Modeling the Benchmark Active Control Technology Wind-Tunnel Model for Application to Flutter Suppression

MODELING THE BENCHMARK ACTIVE CONTROL TECHNOLOGY WIND- TUNNEL MODEL FOR APPLICATION TO FLUTTER SUPPRESSION * Martin R. Waszak, NASA Langley Research Center, Hampton, Virginia Abstract on the BACT enabled the investigation of multivariable flutter suppression. And the availability This paper describes the formulation of a model of the of truly multivariable control laws provides an dynamic behavior of the Benchmark Active Controls opportunity to evaluate the effectiveness of a Technology (BACT) wind-tunnel model for [3] controller performance evaluation (CPE) tool used application to design and analysis of flutter to assess open- and closed-loop stability and suppression controllers. The model is formed by controller robustness when applied to multivariable combining the equations of motion for the BACT systems. An underlying requirement of all these wind-tunnel model with actuator models and a model objectives is the availability of a mathematical model of wind-tunnel turbulence. The primary focus of this of the BACT dynamics.

paper is the development of the equations of motion A mathematical model is the basis for nearly all from first principles using Lagrange’s equations and control design methods, therefore an appropriate model the principle of virtual work. A numerical form of is essential. The importance of having a good model the model is generated using values for parameters of the dynamic behavior cannot be overstated and the obtained from both experiment and analysis. A model must be developed with a mind toward the needs unique aspect of the BACT wind-tunnel model is that of control law design. In addition, appropriate models it has upper- and lower-surface spoilers for active are required to accurately assess system performance control. Comparisons with experimental frequency and robustness. Extensive analysis and simulation are responses and other data show excellent agreement and usually required before controller implementation to suggest that simple coefficient-based aerodynamics are assure that safety is not compromised. This is sufficient to accurately characterize the aeroelastic especially true in the area of aeroservoelastic testing in response of the BACT wind-tunnel model. The which failure can result in destruction of the wind- equations of motion developed herein have been used tunnel model and damage to the wind-tunnel.

to assist the design and analysis of a number of flutter Mathematical models for control law synthesis must suppression controllers that have been successfully characterize the salient dynamic properties of the implemented.

system. One of the most important properties to Introduction accurately model is the frequency response in the Active control of aeroelastic phenomena, especially vicinity of the key dynamics over the anticipated range in the transonic speed regime, is a key technology for of operating conditions. In the case of flutter [1] future aircraft design. The Benchmark Active suppression, the key dynamics occur near the flutter Controls Technology (BACT) project is part of NASA frequency and the operating conditions correspond to a Langley Research Center’s Benchmark Models wide range of dynamic pressures and Mach numbers [1,2] Program for studying transonic aeroelastic representing both stable and unstable conditions. Also phenomena. The BACT wind-tunnel model was important are the key parametric variations associated developed to collect high quality unsteady aerodynamic with the system and the uncertainties associated with data (pressures and loads) near transonic flutter the assumptions and limitations of the analysis tools conditions and demonstrate flutter suppression using and other data used to build the model.

spoilers. Accomplishing these objectives required the The development of the model of the dynamic design and implementation of active flutter behavior of the BACT presented herein was motivated suppression. The multiple control surfaces and sensors by several factors. A primary motivation was based on the fact that the tool normally used at NASA Langley * [4] Aerospace Research Engineer, Senior Member AIAA.

to model aeroelastic systems, ISAC , is unable to model spoilers. Since the demonstration of flutter Copyright © 1996 by the American Institute of suppression using spoilers was a key objective of the Aeronautics and Astronautics, Inc. No copyright i s asserted in the United States under Title 17, U.S. Code.

BACT project an alternative modeling approach was The U.S. Government has a royalty-free license t o needed. Another motivation for the particular exercise all rights under the copyright claimed herein for modeling approach taken here was the desire to assess Governmental purposes. All other rights are reserved b y the impact of model uncertainty and parametric the copyright owner.

AIAA Paper No. 96-3437 - AIAA Atmospheric Flight Mechanics Conference variations on control system design, performance, and robustness.

Trailing Edge By developing the model from first principles using Control 0.45c appropriate idealizations of the structural and 14.4 in.

Upper aerodynamic characteristics, a model was developed that Spoiler includes spoiler controls and explicitly contains the key physical parameters. The analytical and parametric 32 in.

nature of the model also provides physical insights not 0.30c 9.6 in.

readily obtained from purely numerical models (such as those produced by ISAC and similar tools). While this 0.15c paper emphasizes the development of the equations of 2.4 in.

motion of the BACT wind-tunnel model, a complete 0.25c model is presented including actuator and turbulence 4 in.

models, and validation of the resulting numerical model.

c = 16 in.

The BACT Wind-Tunnel Model The BACT wind-tunnel model is a rigid, rectangular, wing with an NACA 0012 airfoil [5] section. A drawing of the model is shown in Figure 1. It is equipped with a trailing-edge control Figure 1 - BACT Wing Section Diagram surface, and upper- and lower-surface spoilers that can be controlled independently via hydraulic actuators.

Turntable The wind-tunnel model is instrumented with pressure transducers, accelerometers, control surface position sensors, and hydraulic pressure transducers. The Fore-Aft Stiffening Central Beam accelerometers are the primary sensors for feedback control and are located at each corner of the wing.

The wing is mounted to a device called the Pitch and [5] Plunge Apparatus (PAPA) which is designed to Pitch and permit motion in principally two modes -- rotation (or Plunge pitch), and vertical translation (or plunge). A drawing Rod System of the PAPA is shown in Figure 2. The mass, inertia, and center of gravity location of the system can be controlled by locating masses at various points along the mounting bracket. The stiffness properties "Chevron" can be controlled by changing the properties of the Mounting Bracket rods. The PAPA is instrumented with strain gauges to measure normal force and pitching moment and is mounted to a turntable that can be rotated to control Figure 2 - PAPA Diagram the wing angle of attack.

[8] problem in aeroelasticity. The difference is The combination of the BACT wing section and primarily the complexity of aerodynamic behavior and PAPA mount will be referred to as the BACT system.

presence of additional structural modes. The finite The BACT system was precisely tuned to flutter span and low aspect ratio of the BACT wing within the operating range of the Transonic Dynamics [6] introduce significant three dimensional flow effects.

Tunnel (TDT) at NASA Langley Research Center in Higher frequency structural degrees of freedom are which the system was tested. The range of Mach associated with the PAPA mount and the fact that the numbers and dynamic pressures over which flutter wing section is not truly rigid.

occurs permits the study of transonic aeroelastic The control surfaces also introduce complexities not phenomena. More detailed descriptions of both the typically reflected in the classical 2-DOF system. The BACT wing section and the PAPA mounting system mass and inertia of the control surfaces and potential can be found in References 5 and 7.

flexibility in their support structures introduce inertial The BACT system has dynamic behavior very coupling effects. The finite span of the control similar to the classical two degree-of-freedom (2-DOF) American Institute of Aeronautics and Astronautics surfaces and their close proximity to each other also introduce significant aerodynamic effects. All these K h issues influence the development of the equations of h motion but, as will be seen, do not force the K θ θ abandonment of the 2-DOF system structure.

Aeroelastic Equations of Motion m , I 1 1 K Lagrange’s equations were used to derive the δ e m , I 2 2 equations of motion for the BACT system and the principle of virtual work was used to obtain δ e expressions for viscous damping and the generalized h e aerodynamic forces. Lagrange’s equations and the e δ principle of virtual work provide a simple and straight δ c forward method for deriving the equations of motion [9] for aeroelastic systems. Lagrange’s equations readily allow one to represent motions relative to a Figure 4 - Structural representation showing trailing moving frame. The principle of virtual work has the edge control only.

advantage of automatically accounting for the forces of nature of the motion of the BACT system. The wing constraint and thereby greatly simplifying the section and each control surface are assumed to be determination of generalized forces.

rigid in both the spanwise and chordwise directions.

The basic requirement for applying Lagrange’s This assumption is supported by the fact that the equations is that the velocity of each point in the body wing and control surfaces were constructed to be as be represented in an inertial frame. The efficient rigid as possible.

application of Lagrange’s equations can be facilitated It is also assumed that the motion is limited to two by representing the inertial quantities in convenient degrees of freedom -- pitch and plunge. This coordinate systems and selecting an appropriate set of assumption implies that the other structural modes of generalized coordinates.

the BACT system are insignificant. Investigation of Coordinate Axes and Generalized Coordinates the structural vibration characteristics of the PAPA The BACT system can be idealized as a collection of mount with a very similar wing model was shown to four rigid bodies corresponding to each of the three [10] support this assumption. The next lowest control surfaces and the remaining wing/PAPA frequency for any transverse mode was more than six element. Figure 4 depicts the relevant quantities for times the frequency of the pitch and plunge modes and the wing and the trailing edge control surface. The well outside the frequency range of interest.

spoiler control surfaces were treated in an analogous Based on these assumptions the BACT requires five fashion but are omitted here for ease of discussion.

generalized coordinates, two associated with the pitch There were essentially five coordinate systems used.

and plunge degrees of freedom of the entire wing and One coordinate system is fixed to the wing, moving three are associated with the angular rotation of the with it. Another coordinate system is fixed in inertial control surfaces. The five generalized coordinates space and oriented relative to the turntable to which the therefore are pitch angle, θ , and plunge displacement, BACT system was mounted. It was chosen to coincide h , of the body fixed coordinate axes relative to the with the undeformed position of the body-fixed system.

inertial coordinate axes and the trailing edge, upper-, The three others are fixed to each of the control and lower-spoiler control surface angles, δ , δ , and TE US surfaces and rotated relative to the body-fixed system δ , respectively. The three coordinates, h , θ , and LS about the hinge of each surface.

δ δ ≡ , for the system including only the trailing TE The origin of the inertial coordinate axes is located edge control surface are depicted in Figure 4.

at the shear center of the undeformed position. The The relation between the generalized coordinates and origin of the body fixed coordinate axes coincides with the angle of attack of the wind-tunnel model will be the instantaneous shear center of the system. The important in formulating the generalized aerodynamic origin of the control surface-fixed coordinate axes forces later. Based on the choice of the generalized coincide with the hinge lines.

coordinates the following expression can be used to The generalized coordinates were selected to simplify determine the angle-of-attack.

the derivation of the equations of motion. Their selection is based on some key assumptions about the American Institute of Aeronautics and Astronautics commanded and the actual control surface rotation, δ c ˙ ˙ ( , , ) w x y t θ ( ) ( ) ( ) h t x t l g α θ θ ( , , ) ( ) x y t t = + + + − (1) and δ , respectively.

T U U U 0 0 0 The total potential energy for the BACT system is simply the sum of the gravitational potential and the where θ is the turntable angle, l ( ) x is the distance T strain energy, Eqns (3) and (4), respectively.

from the origin of body fixed coordinate system to the Applying Lagrange’s Equations angle of attack reference, x (positive aft), w is the g Lagrange’s equations can be expressed in the normal perturbation velocity of the local flow field following form.

(positive down relative to the free stream flow), and U is the freestream velocity.

0 d T T U ∂ ∂ ∂   Q − + = (5) q   i Kinetic and Potential Energy ˙ dt q q q ∂ ∂ ∂   i i i The selection of the generalized coordinates allows where the T and U are the kinetic and potential expressions for the kinetic and potential energies to be energy of the system, respectively, q is the i th i formulated. The kinetic energy of the BACT system is generalized coordinate, and Q is the generalized force qi the sum of the kinetic energy of the wing and the three associated with q and includes externally applied i control surfaces.

forces, nonconservative forces, and forces of The kinetic energy of a body is the work required to constraint.

increase its velocity from rest to some value relative to Applying the expressions for the kinetic and an inertial frame. Using the quantities defined in potential energies to Eqn (5) results in the following Figure 4 the kinetic energy expression for the BACT system of equations.

system including only the trailing edge control can be written ˙˙   m s s h K h     0 0   h h h θ δ 1 1         2 2 ˙ ˙ ˙ ˙˙ T m h e I = + + ( ) θ θ s I s K θ θ + 0 0     1 1 1 h θ θ θδ θ     2 2     ˙˙ (2)     s s I K δ δ 0 0 h δ θδ δ δ         1 1 2 2 ˙ ˙ ˙ ˙ ˙ (6) m h e e I + + + + + ( ) ( ) θ δ θ δ 2 2 2 δ 2 2 m   0      Q h       s δ θ = + cos 0 + g Q cos θ       The potential energy of a rigid body consists of two c h θ T θ       terms, the strain energy and the gravitational potential K s δ cos Q h δ δ       δ energy. The gravitational potential is defined relative The terms m , I , and I are the generalized masses of θ δ to a datum, usually the origin of the inertial reference the pitch, plunge, and control surface modes, frame. Using the quantities defined in Figure 4 and respectively. The terms s , s , and s are the assuming the datum to be the origin of the inertial h θ h δ θδ frame the gravitational potential for the BACT system inertial coupling between the various degrees-of- is freedom. These terms are related to the quantities defined in Figure 4 by the following relations.

U m g h e = − + ( sin ) cos θ θ g T 1 1 (3) m m m s m e m e ≡ + ≡ + 1 2 1 1 2 2 h θ m g h e e − + + ( sin sin ) cos θ δ θ T 2 2 δ 2 2 I I I m e m e s m e ≡ + + + ≡ (7) h 1 2 1 1 2 2 2 θ δ δ where g is the gravitational acceleration.

I I m e s I m e e ≡ + ≡ + 2 2 2 2 2 2 δ θδ δ The strain energy is the work done in going from the undeformed reference position to the deformed If one assumes that the control surface stiffness is position. Using the quantities defined in Figure 4 the very large, (i.e., the deformation due to hinge load is strain energy for the BACT system is insignificant) then Eqn (6) can be simplified by eliminating the generalized force associated with the 1 1 1 2 2 2 U K h K K = + + − θ δ δ ( ) (4) control surface, Q .

e h c θ δ δ 2 2 2 Assume that the control surface stiffness is very where K and K are the spring constants associated h θ large so that with the stiffness properties of the PAPA mount and K ε ≈ << 1 , where K is the spring constant representing the δ δ ε flexibility in the structure supporting the actuator.

Eqn (6) can now be approximated by the following Note that the strain energy associated with the control equations after eliminating terms of order ε .

surface is based on the difference between the American Institute of Aeronautics and Astronautics mass matrix from Eqn (9). The other constants, ω h δ δ = (8) c and ω , correspond to the in vacuo vibration θ frequencies for the pitch and plunge modes.

˙˙ m s K 0 h   h       h h θ + External (Aerodynamic) Forces         ˙˙ s I K 0 θ θ h θ θ θ         (9) The externally applied forces are due to s m Q       h h δ aerodynamics and result from the distributed pressures ˙˙ g δ θ = − + + cos       T s s Q applied to the surface of the BACT wing. For the h θδ θ θ       wing the virtual work can be written Note that the value of θ has been assumed to be small c + b δ δ W p x y t z x y t dxdy = ( , , ) ( , , ) (12) so that cos θ is approximately unity. Also note that ∫ ∫ c − 0 the inertial coupling between the wing structure and where b is the wing semi-span, c is the section chord, the control surface is retained in the equations.

p(x,y,t) is the differential pressure distribution over All that remains to complete the equations of the surface of the wing, and δ z(x,y,t) is a virtual motion is to determine expressions for the generalized displacement normal to the wing surface. Figure 5 forces, Q and Q .

h θ depicts the pressure distribution for a chordwise Applying the Principle of Virtual Work section.

The principle of virtual work can be applied to The virtual displacement can be written in terms of obtain expressions for generalized forces. The basic virtual displacements of the generalized coordinates h , advantage of using this method is that the forces of θ , and δ as follows.

constraint are eliminated automatically. In addition, − = + − δ δ δ θ z x y t h t x x t ( , , ) ( ) ( ) ( ) p the principle of virtual work can be used to determine expressions for dissipative forces such as damping.

< x x , 0 (13)  f + The generalized force, Q , can be determined from  qi − > δ δ x x x x ( ) , f f  the following equation.

Substituting this expression in Eqn (12) and W ∂ δ Q = (10) performing the differentiation described in Eqn (10) q i q ∂ δ i results in the following expressions for the generalized aerodynamic forces.

where δ W is the work done on the system by an arbitrary infinitesimal (or virtual ) displacement of the W ∂ δ Q = h generalized coordinates. This virtual work includes h ∂ δ the work done by nonconservative forces (e.g., c + b damping) and external forces. The work done by p x y t dxdy L = − ≡ − ( , , ) ∫ ∫ c − 0 forces of constraint are zero under virtual W δ ∂ displacements.

Q = θ δ ∂ θ Nonconservative (Damping) Forces (14) c + b Structural damping is often characterized as a p x y t x x dxdy M = − − ≡ ( , , )( ) p p ∫ ∫ c − 0 viscous force. Experimental data suggests that this is W δ ∂ a reasonable assumption for the BACT system Q = δ δ ∂ δ undergoing small motions. Viscous forces are those c where the force varies in proportion to the velocity at + b p x y t x x dxdy H = − − ≡ ( , , )( ) f ∫ ∫ δ the point the force is applied but in the opposite x 0 f direction. The following expression was derived to z represent the generalized damping forces for the BACT.

p(x,t) nc m s ζ ω   2 0 Q       h h h θ h = − (11)       nc s I ζ ω 0 2 Q   h θ θ θ θ     θ   c x The constant terms in Eqn (11) were chosen so that the damping coefficients associated with the plunge and c − x x p f pitch modes, ζ and ζ , respectively, correspond to h θ those obtained from experiment. The matrix Figure 5 - Pressure distribution over wing section.

premultiplying the diagonal damping matrix is the American Institute of Aeronautics and Astronautics Where L , M , and H are the lift, pitching moment e p δ C C − −       Q L L   h α 0 qS qS θ = about the reference point, x , and the control surface     +   T p e cC cC Q M M   θ α     0   moment about its hinge-line, respectively. The C C − − ˙˙ l reference point, x , was chosen to correspond to the     qSc h L L ˙ ˙ α α p +     cC c C ˙˙ l U 2 shear center (i.e., the origin of the body fixed M M θ   ˙ ˙ α α   0   coordinate system).

c   There are several ways in which the integrated C C C − − − l C +

L L L L L ( )

˙ ˙   α α q α   qS h U 2 surface pressures can be approximated in practice. If +   (17)   c ˙ U θ   computational aerodynamic analysis results are 0   cC c C C C − + l M M M M

( )

q ˙ α α α   U 2 available the integration can be approximated by using   0 the pressures on the computational grid. If C − C C −   − 0 L h      L L qSc ˙   α ˙ δ δ experimental force and moment data are available it can qS qS δ + +   + δ       cC C cC 0 M θ M M U 2 ˙   be used directly allowing for potential differences in the α     0 δ     δ   moment reference point.

c   A common method of approximating the − − C C L L   ˙ ˙ α α w   qS g U 2 aerodynamic forces is to use stability and control −     c w U g   derivatives. The aerodynamic forces are represented as 0   C cC M M α ˙ α   U 2 a linear function of angle-of-attack and control surface   0 deflection and their rates as shown in Eqns (15) and Complete BACT Equations of Motion (16). Descriptions of each coefficient are presented below in Table 2. A comparable expression for the Combining the generalized forces, Eqns (11) and hinge moment has been omitted since the need for the (17), with Eqn (9) results in the complete equations of generalized force associated with hinge moment in the motion for the BACT system. The general form of the equations of motion was eliminated in Eqn (9). equations can be expressed as L qSC = L ˙˙ ˙ M M q D D q K K q − + − + −

( ) ( ) ( )

s a s a s a e qS C C C = + + α δ (15) Q Q M g = + + cos θ θ (18)

L L L [

0 α δ T T g T ˙˙ ˙ c  B B B Ew + + + + δ δ δ   ˙ 2 1 0 ˙ ˙ C C C + + + α θ δ  L L L ˙ ˙ q α   δ U 2  where q is the vector of generalized coordinates, w is the vector of disturbance inputs, and the definitions of M qScC = p M the other symbols can be readily determined by comparison with Eqns (9), (11), and (17).

qSc C C C = + + α δ (16)

M M M [

0 α δ Notice that the effect of the aerodynamic forces is to modify the mass, damping, and stiffness properties of c    ˙ ˙ ˙ C C C + + + α θ δ the system. It is this aerodynamic coupling that is the  M M M ˙ ˙ q α   δ U 2  essential feature of aeroelastic systems and leads to the flutter instability.

The coefficient-based approach is a rather simplistic Also note that there are three terms in the equations way to represent the aerodynamic forces and moments, that are constant, assuming the turntable angle is fixed.

but it is quite acceptable for the BACT system as will These terms determine the static equilibrium of the be seen. The need to include more sophisticated system.

aerodynamic modeling approaches such as rational Equilibrium Solution and Perturbation Equations function approximations or time accurate CFD is mitigated by the fact that the reduced frequency for the The static equilibrium of the BACT system is BACT system is relatively low, approximately 0.044.

obtained by setting all time derivatives in Eqn (18) to Using the expression for angle-of-attack from zero and solving for the generalized coordinates. Doing Eqn (1) in Eqns (15) and (16) the expression for the so results in the following expression.

generalized applied external forces in Eqn (17) can be 1 − e

obtained. q K K Q Q = − ( ) + θ

(

s a T T 0 0 (19) + + cos M g B θ δ

)

g T 0 0 American Institute of Aeronautics and Astronautics The subscript on the generalized coordinate vector is The mass and inertia parameters were obtained by used to indicate static equilibrium. The subscript on measuring the mass, stiffness, and damping properties the control input is used to denote its static value of the various components of the BACT system. The (i.e., bias). geometric parameters (e.g., centers of gravity, shear The generalized coordinates can be expressed as the center, sensor locations, and aerodynamic reference sum of the static (or equilibrium) part, q , and a quantities) were also obtained directly from ˜ perturbation part , q , measurement of the BACT and PAPA components.

The mass, stiffness, and damping parameter values ˜ q q q = + (20) used in the numerical form of the BACT equations of motion are presented in Table 1. The center of gravity The control input can be expressed as the sum of the and shear center were nearly coincident and located at bias or static part, δ , and the time varying or the mid-chord point of the wing.

˜ dynamic part, δ .

Table 1 - Mass, Stiffness, and Damping Parameters ˜ δ δ δ = + (21) S y m b o l Description Value U n i t s Substituting Eqns (20) and (21) into Eqn (18), using m mass (plunge 6.08 slug the fact that q , θ , and δ are constant and 0 T 0 generalized mass) eliminating the constant terms by using Eqn (19) I 2 pitch inertia 2.80 results in the perturbation equations of motion for the θ slug-ft (pitch generalized mass) BACT system as shown in Eqn (22).

˙˙ ˙ − − 1 1 K   plunge stiffness 2686 lb/ft     ˜ ˜ q q     ( ) ( ) ( ) ( ) M M K K M M D D − − − − − − h s a s a s a s a =       ˙ ˜ ˜     q I q   0       K pitch stiffness 3000 ft-lb θ − − 1 1     ˙˙ ˙ − − − − ( ) ( ) M M B M M B ˜ ˜ s a s a 2 1 + + δ δ ω     in vacuo plunge frequency 21.01 rad/sec h     0 0     (22) ω in vacuo pitch frequency 32.72 rad/sec − − 1 1 θ     − − − − ( ) ( ) M M B M M E ˜ s a s a 0 + + w δ     ζ     0 0 plunge damping ratio 0.0014 -     h ζ pitch damping ratio 0.0010 - While the form of the equations that appear in θ Eqn (18) describe the complete motion of the system, s plunge - pitch coupling 0.0142 slug-ft h θ it is the form of the equations of motion presented in Eqn (22) that are most readily applicable to typical s plunge - TE coupling 0.00288 slug-ft h δ TE control system design methods. Note that even though s 2 pitch - TE coupling 0.00157 θδ slug-ft Eqn (22) was derived for a single control surface, TE extension to multiple control surfaces is straight s plunge - US coupling 0.00039 slug-ft h δ US forward since there is no inertial coupling between the various control surfaces. There is, however, s 2 pitch - US coupling 9.8e-05 θδ slug-ft US aerodynamic coupling between the control surfaces due to their close proximity to each other. The form of the The static aerodynamic parameters were determined aerodynamic force expressions allows this coupling to from experimental data obtained from a previous wind- be approximated within the stability and control tunnel test in which the BACT wing was mounted on derivative terms. This can be done by altering the a force and moment balance. Force and moment data derivative values to account for control surface biases.

for various angles of attack and control surface Perturbation effects, however, are ignored in this positions were used to compute most of the stability approach.

and control derivatives using finite differences.

Numerical Model Parameters Sufficient experimental data was only available to quantify the trailing edge control surface and upper- A numerical form of the equations of motion of the spoiler aerodynamic characteristics. Available data for BACT system was obtained by substituting numerical the lower-spoiler was not complete enough to values for each parameter in the equations of motion characterize its aerodynamics. In addition, there was developed above. Most of the parameter values were little data to account for aerodynamic coupling between obtained from experimental data but some of the the spoiler and trailing edge control. Therefore, the aerodynamic data was obtained from numerical analysis numerical model is limited to the trailing edge and using computational aerodynamics.

American Institute of Aeronautics and Astronautics upper-spoiler surfaces with no aerodynamic coupling Table 2 - Aerodynamic Parameters between controls (e.g., spoiler blanking the trailing S y m b o l Description Value U n i t s edge control).

C lift at zero angle of 0 - L0 The dynamic derivatives (e.g., C and C ) were M L attack ˙ q α C pitching moment at 0 - obtained from computational aerodynamic analysis.

M0 zero angle of attack Analytical values were used because they were C lift curve slope 4.584 per rad available from models previously generated using L α ISAC. However, the dynamic derivatives associated C moment curve slope 1.490 per rad M α with control surface deflection rate were not available C plunge damping due to -3.1064 per rad L from these models. The parameters that were not ˙ α angle-of-attack rate available were assumed to be zero.

C plunge damping due to 2.5625 per rad L The numerical values for the static and dynamic q pitch rate stability and control derivatives are presented in C pitch damping due to -2.6505 per rad M ˙ Table 2. These values are only valid at a single Mach α angle-of-attack rate number, 0.77, and a single dynamic pressure, 143 psf.

C pitch damping due to -0.4035 per rad M q The moment coefficients are referenced relative to the pitch rate shear center that coincides with the mid-chord point of TE lift effectiveness 0.63 per rad C L δ the wing. Finally, note that the expression for the TE generalized aerodynamic forces, Eqn (17), requires the C TE rate lift 0 per rad L ˙ δ TE effectiveness selection of an angle of attack reference point, l (i.e., C TE moment -0.0246 per rad the distance between the shear center and the point at M δ TE effectiveness which the angle of attack of the wing is measured). A C TE rate moment 0 per rad parametric study found that the best correlation M ˙ δ TE effectiveness between the numerical model and experimental data C US lift effectiveness 0.22 per rad resulted when the reference point was chosen to be the L δ US aerodynamic center.

C US rate lift 0 per rad L ˙ δ US Actuator and Turbulence Models effectiveness and the Output Equation C US moment 0.0573 per rad M δ US effectiveness The equations of motion alone are not sufficient to C US rate moment 0 per rad M describe the dynamic behavior of the BACT system.

˙ δ US effectiveness While the 2-DOF system structure is sufficient to l distance between shear -0.175 % c describe the basic aeroelastic properties of the BACT center and aerodynamic system, additional elements are necessary to develop a center model suitable for control system design.

S planform area 3.55 2 ft The relative magnitude of the dynamic response is c mean aerodynamic 1.33 ft determined by the nature of the disturbance chord environment. This influences the control activity required to achieve the desired level of closed-loop It is the combination of the actuator models, the performance. Therefore, a characterization of the turbulence model, output equation, and the aeroelastic turbulence environment in the TDT is needed, i.e., a equations of motion that will determine the degree to turbulence model. The ability of the control surfaces which a control system will be able to achieve a to produce the desired activity is dependent on the desired level of performance and robustness.

dynamic response characteristics of the actuators Actuator Models including bandwidth, position and rate limits, and other nonlinearities. Therefore, characterizations of the Actuator models of the BACT wind-tunnel model actuator dynamics are also needed, i.e., actuator were obtained from experimental data using a simple [11] models. Finally, a set of measurement signals is parameter estimation process. The process selected required to provide the basis for feedback control. An the parameters of the second order actuator model output equation relating the generalized coordinates to shown in Eqn (23) to minimize the frequency response the measurement variables is therefore required.

error over the frequency range of interest.

American Institute of Aeronautics and Astronautics speed, V , corresponds to a different Mach number.

δ ω ( ) s k No data at the appropriate operating conditions is = (23) 2 2 δ ( ) s available. A reference speed close to the airspeed ζω ω s s c + + 2 associated with a particular test condition was used for Here k is a gain, and ω and ζ are frequency and analysis and design purposes.

damping, respectively. The parameter values Output Equation resulting from the parameter estimation process are The BACT system has four accelerometers, one presented in Table 3.

mounted in each corner of the rectangular wing. These Table 3 - BACT Actuator Model Parameters accelerometers sense vertical acceleration measured in S y m b o l Description Value U n i t s g’s, positive up (opposite to the sign convention for k TE actuator gain 1.02 deg/deg plunge, h ). The acceleration at any point on the TE BACT wing, excluding control surfaces, has two TE damping ratio 0.56 - ζ TE components as shown in Eqn (25).

ω TE frequency 165.3 rad/sec TE ˙˙ ˙˙ k US actuator gain 1.16 deg/deg h d − + θ

( )

US ˙˙( ) z x = (25) US damping ratio 0.85 - ζ g US ω US frequency 164.0 rad/sec US where h and θ are the generalized coordinates, d is the k LS actuator gain 1.09 deg/deg chordwise distance from the reference point to the LS shear center, and g is the gravitational acceleration.

LS damping ratio 0.76 - ζ LS Table 5 presents the chordwise distance (positive aft) ω LS frequency 168.4 rad/sec LS between the shear center and the accelerometer for Turbulence Model each of the four accelerometers -- leading edge inboard (LEI), leading edge outboard (LEO), trailing edge A model of the turbulence environment within the inboard (TEI), and trailing edge outboard (TEO).

TDT was developed using power spectrum data from reference 12. The model structure is that of a Dryden Table 5 - Accelerometer Locations spectrum with the parameters adjusted to approximate Accelerometer d d d d the desired power spectral density. Equation (24) LEI LEO TEI TEO Location shows the form of the turbulence model and Table 4 Distance (ft) -0.599 -0.599 0.433 0.420 presents a range of values for the turbulence model parameters.

All the components described above were combined   π 2 to form the complete numerical model of the BACT s +   system. The following section addresses a variety of w s ( ) β π αβ   2 g = (24) analyses that were performed to assess the accuracy and s ( ) η γ   g π π 4 4 validity of the model.

s s + +   γ γ   Validation of Numerical Model There are many ways to assess various aspects of 2 2 π L 2 π L 2     t t where β β = and γ γ = . the validity of the BACT numerical model. In this p p     V V section a few comparisons are made between the properties of the numerical model and the actual BACT Note that the parameter values are based on data wind-tunnel model. These assessments can be broken collected in an air medium, not the medium in which down into two categories -- static properties and the BACT was tested (i.e., R-12) and so the reference dynamic properties. The static properties assess the Table 4 - TDT Turbulence Model Parameters characteristics of the equilibrium solutions. The dynamic properties assess the key response Reference Speed (fps) characteristics of the system in the context of flutter S y m b o l 1 0 0 2 0 0 3 0 0 4 0 0 behavior and control system design.

α 0.01 0.025 0.007 0.082 Static Properties β 0.477 0.475 0.521 0.667 p The equilibrium position (pitch and plunge) of the γ 0.546 0.464 0.497 0.533 p BACT system depends on the turntable angle and wind- L 3.261 3.71 3.391 4.163 tunnel operating conditions and represents a balance t American Institute of Aeronautics and Astronautics between the elastic and aerodynamic forces acting on the maximum level of control activity depends directly the wing. Good agreement between the equilibrium on the level of response of the wind-tunnel model to position of the wind-tunnel model and the equilibrium turbulence. Each of these properties will be reviewed solution of the numerical model would indicate that the and compared with experimental data (and numerical stiffness (structural and aerodynamic) and control models generated with ISAC where possible) to assess surface effectiveness properties are well modeled. the validity of the numerical model.

Two variables that were recorded during the wind- Flutter Properties tunnel tests are turntable angle and pitch angle (plunge The BACT wind-tunnel model experienced flutter in position was not measured). In addition, the test the TDT at a dynamic pressure of approximately 148 conditions (i.e., Mach number, dynamic pressure, and psf at a Mach number of 0.77. The flutter dynamic control surface positions) were recorded. Using the pressure for the numerical model is 150.8 psf, an later quantities in Eqn (19), one can determine the difference of 1.9 percent. ISAC generated models equilibrium pitch and plunge position of the BACT indicate flutter occurs between 156 and 163 psf.

system for comparison with experiment.

The flutter frequency of the BACT wind-tunnel Table 6 presents the computed and measured pitch model is approximately 4 hertz. At the same operating angle for a small representative set of test conditions.

condition the flutter frequency of the numerical model The error in equilibrium pitch angle is less than 5 is 4.16 hertz, a 4.0 percent difference. ISAC generated percent for all but one point. In addition, the trends are models indicate the flutter frequency to be consistent; increasing the turntable angle increases the approximately 4.22 hertz.

pitch angle, deflecting the upper-spoiler decreases the In terms of the flutter dynamic pressure and pitch angle, and deflecting the trailing edge control frequency at the Mach number for which aerodynamic downward decreases the pitch angle.

data was available, the numerical model of the BACT The size of the pitch angle increments due to system gives excellent results.

turntable angle and spoiler deflection are also very Transfer Function Comparisons similar. The main discrepancy is in the pitch angle increment due to trailing edge control deflection. The One of the most important measures of model pitch angle change due to trailing edge control from the fidelity for control system design is the frequency experimental data is five times higher than from the response. In order to effectively design a control numerical model. system to stabilize a flutter mode the design model must accurately characterize the dynamic behavior of Dynamic Properties the aeroelastic system over a fairly wide range of A major concern in using the numerical model for dynamic pressures from stability to neutral stability to control system design is the ability of the model to instability.

accurately represent the transition to flutter including Figures 6 and 7 show comparisons between the the frequency and dynamic pressure of the flutter onset.

frequency responses for the numerical model and the Another concern is the fidelity of the model from a actual wind-tunnel model. The operating condition frequency response perspective because of the corresponds to subsonic Mach number and a dynamic relationship between the frequency response and the pressure of 125 psf, well below flutter. In Figure 6 structure of the control system. Finally, the level of response due to turbulence is an important issue since Table 6 - Static Equilibrium Position Comparison Mach Dynamic θ δ δ θ θ ∆θ T TE US exp model Number Pressure (psf) (%) (deg) (deg) (deg) (deg) (deg) .65 112 1.6 0 0 2.1 2.17 3.33 .65 115 1.6 0 - 1 0 2.0 2.05 2.50 ∆θ ( δ ) 0.1 0.12 US .70 126 1.6 0 0 2.4 2.28 -5.00 .70 126 1.6 1 0 0 2.0 2.20 10.0 ∆θ ( δ ) 0.4 0.08 TE .77 120 1 . 4 0 0 2.0 1.95 -2.50 .77 120 4 . 5 0 0 6.0 6.27 4.50 ∆θ ( θ ) -4.0 -4.32 T American Institute of Aeronautics and Astronautics 0.3 the output is the trailing edge accelerometer and the input is the trailing-edge control. In Figure 7 the 0.2 output is the trailing-edge accelerometer and the input is the upper spoiler control. An ISAC-based model is 0.1 also presented in Figure 6 for comparison. Recall that ISAC cannot model spoilers and so no ISAC Magnitude (g/deg) 0 2 4 6 8 10 12 comparison can be made in Figure 7. The frequency response comparisons for the other accelerometers are comparable to those shown here.

There is excellent correlation between the experimentally obtained frequency responses and those -100 Phase (deg) of the numerical model. The model clearly captures the key aspects of the dynamic response of the BACT -200 0 2 4 6 8 10 12 system at the subcritical dynamic pressure of 125 psf.

Frequency (Hz) There are, however, slight discrepancies in the Figure 6 - Frequency response comparison for trailing- frequency of the magnitude peak near 3.5 hertz and in edge inboard accelerometer due to trailing edge control: the phase characteristics of both responses.

q = 125 psf.

Open-Loop RMS Accelerations (solid - exp, dash - model, dot - ISAC) It is important for the numerical model to accurately 0.6 characterize the response of the system to disturbances since disturbance response determines the control 0.4 activity required to achieve the desired level of closed- loop performance. The disturbance source for the 0.2 BACT system is wind-tunnel turbulence. One measure Magnitude (g/deg) of the degree to which the numerical model 0 2 4 6 8 12 characterizes the turbulence is rms acceleration.

Table 7 presents a comparison of the rms trailing edge inboard accelerations at two dynamic pressures.

The comparison is based on normalizing the dynamic pressure by the associated flutter dynamic pressure, q .

f -100 Phase (deg) Normalization is needed because of the differences in -200 the flutter dynamic pressure for the experimental data 0 2 4 6 8 10 12 and the numerical model. The reference speed used to Frequency (Hz) scale the turbulence model is 400 fps and is consistent Figure 7 - Frequency response comparison for trailing- with the speed of the flow in the wind-tunnel.

edge inboard accelerometer due to upper spoiler control: Note that there is a Mach number mismatch q = 125 psf.

between the experimental data and the model-based data (solid - exp, dash - model) since the aerodynamic parameter values in the numerical model are based on data collected at Mach Table 7 - Comparison of RMS Trailing-Edge 0.77 and the experimental data was obtained for Mach Acceleration numbers of 0.63 and 0.71, respectively. The good RMS Trailing Edge agreement in the response level implies that the Acceleration (g) numerical model can be used to assess rms response.

q (psf) Experiment Model % Error norm Comments 0.75*q 0.0207 0.0188 -9.2 f Based on the accuracy of the flutter properties, the 0.90*q 0.0340 0.0350 2.9 subcritical frequency responses, and rms disturbance f response, it is reasonable to expect the model to could deviate from the actual system despite the accurately characterize the dynamic response of the similarities presented above.

BACT system over a wide range of operating In addition, the discrepancies identified above should conditions and is appropriate for control system design.

be taken into account during control system design and Nevertheless, the dynamic behavior of the numerical dynamic analysis. In particular, the static analysis model near flutter has not been directly verified and American Institute of Aeronautics and Astronautics supports the possibility that the pitch effectiveness of Model for Benchmark Active Controls Testing.

the trailing edge control surface in the numerical model AIAA Paper No. 91-1011.

may be somewhat low. There are also slight peak [2] Bennett. R.M.; Eckstrom, C.V.; et al..: The frequency and phase differences between the numerical Benchmark Aeroelastic Models Program - model and the experimental data in the frequency Description and Highlights of Initial Results.

response plots that could represent uncertainty in pole NASA TM-104180, Dec. 1991.

and zero locations of the numerical model. [3] Wieseman, C.D.; Hoadley, S.T.; and McGraw, S.M.: On-Line Analysis Capabilities Concluding Remarks Developed to Support Active Flexible Wing The dynamic model of the Benchmark Active Wind-Tunnel Model. Journal of Aircraft . Vol.

Control Technology (BACT) wind-tunnel model 23, No. 1. pp. 39-44. Jan.-Feb. 1995.

presented herein has many advantages over a purely [4] Adams, W.M.: ISAC: A Tool for numerically derived model. It is analytical and Aeroservoelastic Modeling and Analysis.

parametric in nature and therefore lends itself to NASA TM-109031. December, 1993.

sensitivity and uncertainty analysis. Since the [5] Rivera, Jr., J.A.; Dansberry, B.E.; et al..: aerodynamic effects are represented in derivative form NACA 0012 Benchmark Model Experimental experimental data can readily be substituted for Flutter Results with Unsteady Pressure analysis-based data. A major advantage of the Distributions. AIAA Paper No. 92-2396. 33rd modeling approach is that it allows experimental AIAA/ASME/ ASCE/AHS/ASC Structures, stability and control derivative data to be used to model Structural Dynamics, and Materials Conference.

spoiler effects not possible with the traditional Dallas, TX, April 1992.

modeling method. The modular form of the model [6] Baals, D.D. and Corliss, W.R.: Wind Tunnels also allows various components of the model to be of NASA. pp. 79-81. NASA SP-440.

modified or replaced. This is very useful in cases [7] Rivera, Jr., J.A.; Dansberry, B.E.; Durham, where actuator models and turbulence models are M.H.; Bennett, R.M.; and Silva, W.A.: modified or updated.

Pressure Measurements on a Rectangular Wing ® ® A Matlab /Simulink implementation of the wind- with a NACA 0012 Airfoil During tunnel model, actuator models, turbulence model, and Conventional Flutter. NASA TM-104211, July digital controller effects has been developed for the 1992.

purpose of evaluating and analyzing the dynamic [8] Bisplinghoff, R.L. and Ashley, H.: Principles behavior of the BACT system. It has been used by of Aeroelasticity. Dover Publications, Inc., New several researchers to assist in the design and analysis York. 1962.

of flutter suppression controllers. A variety of [9] Waszak, M.R. and Schmidt, D.K.: Flight classical, H , μ -synthesis, neural network, and ∞ Dynamics of Aeroelastic Vehicles. Journal of adaptive controllers have been designed using the Aircraft . pp. 563-571. Vol. 25, No 6. June numerical model and have been successfully tested in 1988.

the Langley Transonic Dynamics Tunnel.

[10] Dansberry, B.E.; Durham, M.H.; Bennett, The BACT model and test data are also being R.M.; Turnock, D.L.; Silva, W.A.; and Rivera, developed as a case study package for educational use.

Jr., J.A.: Physical Properties of the Benchmark The relatively simple structure of the BACT system Models Program Supercritical Wing. NASA coupled with the availability of extensive and detailed TM-4457. September 1993.

experimental data make the BACT an excellent [11] Waszak, M.R. and Fung, J.: Parameter candidate for the study of dynamics and control of Estimation and Analysis of Actuators for the aeroelastic systems.

BACT Wind-Tunnel Model. AIAA Paper No.

96-3362. AIAA Atmospheric Flight Mechanics Acknowledgment Conference. San Diego, CA. July 29-31, The author wishes to acknowledge Rob Scott, Sheri 1996.

Hoadley, Carol Wieseman, Robert Bennett, Robert [12] Sleeper, R.K.; Keller, D.F.; Perry III, B.; and Sleeper, and the entire BACT team without whose help Sanford, M.C.: Characteristics of Vertical and this work could not have been accomplished.

Lateral Tunnel Turbulence Measured in Air in References the Langley Transonic Dynamics Tunnel.

[1] Durham, M.H.; Keller, D.F.; Bennett, R.M.; NASA TM-107734. March 1993 and Wieseman, C.D.: A Status Report on a American Institute of Aeronautics and Astronautics