1 American Institute of Aeronautics and Astronautics

Formulation of an Aircraft Structural Uncertainty Model for Robust Flutter Predictions

Brian P. Danowsky* and Dr. Frank R. Chavez† Iowa State University, Ames, IA, 50012

and

Marty Brenner‡ NASA Dryden Flight Research Center, Edwards, CA, 93560

Flutter is a dynamic instability that aerodynamic vehicles encounter in atmospheric flight. The interaction between structural elastic, structural inertial and aerodynamic forces may cause the flexible vehicle to undergo divergent oscillations, at which point flutter is encountered. Undesirable effects of this behavior include difficult controllability, structural fatigue and even catastrophic structural failure. This point of instability is dependant on many factors including the structural properties, structural geometry, aerodynamic shape and the flight condition. Since these factors may influence the flutter point in a sensitive manner investigation of uncertainty in these properties is warranted. A modern method to investigate system uncertainty is with the use of robust stability. These modern techniques are used to analyze the uncertainty in structural properties (mass and stiffness properties) of a wing in flight and the effect these uncertainties have on the flutter point. Recent use of these robust stability techniques on the flutter problem have focused on uncertainty in the natural structural modal frequencies. The uncertainties in the modal frequencies are also typically assumed independent. Uncertainties in the natural structural mode shapes have not been explored in complete detail. By including uncertainty in the structural mode shapes the robust flutter margins will be much less conservative. A complete structural uncertainty model for robust flutter prediction is constructed. Robust flutter margins are found for a fictitious wing with uncertainties in wing mass and stiffness properties, using the structured singular value. Since the robust flutter margins include uncertainty in the structural mode shapes, as well as the structural mode frequencies, they are least conservative estimates. The uncertainties in many structural properties on the wing are investigated and the effect that they have on the flutter point is determined. The formulation presented herein can be applied to a wide array of problems concerning the sensitivity of the flutter solution.

Nomenclature AIC = aerodynamic influence coefficient matrix AIC* = aerodynamic force matrix portion which is not a function of mode shapes ANOM = nominal aeroelastic dynamic state matrix Aη = coefficient of η in governing aeroelastic equation of motion

Aη = coefficient of η in governing aeroelastic equation of motion

Aη = coefficient of η in governing aeroelastic equation of motion

xA = coefficient of x in governing aeroelastic equation of motion a = aerodynamic portion of governing aeroelastic equation of motion cP = pressure coefficient vector * MS Graduate Student, Department of Aerospace Engineering, 2271 Howe Hall, Student member of AIAA. † Professor, Department of Aerospace Engineering, 2271 Howe Hall, Professional member of AIAA ‡ Engineer, NASA DFRC PO Box 273 MS 4840D, Professional member of AIAA.

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1853

Copyright © 2004 by Brian Danowsky. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2 American Institute of Aeronautics and Astronautics

c = mean chord length ∆ = plant uncertainty operator δH = uncertainty in mode shape matrix δJ = uncertainty in J δΩ = uncertainty in Ω δV = allowable variation to nominal velocity E = plate modulus of elasticity E* = equivalent modulus of elasticity for plate F = external aerodynamic forces Γ* = equivalent modulus of density for plate H = matrix of retained structural mode shapes H0 = nominal mode shape matrix H∞ = transfer function operator space with finite infinity norm, analytic in right half of the complex plane h = displacement vector consisting of z translations, x rotations, and y rotations for each node point J = structural parameter value (i.e. equivalent modulus of elasticity or density) J0 = nominal value of J j = square root of -1 K = modal stiffness matrix KPlate = individual plate stiffness matrix K*

Plate = stiffness matrix portion which is only a function of geometry KWing = stiffness matrix for entire fixed cantilevered wing k = non-dimensional reduced frequency L = left lag state coefficient of AIC* rational function approximation λi = ith lag state value M = nominal plant aeroelastic state matrix with uncertainty defined M = modal mass matrix MPlate = individual plate mass matrix M*

Plate = mass matrix portion which is only a function of geometry MWing = mass matrix for entire fixed cantilevered wing µ = structured singular value η = modal coordinate representation Ω = mode frequency matrix Ω0 = nominal value of Ω ω = frequency of oscillation P = nominal plant operator pi = ith order coefficient of velocity for polynomial fit to atmospheric density Q = aerodynamic force matrix q∞ = dynamic pressure R = right lag state coefficient of AIC* rational function approximation Ri = ith order coefficient of AIC* rational function approximation ρ = plate density ρ(U∞) = atmospheric density as a function of freestream velocity SDiag = diagonal matrix of aerodynamic panel areas Tas = transformation matrix from structural coordinates to aerodynamic coordinates T(jk) = matrix of lag states for AIC* rational function approximation TXas = transformation matrix from structural coordinates to the derivative of aerodynamic coordinates t = plate thickness U∞ = freestream velocity V0 = nominal velocity ν = Poisson’s ratio for plate material WJ = parameter uncertainty weighting parameter WV = velocity external weighting parameter w = downwash vector

3 American Institute of Aeronautics and Astronautics

w = vector of uncertainty feedback signals linearly related to y by ∆ x = unsteady aerodynamic lag state vector y = vector of uncertainty feedback signals

I. Introduction Flutter is a phenomenon that occurs in aerospace vehicles when aerodynamic, structural elastic and inertial

forces interact in such a way that oscillations of the vehicle may become divergent. This divergent oscillation can cause unstable controllability characteristics, quicker fatigue time and possible structural failure. The flutter problem is becoming more prevalent as we continue to produce faster, more high performance aircraft.

This problem has been apparent since the early days of World War II when planes became faster, and the monoplane began to replace the biplane design. Aircraft became lighter, less rigid and were capable of higher performance missions. We are still following this trend today as we push the envelope for lighter, faster, more multipurpose high-performance aircraft. Relevant thin and flexible wing aircraft susceptible to flutter include the Active Aeroelastic Wing aircraft and the Helios at NASA Dryden Flight Research Center. Morphing wing technology is another area in which further research on flutter uncertainty would provide a great benefit.

Although flutter has been addressed in previous research,9-11 there is a need for further development concerning this problem. Accuracy in predicting the onset of flutter is dependant on assumptions made and methods used. Sources of uncertainty in the predictions need more thorough investigation. Perturbations in structural properties, such as mass and stiffness properties can significantly affect the conditions under which flutter is predicted. A simple, complete, and accurate model for robust flutter prediction is desired for the higher performance, multi-mission aircraft of today and tomorrow.

Recently, robust flutter prediction methods have focused on the uncertainty in predictions resulting from uncertainty in the structural mode frequencies only. The uncertainties in the modal frequencies are also typically assumed independent. Uncertainty in structural mode shapes has not been addressed in sufficient detail. By including uncertainty in the structural mode shapes the flutter predictions become much less conservative.

As demonstrated in this work, structural mode shape variation does in fact affect solution uncertainty to a degree that warrants inclusion into the formulation. The plot displayed in Fig. 1 demonstrates this importance. It is observed that when uncertainty in the structural mode shape is not accounted for the predicted flutter boundary is significantly different than when the uncertainty in mode shape is included. Development of a complete structural uncertainty model, including structural mode shape uncertainty as well as structural mode frequency uncertainty, to predict robust flutter boundaries, is the main goal of this research. Applicability of this method is demonstrated with the use of a fictitious fixed cantilever wing.

4 American Institute of Aeronautics and Astronautics

II. Robust Stability Robust stability deals with the stability of the interconnections of stable operators.16-18 The Small Gain Theorem

serves as a basis for the determination of stability of the interconnections of stable operators. Robust stability is analyzed with respect to a set of perturbations by direct use of the small gain theorem.

Uncertainty in an analytical plant model is described by these perturbations. The small gain theorem states that a closed-loop feedback system of stable operators is internally stable if the loop gain of those operators is stable and bounded by unity (Fig. 2).10 In Fig. 2 the operators P and ∆ are stable transfer function operators. The closed loop system is well-posed and internally stable if 1P ∞∆ < . This condition also ensures that a unique output, y, will exist for any input, u.

The operator ∆ represents perturbations to the nominal plant P. The true plant is assumed to be a known nominal plant value with perturbations that represent the plant’s uncertainty.

The space H∞, which the operators belong to, is defined by Eq. (1) below. This space defines all stable transfer function matrices that are analytic in the right half of the complex plane.

: Re( ) 0P P is analytic in s and P∞ ∞= > < ∞H (1)

The infinity norm is defined by Eq. (2) below.

Flutter Points: Mach Number vs. Altitude

18000

19000

20000

21000

22000

23000

24000

25000

26000

27000

28000

29000

30000

31000

32000

33000

34000

35000

36000

37000

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mach Number

Alti

tude

, ft.

Figure 1: Flutter boundary comparison of nominal wing properties, perturbed wing densities accounting for structural mode shape variation, and perturbed wing densities not accounting for structural mode shape variation

Figure 2: Block diagram of the small gain theorem

∈∆,P H∞

P

u y

∆

5 American Institute of Aeronautics and Astronautics

( )sup ( )P P jω

σ ω∞ = (2)

In the above equation σ represents the maximum singular value. By direct use of the small gain theorem and properties of the norm it can be shown that the interconnection is

robustly stable if Eq. (3) is satisfied.12,16-18

1

P ∞

∞

<∆

(3)

Although the small gain theorem guarantees stability of the system it is overly restrictive since the structure of the uncertainty is not defined. If the uncertainty structure is known and defined a less conservative measure, which is based on the small gain theorem, can be utilized. That method is µ, the structured singular value (Eq. 4).16-18

1( )

min ( ) : det( ) 0P

I Pµ

σ∆∈

=∆ − ∆ =

∆

(4)

The operator set ∆ has a certain structure associated with it. By applying the structured singular value, only sets with that certain defined structure (∆) will be considered, rendering a less conservative measure of robust stability. Given the system in Fig. 2, the plant P is robustly stable with respect to the set ∆ if and only if Eq. (5) is satisfied.16-

18

1

( )Pµ∞

<∆

(5)

This equation is equivalent to the small gain theorem (Eq. 3) if the operator set ∆ has no defined structure.

III. Wing Structural Model The wing used in analysis comprised of a constant thickness fixed cantilevered structure. The wing structure,

completely contained on a constant plane is displayed in Fig. 3. The structural model is entirely

composed of quadrilateral plate elements. Fig. 4 depicts a two-dimensional quad-rilateral plate with constant thickness and four nodes. The degrees of freedom consist of three translations (Tx,Ty,Tz) and three rotations (Rx,Ry,Rz) at each node for a total of twenty-four degrees of freedom per element.14

For the purposes presented here it was conclusive that the translations in the x and y directions (i.e. in the plane of the wing structure) would be small in comparison to translations in the z direction (i.e. out of plane of the wing structure). This also held true for rotations in the z direction being much smaller than those in the x and y directions. These small degrees of freedom could be neglected. This is a common simplification allowing only out of plane

Figure 3: Fixed cantilevered wing structural element coordinates

Freestream Flow Direction

6 American Institute of Aeronautics and Astronautics

wing twisting and bending. By fixing these degrees of freedom at each node a reduced model, with only twelve degrees of freedom per element is produced.

The individual stiffness and mass matrices for each plate are dependant on both material properties and geometry. The material properties, being constant, can be factored out of the elemental matrices. Eqs. (6) and (7) display the mass and stiffness matrices respectively with the material properties factored out.

K*Plate and M*

Plate are matrices that are simply functions of geometry. ρ is the plate density, t is the plate thickness; E is the modulus of elasticity for the plate and ν is Poisson’s ratio.

Plate PlateM tρ= *M (6)

2

3

12(1 )Plate Plate

EtK

ν=

−

*K (7)

The material properties together form two constants, one for the stiffness matrix and one for the mass matrix. These can be referred to the as the equivalent modulus of elasticity and the equivalent modulus of density respectively (Eqs. 8 and 9).

*

2

3

12(1 )

EtE

ν=

− (8)

* tρΓ = (9)

This model for a quadrilateral plate with constant thickness was taken directly from Przemieniecki.14

From finite element theory we have the governing equation of unforced motion for the wing with given mass and stiffness matrices (Eq. 10).

0Wing Wing+ =M h K h (10)

Note that structural damping is not accounted for. The displacement vector, h, is organized as follows.

TTT T

z x yT R R= ⎡ ⎤⎣ ⎦h (11)

The quantities Tz, Rx and Ry represent vectors of each nodal z translation, nodal x rotation and nodal y rotation respectively. .Modal analysis was performed on this finite element model to produce nominal structural mode frequencies and nominal structural mode shapes.

IV. Unsteady Aerodynamic Model The equation of motion representing the unforced structural dynamics represented in Eq. (10) must be expanded

to account for forces resulting from aerodynamics while the wing is in a freestream flow (Eq. 12).

Figure 4: Quadrilateral plate element with constant thickness

Rz

Rx

Ry

Tz

Ty

Tx

nodes

7 American Institute of Aeronautics and Astronautics

, ( , , , ...)Wing Wing Aero SF+ =M h K h h h h (12)

The aerodynamic forces represented by FAero,S can be represented by,

[ ], Diag PAero S saq S∞=F T c . (13)

The diagonal matrix, SDiag, is a matrix of each aerodynamic panel characteristic area. The other quantities in this equation are the dynamic pressure, q∞, and the panel pressure coefficient vector cP.

The calculations done to determine the wing aerodynamic forces are done in a different coordinate system than the structural modal calculations. The matrix, Tsa is a coordinate transformation matrix that converts the forces in the aerodynamic domain the structural domain. This and other resulting transformation matrices are calculated using equivalent energy methods.14

This pressure coefficient vector is a function of the local downwash and the aerodynamic influence matrix.

[ ]( , )p AIC k Mach=c w (14)

The complex valued aerodynamic influence matrix (AIC), which is a function of Mach number and reduced frequency (Eq. 15), is calculated independently. The unsteady doublet lattice method, based on oscillatory airflow, was used to generate this varying matrix.5

c

kU

ω

∞

= (15)

The local downwash vector is defined in Eq. (16). The matrix TXas is a transformation matrix that transforms from structural coordinates to the derivative of the aerodynamic coordinates. Tas is a transformation matrix from structural coordinates to aerodynamic coordinates. The quantity Hη represents a modal approximation to h in Eq. (12). The matrix H is a matrix of the retained natural structural mode shapes where η is the modal coordinate representation. The mean chord length is represented by c .

1

Xas asjkc

= +w T Hη T Hη (16)

With all these quantities in place the governing equation of motion (Eq. 12) takes on the following form in Eq. (17).

Mη + Kη = Qη (17)

The matrices M and K are modal mass and stiffness matrices (Eqs. 18 and 19).

TWingM = H M H (18)

TWingK = H K H (19)

The aerodynamic matrix, Q, is defined in Eq. (20).

[ ][ ]( , )Tsa Diag Xas as

jkq S AIC k Mach

c∞= +⎡ ⎤

⎢ ⎥⎣ ⎦Q H T T T H (20)

8 American Institute of Aeronautics and Astronautics

Flutter points can be calculated from Eq. (17) using the V-g method for flutter prediction.4

V. Linear State Space Model In order for the uncertainty model (Fig. 2) to be constructed Eq. (17) must be represented as a linear state space

system. Since the aerodynamic influence matrix (AIC) is not real-rational an approximation as such must be formulated.

A. Unsteady Aerodynamic Rational Function Approximation A state-space approximation for the aerodynamic influence matrix must be obtained in order to formulate a

complete state-space model. A complete state space model is required for subsequent robust stability analysis. Since mode shape variation is being considered, it is desired to remove the mode shape dependence from the aerodynamic influence matrix Q. A new quantity, AIC* is introduced which is not a function of the mode shapes.

[ ] [ ][ ]*( , )sa Diag Xas as

jkAIC S AIC k Mach

c= +⎡ ⎤

⎢ ⎥⎣ ⎦T T T (21)

Note that the matrix AIC* contains all the dependence on the characteristic frequency k and the Mach number and is not real-rational in k. A state space realization cannot be formulated if this is not rational so an approximation must be implemented. A rational function approximation (RFA), originally developed by Roger15 is used to approximate this matrix as a function of k so that a state space realization can be formulated. Other recent modifications have been made to this method for rational function approximation.8 The RFA is displayed in Eq. (22).

[ ]* 20 1 2( ) ( )AIC jk jk k jk≅ + − +R R R LT R (22)

The quantity T(jk) is a matrix of specified lag states (Eq. 23). The lag states are represented by λ1,…, λr. The real valued coefficient matrices R0, R1, R2, L and R are found using a least squares algorithm. The lag states, λi, are quantities supplied prior to this least squares fitting. They are chosen so that when the V-g calculated flutter point is determined, utilizing the approximated aerodynamics defined by Eq. 22, it will match the flutter point determined using the full, non-approximated aerodynamics in Eq. 21. Constrained optimization techniques were used to achieve this.13

( ) 11( ) ( , , )rjk jk jkI diag λ λ

−= +T … (23)

The R0, R1, R2, L and R coefficients that comprise the RFA are dependant on Mach number; therefore different coefficients will result from different Mach numbers.

B. Approximation of the Atmospheric Density as a Function of Freestream Velocity The atmospheric density is dependant on the Mach number and the freestream velocity by the standard

atmosphere§. In order to guarantee that this density is correct when the flutter analysis is performed the standard atmosphere must be utilized since the Mach number is fixed. The density is directly related to the velocity through the standard atmosphere when the Mach number is fixed. It is desired to determine a polynomial approximation of this density as a function of freestream velocity.

A function for ρ is introduced which is a function of freestream velocity (Eq. 24).11 This function, which approximates the air density, is dependant on the Mach number. It has been shown that a third order polynomial fit works very well for density values below the stratosphere.

2 30 1 2 3( )U p p U p U p Uρ ∞ ∞ ∞ ∞≅ + + + (24)

§ Since M = V/a and a = (γRT)1/2. T is dependant on altitude, which in turn defines the value for ρ.

9 American Institute of Aeronautics and Astronautics

The coefficients p0, p1, p2, and p3 result from the polynomial fit. These values are dependant on Mach number so different coefficients will result from different Mach numbers.

C. State Space Representation A linear state space model of the entire aeroelastic equation of motion can be constructed which is dependant

solely on the freestream velocity. The coefficients resulting from the unsteady aerodynamic RFA and the atmospheric density approximation will make this representation also dependant on a fixed Mach number.

The state vector is represented by Eq. (25).

[ ]TT T T=z η η x (25)

The quantity x in this state vector represents the unsteady aerodynamic lag states resulting from the state space representation of Eq. (22). With the above state vector defined, the state space system is represented by,

NOM=z A z . (26)

For a fixed Mach number value the matrix ANOM will be dependant solely on freestream velocity. Observation of the system poles as a function of the freestream velocity will reveal the nominal flutter point, identical to that obtained using the V-g method.

VI. Application of Uncertainty In order to determine the robust stability of the entire system with parametric uncertainty applied the uncertainty

in the parameters must be added to the model. An allowable variation in the freestream velocity must also be added to the model so that robust flutter margins, defined by the flutter velocity, can be found.

A. Allowable Variation in the Freestream Velocity The freestream velocity is represented by a nominal velocity, V0, and a small allowable variation to that velocity,

δV (Eq. 27).

0U V Vδ∞ = + (27)

Replacing every occurrence of U∞ in the model will result in a non-linear dependence on the parameter δV (6th order polynomial dependence). References 6 and 11 describe a method to represent polynomial dependant uncertainty in a form that is suitable for robust stability analysis. This method involves the introduction of a series of coupled feedback signals. Defining matrices as a product of two matrices, using a singular value approach, is also required to apply this method.

Recall the equation of motion for the wing aeroelastic dynamics (Eq. 17). If this equation is pre-multiplied by the modal mass matrix (Eq. 18) it can be defined by Eq. (28).

2 a+ Ω =η η (28)

The right hand side of this equation, a, is a function of the states (Eq. 25) and their derivatives. Coefficients of like states must be separated to yield Eq. (29). This is simply a rearrangement of Eq. (28).

xA A A Aη η η= + +η η η x (29)

These coefficients are functions of the velocity variation parameter δV. Taking the Aη coefficient on the left hand side and expressing it in terms of δV yields Eq. (30).

10 American Institute of Aeronautics and Astronautics

( )( )

2 3

0 1 2 3

0 1 2 3

A A VA V A V A

A V A V A VA

η η η η η

η η η η

δ δ δ

δ δ δ

= + + +

= + + +

η η η η η

η η η η (30)

Defining the matrix 3Aη as a product of two matrices ( 3 3 3L RA A Aη η η= ), introduce new feedback signals 3 3L Ry Aη= η and 3 3L Lw V yδ= I . Defining the matrix 2 3[ ]LA Aη η as a product of two matrices

( 2 3 2 2[ ]L L RA A A Aη η η η= ), another set of signals can now be introduced as 2 2 2 3L Ra Rb Ly A A wη η= +η and 2 2L Lw V yδ= I . This process is repeated until Eq. (30) is in a linear form. If all coefficients in Eq. (29) are represented

in the same manner the equation of motion will be in a suitable form to apply robust stability analysis with an allowable variation in the freestream velocity.

B. Uncertainty in Structural Parameters The wing is composed of several quadrilateral plate elements. The structural properties of each plate element

form two quantities: the equivalent modulus of elasticity (Eq. 8) and the equivalent modulus of density (Eq. 9). These quantities have a certain measure of uncertainty. Define any plate modulus (either elasticity or density) as a nominal value plus some unknown uncertainty (Eq. 31).

0J J Jδ= + (31)

J in this equation represents any modulus for any plate. The wing structural mode shapes and structural mode frequencies are dependant on this J parameter (Eqs. 32 and 33).

0 0 JJ

δ δ∂

= + = +∂

HH H H H (32)

0 0 JJ

δ δ∂Ω

Ω = Ω + Ω = Ω +∂

(33)

Note that only the first order dependence is considered since the δJ parameter is assumed to be relatively small. The first order eigenvector and eigenvalue derivatives ( J∂ ∂H and J∂Ω ∂ ) were calculated analytically using a method derived by Friswell.7

Replacing H and Ω from Eqs. (32) and (33) in the equation of motion will result in a model with 2nd order uncertainty in δJ, 6th order uncertainty in δV, and all the resulting mixed uncertainty parameters (δVδJ, δV2δJ, etc.). This results in a model with 21 different powers of pure and mixed uncertainties. Since δJ is considered small the 2nd order terms involving this parameter can be neglected. Applying this simplification will yield a model with only 14 different pure and mixed uncertainties.

Since the uncertainty is still polynomial dependant with the two parameters involved (δV and δJ), the method described above can be used to decompose this system into a form suitable fro robust stability analysis.6,11 This will result in a model suitable for Linear Fractional Transformation (LFT) formulation.

A model was constructed representative of Fig. 2 with the nominal dynamics in the operator P and all the uncertainty (both δV and δJ) in the operator ∆.

VII. M-∆ Model Formulation With the equation of motion in a suitable form, the M-∆ framework can be constructed with both parametric

uncertainty in the wing structural properties and the allowable velocity variation. Define a signal vectors y and w, which are vectors of all the coupled feedback signals formed upon separation of

the uncertainty parameters. These vectors are organized so that velocity uncertainty and structural parameter uncertainty is separated. The following equations are formed (Eqs 34 and 35)

11 American Institute of Aeronautics and Astronautics

11 12

21 22

=⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

M My w

M Mz z (34)

0

0

V V

J J

V

J

δ δ

δ δ

δ

δ=

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

w = ∆y

w yI

w yI

(35)

Representation of the model in the form of Fig. 2 is displayed in Fig. 5.

Figure 5: M-∆ block diagram with uncertainty in the velocity and structural parameters The matrix ZZM will be identical to ANOM in Eq. (26) with the freestream velocity value set at V0. This matrix

represents the nominal dynamics with no uncertainty present. In order for robust stability analysis to be applied this matrix must be stable (i.e. the nominal velocity V0 must be a velocity at which the nominal wing is stable and not in flutter).

The model in Fig. 5 can be transformed to a form represented by Fig. 2 by incorporating a feedback loop between z and z with an integral operator. Addition of this integral operator will make the resulting lower operator (P in Fig. 2) frequency dependant.

The nominal flutter margin can be calculated using the M-∆ model as well. If no uncertainty is applied in the structural parameters the resulting model will be similar with the ∆ uncertainty matrix containing only the allowable variation in the freestream velocity. The M matrix will differ subject to this as well. Methods to calculate the nominal and robust flutter margins will differ since uncertainty is in multiple versus single parameters.9

VIII. Calculation of Robust Flutter Margins Using the M-∆ Uncertainty Model

A. Nominal Flutter Margin Initially, the nominal flutter margin can be calculated using the M-∆ uncertainty model. The results of this

calculation are compared against the results using the classical V-g method for flutter prediction to verify the validity of the M-∆ model.

By accounting for velocity variation only, the ∆ matrix will be a diagonal matrix with the same δV value as each diagonal element. Since uncertainty is only in a single parameter the method to determine the nominal flutter margin will not involve the calculation of the structured singular value. A simple algorithm, developed by Lind and Brenner,9 to determine the nominal flutter margin with this model in place is described below.

Given the plant M at a stable nominal velocity V0 and the operator ∆ as in Fig. 5, the nominal flutter margin is determined by iteratively changing initial bounds on the ∆ matrix until the difference between them is within some ε > 0. The ∆ matrix is only dependant on one parameter, so define initial bounds for this parameter: δVupper and δVlower. These values are iteratively varied so that the smallest δV causing instability in the M-∆ model is determined. This

00

V

J

δ

δ

∆∆

V V V J VZ

J V J J JZ

Z V Z J ZZ

δ δ δ δ δ

δ δ δ δ δ

δ δ

M M MM M MM M M

Jδy

VδyVδw

Jδw

z z

12 American Institute of Aeronautics and Astronautics

iterative method was directly extracted from Ref. 9. Results of this and the comparison to the classical V-g method for flutter prediction are displayed in Table 1.

By observing the poles of the ANOM matrix (Eq. 26) it was noted that the flutter point occurs with a decrease in velocity. This abnormal stability behavior is opposite of what is expected and is dependant on this case. At low velocities, close to zero, the wing will be in an unstable region. This is a consequence of the constant Mach number in the aerodynamics. By fixing this Mach number when applying the V-g method the temperature and subsequently the altitude and air density are allowed to vary in order to find a solution matching the dynamic pressure. At low velocities the temperature will be relatively low to compensate for the fixed Mach number. This will drive the altitude very high and the atmospheric density very low into regions that are unsuitable and not physically valid. This fact limits the velocity range that the complete state space system is valid. The system is only physically valid for a limited range of freestream velocities.

At velocities above the calculated flutter instability the wing is stable. This is due to the fact that this matrix is Mach number dependant. This being the case, the algorithm supplied by Lind was modified slightly. The nominal velocity, V0, will be at a stable region above the flutter instability.

It is evident that the nominal flutter margin calculated using the M-∆ model is very close in comparison to the flutter point calculated using the classical V-g method. This verifies that the model was constructed accurately. The approximations to the air density and aerodynamic influence matrix RFA are verified. The minimal error between the two values in Table 1 can be attributed to the accuracy of the aerodynamic rational function approximation.

B. Robust Flutter Margins Accounting for Structural Parameter Uncertainty The robust flutter margins, resulting from uncertainty in the wing structural properties requires a more

complicated approach since there is uncertainty in more than one parameter. By use of the structured singular value (µ) and knowledge of the uncertainty structure the robust flutter margins are calculated in a least conservative manner.

An algorithm, again developed by Lind,9 was used to calculate the robust flutter margins. The structured singular value is given a unity norm condition for stability (Eq. 36). This results from a unity norm bound condition on the uncertainty operator ∆ ( 1∞∆ < ). The system in Fig. 2 is stable if and only the structured singular value is less than unity.

( ) 1Pµ < (36)

Recall the ∆ matrix in Eq. (35). If the above condition is implied this matrix must have a unity norm to guarantee stability. If this is true then the variation in the freestream velocity will be limited to be less than one. In order to allow velocity variation values of higher magnitude a weighting must be applied to the model to account for this. It can be shown that applying an external weighting WV, to the nominal plant P, as in Eq. (37), will allow the uncertainty to have a magnitude greater than one and the magnitude of this weighting will in fact be the robust margin if µ( P ) = 1.9

0

0

VWP P

I=

⎡ ⎤⎢ ⎥⎣ ⎦

(37)

Weighting on the δJ uncertainty operator cannot be applied externally. If it is desired to allow uncertainty in this operator to exceed a magnitude of one the model must be updated internally by applying Eq. (38).

Table 1: Nominal flutter margin comparison M-∆ Flutter Margin V-g Flutter PointVelocity (fps) 331.02 330.87 Dynamic pressure (lbs/ft3) 117.96 117.39 Temperature (R) 506.79 506.33 Altitude (ft.) 3337.5 3466.9 Air density (slugs/ft3) 2.15E-03 2.15E-03 Characteristic frequency 0.1617 0.1605 Frequency of oscillation (rad/s) 1.2295 1.2204

13 American Institute of Aeronautics and Astronautics

0 JJ J W Jδ= + (38)

The weighting parameter WJ, which is the maximum allowable magnitude of uncertainty in the structural parameter J, will influence the M matrix internally and cannot be applied in a manner similar to Eq. (37).

With the above in place an algorithm that varies the weighting parameter, WV, was utilized. This weighting factor is iteratively varied until µ(P) is within a prescribed margin of error of one. When µ( P ) = 1 the robust flutter velocity is VRob = V0 + WV.9 This algorithm was also directly extracted from Ref. 9.

The wing is composed of 36 quadrilateral plates with two specific structural properties per plate; the equivalent modulus of elasticity (Eq. 8) and the equivalent modulus of density (Eq. 9). This structural model therefore has 72 possible parameters for which uncertainty can be applied.

The algorithm was applied with the M-∆ model for each possible structural parameter with uncertainty bounded at 0, 10%, 20% and 30% of the nominal value. This was done computationally.2 Robust flutter margins were found for each parameter. Two select parameters and the resulting robust flutter margins for varying uncertainty bounds are displayed in Tables 2 and 3. Refer to Fig. 3 for the plate locations.

It was noted that the uncertainty in the two parameters on the two plates chosen had the most profound effect on the flutter margin.

A more thorough investigation on the Γ* parameter revealed that uncertainty in this parameter has more effect on plates further away from the wing root. It was also noted that generally the uncertainty in this parameter also had more of an effect toward the trailing edge in the flow direction.

Figure 6 graphically displays the Γ* parameter dependence on plate location. The data displayed is in series corresponding to chord wise plate location. Each data set displayed is representative of the plates along the local chord line starting at the leading edge and following to the trailing edge. Subsequent series represent the chord wise plate locations starting from the wing root and following to the wing tip.

A similar investigation was done with the E* parameter. It was evident that the effect of this parameter was not as significant in relation to span-wise location. There seemed to be an increase in effect as the chord-wise location was increased downstream but this was still marginal.

The most profound effect of this parameter was at plate number 4. This plate is on the trailing edge of the fixed wing root. An uncertainty in the stiffness in this location produces a robust flutter margin of the largest magnitude; therefore the flutter instability is the most sensitive to this parameter. Figure 7 graphically displays the E* parameter dependence on plate location.

Although the most sensitive parameter was the equivalent modulus of elasticity on plate number 4 it is conclusive from the Figs. 6 and 7 that the equivalent modulus of density has more of a profound effect on the wing as a whole. The flutter point is more sensitive to this parameter on most of the plates.

It is possible to investigate a number of different parameter uncertainties simultaneously. The M-∆ model must be formulated internally to account for this. If this is done there will be additional independent δJ uncertainty blocks

Table 2: Robust flutter margins with uncertainty in E* in plate number 4 % Uncertainty ~0 10 20 30

Vrob, fps 331.11 332.58 333.84 335.00 ωrob, rad/s 1.2299 1.239 1.2476 1.2555

ρair,rob, slugs/ft3 2.16E-03 2.24E-03 2.31E-03 2.38E-03 altitude, ft 3260.7 2000.8 918.11 -87.184

Trob, R 507.06 511.55 515.42 519 krob 0.1617 0.1622 0.1627 0.1631

qrob, lbs/ft2 118.29 123.92 128.92 133.72

Table 3: Robust flutter margins with uncertainty in Γ* in plate number 36 % Uncertainty ~0 10 20 30

Vrob, fps 331.70 332.39 333.35 334.28 ωrob, rad/s 1.2337 1.2381 1.2437 1.2506

ρair,rob, slugs/ft3 2.19E-03 2.23E-03 2.28E-03 2.34E-03 altitude, ft 2754.3 2161.8 1340.7 539.73

Trob, R 508.87 510.98 513.91 516.77 krob 0.1619 0.1621 0.1624 0.1628

qrob, lbs/ft2 120.53 123.19 126.95 130.71

14 American Institute of Aeronautics and Astronautics

present in the model (Fig. 5). The ∆ block will be composed of the variation block for the freestream velocity and a number of uncertainty blocks representing the uncertainty in the multiple parameters chosen.

A similar manner to investigate uncertainty may involve simultaneous uncertainty in the same parameter in many plates. If this is done the analytical eigenvector and eigenvalue derivatives7 will be derived based on this.

IX. Concluding Remarks A complete structural uncertainty

model has been constructed for a fictitious fixed cantilever wing to predict robust flutter boundaries. The model includes uncertainty in structural mode shape, as well as structural mode frequency. Eigenvector and eigenvalue derivatives have been calculated analytically so that errors due to computational inaccuracy with finite differencing techniques are avoided.

The complete structural uncertainty model, constructed herein, will analytically produce robust flutter margins using µ analysis in a least conservative manner. This is due to the fact that uncertainties in the structural mode shapes, as well as structural mode frequencies, are considered.

This model can be applied to a wide array of problems involving parametric uncertainty and the effect it has on the flutter margin. A much more complex structural model (i.e. an entire aircraft) can be studied with uncertainty in many parameters using the formulation described herein.

Acknowledgment Thanks go to the Iowa Space Grant

Consortium and NASA Dryden Flight Research Center for valuable resources and funding to aid this research project. This project would not be possible without the valuable leadership and expertise of Dr. Frank Chavez at Iowa State University and the generous aid and direction of Marty Brenner at NASA Dryden Flight Research Center.

References 1Balan, N. and Mujumar, P.M., “Minimum-State Approximation: A Pure Lag Approach,” Journal of Aircraft, Vol. 35, No. 1,

1997, pp. 161-163.

Figure 6: Robust flutter margin vs. chord-wise plate location for 30% uncertainty in Γ*

Figure 7: Robust flutter margin vs. chord-wise plate location for 30% uncertainty in Ε*

15 American Institute of Aeronautics and Astronautics

2Balas, G.J., Doyle, J.C., Glover, K., Packard, A. and Smith, R., µ-Analysis and Synthesis Toolbox for use with Matlab. The Mathworks, Inc., MA, MUSYN, Inc., MA, 1998.

3Belcastro, C.M., “Parametric Uncertainty Modeling: An Overview,” AACC, 1998. 4Bisplinghoff, R.L., Ashley, H. and Halfman, R.L., Aeroelasticity, Addison-Wesley Publishing, Cambridge MA, 1955. 5Blair, M.A., A Compilation of the Mathematics Leading to the Doublet Lattice Method, Air force Wright Laboratory,

Dayton, OH, WL-TR-92-3028, March 1992. 6Boukarim, G.E. and Chow, J.H., “Modeling Nonliner System Uncertainties using a Linear Fractional Transformation

Approach,” Proceedings of the American Control Conference, Philadelphia PA, June 1998, pp. 2973-2979 7Friswell, Michael I., “Calculation of Second and Higher Order Eigenvector Derivatives,” Journal of Guidance, Control, and

Dynamics, Vol. 18, No. 4, 1994, pp. 919-921. 8Karpel, M and Strul, E., “Minimum-State Unsteady Aerodynamic Approximations with Flexible Constraints,” Journal or

Aircraft, Vol. 33, No. 6, 1996, pp. 1190-1196. 9Lind, R. and Brenner, M., Robust Aeroservoelastic Stability Analysis, Springer-Verlag, London, April 1999. 10Lind, R. and M. J. Brenner, “Robust Flutter Margin Analysis That Incorporates Flight Data,” NASA/TP-1998-206543,

March, 1998, pp. 12-13, 22. 11Lind, R., “Match-Point Solutions for Robust Flutter Analysis,” 43rd AIAA Structural Dynamics and Materials Conference,

Denver, CO, 2002. 12Maciejowski, J.M., Multivariable Feedback Design, Addison-Wesley Publishing, Cambridge MA, 1989. 13Matlab Optimization Toolbox, Ver. 2.2, The Mathworks, Natick, MA, 2002. 14Przemieniecki, J. S., Theory of Matrix Structural Analysis, Dover Publications, 1985. 15Roger, K. L., “Airplane Math Modeling Methods for Active Control Design,” Structural Aspects of Active Controls,

AGARD-CP-228, August, 1977, pp.4-1-4-11. 16Skogestad, S. and Postlethwaite, I., Multivariable Feedback Control Analysis and Design, John Wiley & Sons, England,

1996. 17Zhou, K. and Doyle, J.C., Essentials of Robust Control, Pretence Hall, NJ, 1998. 18Zhou, K., Doyle, J.C., and Glover, K., Robust and Optimal Control, Pretence Hall, NJ, 1996.