are the most accurate, many flow features do not depend on a precise

evaluation of the viscous and turbulent terms. For example, if the thickness of a wing section is small and there is no boundary layer separation, the viscosity has little effect on the flow fields. If the viscous terms are removed from Navier-Stokes equations, the equation becomes an Euler equation. Furthermore, if the flow around the aircraft wing is irrotational and the perturbation is small, the TSD theory can be applied.

The TSD equation written in a conservative form is given by3

0 1 2 0f f ft x y z

3f∂ ∂∂ ∂+ + + =

∂ ∂ ∂ ∂ (1)

where

( )2 20 1 2, , 1 , t x x x y y xf A B f E F G f H f3 zφ φ φ φ φ φ φ= − − = + + = + =φ

The above equations are given in a physical coordinate system, (x, y, z, t). φ is the disturbance velocity potential. The coefficients A, B, and E are defined as

(2) 2 2, 2 , 1A M B M E M= = = − 2

Several choices are available for the definitions of F, G and H depending on how the TSD equation is derived. The coefficients used in this paper are


2

( ) ( ) ( )2 21 11 , 3 , 12 2

2F M G M H Mγ γ γ= − + = − = − − (3)

where M is the free stream Mach number and γ is the ratio of specific heats. The TSD equation is solved using a time-accurate AF (Approximate Factorization) method. The AF algorithm

consists of a Newton linearization procedure coupled with an internal iteration technique. The solution process involves two steps. First, a time linearization step is performed to determine an estimate of the potential field. Second, internal iterations are performed to minimize linearization and factorization errors.3

B. Structural Dynamic Equation The aeroelastic equation of motion can be formulated by Hamilton Theorem for elastic models and is written in

matrix form as follows:5

( ){ } ( ){ } ( ){ } ( ){ }g g gM q t C q t K q t Q t⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ + =⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (4)

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 forces as follows:

( ) ( ) ( )2 212 , , , i r i pS

Q t U c h x y C x y t dSρ∗

∗ ∗ ∗ ∗∞= ∆∫ ∗ (5)

where subscript ‘i’ indicates the influence mode and S* is the non-dimensional plane area of wing. 2 212 rU cρ ∞

is multiplied to make the dimensional force term because the inside of the integral is non-dimensional. In Eqs. (4) and (5), symbol ‘t’ represents physical time, hence one must pay attention to the transition from non-dimensional time in the unsteady aerodynamics into physical time in the structural dynamics. The solution of Eq. (5) is obtained from the 2 x 2 Gauss numerical integration method. The structural damping ratio is generally assumed to be 0.005 - 0.02.

Ordinary differential equations such as Eq. (4) can be reduced to the state vector forms for efficient numerical calculations. That can be written as

( ){ } [ ] ( ){ } [ ] ( ){ }x t A x t B u t= + (6)

where

[ ][ ] [ ]

[ ][ ]

1 1

g g g g g

0 0,

IA

M K M C M− −

⎡ ⎤⎢ ⎥=

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− −⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦1B −

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

( ){ } ( ){ }( ){ } ( ){ } { }

( ){ }0

, q t

x t u tQ tq t

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪= = ⎪⎨ ⎬ ⎨ ⎬

⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭

A fifth-order Runge-Kutta method is used to solve the equation of motion.

III. Numerical Results and Discussions

A. Verification of Aeroelastic Analysis System The present aeroelastic analysis system using the TSD equation is verified with the flight test data of the F-16

wing model, which are obtained from Ref. 4. The appendix in the reference presents three mode shape data (wing bending, torsion and forward wing torsion modes). Figure 1 shows the finite element model and TSD grid for the aerodynamic analysis. Each aerodynamic grid of x and y direction is clustered on the near leading and trailing edge as well as the hinge part. The wing model has a shape of a NACA 64A204 airfoil. To derive the mode spline, the


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wing part should be divided into three parts; the wing, the control surface and the launcher. After obtaining the mode spline, the divided parts are re-united in the aerodynamic coordinate.

The present aeroelastic results using the TSD equation estimate the flutter velocity and frequency accurately by comparison with the DLM results and the flight test data. The flutter velocity and frequency data by the experiment and analysis are arranged in Table 1. The flutter speeds are expressed by the knots calibrated airspeed. The present aeroelastic result using the TSD equation shows the closer flutter speed to flight test than the DLM analysis on the wing-bending mode.

X (inch)

Y(in

ch)

300 350 400

50

100

150

X

Y

0 0.5 10

0.5

1

a) Finite element model b) Aerodynamic grid

W

Forwa

B. TransoOne of

aircraft maeroelasticjoint parts

The prtrainer devhorizontal50 aircrafconfigurat

Figure 1. Finite element model and aerodynamic grid of the F-16 wing model.

Table 1. Comparisons of flutter velocity and frequency between experiment and analyses.

Vibration analysis Flutter analysis (DLM)4 Flight test4 Present

(TSD)

Mode Fn (Hz)

Vf (KCAS)

Ff (Hz)

Vf (KCAS)

Ff (Hz)

Vf (KCAS)

Ff (Hz)

ing-bending 9.191 745.2 9.37 585 9.5 628.2 9.29 Torsion 9.964 - - - - - -

rd wing-torsion 10.246 442.4 10.17 - - - -

nic Flutter Analysis of the Aircraft Model the strengths of the TSD equation in the aeroelastic analysis is that it alone can be applied to a realistic odel considering all wings and control surfaces. The application using the Euler and Navier-Stokes programs would be impossible because of the impracticable computing time and mesh treatments on

between control surface and the remaining parts. esent aeroelastic program is applied to the T-50 aircraft model. The T-50 is a supersonic advanced jet eloped by KAI (Korea Aerospace Industries, Ltd.). The aircraft model includes the body, main wings,

tails, vertical fin, launchers and several control surfaces. To investigate the flutter characteristics of the T-t model with external stores in the transonic regime, the flutter analysis is performed for two ions including launchers on wing tips; one is the model without the inboard and outboard stores, and the

a) Model without stores b) Model with stores Figure 2. External configurations.


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other has fuel tanks on the inboard stores and bombs on the outboard stores, respectively. The configurations of the aircraft models for the flutter analysis are described in Fig. 2.

A finite element model for structural analysis and a surface grid for aerodynamic analysis are illustrated in Fig. 3. The mode shapes and natural frequencies of the aircraft models are obtained by the structural analysis with MSC/NASTRANTM. In the aerodynamic analysis, the airfoil of the main wing has the shape of NACA 64 series, and each of the horizontal and vertical tails has a biconvex airfoil while the body is assumed to be plate. An aerodynamic analysis for the vertical fin is performed in different aerodynamic domains of horizontal lifting surfaces on the assumption that the aerodynamic interferences between the vertical fin and horizontal lifting surfaces are small. Even though the external stores are not modeled aerodynamically, the structural influences of the stores are considered.

Figure 4 shows the steady pressure contours on the upper surface of the horizontal plane and on the vertical fin at Mach numbers 0.80 and 0.95. The steady pressure contours of the vertical fin on both right and left sides are same results because of the biconvex airfoil. There are the shock waves on the horizontal tail and vertical fin at Mach number of 0.95.

Generally, the aeroelastic analysis requires a data transfer stage between the structural and aerodynamic grid points because they have been subject to different engineering considerations. In this study, the mode shapes on the structural grid are interpolated to the aerodynamic grid using the Infinite Plate Spline (IPS) method. For efficient data interpolation, about 500 structural points are selected among the finite element grid points. It is also considered that the two physical systems have different length scales. Figure 5 shows the interpolated mode shapes of the first

XY

Z

a) Finite element model b) Aerodynamic model

Figure 3. Structural and aerodynamic modeling of the T-50 supersonic jet trainer.

a) M = 0.80, α0 = 0º b) M = 0.95, α0 = 0º Figure 4. Steady pressure contours.

a) Symmetric bending mode b) Anti-symmetric torsion mode Figure 5. Structural mode shapes of the aircraft model.


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symmetric bending and anti-symmetric torsion modes in the aerodynamic coordinate. If there is a discontinuity, such as a flap or a rudder, then the main wing and vertical fin structures are divided into several parts, such as the control surface and the remaining part of the wing. Therefore, each part must be transferred independently and then superposed. The mode shapes and natural frequencies of the aircraft model have been obtained via free vibration analysis using MSC/NASTRANTM, and 40 elastic modes are applied to the flutter analysis. The time responses are obtained from the aeroelastic analysis for the aircraft model in Fig. 2. Figure 6 presents the 4th modal responses of the aircraft model with inboard and outboard stores at Mach number of 0.9 on the below and above flutter speed, respectively. An interpolation of damping ratios between two dynamic pressures corresponding to convergent and divergent responses is used to predict the flutter points. The damping ratio and frequency of each aeroelastic time response are calculated by the moving block method6 and optimum theory. The comparisons of flutter velocity and frequency boundaries for each aircraft model in Fig. 2 are described in Fig. 7. At all Mach numbers in the transonic regime, the flutter frequencies of the aircraft model without the stores are about 40 Hz, while those of the aircraft with the stores are about 10 Hz. Although the flutter velocities of the aircraft model without the stores are lower than those of the aircraft model with the inboard and outboard stores at below Mach number of 0.93, the flutter velocity boundary is turned up at above the Mach number.

Because it is not a general operating range over low supersonic regime for the aircraft with external stores, the aeroelastic stability is focused up to the transonic regime. In this sense, the result shows that the aeroelastic stability for the aircraft model with external stores is higher than that without stores.

a) Aeroelastic response below flutter speed

b) Aeroelastic response above flutter speed Figure 6. Aeroelastic simulation results at Mach 0.9.

Mach No.

0.7 0.8 0.9 1.0 1.1 1.2

Nor

mal

ized

Vel

ocity

0.6

0.7

0.8

0.9

1.0

1.1

model without storesmodel with stores

Mach No.

0.7 0.8 0.9 1.0 1.1 1.2

Freq

uenc

y (H

z)

0

20

40

60

model without storesmodel with stores

a) Flutter velocity boundary b) Flutter frequency boundary Figure 7. Comparisons of the flutter boundary for each aircraft model.

IV. Conclusion The aeroelastic analysis program has been developed using the TSD theory. To verify the present program, the

aeroelastic results of the F-16 wing model were compared with the public flight test and other aerodynamic analysis results. In the transonic and low supersonic regime, the flutter analysis of the T-50 aircraft models was performed. To investigate the flutter characteristics of the T-50 aircraft model, two external configurations of the aircraft model were considered. The flutter analysis results showed that the aeroelastic stability of each aircraft model was reversed at Mach number of 0.93.

Acknowledgments This research was sponsored by the Korea Aerospace Industries, Ltd. and benefited from discussions at the

Agency for Defense Development. The authors thank for the supports.


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References 1Kwon, H. J., Park, S. H., Lee, J. H., Kim, Y. S., Lee, I., and Kwon, J. H., “Transonic Wing Flutter Simulation Using Navier-

Stokes and k-w Turbulent Model,” 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Austin, Texas, Apr. 18-21, 2005, AIAA-2005-2294.

2Kim, D. H., Park, Y. M., Lee, I., and Kwon, O. J., “Nonlinear Aeroelastic Computations of a Wing/Pylon/Finned-Store Using Parallel Computing,” AIAA Journal, Vol. 43, No. 1, 2005, pp. 53-62.

3Batina, J. T., “Unsteady Transonic Algorithm Improvements for Realistic Aircraft Applications,” Journal of Aircraft, Vol. 26, No. 2, 1989, pp. 131-139.

4Denegri, C. M., Jr., “Limit Cycle Oscillation Fight Test Results of a Fighter with External Stores,” 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference and Exhibit, Atlanta, Georgia, Apr. 3-6, 2000, AIAA-2000-1394.

5Yoo, J. H., Kim, D. H., Kwon, H. J., and Lee, I., “Nonlinear Aeroelastic Simulation of a Full-Span Aircraft with Oscillating Control,” Journal of Aerospace and Engineering, Vol. 18, No. 3, 2005, pp. 156-167.

6Bousman, W. G., and Winkler, D. J., “Application on the Moving-Block Analysis,” 22nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Atlanta, Georgia, Apr. 6-8, 1981, AIAA-1981-653.


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