A z Transform Discrete-Time State Space Formulation

for Aeroelastic Stability Analysis

Alexandre Noll Marques∗∗

Instituto Tecnologico de Aeronautica, CTA/ITA, Sao Jose dos Campos, SP, Brazil

Joao Luiz F. Azevedo†

Instituto de Aeronautica e Espaco, CTA/IAE/ALA, Sao Jose dos Campos, SP, Brazil

In many complex aerospace applications, it is common to use numerical tools in orderto evaluate the unsteady aerodynamic behavior. The present work presents an alternateformulation for the state space representation of aeroelastic systems based on digital con-trol theory that is shown to be more effective and accurate for the coupling of numericalsolutions with such systems. The application of the z transform allows for direct frequencydomain representations of the aerodynamic solutions without the need for approximatingmodels, as generally occurs in other state space formulations. This fact makes this newmethodology also a more straightforward procedure for aeroelastic analyses. A typicalsection model of a NACA 0012 airfoil at transonic speeds is used as test case in order toassess the correctness and accuracy of the proposed formulation. The present results arecompared with data obtained from continuous-time state space formulations and throughthe direct integration of the aerodynamic and structural dynamic equations.

I. Introduction

The lack of accurate aerodynamic analytical models for complex flow situations makes the solution ofrealistic aeroelastic problems a very difficult task. Over the years, approximate solutions of the aerody-

namic potential theory,1–3 and, more recently, numerical solutions obtained with CFD solvers4–11 have beenused in order to represent the aerodynamic response to generic structural behavior in order to decouple theaerodynamic and structural effects. Such aerodynamic data are generally modeled with rational functions12

or interpolating polynomials13–15 in a convenient fashion so as to represent the aeroelastic system through astate space formulation. Some of the most common state space formulations used nowadays for aeroelasticstability analyses are described in detail in the following sections.

The present work presents an alternate approach for the state space representation of aeroelastic systemsbased on discrete-time control techniques. The main idea behind this approach is to avoid the need for anapproximate model of the aerodynamic responses. This goal is achieved by calculating the aerodynamiccharacteristics in the frequency domain with the direct use of the z transform. The complete formulation ofthe new state space representation is also presented. Finally, a typical section model of a NACA 0012 airfoilat transonic flow conditions is used as test case for a detailed comparison among the different state spaceformulations and direct integration results.

The CFD solver employed for evaluating the aerodynamic responses contained in the present work isthe same presented in Refs. 11, 16, and 17. It is based on the 2-D Euler equations, which represent two-dimensional, compressible, rotational, inviscid and nonlinear flows. Therefore, it is completely capable ofcapturing the shock waves present in transonic flows. These equations are discretized in space with a cellcentered, finite volume scheme, and they are advanced in time using a second-order accurate, 5-stage, explicit,hybrid Runge-Kutta scheme.

∗M.S. Student, Division of Aeronautical Engineering. Student Member AIAA.†Currently, Director for Space Transportation and Licensing, Brazilian Space Agency. Associate Fellow AIAA.

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25th AIAA Applied Aerodynamics Conference25 - 28 June 2007, Miami, FL

AIAA 2007-3800

Copyright © 2007 by A.N. Marques and J.L.F. Azevedo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

II. Theoretical Formulation

A general aeroelastic system is characterized by aerodynamic, elastic and inertial forces dynamicallyinteracting with structural deformations. It is very common, therefore, to represent the aerodynamic effectsexclusively through the resulting forces and moments acting on the structure as a forcing term. The metho-dology proposed in the present work is focused on the adequate representation of the aerodynamic operatorfor complex flow situations. Hence, it is instructive to apply such methodology over simple structural mo-dels in order to avoid further complications that might hide the behavior of the aerodynamic model. Thestructural model considered in the present work is the typical section, which is widely known and reportedin the literature.2,9 The dynamic system represented in the typical section is a rigid airfoil with two degreesof freedom, plunge and pitch, as shown in Fig. 1. As presented in Fig. 1, α is the pitch mode coordinate,positive in the nose-up direction, and h is the vertical translation, positive downwards. Moreover, c is theairfoil chord length, while b is its semi-chord length. Finally, xα is the distance from the elastic axis to thecenter of mass and ah is the distance from midchord to the elastic axis, both nondimensionalized using thesemi-chord length (b).

c

b ba bh

x bα

h

α

Figure 1. Typical section configuration.

The governing equation for the motion of such dynamic system is given by

[M ] {η(t)}+ [K] {η(t)} = {Qa(t)} , (1)

where the generalized mass and stiffness matrices are, respectively,

[M ] =

[1 xα

xα r2α

]and [K] =

[ω2

h 00 r2

αω2α

], (2)

and the generalized coordinate and force vectors are

{η(t)} =

{ξ(t)α(t)

}and {Qa(t)} =

{Qah(t)

mbQaα(t)

mb2

}. (3)

In Eq. (2), rα denotes the airfoil dimensionless radius of gyration about the elastic axis, m is the airfoil mass,and ωh and ωα are the uncoupled natural circular frequencies of the plunge and pitch modes, respectively.Furthermore, in Eq. (3), ξ is the plunge mode coordinate, where ξ = h/b.

It is convenient to nondimensionalize the aeroelastic equation in order to represent more general situati-ons. This is accomplished with the procedure proposed by Oliveira,9 using a reference circular frequency tonondimensionalize the time variable,

t = tωr. (4)

With the application the chain of rule for the time derivatives,

d(.)dt

=d(.)dt

dt

dt= ωr

d(.)dt

, (5)

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Eq. (1) becomes[M ] {η (t)}+

[K

] {η (t)} ={Qa (t)

}, (6)

where [K

]=

1ω2

r

[K] and{Qa (t)

}=

1ω2

r

{Qa (t)} . (7)

As previously mentioned, the main objective of the present study is to efficiently determine the generalizedaerodynamic force vector,

{Qa (t)

}, for an arbitrary structural behavior. However, due to the nonlinearities

of the aerodynamic equations, it is very difficult to obtain a general expression for the aerodynamic response.However, this problem is simplified by extending linearity concepts present in the formulation of the potentialaerodynamic equations. As presented in Refs. 2 and 12, the linear aerodynamic responses can be individuallydetermined for each mode, and then superposed for more general responses. Based on these ideas, Oliveira9

proposed the assumption of linearity of the aerodynamic response in the transonic regime with regard tothe modal motion. As there are no rigorous linearization procedures involved, there are no guarantees thatsuch assumption holds. But, it is natural to expect this sort of linear hypothesis to be valid at least forsmall amplitudes. Actually, as it is shown in section VI, there is a certain amplitude range in which thishypothesis holds. Furthermore, it is important to emphasize that the aeroelastic phenomena analyzed inthe present work are restricted to small amplitude motions. Therefore, when considering flutter, which ischaracterized by diverging oscillations, it is possible that only the onset of this phenomenon can be identifiedwith such approach. There may occur situations in which the aerodynamic nonlinear behavior suppressesthe diverging oscillations creating limit cycle oscillations, and the present formulation may not correctlycapture such cases. Nevertheless, for safety reasons, most designers are interested in the flutter onset point.

As a consequence of the linearity assumptions, it is possible to determine the aerodynamic response to ageneral structural behavior from the convolution of an impulsive or indicial aerodynamic solution.2,16–18 Theconvolution operation, however, is more easily handled in the frequency domain, in which it is representedby a simple multiplication operation. Hence, the aerodynamic forces linearized with regard to the modaldisplacements can be written, in the frequency domain, as

L(κ) = LIh(κ)h(κ) + LIα(κ)α(κ), (8)

My(κ) = MIh(κ)h(κ) + MIα(κ)α(κ). (9)

In Eqs. (8) and (9), the reduced frequency κ is used to represent the frequency domain. Moreover, Ldesignates the lift force, and My the pitching moment with respect to the elastic axis, positive in the nose-updirection. The h(κ) and α(κ) variables represent the amplitudes of the frequency content of motions inplunge and pitch modes, respectively, while the LIh

(κ), LIα(κ), MIh(κ), and MIα(κ) coefficients indicate

complex numbers which designate the magnitude and phase delay of the generalized forces with respect tounitary-amplitude harmonic motions of the respective structural mode. If one considers the usual definitionsof aerodynamics, it follows that the lift and moment coefficients are given, respectively, by

C`(κ) =L(κ)

12ρ∞U2∞c

=LIh

(κ)h(κ)12ρ∞U2∞c

+LIα(κ)α(κ)

12ρ∞U2∞c

, (10)

Cm(κ) =My(κ)

12ρ∞U2∞c2

=MIh

(κ)h(κ)12ρ∞U2∞c2

+MIα(κ)α(κ)12ρ∞U2∞c2

. (11)

Following the generalized coordinate notation of Eqs. (3) and (7), the generalized forces are finally given by

Qah(κ) = − L(κ)mbω2

r

= −ρ∞U2∞

mω2r

[C`Ih

(κ)ξ(κ)

2+ C`Iα

(κ)α(κ)]

, (12)

Qaα(κ) =My(κ)mb2ω2

r

=ρ∞U2

∞mω2

r

[CmIh(κ)ξ(κ) + 2CmIα

(κ)α(κ)] , (13)

or{Qa(κ)

}=

(U∗)2

πµ[A(κ)] {η(κ)} , (14)

where the mass ratio is defined asµ =

m

πρ∞b2, (15)

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the characteristic speed is

U∗ =U∞bωr

, (16)

and the aerodynamic influence coefficient matrix is given by

[A(κ)] =

[−C`Ih

(κ)/2 −C`Iα(κ)

CmIh(κ) 2CmIα

(κ)

]. (17)

Moreover, ρ∞ and U∞ designate, respectively, the undisturbed flow density and speed.There are different approaches which can be used to determine the desired aerodynamic coefficients. As

there is no closed-form solution for the compressible subsonic and transonic unsteady aerodynamic forces,this work is aimed at numerical methods in which the impulsive solution is known for a determined numberof discrete reduced frequency values. However, the calculation of aerodynamic responses in the frequencydomain is beyond the scope of the present paper, and it is assumed that this information is known a priori.Just to name a few possibilities for the calculation of such responses, for the compressible subsonic regime,there are numerical methods for the solution of integral equations, tabulated results,1,3 and approximateformulae19 for the linearized potential equations.2 More realistic solutions, especially for the transonicregime, can be achieved with modern CFD solvers, either with the solution of harmonic oscillations,5,7 or bydirect and indirect determination of impulsive or indicial responses.4,6, 8, 10,11 All the aerodynamic responsespresented here are obtained by exciting the aerodynamic system on a frequency range of interest throughthe individual application of a smooth pulse motion in both modes.11,16,17

III. Continuous-Time State Space Formulation

A. General Formulation

A state space representation of a system corresponds to the description of the system dynamics in terms offirst-order differential equations, which may be combined into a first-order vector-matrix differential equa-tion.20 In the present case, this formulation can be achieved by defining9

{x1 (t)} = {η (t)} , (18){x2 (t)} = {η (t)} = {x1 (t)} , (19)

{x (t)} =

{{x2 (t)}{x1 (t)}

}, (20)

where {x (t)} is the system state vector. Hence, the governing equation of motion becomes[M

]{x (t)}+

[K

]{x (t)} = {q (t)} , (21)

where [M

]=

[[M ] [02×2]

[02×2] [I2×2]

],

[K

]=

[[02×2]

[K

]

− [I2×2] [02×2]

], (22)

and

{q (t)} =

{ {Qa (t)

}

{02×1}

}. (23)

If one considers that the aerodynamic forces can be conveniently represented in the frequency domain,the system can be more easily studied in the Laplace domain. Applying the Laplace transform to Eq. (21),one obtains

s[M

]{X (s)}+

[K

]{X (s)} =

{Q (s)

}, (24)

wheres =

s

ωr(25)

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is the dimensionless Laplace transform complex variable. The idea behind this procedure is to evaluate theaerodynamic influence coefficient matrix over a reduced frequency range of interest and, by making use ofthe analytical continuation principle,21 to extend such result to the entire complex plane.

As previously discussed, the available aerodynamic responses in the frequency domain consist in setsof numerical values, which are not convenient for the solution of Eq. (24). Therefore, it is necessary toapproximate these data using interpolating polynomials. There is a number of different polynomials reportedin the literature,12,13,22,23 from which the ones suggested by Roger13 and Eversman and Tewari22,23 areselected by the authors for implementation and tests. This choice is based on the fact that such polynomialsare the most commonly used in applications similar to the ones intended in the present work, and that theyare conveniently constructed for the formulation of aerodynamic state variables. Moreover, although thesepolynomials are very similar, they result in different aerodynamic state variables. Hence, the state spaceproblem is slightly different for each of the polynomials used. The adopted approximating polynomials, theinterpolating procedure, and the resulting state space formulations are all presented in detail in the followingsubsections.

Regardless of which polynomial is used for the representation of the aerodynamic response in the Laplacedomain, the resulting state space representation of the typical section aeroelastic system has always the form

{χ (t)} = [D] {χ (t)} . (26)

{χ (t)} is the new state vector that results from the addition of the aerodynamic state variables,

{χ (t)} =

{η (t)}{η (t)}{ηa (t)}1

...{ηa (t)}na

, (27)

where na is the number of added aerodynamic state variables. Moreover, [D] is the stability matrix of thesystem, which can be written as

[D] =

[[Dηη] [Dηa][Daη] [Daa]

], (28)

where

[Dηη] =

[−[M]−1[C] −[M]−1[K]

[I2×2] [02×2]

], (29)

[M] = [M ]− 1πµ

[A2] , (30)

[K] =[K

]− (U∗)2

πµ[A0] , (31)

[C] = −U∗

πµ[A1] , (32)

and the [An] matrices come from the interpolating polynomial representations of the aerodynamic forces,as presented next. The other terms of the stability matrix also depend on the choice of the interpolatingpolynomial. Finally, the aeroelastic stability analysis can, then, be reduced to the classical eigenvalue problemfor each value of the U∗ parameter,

([D]− s [IN×N ]] {χ (s)} = {0N×1} , (33)

where N = 2na + 4.

B. Roger Polynomial

The polynomial proposed by Roger13 is the same used by Oliveira9 and Abel.14 It is given, already in theLaplace domain, by

[A (¯s)] = [A0] + [A1] ¯s + [A2] ¯s2 +nβ∑

n=1

([A(n+2)

] ¯s¯s + βn

), (34)

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where the βn’s introduce the aerodynamic lags with respect to the structural modes. Moreover, nβ is thenumber of added poles and [An] are the approximating coefficient matrices. The βn and [An] variables aredetermined by an optimized least squares approximation method, in which the nonlinear lag parameters areinteractively evaluated with a simplex search method,24 while the linear coefficients are given by a simpleleast square fitting at each step.22 It is important to emphasize that, in Eq. (34), the complex Laplacevariable is nondimensionalized such that

¯s =sb

U∞. (35)

This approach is followed in order to maintain consistency with respect to the definition of the reducedfrequency. After determining the desired coefficients, the switch back to the nondimensionalization presentedin Eq. (25) is accomplished by replacing

¯s =s

U∗ . (36)

The introduction of the aerodynamic lags makes it possible to define new aerodynamic state variables,as follows

{ηa (s)}n =s

s + U∗βn{η (s)}

= L{{η (t)}}L{

e(−U∗ tβn)}

. (37)

Therefore, in this case, the number of added aerodynamic state variables is the same as the number of addedpoles, i.e., na = nβ . In the time domain, Eq. (37) yields

{ηa (t)}n =∫ t

0

{η(τ)} e[−U∗βn(t−τ)]dτ

= {η (t)} − U∗βn

∫ t

0

{η(τ)} e[−U∗βn(t−τ)]dτ , (38)

or, in the differential form,

{ηa (t)}n = {η (t)} − U∗βnd

dt

∫ t

0

{η(τ)} e[−U∗βn(t−τ)]

= {η (t)} − U∗βn {ηa (t)}n . (39)

Furthermore, substituting Eqs. (34) and (37) into Eq. (14), the generalized aerodynamic force vector is givenby

{Qa (s)

}=

(U∗)2

πµ

([A0] +

s

U∗ [A1] +s2

(U∗)2[A2] +

nβ∑n=1

[A(n+2)

] {ηa (s)}n

). (40)

In the time domain, the inverse Laplace transform of Eq. (40) results in

{Qa (t)

}=

(U∗)2

πµ

([A0] {η (t)}+

1U∗ [A1] {η (t)}+

1(U∗)2

[A2] {η (t)}+

nβ∑n=1

[A(n+2)

] {ηa (t)}n

). (41)

Finally, introducing Eq. (40) into (24), the remaining terms of the stability matrix of the resulting systemare

[Dηa] =

[(U∗)2

πµ [M]−1 [A3](U∗)2

πµ [M]−1 [A4] . . . (U∗)2

πµ [M]−1[A(na+2)

]

[02×2] [02×2] . . . [02×2]

], (42)

[Daη] =

[I2×2] [02×2][I2×2] [02×2]

......

[I2×2] [02×2]

(2na×4)

, (43)

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[Daa] =

−U∗β1 [I2×2] [02×2] . . . [02×2][02×2] −U∗β2 [I2×2] . . . [02×2]

......

. . ....

[02×2] [02×2] . . . −U∗βnβ[I2×2]

. (44)

C. Eversman and Tewari Polynomials

The polynomial originally proposed by Eversman and Tewari22,23 is very similar to the Roger polynomial.In the Laplace domain, it is written as

[A (¯s)] = [A0] + [A1] ¯s + [A2] ¯s2 +nβ∑

n=1

([A(n+2)

] 1¯s + βn

), (45)

where, once again, the βn’s introduce the aerodynamic lags with respect to the structural modes. Similarly,the variables of the polynomial are also determined by the same optimized least squares approximationmethod as in the Roger polynomial case. The formulation leads to the following definition of the aerodynamicstate variables

{ηa (s)}n =1

s + U∗βn{η (s)}

= L{{η (t)}}L{

e(−U∗ tβn)}

, (46)

which, in the time domain, can be written as

{ηa (t)}n =∫ t

0

{η(τ)} e[−U∗βn(t−τ)]dτ (47)

or, in the differential form,

{ηa (t)}n =∫ t

0

{η(τ)} e[−U∗βn(t−τ)]dτ

= {η (t)} − U∗βn

∫ t

0

{η(τ)} e[−U∗βn(t−τ)]dτ

= {η (t)} − U∗βn {ηa (t)}n . (48)

Similarly to what occurs with the Roger polynomial, the number of augmented states is equal to the numberof added poles, i.e., na = nβ .

Substituting Eqs. (45) and (46) into Eq. (14), the generalized aerodynamic force vector is given onceagain by Eq. (40). However, due to the different definition of the aerodynamic state variables, the stabilitymatrix is slightly different, namely,

[Dηa] =

[(U∗)3

πµ [M]−1 [A3](U∗)3

πµ [M]−1 [A4] . . . (U∗)3

πµ [M]−1[A(na+2)

]

[02×2] [02×2] . . . [02×2]

], (49)

[Daη] =

[02×2] [I2×2][02×2] [I2×2]

......

[02×2] [I2×2]

(2na×4)

, (50)

while the [Daa] term is exactly the same one given by Eq. (44).Reference 23 suggests that this model of aerodynamic state variables provides a better representation of

the aerodynamic delays than the one suggested by Roger.13 Furthermore, Eversman and Tewari22 identifieda repeated pole problem that may occur with both formulations. This problem consists in the occurrenceof poles very close to each other. Whenever this happens, the linear [An] coefficients corresponding to these

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lag parameters are very close in magnitude, very large, and of opposite signs. Such phenomenon makes thesubsequent eigenvalue problem poorly conditioned. In order to overcome such difficulty, Ref. 22 suggests aslight modification to the lag terms in such situations. Consider, for example, that

β2 = β1 + ε, (51)[A4] = − ([A3] + [ε]) , (52)

where [ε] is a 2 × 2 matrix whose elements are significantly smaller in magnitude than the correspondingelements of [A3]. Then, in the time domain, the addition of both terms yields

[A3] e−β1 t + [A4] e−β2 t = [A3] e−β1 t − ([A3] + [ε]) e−(β1+ε)t

= [A3] e−β1 t(1− e−εt

)− [ε]e−β1 te−εt. (53)

If a Taylor series expansion is used for the eεt exponential, then

e−εt = 1− εt +12

(εt)2 +O((εt)3

). (54)

Since ε is small, it is reasonable to neglect the higher order terms. Therefore, Eq. (53) results in

[A3] e−β1 t + [A4] e−β2 t ≈ [A3] εte−β1 t − [ε]e−β1 t, (55)

or, in the Laplace domain,

[A3]1

¯s + β1+ [A4]

1¯s + β2

≈ −[ε]1

¯s + β1+ [A3]

ε

(¯s + β1)2 . (56)

The same type of analysis can be extended for the case of three, four, or actually any number of polesthat occur very close to each other in the original formulation. With the present approach, and for a givenunsteady aerodynamic data set, the number of aerodynamic state variables added to the model remains thesame, while the number of poles is reduced. Eversman and Tewari22 demonstrate that this procedure doesnot alter the fitting accuracy of a given polynomial, but it results in a better conditioned eigenvalue problem.Therefore, in many cases, the following approximating polynomial is a more useful model than the originalone given in Eq. (45).

[A (¯s)] = [A0] + [A1] ¯s + [A2] ¯s2 +nβ∑

n=1

([A(n+2)

] 1¯s + βn

)

+nβ∑

n=(nβ1+1)

([A(n+N2+2)

] 1(¯s + βn)2

)

+nβ∑

n=(nβ2+1)

([A(n+N3+2)

] 1(¯s + βn)3

)

+nβ∑

n=(nβ3+1)

([A(n+N4+2)

] 1(¯s + βn)4

)+ . . . (57)

withN2 = nβ − nβ1 , N3 = 2nβ − nβ1 − nβ2 , N4 = 3nβ − nβ1 − nβ2 − nβ3 , . . . (58)

where nβ is the total number of poles and it is assumed that

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β1, . . . , βnβ1are non-repeated poles,

β(nβ1+1), . . . , βnβ2are poles that occur twice,

β(nβ2+1), . . . , βnβ3are poles that occur three times,

and so on.

Notice that, in this case, the number of aerodynamic states is

na = nβ + (nβ − nβ1) + (nβ − nβ2) + · · · . (59)

Furthermore, Eversman and Tewari22 also argue that this alternate formulation may significantly reduce thecomputational time required for the optimization procedure, since the total number of poles is decreased forthe same number of aerodynamic states.

In this case, the first nβ added aerodynamic state variables have the same form presented in Eq. (46),while the subsequent N2 are given by

{ηa (s)}(n+N2)=

1s + U∗βn

{ηa (s)}n

= L{{ηa (t)}n}L{

e(−U∗ tβn)}

,

n = (nβ1 + 1) , (nβ1 + 2) , . . . ,nβ , (60)

or, in the time domain,

{ηa (t)}(n+N2)=

∫ t

0

{ηa(τ)}n e[−U∗βn(t−τ)]dτ ,

n = (nβ1 + 1) , (nβ1 + 2) , . . . ,nβ . (61)

Similarly, the following (N3 −N2) aerodynamic state variables are given by

{ηa (s)}(n+N3)=

1s + U∗βn

{ηa (s)}(n+N2)

= L{{ηa (t)}(n+N2)

}L

{e(−U∗ tβn)

},

n = (nβ2 + 1) , (nβ2 + 2) , . . . ,nβ . (62)

The remaining aerodynamic state variables follow the same pattern. One may notice that there are twoaerodynamic state variables for each double multiplicity pole. The first of these state variables is a functionof the structural modes, while the second one is a function of the corresponding first one. In a triplemultiplicity pole, there is a third aerodynamic state variable which, in turn, is a function of the secondaerodynamic state variable associated with that particular pole, and so forth.

Hence, for instance, the differential form of the second aerodynamic state variable in the case of doublemultiplicity poles can be written as

{ηa (t)}(n+N2)=

∫ t

0

{ηa(τ)}n e[−U∗βn(t−τ)]dτ

= {ηa (t)}n − U∗βn

∫ t

0

{ηa(τ)}n e[−U∗βn(t−τ)]dτ

= {ηa (t)}n − U∗βn {ηa (t)}(n+N2),

n = (nβ1 + 1) , (nβ1 + 2) , . . . ,nβ . (63)

A similar type of reasoning would apply to the cases of three or higher multiplicity poles, according to theprevious discussion.

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In the present case, the stability matrix of the aeroelastic problem is composed by

[Dηa] =

(U∗)3

πµ [M]−1[A1

](U∗)4

πµ [M]−1[A2

](U∗)5

πµ [M]−1[A3

]. . .

[02×2nβ

] [02×2(nβ−nβ1)

] [02×2(nβ−nβ2)

]. . .

, (64)

[Daη] =

[ [02nβ×2

][I][

02(na−nβ)×2

] [02(na−nβ)×2

]]

, (65)

[Daa] =

[R1][02nβ×2(nβ−nβ1)

] [02nβ×2(nβ−nβ2)

]. . .[

02(nβ−nβ1)×2nβ

][R2]

[02(nβ−nβ1)×2(nβ−nβ2)

]. . .[

02(nβ−nβ2)×2nβ

] [02(nβ−nβ2)×2(nβ−nβ1)

][R3] . . .

......

.... . .

+

[02×2] [02×2] [02×2] . . .

[I2×2] [02×2] [02×2] . . .

[02×2] [I2×2] [02×2] . . ....

.... . .

...

, (66)

where[A1

]=

[[A3] [A4] . . .

[A(nβ+2)

] ], (67)

[A2

]=

[ [A(nβ1+N2+3)

] [A(nβ1+N2+4)

]. . .

[A(nβ+N2)

] ], (68)

[A3

]=

[ [A(nβ2+N3+3)

] [A(nβ2+N3+4)

]. . .

[A(nβ+N3)

] ], . . . (69)

[I]

=

[I2×2][I2×2]

...[I2×2]

(2nβ×2)

, (70)

and

[R1] = −U∗

β1 [I2×2] . . . [02×2]...

. . ....

[02×2] . . . βnβ[I2×2]

, (71)

[R2] = −U∗

β(nβ1+1) [I2×2] . . . [02×2]...

. . ....

[02×2] . . . βnβ[I2×2]

, (72)

[R3] = −U∗

β(nβ2+1) [I2×2] . . . [02×2]...

. . ....

[02×2] . . . βnβ[I2×2]

, . . . (73)

Noticing that, for a given number of augmented aerodynamic states, a multiple pole approximation ismore efficient in the optimization process because of the reduction in lag parameters, Eversman and Tewari22

extended this idea and suggested a third version for the approximating polynomial, which is given by

[A (¯s)] = [A0] + [A1] ¯s + [A2] ¯s2 +na∑

n=1

([A(n+2)

] 1(¯s + β)n

). (74)

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Therefore, in this case, there is only one lag parameter β, which occurs with multiplicity na. The firstaerodynamic state variable is represented by Eq. (46), while the remaining are similar to the definition ofEq. (60). Consequently, the stability matrix components are

[Dηa] =

[(U∗)3

πµ [M]−1 [A3](U∗)4

πµ [M]−1 [A4] . . . (U∗)(na+2)

πµ [M]−1[A(na+2)

]

[02×2] [02×2] . . . [02×2]

], (75)

[Daη] =

[I2×2] [02×2][02×2] [I2×2][02×2] [02×2]

......

[02×2] [02×2]

(2na×4)

, (76)

[Daa] = −U∗β [I2na×2na] +

[ [02×2(na−1)

][02×2][

I2(na−1)×2(na−1)

] [02(na−1)×2

]]

. (77)

In the present work, all four approximating models discussed are employed in a specific test case in orderto address which one is more suitable for the complete aeroelastic analysis. This evaluation involves thecompromise between fitting accuracy and conditioning of the eigenvalue problem.

IV. Discrete-Time State Space Formulation

Instead of approximating the discrete-time aerodynamic responses in order to make them suitable forapplication in Eq. (24), as presented in the previous section, another alternative is to represent every othertime-dependent variable in a sampled, or discrete, manner. By doing so, it is possible to convert continuous-time state space equations into discrete-time state space equations and use digital control theory for thestability analysis.20 The time step required for an accurate solution of the aerodynamic behavior must besmall compared with the significant time constants of the system, which guarantees that no considerableerror is introduced by the discretization procedure.

As indicated in Ref. 20, the solution of a continuous-time state space system, such as the one representedin Eq. (21), is given by

{x ((n + 1)∆t)} = e([W ]∆t){x (n∆t)}

+ e([W ](n+1)∆t)

∫ (n+1)∆t

n∆t

e(−[W ]τ) {qx(τ)} dτ , (78)

where[W ] = −

[M

]−1 [K

], {qx (t)} =

[M

]−1

{q (t)} . (79)

It is very important to emphasize that the nondimensionalization of the aeroelastic system time step, ∆t, isnot necessarily the same used for the time step of the flow solver. For example, according to the nondimen-sionalization of the flow variables generally used in CFD solvers, the correspondence between the aeroelasticand the aerodynamic time steps is given by

∆t = (∆t)a∞2b

,

∆t = (∆t)ωr =U∞2b

a∞b

∆t

U∗ =2M∞U∗ ∆t, (80)

where ∆t designates the flow solver time step. Considering that both time steps are small enough, it isreasonable to assume {

Qa (t)}

={Qa (n∆t)

}, n∆t ≤ t < (n + 1)∆t, (81)

and, consequently,

{q (t)} = {q (n∆t)} , n∆t ≤ t < (n + 1)∆t, (82){qx (t)} = {qx (n∆t)} , n∆t ≤ t < (n + 1)∆t. (83)

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Hence, as demonstrated in Ref. 20, Eq. (78) can be rewritten as

{x ((n + 1)∆t)} = [G (∆t)] {x (n∆t)}+ [H (∆t)] {qx (n∆t)} , (84)

where

[G (∆t)] = e([W ]∆t), (85)

[H (∆t)] =∫ ∆t

0

e([W ]τ)dτ = [W ]−1[e([W ]∆t) − [I4×4]

]. (86)

Moreover, the sampling of the state and load vectors, {x (t)} and {qx (t)}, results in the following sequences,respectively,

{x[n]} = {x ((n− 1)∆t)} , (87){qx[n]} = {qx ((n− 1)∆t)} . (88)

Therefore, the discrete version of Eq. (84) is

{x[n + 1]} = [G (∆t)] {x[n]}+ [H (∆t)] {qx[n]} . (89)

The z transform stands for discrete-time sequences as an operation similar to the Laplace transformfor the continuous-time functions.20 Therefore, instead of using approximating polynomials for the discreteaerodynamic response, it may be more convenient to perform the frequency domain analysis with the ap-plication of the z transform, or z domain analysis. The present formulation is based on the one-sided ztransform, defined as

Y (z) = Z {y[n]} =∞∑

n=0

y[n]z−n, (90)

where Y (z) is the z transform of the discrete sequence y[n]. Ogata20 shows that, for an equivalent sampledfunction, i.e.,

y(t) =

{y[n + 1], t = n∆t,0, otherwise,

(91)

the Laplace transform, Y (s), is formally the same as the z transform, Y (z), for the values of s that respect

s =1

∆tln z. (92)

Furthermore, it is important to notice that the definition of the z transform itself results in a polynomialin the z domain. This fact facilitates the construction of the state space model, avoiding the need for apolynomial fitting process.

One very important theorem concerning the z transform is the shifting theorem.20 Such theorem statesthat, if y[n] = 0 for n < 0, then

Z {y[n + j]} = zjY (z). (93)

Consequently, the z transform of Eq. (89) yields

z{X(z)} = [G (∆t)] {X(z)}+ [H (∆t)] {Qx(z)} (94)

Thus, the formulation of the discrete aeroelastic problem depends on the determination of the forcing term,{Qx(z)}, in the z domain. Moreover, Oppenheim and Schafer18 show that the z transform also presents theconvolution theorem property, which states that the z transform of the convolution sum of two sequences isidentical to the multiplication of their individual z transforms, i.e.,

Z {y1[n] ∗ y2[n]} = Z

∞∑

j=−∞y1[j]y2[n− j]

= Y1(z)Y2(z). (95)

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Therefore, the discrete-time, z domain, equivalent statement of the hypothesis adopted in Eq. (14) is

{Qa(z)

}=

(U∗)2

πµ[A(z)] {η(z)} , (96)

where

[A(z)] =

[−C`Ih

(z)/2 −C`Iα(z)

CmIh(z) 2CmIα(z)

]. (97)

Hence,

{Qx(z)} =[M

]−1{ {

Qa(z)}

{02×1}

}=

(U∗)2

πµ

[M

]−1{

[A(z)] {η(z)}{02×1}

}. (98)

As explained in Ref. 18, a direct consequence of the convolution property is that the sought z transformof the aerodynamic coefficients, which constitute the terms of the aerodynamic influence coefficient matrix,are given in the following form

C`Ih(z) =

ClhOUT(z)

hIN (z), CmIh

(z) =CmhOUT

(z)hIN (z)

,

C`Iα(z) =

ClαOUT(z)

αIN (z), CmIα

(z) =CmαOUT

(z)αIN (z)

, (99)

where C`hOUT(z), CmhOUT

(z), C`αOUT(z), and CmαOUT

(z) are the z transforms of the time histories of ae-rodynamic responses in terms of the corresponding coefficients, while hIN (z) and αIN (z) are the z transformsof the input excitations in the plunge and pitch modes, respectively. For the sake of simplicity, henceforththe development of the aerodynamic state variables is only demonstrated for C`h

, and the results are thengeneralized for the other terms of the aerodynamic coefficient matrix.

Therefore, according to the suggestions of Ref. 20, by defining the auxiliary function

Yh(z) =ξ(z)ah(z)

, (100)

where

ah(z) = 1 + ah1z−1 + ah2z

−2 + . . . + ah(nT−1)z−(nT−1), (101)

ahn =hIN [n + 1]

hIN [1], (102)

and nT is the total number of points that constitute the aerodynamic response sequence, it is possible toconstruct the following aerodynamic state variables

Xah1(z) = z−(nT−1)Yh(z),Xah2(z) = z−(nT−2)Yh(z),

...Xah(nT−1)

(z) = z−1Yh(z). (103)

These new states relate to each other by

zXah1(z) = Xah2(z),zXah2(z) = Xah3(z),

...zXah(nT−2)

(z) = Xah(nT−1)(z), (104)

and Eq. (100) results in

zXah(nT−1)(z) = −ah1Xah(nT−1)

(z)− ah2Xah(nT−2)(z)− . . .− ah(nT−1)

Xah1(z) + ξ(z). (105)

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Hence, Eqs. (104) and (105) can be rewritten in matrix form, as follows

z {Xah(z)} =

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 1−ah(nT−1)

−ah(nT−2)−ah(nT−3)

. . . −ah1

{Xah(z)}+

[0 0 . . . 0 1

]T

ξ(z), (106)

where the z transform of the aerodynamic state vector is given by

{Xah(z)} =

Xah1(z)Xah2(z)

...Xah(nT−1)

(z)

. (107)

Furthermore, from Eq. (99), one can obtain that

C`Ih(z) =

C`hOUT(z)

hIN (z)=

C`hOUT[1]z−1 + . . . + C`hOUT

[nT ]z−nT

hIN [1]z−1 + . . . + hIN [nT ]z−nT

=−2

[bC`h0 + bC`h1z

−1 + . . . + bC`h(nT−1)z−(nT−1)

]

ah(z), (108)

where

bC`hn = −C`hOUT[n + 1]

2hIN [1]. (109)

Then, neglecting for now the term related to the pitch mode in Eq. (10), one can write that

C`h(z) = C`Ih

(z)ξ(z)2

= −bC`h0ξ(z)

− ξ(z)ah(z)

[(bC`h1 − ah1bC`h0) z−1 + . . . +

(bC`h(nT−1)

− ah(nT−1)bC`h0

)z−(nT−1)

]. (110)

Therefore, the lift coefficient due to the plunge mode is given in the z domain by

C`h(z) = −

[cC`h(nT−1)

cC`h(nT−2). . . cC`h1

]{Xah(z)} − bC`h0ξ(z), (111)

wherecC`hn = bC`hn − ahnbC`h0 . (112)

From such formulation, it is straightforward to deduce that the Cmh(z) depends on the same aerodynamic

state vector {Xah(z)}. Furthermore, in an analogous way, it is possible to define similar concepts for thepitch mode. Therefore, it is possible to put all these definitions together in order to construct a general statespace representation for the generalized aerodynamic forces in the z domain. First of all, one can define

aα = 1 + aα1z−1 + aα2z

−2 + . . . + aαnT−1z−(nT−1), (113)

aαn =αIN [n + 1]

αIN [1], (114)

bC`αn = −C`αOUT[n + 1]

αIN [1], (115)

bCmαn =2CmαOUT [n + 1]

αIN [1], (116)

bCmhn =CmhOUT

[n + 1]hIN [1]

, (117)

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and

{Y a(z)} =

[1

ah(z) 0

0 1aα(z)

]{η(z)}, (118)

{Xa1(z)} = z−(nT−1){Y a(z)},{Xa2(z)} = z−(nT−2){Y a(z)},

...{Xa(nT−1)(z)

}, = z−1{Y a(z)}. (119)

These aerodynamic states relate to each other by

z {Xa1(z)} = {Xa2(z)} ,z {Xa2(z)} = {Xa3(z)} ,

...z

{Xa(nT−2)(z)

}=

{Xa(nT−1)(z)

}, (120)

and

z{Xa(nT−1)(z)

}=

[− [

A(nT−1)

] − [A(nT−2)

]. . . − [

A1

] ]{Xa(z)}+ [I2×2] {η(z)}, (121)

where the z transform of the aerodynamic state vector is given by

{Xa(z)} =

{Xa1(z)}{Xa2(z)}

...{Xa(nT−1)(z)

}

, (122)

and[An

]=

[ahn 00 aαn

]. (123)

Finally, it is possible to describe the forcing term, {Qx(z)}, through the new state space representation, asfollows,

{Qx(z)} = [Aa]{Xa(z)}+ [Ax]{X(z)}, (124)

where

[Aa] =(U∗)2

πµ

[M

]−1[

[B][02×(2nT−2)

]]

, (125)

[Ax] =(U∗)2

πµ

[M

]−1[

[02×2] [B0][02×2] [02×2]

], (126)

[B] =

[BC`(nT−1)

] [BC`(nT−2)

]. . . [BCl1 ][

BCm(nT−1)

] [BCm(nT−2)

]. . . [BCm1 ]

, (127)

[BC`n ] =[

bC`hn − ahnbC`h0 bC`αn − aαnbC`α0

], (128)

[BCmn ] =[

bCmhn − ahnbCmh0 bCmαn − aαnbCmα0

], (129)

and

[B0] =

[bC`h0 bC`α0

bCmh0 bCmα0

]. (130)

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Hence, Eq. (124) describes a state space representation in the z domain that is adequate for application inEq. (94). The final expression for the state space problem is, then, given by

([D

]− z [IN×N ]] {χ(z)} = {0N×1} , (131)

where

[D

]=

[ [Dxx

] [Dxa

][Dax

] [Daa

]]

, (132)

[Dxx

]= [G(∆t)] + [H(∆t)] [Ax], (133)[

Dxa]

= [H(∆t)] [Aa], (134)

[Dax

]=

[ [0(2nT−4)×2

] [0(2nT−4)×2

]

[02×2] [I2×2]

], (135)

and

[Daa

]=

[02×2] [I2×2] [02×2] . . . [02×2][02×2] [02×2] [I2×2] . . . [02×2]

......

.... . .

...[02×2] [02×2] [02×2] . . . [I2×2]

− [A(nT−1)

] − [A(nT−2)

] − [A(nT−3)

]. . . − [

A1

]

. (136)

Here, N designates the total number of states, which is given by N = (2nT + 2). The state vector, {χ}, isformed by all the structural and aerodynamic state variables, and its z transform is defined as

{χ(z)} =

{{X(z)}{Xa(z)}

}. (137)

Although this discrete-time formulation does not require the construction of approximating polynomials,it has one drawback. The above formulation indicates that the number of augmented aerodynamic statevectors is the same as the number of the time steps needed to accurately represent the unsteady aerodynamicresponse. Therefore, the size of the eigenvalue problem is much larger than in the continuous-time formula-tion. The number of state variables may be hundreds or thousands of times larger in the discrete-time case.However, since in well-constructed root locus analyses the eigenvalues vary little from each other for twoconsecutive values of the root locus parameter, there are numerical schemes that can solve for the desiredeigenvalues quite rapidly. The greatest limitation, then, is the memory size. In any event, this difficulty canbe overcome by choosing an appropriate number of points in order to represent the aerodynamic response fora certain frequency range of interest. Moreover, since matrix

[D

]has very few nonzero elements, memory

usage can be also reduced by employing techniques for construction of sparse matrices.

V. Direct Integration

A more straightforward manner of solving the aeroelastic problem is the simultaneous direct integrationof the structural dynamic equations together with the flow unsteady equations. This may seem as a verynatural procedure, but it turns out to be extremely costly as well, specially if complex, both aerodynamicallyand structurally, cases are concerned. It is instructive, however, to compare the results given by the proposedstate space formulations with direct integrations of the aeroelastic problem. For the two-dimensional Eulerequations coupled with a simple typical section model, the computational cost, although much higher thanfor the state space models, is not prohibitive. Therefore, a new version of the flow solver coupled with thestructural dynamic equations is developed for validation purposes.

Before proceeding to the numerical discretization of the structural dynamic equations, it is very importantto adequate them to the nondimensionalization of the flow solver, such that both integrations advance at thesame pace. The CFD code used in the present work11,16 applies the time nondimensionalization indicatedin the previous section. Thus, using the same nondimensionalization, Eq. (1) becomes

ξ(t)

+ xαα(t)

+(

2M∞U∗

ωh

ωr

)2

ξ(t)

= −4M2∞

πµC`

(t), (138)

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xαξ(t)

+ r2αα

(t)

+(

2M∞rα

U∗ωα

ωr

)2

α(t)

=8M2

∞πµ

Cm

(t), (139)

where M∞ denotes the undisturbed flow Mach number.The numerical integration of the structural dynamic equations is implemented following the suggestions

in Ref. 25, with a very simple second-order accurate, centered difference scheme for the second derivativeterms. Hence, the discretization yields

ξ((n + 1)∆t

) ≈ ξ[n + 1]− 2ξ[n] + ξ[n− 1](∆t

)2 =⇒

ξ[n + 1] ≈ (∆t

)2ξ((n + 1)∆t

)+ 2ξ[n]− ξ[n− 1], (140)

and, similarly,α[n + 1] ≈ (

∆t)2

α((n + 1)∆t

)+ 2α[n]− α[n− 1]. (141)

The substitution of Eqs. (140) and (141) into Eqs. (138) and (139) yields, after some manipulation,

ξ[n + 1] =

[r2α

x2α − r2

α

(2M∞U∗

ωh

ωr∆t

)2

+ 2

]ξ[n]

− xαr2α

x2α − r2

α

(2M∞U∗

ωα

ωr∆t

)2

α[n]

+4

(M∞∆t

)2

πµ (x2α − r2

α)(r2αC`[n] + 2xαCm[n]

)

− ξ[n− 1], (142)

and

α[n + 1] =

[r2α

x2α − r2

α

(2M∞U∗

ωα

ωr∆t

)2

+ 2

]α[n]

− xα

x2α − r2

α

(2M∞U∗

ωh

ωr∆t

)2

ξ[n]

− 4(M∞∆t

)2

πµ (x2α − r2

α)(xαC`[n] + 2Cm[n])

− α[n− 1]. (143)

Therefore, after having determined the values of C` and Cm at a given time step, the code uses Eqs. (142)and (143) for evaluating the new position of the airfoil. Then, it returns to the flow solver algorithm andthis interactive process goes on until enough information is gathered in order to obtain the damping andfrequency characteristics of the response. It is relevant to emphasize that one simulation is required foreach value of the characteristic speed parameter, U∗. Thus, this approach requires as many simulations asnecessary in order to bracket the flutter condition, with converging and diverging situations, until the desiredaccuracy is obtained.

VI. Results and Discussion

The main test case considered in the present paper considers the aeroelastic stability of a NACA 0012airfoil at M∞ = 0.80 and zero initial angle of attack. The structural parameters which define the problemare ah = −2.0, xα = 1.8, rα = 1.865, µ = 60, ωα = 100rad/s, ωh = 100rad/s, and ωr = ωα is used asreference. This case is also reported in Ref. 8, in which the same sort of stability analysis is performedwith aerodynamic data obtained through a numerical scheme very similar to the one used in the presentwork. The impulsive aerodynamic response of both modes, in terms of the generalized force coefficients,and already in the frequency domain, are presented in Figs. 2 and 3. It is important to notice that theseresults are based on a pitching motion that occurs around the quarter-chord point of the airfoil, and that the

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moment coefficients are given with respect to that point. Since the elastic axis is located at another point,before proceeding to the aeroelastic stability analyses, the following transformations are necessary2,26

Cmα= Cmα

+ rC`α− rCmh

− r2C`h(144)

Cmh= Cmh

+ rC`h(145)

C`h= C`h

(146)C`α = C`α − rC`h

, (147)

wherer =

ah

2− 0.25, (148)

and the (∼) variables denote the adapted coefficients. The results in Figs. 2 and 3 present the frequencycontent of the response up to κ = 3.00 because this is about the higher reduced frequency involved in thecases considered for the construction of the root locus stability analyses subsequently shown. The resultsobtained by the authors are those represented by EP, which stand for exponentially-shaped pulse, since thisis the type of excitation imposed to the aerodynamic system.17 Figures 2 and 3 also include the aerodynamicresponses to harmonic (H) motions at different reduced frequencies in both modes. Such responses are shownin order to address the linearity question brought up in a previous section of this paper. Reference 18 showsthat obtaining impulsive responses from smooth excitations is only possible if the linearity assumptions,that are made in this work, hold. Therefore, the agreement between EP and H data is a proof that suchhypotheses are valid for small amplitudes. Furthermore, although Ref. 8 offers comparison data only toreduced frequencies up to κ = 1.00, the present results match the literature (Lit.) values very closely in thatrange. This agreement is another assessment of the correctness of the present results. Further validation ofthe numerical tool here used can be found in Refs. 16 and 17.

X X X X XX

XX X

X XX

XX

X

X

X

X

κ

CL

0 0.5 1 1.5 2 2.5 3-25

-20

-15

-10

-5

0

5

EPHLit.

X

Real

Imaginary

(a) C` frequency domain response.

X X XX X

XX

X X

X X XX

X

X

X

X

X

κ

Cm

0 0.5 1 1.5 2 2.5 3

-2

0

2

4

6 EPHLit.

X

Real

Imaginary

(b) Cm frequency domain response.

Figure 2. Low reduced frequency response of a NACA 0012 airfoil at M∞ = 0.80 and α0 = 0 to an impulsiveinput in the plunge mode.

The four interpolating polynomials discussed in the paper are applied in order to conveniently appro-ximate the aerodynamic results. The authors use 1, 2, 3, 4, 6, and 10 augmented state variables in everycase, except for the second form of the Eversman and Tewari (E and T - 2) polynomial, which requires theoccurrence of very close poles. Such situation only happened in the case of 4 added states, when a doublepole is observed. The fitting errors for all approximating models are presented in Fig. 4(a). Here, the fittingerror measure corresponds to the mean quadratic relative error, considering the four aerodynamic coefficientsin the average, i.e.,

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X

XX X X X X X X

X

X

XX

X

X

XX

X

κ

CL

0 0.5 1 1.5 2 2.5 3-6

-4

-2

0

2

4

6

8

10

12

14

EPHLit.

X

Real

Imaginary

(a) C` frequency domain response.

XX

X

X XX X

XX

XX

X

XX

X

X

X

X

κ

Cm

0 0.5 1 1.5 2 2.5 3

-3

-2

-1

0

EPHLit.

X

Real

Imaginary

(b) Cm frequency domain response.

Figure 3. Low reduced frequency response of a NACA 0012 airfoil at M∞ = 0.80 and α0 = 0 to an impulsiveinput in the pitch mode.

F.E. =1

4nK

{nK∑

i=1

[∣∣∣∣∣(C`h

(κi)−C∗`h(κi)

)2

R2`hi

∣∣∣∣∣ +∣∣∣∣(Cmh

(κi)−C∗mh(κi))2

R2mhi

∣∣∣∣

+∣∣∣∣(C`α (κi)−C∗`α

(κi))2

R2`αi

∣∣∣∣ +∣∣∣∣(Cmα (κi)−C∗mα

(κi))2

R2mαi

∣∣∣∣]}

, (149)

where

R`hi=

{C`h

(κi) , if C`h(κi) 6= 0,

1.0, otherwise,(150)

and Rmhi, R`αi , and Rmαi are defined in a similar manner. Moreover, the (∗) coefficients are those given by

the approximate model, and nK is the number of samples used to represent the frequency range of interest.Opposite to what Ref. 23 states, in the present case, the first form of the Eversman and Tewari (E and T -1) polynomial does not necessarily result in a better fitting than the Roger polynomial. Actually, there aresituations in which the inverse happens. However, the overall errors resulting from both models are verysimilar. In the only case in which the second form of the Eversman and Tewari polynomial could be used,the error is the same as for the first form, as expected. Nevertheless, the third form of the Eversman andTewari (E and T - 3) resulted in higher errors regardless of the number of augmented states. This resultis probably a consequence of the smaller flexibility in the pole positioning. However, this last formulationindeed showed a much more rapid convergence of the optimization procedure, which is the main justificationfor its use.

It can be argued, however, that the computational cost of the optimization procedure is not significant,at least for current day computers. Hence, it does not constitute a relevant parameter for the choice of themost adequate model. The really important influence in this selection is the effect of the polynomial modelin the solution of the eigenvalue problem that defines the system stability. Figure 4(b) presents the error inthe solution of the eigenvalue problem for different number of augmented states. This error measure is basedon the determinant of ([D]− λ [IN×N ]), scaled by the largest element of the [D] matrix. Here, λ designatesthe eigenvalue corresponding to the pitch mode when Q∗ = 0.1, which is the first point of the root locus tobe presented in the forthcoming paragraphs. Q∗ stands for the characteristic dynamic pressure, and it isdirectly related to U∗ by

Q∗ =(U∗)2

µ. (151)

Figure 4(b) indicates that, as the number of added state variables is increased, the eigenvalue problembecomes more difficult to be solved due to ill-conditioning issues. Therefore, considering both aspects

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presented in Fig. 4, the authors chose the first form of the Eversman and Tewari polynomial with six poles forthe remainder of the aeroelastic stability analyses here reported. Figures 5 and 6 show comparisons betweenthe discrete CFD results and the chosen approximating (AP) model for the aerodynamic coefficients. Theagreement between these data is extremely good, except for some small regions in which abrupt oscillationsof the CFD results are smoothed by the polynomial. However, even in these regions, the fitting error isrelatively small.

Number of added state variables

Erro

r

1 2 3 4 5 6 7 8 9 10

10-3

10-2

RogerE and T - 1E and T - 2E and T - 3

(a) Fitting error for different interpolating polynomials.

Number of added state variables

Erro

r

1 2 3 4 5 6 7 8 9 1010-12

10-10

10-8

10-6

10-4

10-2

100

102

104

106

RogerE and T - 1E and T - 2E and T - 3

(b) Error in the solution of the first point of the eigenvalueproblem.

Figure 4. Error parameters that influence the choice of the interpolating model.

κ

CL

0 1 2 3

0

5

10

15

20

25

CFD resultAP

Real

Imaginary

(a) C` frequency domain response.

κ

Cm

0 1 2 3

-2

0

2

4

6 CFD resultAP

Real

Imaginary

(b) Cm frequency domain response.

Figure 5. Low reduced frequency response of a NACA 0012 airfoil at M∞ = 0.80. and α0 = 0 to an impulsiveinput in plunge mode. Comparisons between CFD results and approximated data.

Finally, the stability analysis is concluded with the construction of a root locus graph, defined by thesolution of the eigenvalue problem given by Eq. (33), for a continuous-time formulation, or Eq. (131), for adiscrete-time one. It is relevant to notice that, in the discrete-time case, the eigenvalue problem is writtenin terms of the z variable, which is defined in a different manner than the s variable. However, they canare related by Eq. (92). As the root locus is presented in terms of the nondimensional s variable, the realrelationship used is

s =s

ωr=

1∆t

ln(z) =U∗

2M∞∆tln(z). (152)

Figures 7 and 8 present the root locus of the aeroelastic problem in question. The characteristic dynamicpressure parameter varies from Q∗ = 0.0 up to 1.0 in ∆Q∗ = 0.1 intervals. Each plotted point corresponds to

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κ

CL

0 1 2 3-4

-2

0

2

4

6

8

10

12

14

CFD resultAP

Real

Imaginary

(a) C` frequency domain response.

κ

Cm

0 1 2 3

-3

-2

-1

0

CFD resultAP

Real

Imaginary

(b) Cm frequency domain response.

Figure 6. Low reduced frequency response of a NACA 0012 airfoil at M∞ = 0.80. and α0 = 0 to an impulsiveinput in pitch mode. Comparisons between CFD results and approximated data.

one of these values. The figures include the solutions obtained using the continuous-time Laplace transform(LT), discrete-time z transform (zT), and direct integration (DI) approaches, as well as the numerical directintegration results given in Ref. 8 (Lit.). The literature data, however, are only available for Q∗ = 0.2, 0.5,and 0.8. Additionally, the results for the first two approaches are determined from the eigenvalue problemspreviously mentioned, while damping and frequency characteristics of the direct integration responses areestimated using the modal identification technique of Bennet and Desmarais.27

XXXXXXXXXX

XXX X X X X X X X

σ/ωα

ω/ω

α

-0.25 -0.2 -0.15 -0.1 -0.05 0

1

2

3

4

5

LTzTDILit.

X

Second Mode

First Mode

Figure 7. Laplace domain root locus of the aeroelastic stability problem in question. Comparison amongcontinuous-time, discrete-time and direct integration calculations.

Before proceeding to further analyses, it is instructive to confirm that the employed aerodynamic res-ponses are really sufficient for the correct solution of the present problem. For that, one should consider thehigher reduced frequency content necessary for the construction of the root locus. The reduced frequency

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X

X

X

X

X

X

X

X

X

X

σ/ωα

ω/ω

α

-0.01 0 0.01 0.02 0.030.7

0.8

0.9

1

1.1

LTzTDILit.

X

(a) First aeroelastic mode curve.

X

XXX

X

XX

X

X

X

σ/ωα

ω/ω

α

-0.25 -0.2 -0.15 -0.1 -0.05 05.28

5.3

5.32

5.34

5.36

5.38

LTzTDILit.

X

(b) Second aeroelastic mode curve.

Figure 8. Individual view of each mode root locus.

relates to the dimensionless frequency ω/ωα by

κ =ωb

U∞=

1U∗

ω

ωα. (153)

Thus, the smaller the dynamic pressure, the higher is the relevant reduced frequency range. Moreover, k isdirectly proportional to ω/ωα, which indicates that the aeroelastic mode with the highest natural frequencydetermines the maximum range of reduced frequency in which the aerodynamic response must be known. Inthe present case, the second aeroelastic mode presents the highest natural frequency, ω/ωα ≈ 5.3, and theminimum value of U∗ considered is

U∗min =

√Q∗

minµ =√

0.1× 60 ≈ 2.45. (154)

Then, κmax ≈ 2.16. As the aerodynamic response is known for reduced frequency values up to 3.00, itrepresents the needed aerodynamic information for the stability analysis in question.

Furthermore, it is evident that the first aeroelastic mode curves and points obtained with the differentmethodologies agree much better than in the second aeroelastic mode. The authors believe that this occursfor two main reasons. First of all, the modal identification technique used in the evaluation of damping andfrequency characteristics of the direct integration data (DI and Lit.) is based on a curve fitting procedureapplied simultaneously for both aeroelastic modes. It turns out that the time domain response of the mostdamped aeroelastic mode dies out much more rapidly than the response of the least damped aeroelastic mode.Hence, the least damped mode, which in the present case is the first aeroelastic mode, is favored in the curvefitting. The authors attempt to reduce this effect by using different sets of direct integration solutions forthe determination of the frequency and damping characteristics of each aeroelastic mode. According tothis approach, the most damped aeroelastic mode characteristics are evaluated with the initial portion ofthe direct integration solution, while the least damped aeroelastic mode characteristics are calculated usingthe whole solution. This procedure attenuates the damping difference effect, but it does not eliminate it.Moreover, the polynomial approximation for the aerodynamic coefficients produces the largest errors in thepitch mode responses for reduced frequency values between κ = 0.5 and 1.0 (see Figs. 5 and 6). Accordingto Eq. (153), this means that these errors affect the continuous-time (LT) pitch mode root locus estimatesmainly in the range 0.5 ≤ Q∗ ≤ 1.8. As the plunge mode natural frequency is smaller, the influence of sucherrors on this mode is restricted to the range Q∗ < 0.1.

It is also important to notice that the discrete-time (zT) root locus is the one which is closer to the DIresults. This means that the z domain state space formulation suggested by the present authors is capable ofadequately representing the aeroelastic system and provides an effective tool for determining flutter instability

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points. The larger errors observed in the continuous-time formulation are probably caused by inaccuraciesthat are intrinsic to the polynomial interpolation procedure. Moreover, the solution of the large eigenvalueproblem given by the z transform formulation has shown to be more practical, efficient, and accurate thanusing the continuous-time approach with its requirement for data approximation. Additionally, the authors’experience has shown that the large size of the aeroelastic stability matrix in the discrete-time case does notlead to ill-conditioning problems. Finally, as it is shown in Figs. 7 and 8, the literature data corroborate thequality of the results contained in the present paper. Even in the second aeroelastic mode case, in which thediscrepancies are the largest, the relative differences are smaller than 1%, although the provided axis scalesmay seem to indicate otherwise.

At last, the flutter condition can be identified as the point at which one of the curves crosses the imaginaryaxis. A point-to-point linear interpolation of the results presented in Figs. 7 and 8 indicates the flutterfrequency. Furthermore, Fig. 9 shows the damping behavior of the first aeroelastic mode in relation to thecharacteristic dynamic pressure and characteristic speed. The same sort of interpolation of such results yieldsthe flutter characteristic dynamic pressure and characteristic speed. The flutter points acquired with thedifferent approaches are shown in Table 1, in which the flutter prediction of Ref. 8 is also included. Actually,the literature value is given by a quadratic interpolation using the offered points. As can be seen, despite thementioned discrepancies, the flutter predictions given by all methods are very similar, and the differencesare not greater than 3% in any of the considered parameters.

X X

X

X

X

X

X

X

X

X

Q*

σ/ω

α

0 0.2 0.4 0.6 0.8 1

-0.01

0

0.01

0.02

0.03 LTzTDIX

(a) Characteristic dynamic pressure influence.

X X

X

X

X

X

X

X

X

X

U*

σ/ω

α

0 2 4 6 8

-0.01

0

0.01

0.02

0.03 LTzTDIX

(b) Characteristic speed influence.

Figure 9. First aeroelastic mode damping variation with respect to the characteristic dynamic pressure andcharacteristic speed.

Table 1. Flutter points.

Method Q∗f U∗f ω/ωα

DI 0.48 5.38 0.90LT 0.47 5.32 0.90zT 0.50 5.47 0.91Lit. 0.50 5.48 0.91

Another way to construct a root locus based on the discrete-time results is in terms of the real andimaginary parts of the eigenvalues in the z domain. This is shown in Fig. 10. However, due to the differentdefinition of the z variable, the stability analysis is slightly different. As explained in Ref. 20, in this case,the imaginary axis of the usual Laplace domain graph is mapped onto the unitary circle around the origin inthe z domain, while the left side of the s domain is mapped inside this circle and the right side is representedoutside. Therefore, as long as the eigenvalues remain inside the unitary circle in the z domain, the system

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is stable, and the instability condition is given by the point at which one of the eigenvalues crosses thatcircle. Naturally, the result of the stability analysis is the same, no matter which root locus representation isused. Additionally, it is important to emphasize that the z domain root locus representation is quite unusualbecause the constant time step is the one used in the CFD solver

(∆t

), while the aeroelastic formulation

time step (∆t) is different for each value of Q∗, or U∗. Therefore, each point in this root locus correspondsto a different time step. However, this has no effect on the stability analysis.

Re(z)

Im(z

)

0.998 0.9985 0.999 0.9995 10

0.01

0.02

0.03

0.04

0.05

First modeSecond mode

Figure 10. z domain root locus of the aeroelastic stability problem in question.

The aeroelastic analysis methodologies proposed by the authors have been tested for a wide range ofsubsonic and transonic situations and proved to provide adequate results for flutter prediction at all cases.16

Finally, it is possible to analyze the effects of the Mach number variation on the flutter phenomenon for theaeroelastic system considered in the present work. These effects are demonstrated in terms of characteristicflutter dynamic pressure in Fig. 11(a), characteristic flutter speed in Fig. 11(b), and flutter frequency inFig. 12. For all cases, the initial angle of attack is α0 = 0 deg.

As one can observe, from M∞ = 0.75 on, the flutter characteristic dynamic pressure and speed experiencea more accentuate drop. This is a reflection of the strong aerodynamic lag introduced by the presence ofshock waves in the transonic regime, which occurs in that Mach number range. This type of phenomenonis known as “transonic dip”, and it is one of the greatest motivations to employ sophisticated aerodynamicmodels in transonic flutter predictions, since the use of linear theories may unsafely neglect this behavior andoverpredict the flutter speed. It is also interesting to notice that the flutter frequency decays considerablywith the increase of the Mach number. Nevertheless, within the scope and objectives of the present work,the most relevant aspect of such results is the good agreement between state space solutions and directintegration simulations. This demonstrates that it is possible to effectively and accurately predict flutteronset points with the approximate models proposed in the present paper, which present large computationalsavings when compared to direct integration schemes.

VII. Conclusions

The authors present a complete description of different state space formulations for the representationof aeroelastic systems, including a new discrete-time, z domain model adequate for aerodynamic unsteadyresponses obtained with numerical methods. Initially, some of the most usual approximate models for theaerodynamic impulsive responses in the frequency domain are tested and compared. These approximatemodels supply the continuous-time state space formulations with the required unsteady aerodynamic infor-mation necessary for the construction of root locus stability analyses. This procedure is, then, compared

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X

X

X

X

X

X

X

X

M∞

Q* f

0.5 0.6 0.7 0.80

1

2

3

LTzTDIX

(a) Flutter characteristic dynamic pressure

X

X

X

X

X

X

X

X

M∞

U* f

0.5 0.6 0.7 0.8

4

6

8

10

12

14

16

LTzTDIX

(b) Flutter characteristic speed

Figure 11. Flutter characteristic dynamic pressure and characteristic speed variations with M∞.

X

X

X

X

X

X

X

X

M∞

ωf/ω

α

0.5 0.6 0.7 0.8

0.8

1

1.2

1.4

1.6

1.8

2

LTzTDIX

Figure 12. Flutter frequency variation with M∞.

with the discrete-time formulation results. Comparisons among these results, and the ones provided with thedirect integration of the structural dynamic and aerodynamic equations, demonstrate that the small fittingerrors introduced by the approximating polynomials lead to some inaccuracies in the final stability analysis.The discrete-time formulation avoids such errors with the direct application of the z transform to the timedomain discrete aerodynamic data numerically obtained. Hence, the latter method has shown to be moreaccurate than the previous state space representations.

Furthermore, the approximate models used with the continuous-time formulations do not allow for a largenumber of aerodynamic state variables in the representation of the unsteady behavior, because this leadsto ill-conditioning issues in the final eigenvalue problem. Therefore, although more accurate continuousrepresentations of the aerodynamic responses are feasible through the addition of lag parameters, suchaddition compromises the overall robustness of the aeroelastic results. On the other hand, the large numberof aerodynamic state variables, which result from the z domain state space representation, does not affect theeigenvalue problem conditioning. Moreover, although this last method leads to quite large stability matrices,only some of their terms present nonzero values. Thus, memory requirements can be lessened with the use

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of sparse matrix construction techniques.Therefore, the paper shows that not only a z transform discrete-time state space representation of aero-

elastic systems is possible, but it also is a very effective and accurate formulation. Furthermore, althoughoptimization techniques facilitate the whole interpolating process, it still requires a quite annoying amount ofwork, specially if recurrent applications are considered. Since the representation of the aerodynamic responsein the frequency domain can be evaluated with the application of the z transform, this new approach alsoseems to be a more straightforward method for standard parametric studies, which are typical of aeroelasticanalyses. It is also important to emphasize that the discrete-time state space formulation can also be veryimportant in the field of aeroservoelasticity, because it allows for the determination of control laws suitableto digital control systems.

Finally, it is relevant to notice that the aeroelastic analysis methodologies are successfully applied totypical section models involving subsonic and transonic flow configurations. The flutter points identified usingthe state space formulations are generally very similar to each other, and also similar to values predicted bythe use of direct integration approaches. Hence, this demonstrates that the models proposed in the presentwork for efficient aeroelastic analyses are perfectly capable of offering good results even in the transonicregime.

Acknowledgments

The authors gratefully acknowledge the partial support for this research provided by Conselho Nacionalde Desenvolvimento Cientıfico e Tecnologico, CNPq, under the Integrated Project Research Grants No.501200/2003-7 and 312064/2006-3. This work is also supported by Fundacao de Amparo a Pesquisa doEstado de Sao Paulo, FAPESP, through a MS Scholarship for the first author according to FAPESP ProcessNo. 04/11200-1. Further support from FAPESP is provided through Process No. 04/16064-9.

References

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19Lin, J. and Iliff, K. W., “Aerodynamic Lift and Moment Calculations Using a Closed-Form Solution of the PossioEquation,” NASA TM–2000–209019, NASA Dryden Flight Research Center, Edwards, CA, April 2000.

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