Closed-loop aeroservoelastic design for flutter ...

Method validation for aeroservoelastic analysis Ruxandra Botez, Iulian Cotoi, Alexandre Doin, Djallel Biskri Ecole de technologie superieure, 1100 Notre Dame West, Montreal, Que., Canada, H3C-1K3 ABSTRACT

Aeroservoelasticity Method ADAM developed by 2 AFWAL, Flight Dynamics Laboratory , Interaction of Structures, Aerodynamics, and Controls ISAC 3 developed by NASA Langley , and Flexible Aircraft 4 Modeling Using State Space FAMUSS . These analysis tools are still in the phase of research and development and there are still many ways for their improvement.

This paper describes a method in aeroservoelastic theory. Most of active flutter suppression techniques only take the aeroelastic interactions between aircraft structure and aerodynamic forces into account. But the aeroelastic dynamics of control surfaces as well as control feedback change flutter speeds and frequencies and then aircraft stability. Thus our point here is first to adjoin aeroelastic dynamics of control surfaces, and further to incorporate the closed loop composed of sensor, control, gain and actuator transfer functions. In this way the method will be able to show the interactions between control modes and aeroelastic modes (open loop) and the influence of control feedback on the stability of all dynamic modes (closed loop). In order to validate this method, we have tested it on an aircraft test 1 model developed by NASA and on which several aeroelastic analyses were experimented. The comparative study between the results given by our method programmed in Matlab and the ones provided by STARS is presented.

During the early 1970’s, the NASA Langley Research Center did not have access to an analysis tool to investigate aeroservoelastic interactions. The Interaction of Structures, Aerodynamics, and Controls (ISAC) system of program modules was developed during the 1970’s to provide this missing analysis capability and continues to be an effective tool. The ISAC system of programs has been or is currently being used in support of numerous projects such as : DAST ARW-1 and ARW-2, DC-10 wind tunnel flutter model, Generic X-wing feasibility studies, Analyses of elastic, oblique – wing aircraft, Active Flexible Wing (AFW) wind tunnel test program, Generic hypersonic vehicles, Benchmark active controls testing project, High speed civil transport, etc.

INTRODUCTION Aeroservoelasticity (ASE) is a multidisciplinary study of interactions between structural dynamics, unsteady aerodynamics and the control system. It has been fast emerging as a primary tool in achieving desirable flight characteristics in modern aircraft. ASE design is a challenging problem in that the aircraft dynamics vary with flight conditions, and instability can occur due to control system dynamics interacting with aircraft dynamics at any point in the flight envelope. One of the ASE applications is active flutter suppression, which is an unstable motion caused by an interaction between structural vibrations and the aerodynamic forces that results in the extraction of energy from the airstream. Such unstable excitations can provoke the loss of control of an aircraft, weaken its structure, and even end with the destruction of the aircraft.

ISAC is written in Fortran and will execute, with minimal changes relative to memory management, on many computer systems with a FORTRAN77 compiler that supports namelist input. It is currently running on VAX micro II computers with VMS, SUN Sparc workstations with a UNIX operating system and IBM minicomputers. The Flight Dynamics Laboratory worked on assimilating such a computer procedure, and the researchers realized ADAM (Analog and Digital Aeroservoelasticity Method) computer program.

Further to an extensive bibliographical research, the following aeroservoelastic analysis software tools were found to exist in the aeronautical industry : Structural Analysis Routines STARS developed by 1 NASA Dryden , Analog and Digital

1

Examples in ADAM include the unaugmented X-29 A and two wind tunnels : 1) The FDL model (YF17) tested in the NASA Langley 16 ft transonic dynamics tunnel and 2) The Forward Swept Wing FSW model mounted in the AFIT 5 ft subsonic wind tunnel.

relevant flutter analyses, we should incorporate all of them in our model before running such analyses.

This software program ADAM was prepared using the VAX 11/785 computer with a VMS operating system. The theoretical manual and the user’s 2 guide for ADAM are provided .

Then, if we add flight dynamic variables and control surfaces (mobile aerodynamic surfaces which can modify the aircraft trajectory) to the aeroelastic model, we obtain a new model which describes interactions between structural vibrations, aerodynamic forces, control surfaces and aircraft trajectory. Finally when we increase the system with the elements of the servo loop such as sensors, controllers, gains or actuators, and introduce their transfer function in the aeroelastic model, the augmented system represents the closed-loop dynamics of the aircraft.

Using Pade polynomials to describe aerodynamic forces, we can easily express aeroelastic dynamics (structural elastic modes) under a linear form.

The digital computer program STARS (STructural Analysis RoutineS) has been designed as an efficient tool for analyzing practical engineering problems and or supporting relevant research and development activities. Examples of advanced NASA Dryden Flight Research Center projects analyzed by the STARS code in recent years include the X-29A, F-18 High Alpha Research Vehicle / Thrust Vectoring Control System, B-52 / Pegasus, Generic Hypersonics, National AeroSpace Plane (NASP), SR-71 / Hypersonic Launch Vehicle, and High Speed Civil Transport (HSCT) projects. The program is written in modular form in standard Fortran language to run on a variety of computers, such as IBM RISC / 6000, SGI, DEC, Cray, and personal computer.

An aircraft model was designed by NASA in order to analyze such phenomena. This model is the dynamic system of an aircraft studied in its lateral modes and incorporates aero-structural elements (flexible aircraft), control surfaces (aircraft commands), and control loop (autopilot feedback). With such a model we can compare the results 1 obtained by NASA with our results. The first step is to fit STARS data with our method and to certify the reliability of these data with respect to our method. The second step is then to validate our method in accordance to STARS results.

Another technique for developing a state space model representation of a flexible aircraft for use in 4 an aeroservoelastic analysis was presented .This technique was based on determining an equivalent system to match the transfer function frequency response. The theory has been implemented in a computer code called Flexible Aircraft Modeling Using State Space FAMUSS at McDonnell Douglas Aircraft Company.

NOMENCLATURE

FAMUSS is an interactive program with a batch capability. It has a user friendly interface which uses windows and graphics. It has been used internally at McDonnell Douglas Corporation on their aircraft.

 

natural frequency generalized coordinates as functions of frequency,    e  r    

e r 

generalized coordinates for elastic modes generalized coordinates for rigid modes generalized coordinates for control modes non-dimensional generalized coordinates (with regards to time) non-dimensional generalized coordinates for elastic modes non-dimensional generalized coordinates for rigid modes non-dimensional generalized coordinates for control modes modal inertia or mass matrix modal damping matrix modal damping coefficients

q qe qr

All the above mentioned four computer programs have been developed in the USA, at research laboratories and industrial companies. We would like to develop an aeroservoelastic tool and this time in Canada, based on the theoretical expertise and software tools already existing. This tool will finally be developed based on the bibliographical research regarding the four software above mentioned. Aeroservoelastic stability depends on the adverse interactions between active control systems (servo-control) and the flexible structure of an aircraft (aeroelasticity). As we need to consider all these interactions to lead

q M D r

K Q QI

2

modal elastic stiffness matrix modal generalized aerodynamic forces imaginary part of modal generalized aerodynamic forces matrix

QR

 0

real part of modal aerodynamic forces matrix modal sensor matrix sensor locations true air density reference air density



air density ratio

V V0 VE

c b

true airspeed reference true airspeed equivalent airspeed VE   V V airspeed ratio  E V0 dynamic pressure qd  1 V 2 2 wing chord length semi-chord

k

reduced frequency

Mach

Mach number

 xs

 qd



generalized



(1)

1 1     M   D  VcQ I    K  V 2 QR   0 (9) 4k 2     (2)

Equations (2) and (1) give the value of  :

(3)



(4)

k  c 2V

V   0

V E2 V   0V E E   0V E  V V

(12)

Then, we replace equation (12) into the factor of QI and equation (11) into the factor of QR, and we obtain : 1 1     2 M   D   0 c  VE QI    K   0VE QR   0 4k 2    

(6)

where Q, the modal generalized aerodynamic forces matrix, is usually complex. The real part of Q denoted by QR, is called the “aerodynamic stiffness”, and is in phase with the vibration displacement ; the imaginary part of Q denoted by QI is called the “aerodynamic damping” and is in phase with the vibration velocity.

(13)

In the aim to normalize the airspeeds, we introduce a variable change technique, which will introduce a reference airspeed Vo through the normal reference frequency 0 defined as :

0 = V0/c The

Then, in order to respect this dynamics, we associate QR with  and QI with  . For this

(14)

variable change

technique

concerns the

conversion of the generalized coordinates  in the  domain into Pin the P domain, which may be written as follows :

reason, as the Q matrix is already a factor of , we need to divide QI by , in order to express QI as a factor of  . Thus, the aeroelastic equation (6) becomes:



(11)

If we divide both sides of equation (11) by V, and we use also the definition of  given by equation (2), we will obtain :

The formulation for linear aeroelastic analysis in the case of the Pk flutter method is :



(10)

V 2   0VE2

The P-method comes from the Pk – method. The processing of iterations is the same but only the form in which the equations are presented is different. Actually, the P-method is a non – dimensional form of the Pk – method, where the generalized coordinates, speed and time are normalized.



V E2   2 0 V

from where :

(5)

THE P-METHOD

M  D  1 qd QI    K  qd QR   0

(8)

We replace qd and  given by equations (4) and (8) into equation (7), and we obtain the following equation :

 0

M  D  K  qd Q(k, Mach)  0

kc 2V

 P P    

where

P   0

(15)

(7) By taking into account equation (15), we will derive  with respect to time, and we can write it with

Equation (5) gives  as function of k :

3

respect to the derivative of  with respect to time, then we obtain the following equations :

KQ  1 0c2QRk, Mach  2

P

  0 P

  02P

and

P

(16) LINEARIZATION OF AEROSERVOELASTIC SYSTEMS

Further, we need to express the reduced frequency k for the calculation of the aerodynamic forces through the matrix Q, as function of P parameters  , and  .

Equation (20) describing the dynamics of aeroservoelastic systems offers only semi-linear representations. In fact, all terms related to the aerodynamic forces are presenting strong nonlinearities with respect to the reduced frequency k. Then, the multitude of analysis and modelling algorithms applied to linear systems requires the motivation to obtain a linear aeroservoelastic system.

Firstly, equation (15) gives  :

   P 0

(17)

Equation (2) gives V :

V

VE

Therefore, the idea to convert the unsteady generalized aerodynamic forces matrix Q from the reduced frequency domain k to the Laplace domain s. Generally, the linearization will give new states, called aerodynamic lags which will describe the dependence of Q matrix of the reduced frequency k. Various methods are available to apply this 5,6,7,8 linearization .

(18)



Furthermore, equations (17), (18), (3) and (10) will be replaced into equation (5) in order to obtain :

k

c



2V

c P  c P V0   0   2 VE 2 c VE

We have chosen the simplest one, the Least Squares LS method – because is applied also in STARS – and in this way we will compare the results. Firstly, we have to express the aerodynamic forces Q under the Pade polynomials form :

(19)

k

 P  2

Q k   A0  jkA1   jk  A2  2

Finally we obtain the aeroelastic normalized equation by the P-method, after introducing the P new generalized coordinates vector  with equations (15) and (16), and using the airspeed ratio  given by equation (3) :



 



P P M PP  DP    DQ  P  K P   2 KQ  P  0 P

P

P

P

j  n

jk

 jk   j 1

A2  j (21) j

where Aj are the coefficients of dimension equal to the matrix Q and obtained from the least square LS algorithm, and the  are the aerodynamic lags arbitrarily introduced.

(20)

The optimization algorithms allow to calculate the optimal values of aerodynamic lags which minimize the approximation error between the aerodynamic

P

where M , D , DQ , K and KQ matrices are generated as follows :

forces matrix and the Pade polynomials.

M M P

K  12 K P

0

This new representation of aerodynamic forces is further replaced in the dynamics equation used in the P-k method (6) :

D P  1 D  ri M Pi,i K Pi,i

0

j n   jk 2 M  D  K  qd  A0  jkA1   jk  A2   A2  j   0 (22) j 1 jk   j  

P DQ  1 0c2QI k, Mach  4k

4

s  j and k  b , then we V

We know that

Equation (15) gives the value of 0 :

obtain

jk 

b s V

0 

(23)

 

b   V 20

(24)

s  s  V j b

V0  c 0

(25)

VE  c 0 and

  s

(26)

 j n  ~ ~ ~ M  D  K  qd   A2  j X j   0  j 1 

qd 

(27)

2

2

0

P

1

j n ~ P P ~ P P ~ P P 2  P  M   D   K      A2 j X j   0  j 1 

(29) where



 P  2

(36)

Equation (36) may be written under the following form :

~ P M P  M P  A2 ~ P D P  D P    A1

Equation (19) gives for the reduced frequency k :

2V

0

  1  P 1 2 2  j  n 2 2 1   2 K  0c  A0   0c    A2 j X j   0 2 2  0   j 1 

(28)

V0 

c

P

2

0

We replace VE given by equation (3) into equation (18) and we obtain:

k

(35)

M  12  b A    1 D  12  cb  A 

~ b D  D  q d   A1 V 

~ b K  K  qd   A1 V 

1 1  0V E2   0 c 2 02 2 2 2

Furthermore, we replace qd given by equation (35) and b/V given by equation (32) into equation (27) and into the coefficients given by equation (28) and 2 we divide the equation divided by 0 :

where

~ b M  M  qd   A2 V 

(34)

The dynamic pressure qd is further written :

and, by replacing equations (25) and (26) into equation (24) we obtain :

V

(33)

Equation (3) gives the following VE :

We know that :

  s

(32)

Equation (14) gives the following V0 :

Therefore, we introduce the state vector of aerodynamic modes :

Xj 

(31)

As we know that b = c / 2 , we obtain the following equation :

By replacing equation (23) into equation (22), we find :

  j  n 2 M  D  K  qd  A0  b sA1  b s A2   s A2 j   0 V  V  j 1 s  V  b j  

 P

~ P K P  K P   2 A0 and where :

(30)

5

(37)

A0  1 0c2 A0 2

(aircraft commands), and the control loop (autopilot feedback). Thus, STARS describes the lateral dynamic mode of an aircraft and contains three perfect rigid-body modes (Y translation, X-rotation roll, and Z-rotation yaw about the centre of gravity, PR), two rigid-control modes (aileron and rudder, C), and eight elastic modes (aero-structural vibrations, E).

A1  1 0cbA1 2

P

P

A2  1 0b2 A2 2 P

Aj  1 0c2 Aj 2 P

3  j  2  n

for

If we derive equation (25), we obtain :

X j  V  j X j   b

Before running any simulation on our method, we should check the quality of STARS data and test them to be sure they are appropriate to our method. The first stage was to convert STARS data to our needs, such as unit conversions. Then we ran a free-vibration analysis to certify the correspondence between the mass matrix M and the stiffness matrix K. That analysis determines the natural frequencies of the aero-structural system or elastic modes. The natural frequencies are given by resolving the eigenvalue problem  = eig(M,K), where  is the natural frequencies vector. The results are presented in Table 1 and match exactly with the ones provided by STARS manual.

(38)

We replace equations (16) and (32) into equation (38), we obtain the following equation (39) :

X j  2 0  j X j  0 P

(39)



If we apply the same variable transformation to Xj as the one to the generalized coordinates vector  then we can define the new state vector of the aerodynamic modes :

RESULTS AND DISCUSSION

P X j  1 X j

(40)

0

Please note that the rigid-body and control modes are considered perfectly rigid and totally free. As these modes are free and do not have any stiffness, their natural frequency should be 0 Hz. However we set them to 1 Hz because we need to set a stiffness to these modes to resolve the eigenvalue problem further and avoid divergence in calculations. But their natural frequency needs to be very small in comparison with the elastic modes natural frequency, as we could consider them almost zero. That is why we chose to set them at 1 Hz as shown in Table 1. At this point we have checked STARS data validity and are able to test our method on STARS test bench.

We divide equation (39) by 0, and we consider also the equation (40), then we can rewrite equation (39) as follows : P P P X j   j X j   P

(41)

where

 j P  2  j 

(42)

Finally, the dynamics equation is written under the following matrix form : I 0  0    0

I 0 0 0  0   P   0    ~P ~P 2 P ~P M 0    P    K  D   A3  P P I  1 I 0 I  0  X 1    0     0    0      P     0 I 0  0 0 I  X n  

  0  0

  P    A2n   P   0  X1P    0    P  n I  X n P   

Simulation parameters provided by STARS are the following :

0

2

P

The reference semi-chord length b = 38.89 in, the 3 reference air density at sea level 0 = 1.225 kg/m , the altitude at the sea level Z = 0 ft , the true air density at sea level is  = 0 , the reference Mach number is Mach = 0.9, the reference sound airspeed at sea level is a0 = 340.294 m/s, and the reference airspeed is V0 = Mach*a0 .

(43)

STARS TEST BENCH

At this step, we introduce aerodynamic forces in our dynamic analysis. The goal is now to certify the consistency of all the data together (matrices M, K and Q). For that we run a flutter analysis on elastic modes and check the coherence with STARS manual results. Table 2 shows STARS

With the objective to find a way to validate this 1 method we have chosen to use the STARS aircraft model developed by NASA. This model is very relevant to our subject because it includes aerostructural elements (flexible aircraft), control surfaces

6

manual results given by three different methods (k-method, P-k method, ASE method) and our solution run with the P-method. Considering the scattering in speed of STARS results, we can conclude that our results are quite acceptable for both flutters.

where the elastic modes 2 (Flutter 1) and 7 (Flutter 2) become unstable (positive damping). Table 3 reassembles the aeroelastic results obtained on the non-linearized ATM model with P-k method, the results obtained on the linearized ATM model with the P-k modified method to include the LS formulation (P-k LS method), and the ones obtained by use of the STARS software. Two flutter points were noticed – when the damping of the unstable mode is positive. In this context, two comparisons could be made. The first one concerns the results of two aeroelastic analyses on the ATM model, one non-linear and the other one linear. The second analysis will show the closeness between the aeroelastic linear analysis on STARS by the ASE method and the linear aeroelastic analysis realized with the P-k LS method.

Figures 1 to 3 show the results obtained by applying an aeroelastic nonlinear analysis on the ATM model in open loop by use of the P-k method. Eight elastic modes of the ATM model are represented by the frequency and damping versus the equivalent airspeed VE. The variation of the frequency with the damping is shown in Figure 1, the frequency versus the equivalent airspeed in Figure 2, the damping versus the equivalent airspeed in Figure 3. The damping is shown in Figure 3, where the elastic modes 2, and secondly the mode 4 become unstable (positive damping). From flutter point of view, mode 2 signifies the first mode becoming unstable – where flutter point 1 occurs, and mode 4 signifies the second unstable mode – where flutter occurs for the second time.

When we observe the equivalent airspeeds for the flutter phenomenon, in any case, the results of the Pk LS method are closer to the results of the P-k method than the ones of the ASE method. This conclusion gives in this direction the quality of the linearization that we applied on the ATM model, because the results of the non-linear model (P-k method) and of the linearized model (LS P-k method) are almost similar. Our linearized model diverges of the linearized model conceived with STARS. This divergence may be explained by the precision of each linearization, and on the other hand by the difference of the linearization methods and more particularly of the optimization techniques.

Table 2 shows the aeroelastic results obtained with the P-k method and also the results obtained by STARS. Two flutter points have been found – actually these are the points where the damping becomes positive. As our non-linear analysis has realized on the ATM model by using the P-k method, it will be convenient to compare our results with STARS results equally obtained by non-linear analyses, for example methods k and P-k. Therefore, the first flutter point was found by the P-k method to be very close to the one found by the k and the P-k methods of STARS, in terms of airspeed or frequency. Regarding the second flutter point, we may see a small difference in the airspeed values between our analysis and the one in STARS. This difference could be explained by the fact that the second flutter point appears at supersonic speeds. The equivalent sound speed at the sea level is approximately 680 knots. As the aerodynamic forces are defined in STARS for a reference Mach number 0.9, it could be possible that the ATM modes are ill - conditioned for supersonic speeds. From frequency point of view, the results are more consistent.

From frequency point of view, the assembly of results of different analyses is homogeneous. At this point we have checked STARS data validity and are able to test our method on STARS test bench. CONCLUSIONS An aeroservoelastic non-linear analysis by use of the P-k method has first introduced the ATM model and verified the data pertinence of the ATM model and the results obtained by use of STARS software at NASA Dryden Flight Research Center. In fact, a comparative study of the results of a nonlinear aeroelastic analysis applied on the ATM model by use of the P-k method and the aeroelastic results with STARS revealed a good coherence at the level of flutter prediction in open loop, which appeared at 450 and 800 knots. Following this analysis the ATM model was found to be consistent.

Figures 4 to 6 show the results obtained by applying a linear aeroservoelastic analysis on the Aircraft Test Model ATM linearized by the LS method in open loop by means of the P-k method. The eight elastic modes of the ATM model are represented by their frequency and damping as function of the equivalent airspeed. More specifically, the variation of frequency with the damping may be seen on Figure 4, the variation of the frequency with the equivalent airspeed is given on Figure 5. The damping versus the equivalent airspeed is traced on Figure 6 –

The second study permitted to test the impact of the aerodynamic forces linearization on the flutter

7

5. Tiffany, S. H., Adams, W. M. Jr., 1988. Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamics forces, NASA-TP 2776. 6. Poirion, F., 1995. Modélisation temporelle des systèmes aéroservoélastiques. Application à l’étude des effets des retards, La Recherche Aérospatiale, No. 2, pp. 103-114. 7. Cotoi, I., Botez, R.M., 2001, Optimization of unsteady aerodynamic forces for aeroservoelastic analysis, Proceedings of the IASTED International Conference on Control and Applications CA2001, Banff, Canada, pp. 105-108, June 27-29. 8. Botez, R.M., Bigras, P., 1999, Methods for aerodynamics approximation in the Laplace domain for the aeroservoelastic studies, Libertas Matematica, vol. XIX, pp.171-181.

prediction, in an aeroelastic context firstly, and secondly, in an aeroservoelastic context. Flutter prediction was very little perturbed by linearization of the aerodynamic forces. The aerodynamic forces linearization, which consist of a LS minimization, is not an optimal linearization. The precision of aerodynamic forces approximation may be really increased by augmenting the number of aerodynamic lags on one hand, or by use of the most powerful linearization methods which guarantee a good precision, by decreasing the number of aerodynamic modes. The main limitation of our study was the non – availability of the flight dynamic model applied on the ATM model, and more specifically the aerodynamic forces model associated to it. The partial description of the ATM model was limited to the movement in the lateral plane of the aircraft, added an additional constraint to our study. The description related to the flight dynamics has been largely diminished, following to the simplifications demanded to study the dynamics of the aircraft model in its lateral plane. As future work, the closed loop results were also analysed by a new method of validation. The results of this work will also be presented at the conference and in Table 4. The ATM model is also presented in Figure 7. ACKNOWLEDGEMENTS The authors would like to thank Dr Kajal Gupta at NASA Dryden Research Flight Center for allowing us to use the ATM model in STARS. Many thanks are due to the other members of STARS Engineering group for their continuous assistance and collaboration : Tim Doyle, Can Bach and Shun Lung.

Table 1

Free vibration modal analysis Flutter 1

REFERENCES 1.

Gupta, K.K., 1997. STARS-An integrated multidisciplinary, finite-element, structural, fluids, aeroelastic, and aeroservoelastic analysis computer program, NASA TM 101709. 2. Noll, T., Blair, M., Cerra, J., 1986. ADAM, An aeroservoelastic analysis method for analog or digital systems, Journal of Aircraft, Vol. 23(11). 3. Adams, W.M., Hoadley, S.T., 1993. ISAC : A tool for aeroservoelastic modeling and analysis, NASA Technical Memorandum 109031, pp. 110. 4. Pitt, D.M., Goodman, C.E., 1992. FAMUSS - A new aeroservoelastic modeling tool, AIAA-922395.

Flutter 2

Equivalent airspeed (knots)

Frequency (rad/s)

Equivalent airspeed (knots)

Frequency (rad/s)

STARS - k-Method

443,3

77,4

861,3

147,3

STARS - P-k method

441,7

77,4

863,4

147,1

STARS - LS method

474,1

77,3

728,9

136,2

Our P-k method

441

77,4

757,8

145,6

Table 2

Aeroelastic non-linear analysis Flutter 1 Frequency (rad/s)

Equivalent airspeed (knots)

Frequency (knots)

STARS - k Method

443,3

77,4

861,3

147,3

STARS - P-k Method

441,7

77,4

863,4

147,1

STARS - ASE Method

474,1

77,3

728,9

136,2

P-k Method

441

77,4

757,8

145,6

P-k LS Method

412,7

77,5

813,6

144,2

Table 3 8

Flutter 2

Equivalent airspeed (knots)

Linear aeroelastic analysis

Figure 3

Figure 1

P-k method – Frequency versus Damping

Figure 4

Figure 2

P-k method – Damping versus Equivalent airspeed

P-k method – Frequency versus Equivalent airspeed

9

LS P-k method – Frequency versus Damping

Figure 5

LS P-k method – Frequency versus Equivalent airspeed

Flutter point Equivalent airspeed (knots)

Frequency (rad/s)

STARS - ASE method

263

77,42

P Method

256,1

51

P-LS Method

278,3

51,9

Table 4 closed loop

LS P-k method – Damping versus Equivalent airspeed

Figure 6

0.1s  5 s

Pilot actions

Roll sensor

0.1 s2  11s  10

1372s s2  193s  1372

Aileron actuator

+

Filter

20 s  20

Roll angle Aileron deflection

Plant : 3 rigid modes 8 elastic modes 2 control modes

Gain

1

+ -

Figure 7

20 s  20 Rudder actuator

Rudder deflection Yaw angle

0.1s s  0.1

1372s s2  193s  1372

Filter

Yaw sensor

Aircraft Test Model ATM1

10

Linear aeroelastic analysis in