Stability and Response Problems for Fixed-Wing Aeroelasticity · Stability and Response Problems...

Chapter 6

Stability and Response Problems forFixed-Wing Aeroelasticity

The present chapter some issues on the stability and response analysis of fixed wings are pre-

sented. It has been shown how the linearized aeroelastic problem can be written in generalized

modal co-ordinate and in Laplace domain as1(s2M+ K

)q = qDE(s;U∞,M∞)q + f (6.1)

where M, K, and E are the modal mass, stiffness, and GAF matrices respectively, whereas f is

the Laplace transform of the modal load vector due to, e.g., by a gust or by a control surface

input. The above equation in time domain (assuming zero initial conditions) becomes

Mq+ Kq = qD

∫ t

0Ξ(τ ;U∞,M∞) q(t− τ)dτ + f (6.2)

where Ξ(t;U∞,M∞) is the inverse Laplace transform of the GAF matrix.

It is apparent that the problem of the stability can be iteratively faced solving for s the

characteristic equation associated to Eq. 6.1 (this point will be clarified in Section Secs 6.1.1

and 6.1.2); nevertheless, if a Finite-State approximation is assumed for the GAF matrix (Sec.

5.3), the stability problem can be reduced to a standard eigenproblem associated to a standard

first order linear ODE (see, e.g., Eq. 5.109).

As regards the response problems, a first issue concerns the modeling of the input: indeed, dif-

ferent descriptions of the input distribution depending on the aeroelastic system charachteristics

should be given if the considered input is, for example, a control surface angle (Subsections 6.2.2

and 6.2.3) or a vertical gust profile (Subsection 6.2.4). A second issue in the aeroelastic response

1For the sake of notation simplicity the Lagrange variables and quantities associated to the natural mode ofvibrations are not marked by a bar symbol.

121

122

problems concerns the analysis: these problems can be solved evaluating the Fourier transform

(or Discrete Fourier Transform) of the state space vector q using the Laplace transform of the

input vector f together with the algebraic solution (for all ω) of Eq. 6.1 for s = jω, and then

going back to the time domain (by using inverse-Fourier transforms): this is the procedure that

is widely considered in most of the commercial aeroelastic code. However, if a Finite-State ap-

proximation the GAF matrix is available, the aeroelastic respons problem can be easily recast

in a first order ODE form (Eq. 5.109) and the the solution can be analytically evaluated by the

eigenvector method (see, e.g., Ref. [6]): these issues will be dealt with in the present Chapter

(specifically, see subsection 6.2.3). Finally, another relevant issue for the aeroelastic response is

the classification in static and dynamic response problem. Althought the dynamic analysis of

linear system includes the static one, some response aeroelastic problems are typically studied

directly as static problems (Secs. 6.2.1 and 6.2.2) in order to better understand the physical

phenomenon and also to perform a sequence a local linear problem for searching the nonlinear

equilibrium solution. Finally, two standard dynamic aeroelastic response problems are presented

in Secs. 6.2.3 and 6.2.4.

6.1 Aeroelastic Stability Problems

In the present section the stability analysis for fixed wing is presented first giving some brief

theoretical remark on stability on linear systems (Sec. 6.1.1) and then applying these concepts

to the linear aeroelastic systems associated fixed wings which are integral-differential systems

in the time domain. In Section 6.1.2 two iterative procedures widely used for solving the non-

standard eigenproblem associated to such a system are presented; then, in Section 6.1.3, using

the finite-state approximation for the unsteady aerodynamics, it is shown that the stability

problem is easily reduced to a standard eigenproblem and, finally, in Sec. 6.1.4, the role of the

critical eigenvector for both the unconventional or conventional eigenproblem is shown.

6.1.1 Some issues on stability of linear representations of aeroelastic systems

In the Chapter 5 it has been shown as the linearized system representing the aeroelastic behavior

of a fixed wing is an integral-differential system (see Eqs. 5.1 and 5.3 for the Laplace domain

representation and Eq. 5.2 for the time domain one).

Some basic issues on stability analysis of systems with state-space representation is presented,

for instance, in Ref. [6], whilst, for more detailed theory one can refers to Ref. [1]. It is worth

to report here the main result of this theory concerning the stability of linear and stationary

AEROELASTIC MODELING – L. Morino F. Mastroddi c⃝ 123

representations continuous in time2

• the system is asymptotically stable if and only if all the eigenvalues (or poles if one considers

the representation in terms of transfer function) are on left of the imaginary axis in the

complex plane.

• the system is stable if there are no eigenvalues (or poles) on right of the imaginary axis

and, if there are any on the imaginary axis, they have all geometric molteplicity equal to

one.

The eigenvalues (or poles) sn’s under discussion and the corresponding complex eigenvector

wn’s can be evaluated considering the homogeneous system associated to the (linear) aeroelastic

model given by the Eqs. 5.1 and 5.3, i.e.,[s2nM+ K− qDE(sn;U∞,M∞)

]w(n) = 0 (6.3)

where the matrices M and K are diagonal if the natural mode of vibration are considered as

shape functions for the Lagrange equation (see Chapter 1). It is worth pointing out that the

function

w(n)esnt (6.4)

is a fundamental solution of the homogeneous problem (namely, the problem for f ≡ 0) associated

to the integral-differential problem given by Eq. 6.2. This is can be easily demonstrated taking

the substitution q(t) = w(n)esnt in Eq. 6.2 identically satisfying Eq. 6.3: indeed, substituting

Eq. 6.4 into Eq. 6.2 yields

(s2nM+ K)w(n)esnt = qD

∫ t

0Ξ(t;U∞,M∞)w(n)esn(t−τ)dτ

or [s2nM+ K− qD

∫ t

0Ξ(t;U∞,M∞)e−snτdτ

]w(n) = 0

which is true if and only if Eq. 6.3 is satisfied. Thus, also for the time linear integral-differential

system, the general structure of the the solution qo(t) for the homogeneous problem is still given

by

qo(t) =∞∑n

cnw(n)esnt (6.5)

2The discussion on the possibility to have a finite dimension or an infinite dimension for the state-space systemis a key aspect in aeroelasticity because (see Sec.5.3) the aeroelastic system are integral-differential and then withinfinite dimension: indeed, in the following we will refer as “finite-dimensional” a linear representation with a finitenumber of eigenvalues, and as “infinite-dimensional” a linear representation with a infinite number of eigenvalues.

124

where the constant cn are determined by the initial conditions and the index n span from one

to the number of the poles, which, in this case, are not necessarily related to the dimension

of the state space because of the trascendental nature of the GAF matrix. Thus, the previous

conclusion on stability, that are typically given for linear ordinary differential equations, are

then applicable also for integral-differential ones of type given by Eq. 6.2.

Equation 5.3, for prescribed flight parameter M∞, U∞, and ϱ∞ (i.e., qD), gives the poles of

the system if one take the determinant of the matrix[s2M+ K− qDE(s;U∞,M∞)

]and solve for

s. It is apparent that the obtained characteristic equation is generally a trascendent function

of s because of the nature of the aerodynamic operator E(s;U∞,M∞) and then, the obtained

poles are not finite: this fact is naturally connected with the integral nature of the aerodynamic

operator in the time domain. It also implies that the connected eigenproblem is not standard as

the system matrices depend on the eigenvalues: some iterative procedures used in the standard

aeroelastic codes are then presented in Sec. 6.1.2 whereas in Sec. 6.1.3 it is easily shown as the

associated eigenproblem becomes a standard eigenproblem when the finite-state approximation

for the aerodynamic is taken into account.

6.1.2 Stability via numerical iterations: k and p− k methods

In this section some iterative procedures for the stability analysis of the aeroelastic system

written in the homogeneous form given by Eq. 6.3 will be presented (see, e.g., the applications

of these algorythms in Ref. [2]).

The main differences between the presented iterative approaches is that the first ones (k

method) give all the information relative to the stability margins (i.e., flutter and/or divergence

speed, Mach number, air density, and flutter frequency); the second method (p − k method)

gives also the possibility (within certain approximations) to obtain the roots stability locus as

function of an aeroelastic system parameter (e.g., the air speed U∞).

k method

k method, also known as “american method”, was originally proposed by Theodorsen (1935) on

the basic idea of introducing the aerodynamic loads in frequency domain as a mass term. This

iterative method evaluates the critical values (flutter and/or divergence limits) associates to Eq.

6.3. Then, the Eq. 6.3 can be rewritten in the frequeuncy domain

[−ω2M+ iωD+ (1 + ig)K− qDE(ω;U∞,M∞)

]w = 0 (6.6)


where D represents a possible modal viscous damping whereas g represents a purely artificial

hysteretic damping. Next, recasting the aerodynamic contribution as a “mass term” one has[−[M+

qDω2

E(k;M∞)

]ω2 + iωD+ (1 + ig)K

]w = 0 (6.7)

Then, introducing the reduced frequency k := ωb/U∞ (where b is, for instance, the half-chord

length) and introducing the new complex variable

p1 :=iω√1 + ig

(6.8)

one can finally obtain[[M+

1

2ϱ∞b2

E(k;M∞)

k2

]p21 + Dp1 + K

]w = 0 (6.9)

where a factor 1/√1 + ig multiplying the term Dp1 has been neglected as this term is equal to 1

when the procedure converges. The previous problem represents an eigenproblem for the eigen-

values p1 associated to complex matrix (e.g., the aerodynamic matrix E) and depending to the

parameters M∞, ϱ∞, and k: nevertheless, the parameter k is correlated to the unknown eigen-

value p1 by its own definition and Eq. 6.9. Then, an iterative procedure has to be performed:

for given ϱ∞ and M∞, the square of the eigenvalues

p21 =−ω2

1 + ig≡ −ω

2(1− ig)

1 + g2(6.10)

can be evaluated for several values of the reduced frequency k: the condition Im(p21) = 0 - when

reached for k = kF - implies that g (which is purely artificial) is zero as it has to be. In the

meanwhile, the characteristic Eq. 6.6 is satisfied. Then, for k = kF , the real part of p21 ≡ −ω2F

yields the limit value ωF and, finally, the limit speed U∞F = ωF b/kF can be evaluated.

A numerically more efficient variant of this procedure –also known as K − E method (E is

for efficient, Ref. [2])– can be obtained introducing the new complex variable

p2 :=iU∞√1 + ig

(6.11)

and then recasting Eq. 6.6 as[[M

(k

b

)2

+ϱ∞2

E(k;M∞)

]p22 +

(k

b

)Dp2 + K

]w = 0 (6.12)

where a factor 1/√1 + ig multiplying the term Dp1 has been again neglected as this term is

equal to 1 when the procedure converges. The previous problem represents an eigenproblem for

the eigenvalues p2 associated to complex matrix (e.g., the aerodynamic matrix E) and depending

to the parameters M∞, ϱ∞, and k: nevertheless, the parameter k is correlated to the unknown

126

eigenvalue p2 as both of them depend on U∞ (Eq. 6.11). Then, an iterative procedure has to

be performed: for given ϱ∞ and M∞, the square of the eigenvalues

p22 =−U2

∞1 + ig

≡ −U2∞(1− ig)

1 + g2(6.13)

can be evaluated for several values of the reduced frequency k: the condition Im(p22) = 0 - when

reached for k = kF - implies that g (which is purely artificial) is zero as it has to be. In the

meanwhile, the characteristic Eq. 6.6 is satisfied. Correspondingly, for k = kF , the real part

of p22 ≡ −U2∞F

yields the limit speed U∞F and, finally, the critical flutter angular frequency

ωF = U∞F kF /b can be evaluated.

For the role and meaning of the critical eigenvector qF corresponding to the evaluated critical

eigenvalue sF = iωF that can be easily evaluated by the homogeneous system given by Eq. 6.3,

see Sec. 6.1.4.

p− k method

p− k method, also known as “british method”, was developed by Hassig (1971) on the base of

an idea originally introduced by Frazer and Duncan in introducing the aerodynamic loads in

frequency domain as stiffness and damping terms.3 This iterative method evaluates all the poles

associated to Eq. 6.3 for all choices of the flight parameters U∞, M∞, and ϱ∞. Then, the Eq.

6.3 can be rewritten[s2M+ sD+ K− qD [ER(k;M∞) + iEI(k;M∞)]

]w = 0 (6.14)

where a first approximation has been taken into account not evaluanting the GAF matrix

(specifically, its real and imaginary parts) as function of the unknown poles s (or p , see Section

5.1.5) but as function defined in the dimensionless Laplace sub-domain k = Im(s)ℓ/U∞ (i.e.,

a Fourier domain). This is essentially due to the fact that typically the unsteady-aerodynmic

codes work in frequency domain instead of Laplace domain: note also that this approximation

is weaker as the evaluated eigenvalues are more distant from the imaginary axis. Then, Eq. 6.14

can be recast as[s2M+ s

[D− 1

2ϱ∞U∞b

EI(k;M∞)

k

]+

[K− 1

2ϱ∞U2

∞ER(k;M∞)

]]w = 0 (6.15)

where, within the same order of approximation discussed above, a factor 1/k multiplying

EI(k;M∞) has been used instead of U∞/sb.4 Furtheremore, introducing the velocity vector

3Note that the letter p in the title was historically introduced as representing the dimensional Laplace domainvariable that we have always indicated with s.

4Note also that the term EI(k;M∞)/k is not singular for k → 0 as discussed in Sec. 5.3.


v := sw one can recast Eq. 6.15 in a first order form as

(Aϱ∞,U∞,M∞;k − sI) u = 0 (6.16)

where

Aϱ∞,U∞,M∞;k :=

[O I

−M−1[K− 1

2ϱ∞U2∞ER(k;M∞)

]−M−1

[D− 1

2ϱ∞U∞bEI(k;M∞)

k

] ](6.17)

and

u :=

(wv

)(6.18)

First, note that the 2N×2N matrix A is a real matrix for prescribed values of the parameters U∞,

ϱ∞, M∞, and k and then, for given values of such parameters, one has a standard eigenproblem

of a real matrix. This parametric dependency has been emphasized and, in particular, the

dependency on

k :=Im(s)b

U∞(6.19)

shows that the matrix is dependent on the complex eigenvalues s: this implies that a recorsive

procedure to solve the eigenproblem given by Eq. 6.16 has to be performed as follows.

First, suppose to fix the value of the physical flight parameters U∞, ϱ∞, and M∞: the

procedure can be re-iterated for each value of the parameters in order to obtain, for instance,

the stability scenario given by the root locus. After this position the matrix A depends only on

k: then, the p− k procedure consists of two parts.

• Divergence condition (zero poles)

The matrix A is evaluated for k = 0: if the standard eigenproblem given by Eq. 6.16 yields

an eigenvalues equal to zero, then a divergence condition occurs: indeed, in this case the

condition given by Eq. 6.19 has been also satisfied.

• Evaluation of poles not equal to zero

Consider the possibility to evaluate the aeroelastic pole or eigenvalue which originally, for

flight parameters all equal to zero, was the first structural eigenfrequency: then, one can

consider as first initial guess for the reduced frequency

k(0)1 :=

ω1b

U∞(6.20)

where ω1 is the first angular structural eigenfrequency. Then, by the eigenproblem given by

Eq. 6.16, the eigenvalues s(0)i (i = 1, 2, ..., 2N) can be evaluated. Next, consider the eigen-

value belonging to this set with imaginary part closer to ω1: suppose that this eigenvalue

128

be s(0)1 and consequently one can obtain another estimate for the reduced frequency

k(1)1 :=

Im(s(0)1 )b

U∞(6.21)

If ε represents a satisfactory level of accuracy, whenever the condition∥∥∥k(1)1 − k(0)1

∥∥∥ < ε (6.22)

is satisfied, the pole s(0)1 is the first pole of the aeroelastic sistem. If not, the matrix given

by Eq. 6.17 can be evaluated for k = k(1)1 and the eigenproblem given by Eq. 6.16 can be

performed again: if the eigenvalues are indicated with s(1)i (i = 1, 2, ..., 2N), and with s

(1)1

the eigenvalue with imaginary part closer to ω1, one can obtain another estimate for the

reduced frequency

k(2)1 :=

Im(s(2)1 )b

U∞(6.23)

and then another stop condition for the iteration procedure can be performed considering

a relation similar to Eq. 6.22 changing the role of the indeces.

Once the first pole has been caputered, the second pole can be estimated re-iterating the

procedure starting with the second structural eigenvalues, i.e., considering the reduced

frequency

k(0)2 :=

ω2b

U∞(6.24)

These two steps procedure can be re-performed for any values of the flight parameters U∞, ϱ∞,

and M∞ with in the flight envelope in order to obtain the complete stability scenario. The

approximations considered in the above recoursive resolution show that the obtained root loci

are more approximated as they are closer to the imaginary axis (see, e.g., Refs. [41, 15]).

Again, for the role and meaning of the critical eigenvector vF (or qF considering a second

order form, see Eq. 6.3) corresponding to the evaluated critical eigenvalue sF = iωF that can

be easily evaluated by the homogeneous system given by Eq. 6.16, one can see Sec. 6.1.4.

Next, an example of stability aeroelsatic analysis is presented. In Table 6.1 are shown the first

five numerical eigenfrequencies obtained via Finite Element method (MSC.NASTRAN code)

relative to the the so-called Body-Freedom-Flutter (BBF) configuration, a test case model

configuration introduced to study the coupling between flight dynamics and aeroelasticity, see

Fig. 6.1. The coresponding mode-shape functions are depicted in Figs. 6.2-6.4. This example

of numerical eigenanalysis is considered in the following as basis to perform the aeroelastic


Figure 6.1: Body-Freedom-Flutter aircraft configuration (left Figure) and its Finite ElementModel (right Figure)

Mode n. type freq. (Hz)

1 symmetric bending (B S) 5.832 antisymmetric bending (B AS) 8.833 in-plane bending (B P) 13.454 antisymmetric torsion (T AS) 19.825 symmetric torsion (T S) 20.09

Table 6.1: Table of the first five numerical eigenfrequency of the wing structure of Fig. 6.1obtained via Finite element method

Figure 6.2: First and second numerical mode shapes of the BFF aircraft in Fig. 6.1

Figure 6.3: Third and fourth numerical mode shapes of the BFF aircraft Fig. 6.1

130

Figure 6.4: Fifth numerical mode shape of the BBF aircraft in Fig. 6.1

Figure 6.5: Root loci for the BFF of Fig. 6.1 as function of 5U∞ (M∞ = 0 and ϱ = 1.117 kg/m3)


Figure 6.6: Modification of the root loci of Fig. 6.5 when ϱ∞ = 0.44, 0.74, 1.04, 1.34 kg/m3 andM∞ = 0.

stability analysis. Figure 6.5 shows the root loci with respect to U∞ for a stability analysis

with p − k method performed on the wing configuration presented in Section 1.3.1: using a

modal basis composed by the first five structural modes (see Figs. 6.2-6.4), a p − k procedure

has been performed using the MSC.NASTRAN code (solution sequence 145, Ref. [2]) with a

lifting surface approach known as Doublet Lattice Metthod (DLM). The mach number M∞ = 0

wheras the air density ϱ∞ = 1.117 kg/m3. Note that the critical root associated to the flutter is

that coming from the first torsional anti-symmtric mode (see Tab. 6.1 and Fig. 6.3) Figure 6.6

shows the variations of this root locy when the air density is modified from 0.44 to 1.34 kg/m3,

whereas Fig. 6.7 shows the variation when the M∞ is modified from 0 to 0.6.

6.1.3 Stability via Finite-State approximation

In Section 5.3.3 it has been shown that the linearized equation of aeroelasticity of a fixed wing

can be reduced in a state-space form and, specifically, the free aeroelastic response as

x = Aϱ∞,U∞x (6.25)

where x is the state-space vector (including the lagrangean and the aerodynamic states, see Eq.

5.110) and Aϱ∞,U∞ is the aeroelastic state-space matrix as obtained by the finite-state modeling:

132

Figure 6.7: Modification of the root loci of Fig. 6.5 when M∞ = 0, 0.2, 0.4, 0.6 and ϱ∞ =1.117kg/m3.

this matrix is explicitly dependent by the air speed U∞ and the air density ϱ∞ (see Eqs. 5.108

and 5.110) and implicitly dependent by the Mach number M∞.

As well known by the the theory of the linear ODE, the stability of such a dynamical system

can be completely described studying the corresponding standard eigenproblem

(Aϱ∞,U∞ − snI) u(n) = 0 (6.26)

The real part of the obtained eigenvalues sn will give quantitative indications on the system

stability. The eigenvectors u(n) will be 3N -dimensional and its first N -partition will have the

same meaning of the complex eigenvector as obtained by the k or p − k approach using the

Lagrangean aeroelastic equation in the original second-order form. The physical meaning of

such an eigenvector when corresponding to the critical eigenvalue will be illustrated in Section

6.1.4.

6.1.4 The role of the critical eigenvector in the stability analysis

The aeroelastic stability analysis consists in the resolution of the Eq. 6.3 in term of the critical

(flutter or divercence) parameters (Ucr, Mcr, ϱcr) and the critical eigenvalues scr (scr = 0 for


divergence, and scr = iωcr for flutter): several numerical procedures have been presented to

reach this gol, but, in any case, once the complete aeroelastic matrix corresponding to these

critical parameter has been evaluated, also the corresponding N−dimensional eigenvector wcr

could be evaluated as non trivial solution of the homogeneous algebraic problem given by Eq.

6.3. Note that if the problem were written in first order form, e.g., as in the p− k method, the

critical eigenvector should be 2N -dimensional as given by (Eq. 6.18)

ucr =

(wcr

scrwcr

)(6.27)

whereas, if one used, for example, the finite-state aerodynamics, this vector would be 3N -

dimensional, see Eq. 6.26. The objective of this section is to show the role of this eigenvector

in the system aeroelastic responce in time domain. Indeed, the time response in this critical

condition is given, in term of Lagrange variables and on the base of the associated eigenproblem,

by

qcr(t) =Nts∑n=1

cnesntw(n) (6.28)

where Nts is for the total number of states used for the stability analysis, i.e., it may be 2N if

one considers the p − k method or 3N if one considers the finite-state aerodynamic approach

presented in Sec. 5.3.2; furthermore, cn are the constants depending on the assigned initial

conditions and the sum is theoretically exthended on all the eigenvalues (and eigenvectors)

evaluated at the critical conditions. Because of the critical condition, the system is stable not

asymptotically and then at steady conditions the response will be governed only by the critical

modes (with the eigenvalue couple s1,2 = ±iωcr), i.e.,

qcr(t) ≃ ceiωcrtwcr + C.C. ≡ 2c [wRcr cos(ωcrt)− wIcr sin(ωcrt)] (6.29)

where the real and the imaginary parts of the critical eigenvector have been indicated. Finally,

the aeroelastic response of the system in this condition and in term of the physical displacement

is given by (see, e.g., Eq. 1.4)

ucr(ξα, t) ≃

N∑n

qcrn(t)ψn(ξα) (6.30)

= 2c[wRcr1

cos(ωcrt)− wIcr1sin(ωcrt)

]ψ1(ξ

α) + 2c[wRcr2

cos(ωcrt)− wIcr2sin(ωcrt)

]ψ2(ξ

α)

+ . . . + 2c[wRcrN

cos(ωcrt)− wIcrNsin(ωcrt)

]ψN (ξα)

which can be written - indicating with |wcrn | and wcrn the modulus and the phase of the n−th

complex component of the critical eigenvector - as

ucr(ξα, t) ≃ 2c |wcr1 | cos

(ωcrt+ w

cr1

)ψ1(ξ

α) + 2c |wcr2 | cos(ωcrt+ w

cr2

)ψ2(ξ

α)

+ ... + 2c |wcrN | cos(ωcrt+ w

crN

)ψN (ξα) (6.31)

134

Component Real part Imag. part

1 0.0 + 0.02 0.0777 + 0.18443 0.0 + 0.04 1.0 + 0.05 0.0 + 0.0

Table 6.2: Critical eigenvector for the stability analysis presented in Sec. 6.1.2

Equation 6.31 shows that the free aeroelastic response in critical condition is a combination

(in the space dimension) of the function ψn(ξα) (i.e., the natural structural mode of the struc-

ture) with coefficients that are simply periodic functions (with angular frequency equal to ωcr)

with different phases: the amplitude of this functions and the entity of this phase angles are

clearly regulated by the several components of the critical eigenvector. Then, the role of the

critical eigenvector is that to quantitatively indicate the structural-mode partecipation to the

free aeroelastic response. These issues gives a quite different characherization to the aeroelastic

mode with respect to the structural (undamped) ones.

Table 6.2 shows the critical eigenvector obtained in the stability analysis shown in Fig. 6.5:

the coupling nature of the “aeroelastic mode” is apparent as given by the aeroelastic coupling

between the second and the fourth structural modes.

6.2 Aeroelastic response problems

In the present section first, some typical static aeroelastic response problem will be described

and modeled in the subsection 6.2.1 and 6.2.2. Then, in the subsection 6.2.3 and 6.2.4 some

typical dynamic aeroelastic response problems are described and modeled too.

6.2.1 Static Aeroelastic response of a wing in horizontal flight

The static aeroelastic deformation of an aircraft in a steady horizontal flight is one of the relevant

issue in aeroelasticity. Typically the trim variables, i.e., the angle of attack α, the control surface

angles βi, are problem unknowns together with the elastic Lagrange vector q. For the sake of

simplicity and clearness, we will consider in the following as trim variable only the angle of

attack α, but similar considerations and conclutions could be done taking into account also the

other trim variables of the complete aircraft.

Suppose initially that the aircraft angle of attack α is a prescribred input variable for the

problem (this may physically correspond to wind-tunnel condition of a model). Then, the


normalwash on the wing surface (i.e., the component of the air speed normal to the wing

surface) is naturally imposed to be equal to that of the solid material particle of the wing surface

(aerodynamic boundary condition, see Section 5.1.1), see Eq. 5.13. Then, if one considers a

zero-order discretization for the aerodynamic surface and indicates with j the index relative to

the generic aerodynamic panel, one obtains

χj = vb(ξαj ) · n(ξαj ) =

(−iU∞ +

N∑n

qnψ(n)(ξαj )

)·(nα(ξ

αj ) +

N∑n

∆nn(ξαj )qn +H.O.T.

)(6.32)

where vb(ξαj ) is the body velocity (rigid, −iU∞, plus elastic), and n(ξαj ) is the actual normal

vector on the j-th panel composed by a rigid contribution depending by the imposed angle of

attach, nα(ξαj ), and one due to the linear elastic deformation. Noting that the vector nα(ξ

αj )

can be expressed by

nα(ξαj ) ≃ n0(ξ

αj ) + α

−nz0

0nx0

= n0(ξαj ) + αn1(ξ

αj ) (6.33)

when one considers small values of the angle of attack and negligible value of the diedral an-

gle of the wing, one can rewrite Eq. 6.32, developing the inner product and considering the

dimensionless normalwash χ := χ/U∞, as

χj = −i · n0(ξαj )− α i · n1(ξ

αj ) (6.34)

−N∑

n=1

qni ·∆nn(ξαj ) +

N∑n=1

qnψ(n)(ξαj ) ·

[n0(ξ

αj ) + α n1(ξ

αj )]

U∞+ H.O.T.

where the underlined terms5 are the normalwash contributions due to the rigid body motion

(rigid configuration with zero angle of attack plus non-zero angle of attack) whereas the double-

underlined ones are those induced by the elastic deformation.

Now, if one consider the normalwash (due to the global motion and attitude angle) in steady

condition (q = 0), one has in the linearized case

χstj = −i · n0(ξ

αj )− α i · n1(ξ

αj )−

N∑n=1

qni ·∆nn(ξαj ) = χ0j + α χ1j +

N∑n

E1jn(0) qn (6.35)

where the vectors with entries χ0j and χ1j are implicitly defined and E1jn(0) are the entries of

the normalwash matrix as defined in Sec. 5.1.1 at zero frequency.

5Note that these purely steady terms have been neglected in the linear dynamic analysis (flutter analysis, seeSection 5.1.1) as uninfluent in the stability discussion.

136

Thus, the Lagrange equation of motion 5.1 (and considering also Eqs. 5.3 and 5.4) in time

domain and in steady state condition and after imposing the angle of attack α yield. . .

mnω2n

. . .

− qD E(0)

q = qDE4 E3(0) E2(0) (fχ0 + fχ1 α) (6.36)

where the parametric dependency of the aerodynamic matrices by the mach number M∞ and

the flight speed U∞ has not showed. The Equation 6.36 is a linear algebraic equation which

gives the static deformation vector q once the angle of attack α is prescribed. Note that the

resolution of such a system is possible if (and only if) the system is divergence free, i.e., if

det

. . .

mnω2n

. . .

− qD E(0)

= 0 (6.37)

It is worth to point out that the static-response problem considered above is relative, e.g., to

the case of a wing in a wind tunnel and not to an airplane in flight condition: in this case, the

airplane weight W is an input variable of the problem, and the set of the N elastic Lagrangian

variables together with the angle of attack α are the N+1 unknowns of the problem. The N+1

associated equations are the N Lagrange equations associated to the elastic deformation and the

equilibrium equation associated to the vertical rigid motion: the Lagrangian variable associated

to this motion is known and equal to zero, thus q0(t) ≡ 0 and the variables associated to the

elastic modes are unknown and they are q1, q2, ...,qN . Then, the final algebraic linear sistem is

0 0 ... 0m1ω

21

m2ω22

. . .

mNω2N

− qD

E0,1(0) E0,2(0) ... E0,N (0)E1,1(0) E1,2(0) ... E1,N (0)E2,1(0) E2,2(0) ... E2,N (0)

... ... ... ...EN,1(0)EN,2(0) ... EN,N (0)

q1q2...qN

= qDE4E3(0)E2(0) (fχ0 + fχ1 α) +

W00...0

(6.38)

where the GAF matrix is a (N + 1)×N matrix because it represents now the (static) loading

influence of the N elastic modes on the N +1 equilibrium equations: moroever, also the matrix

E4 in Eq. 6.38 has (N + 1) rows since it performs the projection of the pressure coefficients on

the body surface on the vertical rigid mode and on the N elastic modes as well.6

6It is apparent that the linear system given by Eq. 6.38 can be easily recast in order to be written in thestandard form Ax = b where xT = α, q1, q2, ..., qN.


As further development, also a rigid pitch mode or Langrangean variable could be included in

the aeroelastostatic trim analysis considering also the corresponding Lagrange equation given by

the equilibrium to the rigid pitch rotation. Also in this case a trim unknown like, for example,

the aileron wing-tail angle δWT , have to be included in order to close the problem. Thus, treating

such an angle δWT in a similar way as previously done for the angle α, one can obtain

0 0 ... 00 0 ... 0m1ω

21

m2ω22

. . .

mNω2N

− qD

E−1,1(0)E−1,2(0) ... E−1,N (0)E0,1(0) E0,2(0) ... E0,N (0)E1,1(0) E1,2(0) ... E1,N (0)E2,1(0) E2,2(0) ... E2,N (0)

... ... ... ...EN,1(0) EN,2(0) ... EN,N (0)

q1q2...qN

= qDE4E3(0)E2(0) (fχ0 + fχ1 α+ fχ2 δWT ) +

0W00...0

(6.39)

where a further lagrange equation for the pitch rigid motion (labelled with the index −1) ha

been also included and fχ2 has been defined in a simalar way like fχ1 .

As final comment, it is worth to note that the typical elastostatic displacement and deforma-

tion of an aircraft in steady operative condition may be so high to affect the initial linearizing

hypotheses considered in the present analysis (modal superposition, linearized-potential-flow

discretization, linearized kinematics in modeling the trim variable α, etc.). In this case a itera-

tive step-by-step procedure re-discretizing the structures and the aerodynamics after each step

is recommended.7

Comment on linear v.s. nonlinear analyses for aeroelastic response

The general problem of static/dynamic response of an aircraft in o horizontal flight can be

carried out with several level of model approximation. Indeed if the nonlinear aeroelastic system

is generally modeled as ther nonlinear ordinary differential equation model

x = fU (x) (6.41)

7A practical approach (Ref. [8]) often considered in the aerospace industries is the following: if ∆α(i) representsthe correction to the angle of attack at the i-th iteration, one can assume that this correction follow the law of ageometric series, i.e., be ∆α(i+1)/∆α(i) = R < 1, and then one could extrapolate the final correction due to theelasticity for the angle of attack after the first estimate ∆α(1)

∆α = ∆α(1) 1

1−R(6.40)

138

where f is a generically nonlinear function of x and its parametric dependence by to the generic

flight paramter U is emphasized. When a equilibrium solution xe (namely, a trimmed aeroelas-

tostatic solution) is found for a fixed value of U such that

fU (xe) = 0 (6.42)

a system linearized dynamics can be studied rewriting Eq. 6.41 as

x = A (x− xe) + HOT with A := ∇fU (x)|x=xe(6.43)

and then avoiding the higher order terms (HOT). Thus, three level of dynamical-model fidelity

can be considered (with decreasing fidelity order):

• Fully nonlinear description

It consists of considering as aeroelastic model the Eq. 6.41 (including structural as well

aerodynamic nonlinear description) to be numerically solved by direct time domain simu-

lation.

• Statically nonlinear and dynamically linear description

This constist of numerically evaluating the static solution xe of Eq. 6.42 taking into

account of the overall nonlinear effects and the by studing the linearized dynamic in the

neightbourhood of the equilibrium solution as given in Eq. 6.43 avoiding the HOT.

• Statically linear and dynamically linear description

This constist of numerically evaluating the static solution of Eq. 6.42 not taking into

account the nonlinear nature of the vector fU with respecto to the state space vector x and

with respect to the trim variables (e.g., angle of attack α, control-surface angle δ, etc...)

and, moreover, by studing the linearized dynamic in the neightbourhood of the equilibrium

solution as given in Eq. 6.43 avoiding the HOT.

The third modelling listed above is that currently used in the linear aeroelastic analysis and

presented in the present Chapter.

6.2.2 Static Aeroelastic response of a wing to an aileron step angle: aileroneffectiveness and reversal

In order to face the modeling of the static aeroelastic response due to a control-surface step

angle, let us consider the example of a thin and straight airfoil in a incompressible potential

flow with a trailing-edge control surface. The aeroelastic modeling of this airfoil is the standard


Figure 6.8: Typical section with a aileron.

typical section in steady state condition, i.e., the airfoil is elastically connected with a (vertical)

linear and torsional springs with constants Kθ and Kh located at the airfoil elastic center: it

has a global angle of attack which is the sum of the imposed incidence α and the elastic angle of

torsion θ, see Fig. 6.8. The lift generated with no control surface contribution is applied to one

fourth of the chord c from the leading-edge: thus the static forces equilibrium to the vertical

traslation and the pitch rotation can be written as

Khh = −qDS [κ(α+ θ) + κ1δ] (6.44)

Kθ θ = qDSc [eκ(α+ θ)− κ1e1δ] (6.45)

where h is the plunge state variable with sign opposite to the generated lift, δ is the control

surface angle of deflection, qD is the dynamic pressure, S is the airfoil surface, κ is the lift

slope coefficient for the airfoil, κ1 is the lift slope coefficient for the control surface, e is the

dimensionless distance (with respect to the chord) between one-fourth-chord point and the elastic

center, and e1 the dimensionless distance (with respect to the chord) between one-fourth-control-

surface point and the elastic center. Considering that the total lift changed in sign is represented

by the RHS term in Eq. 6.44 and after an algebraic recasting, the equation 6.45 gives

LtotqDS

= ακ

1− qDqdiv

+ κ1δ

1− qDqrev

1− qDqdiv

(6.46)

with

qdiv :=Kθ

Sκceqrev :=

qdiv1 + e1/e

(6.47)

Indeed, qdiv is the critical divergence dynamical pressure (as apparent noting that this value

for qD make undetermined the value of state variable θ in Eq. 6.45), whereas qrev is another

140

special value for the dynamical pressure: in fact, if the actual value of qD is greater than qrev

the aileron angle δ give to the lift a contribution that is opposite to the desired one as shown by

Eq. 6.46. This phenomenon due to the wing torsional flexibility is known as aileron reversal:

it is apperently not a destructive aeroelastic phenomenon but it can considerably affect the

effectiveness of the control surfaces even if the flight conditions are with qD < qrev but with

an additive contribution to the lift due to δ that is not so significant. It is also apparent be

the elementary example presented above, that the optimal condition to prevent this undesired

effects is the condition e1 = 0, i.e., qdiv ≡ qrev which implies that the flexibility has no effect on

the control surface effectiveness (see Eq. 6.45).

The fact that this phenomena are deeply connected with the flexibility of the wing body is

clearly showed by the following example: if one consider a roll maneuver around the x axis in

flight dynamics (i.e., only rigid body motion), one has the equation

Jx p =1

2ϱ∞U2

∞Sc Cℓ (6.48)

where p = ϕ is the angular roll velocity, Jx is the moment of inertia around the x-axis, and Cℓ

represents the lift contribution useful to perform the roll maneuver: this coefficient, following a

standard flight-mechanics procedure, can be decomposed as

Cℓ = Cℓδδ + Cℓp

pc

2U∞+ Cℓp

pc

2g(6.49)

i.e., in a contribution (for increasing the lift) due the aileron angle δ, a contribution due to the

angular velocity p, and one due to the angular acceleration p. Thus, for a given aileron step

angle δ, in steady state condition (i.e., p = 0, p = pst) one has

0 =1

2ϱ∞U2

∞Sc

(Cℓδδ + Cℓp

pstc

2U∞

)=⇒ pst = − Cℓδ

Cℓp

2U∞c

δ (6.50)

The coefficient Cℓδ is apparently greater then zero because is typically positive the lift variation

due to a positive variation of the aileron δ. Furthermore, if one considers the roll angular velocity

positive accordingly with the x-axis (oriented conventionally like U∞), i.e., positive when the

right wing go up in the roll maneuver, this motion induces a decrement in the angle of attack

of such a wing and then its lift will decrease: this implies that Cℓp < 0, and then the conclution

is that in flight mechanics (no elastic deformation), for a positive step deflection of the aileron

angle δ one always obtain a positive roll velocity pst as indicated by the Eq. 6.50

Next, the general aeroelastic problem of the aeroelasto-static response to a control surface step

angle will be considered in the following. First, the aeroelastic Lagrange equation of motion will

include the dynamics and aerodynamics of the control surface and those of the rigid body motion

which is the result of the maneuver: these two new degrees of freedom can be introduced in a


modal way,8 i.e., considering an aileron shape mode function ψδ(ξα) and a roll-rigid-mode shape

function ψφ(ξα) with associated Lagrange variables δ and the roll angle φ respectively (note

that p = φ). This implies that the motion is assumed to be locally described as a modal-shape

superposition

u(ξα, t) = δ(t)ψδ(ξα) + φ(t)ψφ(ξ

α) +N∑n

qn(t)ψn(ξα) (6.51)

The shape function ψδ(ξα) is zero anywhere but on the control surface domain where it fit

a control surface displacement with an angle of 45 degree; in a similar way, ψφ(ξα) is shape

function fitting a rigid roll rotation of the wing of 45 degree, i.e., if the material co-ordinate

coincide with the carthesian ones at the undeformed configuration,

ψφ(ξα) =

[−(ξ3 − ξ30)

]j + (ξ2 − ξ20)k (6.52)

The reason of using a 45 degree angle is due to the fact that in the chosen hypothesis of linearized

kinematics the tangent can be approximated with the angle and then, see Eq. 6.51, the chosen

Lagrange variable have physically the meaning of aileron and roll angle respectively. Thus, the

2 +N associated Lagrange equations of motion in Laplace domain ares2

mδδ mδφ ... mδn ...mφδ mφφ ... 0 ......

.... . .

mnδ 0 mn...

.... . .

+

mδδω2δδ 0 ... 0 ...

0 0 ... 0 ......

.... . .

0 0 mnω2n

......

. . .

−qD

Eδδ Eδφ Eδ1 ... EδN

Eφδ Eφφ Eφ1 ... EδN

E1δ E1φ E11 ... E1N...

......

......

ENδ ENφ EN1 ... ENN

δφq1...qN

=

100...0

M (6.53)

where the terms of the first column and row of the mass matrix indicate the mass coupling

between the aileron motion and the natural modes (indeed, the rigid roll motion is orthogonal

to the elastic modes if and only if these are the elastic modes of the free structure). This

coupling is assumed to be zero in term of stiffness as shown by the similar terms of the stiffness

matrix, whereas a general coupling is present in term of aerodynamics as shown by the GAF

matrix. Note also that the first Lagrange equation represent the rotational equilibrium around

the aileron hinge: thus, M is (the Laplace transform of) the external moment applied to the

aileron to perform the maneuver, whereas mδδω2δδ represents a concentrated hinge stiffness.

8This is can be always valid in linear analysis.

142

Once a step external moment is applied to the aileron hinge, Eq. 6.53 allows to evaluate the

Laplace transform of the state-space vector; in particular φ and the roll angular velocity p = sφ

can be evaluated and the efficiency and eveluntally a reversal phenomenon can be verified. Note

that using this approach the complete dynamic response can be evaluated, i.e., the transient

up to the steady solution is evaluated: however, as these phenomena are typically static, the

purely static solution could be evaluated from Eq. 6.53 searching directly the static solution,

e.g., writing the system in the first order form and searching static solutions (i.e., p = 0).

6.2.3 Dynamic Aeroelastic response to an aileron step angle: finite-state andnot finite-state approximation

The aeroelastic dynamic response to a control surface step input has always been a relevant

issue in the story of aeroelasticity (see, e.g., Ref. [9]) for the aircraft analysis and design. An

actual example on modeling the aeroelastic response to a control-surface step angle has been

showed at the end of Section 6.2.2. Specifically, the transfer function vector Hcs(s) of the SIMO

aerelastic system relative to the problem shown in such a Section is defined as

Hcs(s) :=(s2M+ K− qDE(s;U∞,M∞)

)−1

100...0

(6.54)

It is worth to point out that the time-response can be obtained taking the inverse Laplace

transform9 of the obtained output vector: this approach is commonly considered in most of the

commercial aeroelastic code (see e.g., Ref. [2]). Nevertheless, the Finite-State aerodynamic for

the approximation for the GAF matrix can be taken into account as shown in Section 5.3.3; the

system can be recast in a first order state-space form as

x = AU∞x + bM(t) (6.55)

where x and AU∞ where defined in the Eq. 5.110, M(t) is the scalar input vector of the applied

hinge moment, whereas b is the input vector defined for this specific response problem as

b :=

OIcsO

with Ics :=

10...

(6.56)

9Note that for s = iω the frequency response function matrix can be equivalently considered in the dynamicanalysis and the Fourier transform instead of the Laplace transform can be applied for the input and outputsignal.


Figure 6.9: Sections characteristics in a swept wing

Comments: influence of sweep angle

As shown by the simple 2-D model presented at the beginning of Section 6.2.2, the aileron

effectiveness is essentially regulated by the torsional stiffness. This comment can be exthended

in 3-D for a wing with zero sweep angle but not if a sweep angle is present. Indeed, in this

case if one denotes with A′A′ a generic wing cross section orthogonal to the wing elastic axis

and with AA a generic cross section streamwise oriented (see Fig. 6.9), one can realize that

the lift increasing due to the aileron essentially induces an upward bending deflection: because

of this deflection, all the points belonging to the section A′A′ moves up of the same quantity

whereas the aerodynamic section AA is such that the leading-edge point A moves up less than

the trailing-edge point A′. The final effect of this lift increasing is that the static angle of attack

is lower than before because of the bending flexibility of the wing. The higher is the sweep angle

Λ the higher is this effect: then, the limit reversal speed will exhibit a trend with respect to the

sweep angle Λ as qualitatively shown in Fig. 6.10.

Neverteless, if one consider now a purely bending static perturbation, all the points belonging

to the section A′A′ moves up of the same quantity whereas the aerodynamic section AA is such

as the leading-edge point A moves up less than the trailing-edge point A′. Thus, the final effect

144

0

Lambda

U_DU_I

U_F

Figure 6.10: Qualitative curves of divergence speed (Udiv), flutter speed (UF ), and limit reversalspeed (Urev) as function of wing sweep angle

of the perturbation due to the bending lagrange variable is that the static angle of attack is

lower than before, i.e., the limit divergence speed increases as the sweep angle Λ increases (see

Fig. 6.10) and it is conversely emphasized by the presence of a negative sweep angle. It is worth

to point out that the mechanism of divergence instability for swept wing is characterized by the

partecipation of bending modes instead of the purely torsional as shown by the simple theory

of the typical section (see Section 6.2.1). This item could be quantitatively confirmed analyzing

the critical eigenvector corresponding to the zero eigenvalue at divergence condition (see Section

6.1.4): indeed, the most relevant component of such a vector should correspond to the assumed

bending mode because of the physical mechanism described above that involves essentially the

bending deformation. In the same Figure 6.10 also the flutter speed as function of the sweep

angle Λ is shown: the presence of the sweep angle induces a stabilizing effect although it is less

sensitive with respect to that happening in static aeroelasticity.

6.2.4 Dynamic Aeroelastic response to a deterministic or stochastic gust

The aerodynamic coupling between the flow conditions induced by an aircraft motion and the

presence of an athmospheric turbulence is a complex issue that could not be analyzed a priori

as a one-way coupled or, ”forced”, phenomenon.


First, one could introduce a total potential function10

φtot = φ + φ′ (6.57)

where φ represents the mentioned average value of the velocity potential due to the flow com-

ponent which does not generate vorticity, whilst φ′ is the contribution due to the atmospheric

turbulence. Thus, the total flow velocity can be obtained by Eq. 6.57 (which, indeed, could not

be written in term of potential function) as

v = ∇φ+ vT (6.58)

with

vT = ∇∗ ×∫∫∫

VζGdV (6.59)

where ζ := ∇× v is the vorticity vector present in the aerodynamic field. Using the Equation

6.58 one can obtain for the boundary condition of the potential problem

vB · n =∂φ

∂n+ vT · n

∂φ

∂n= vB · n− vT · n (6.60)

where vB is the flight speed of the body.

Therefore, assuming the gust problem as a one-way coupling problem, then the turbulence is

supposed to influence the aircraft aerodynamic but not the reverse. This implies the following:

• assume to know the vorticity distribution ζ(x, t) due to the turbulence in all over the fluid

field (e.g., considering the equation of the vorticity evolution);

• then, using th Eq, 6.59, one can evaluate vT and, finally, solve the potential flow with

boundary condition as given by Eq. 6.60 in the flow region where ζ = 0.

10The presence of the turbulence implies the presence of the vorticity in the aerodynamic field: then, if onesupposes for a moment that there exists a velocity potential function φ such as v = ∇φ, and supposing todiscretize, e.g., the bidimentional laplacian of such a function using the finite differences with a constant spacestep h as in the following

1|

4 − 0 − 2|3

one would have in the field

φ1 + φ2 + φ3 + φ4 − 4φ0

h= 0

This means that the value of the potential at the central point φ0 is the arithmetic average of the surroundingmesh point: this implies that the harmonic function, i.e., the function like φ satisfing ∇2φ = 0, cannot ever reacha proper minimum or maximum point.

146

Figure 6.11: Taylor gust model.

Now, in the contest of the above modelling scenario, let us considering the following further

simplifing hypotheses for the modeling of the gust response problem (see Fig. 6.11).

1. Assume that the vorticity ζ is frozen, i.e., in the undisturbed Air Frame of Reference

(AFR)

vT (x, t) = vT (x) (6.61)

that means that vT represents a purely loading contribution for the aircraft aeroelastic

system (i.e., the airplane aerodynamics does not influence the vT distribution). The

above hypothesis can be translated in Body Frame of Reference11 (BFR) in the fact that

the gust profile represented by the function vT move towards (see Fig. 6.11) the airplane

with velocity U∞, i.e.,

vT (xA) = vT (xB − U∞t) (6.62)

where U∞ is the horizontal flight speed and xB and xA are the abscissa of a point of the

gust profile in the BFR and in the AFR respectively (note that these points coincides at

t = 0).

2. one can assume that the velocity vT does not depend upon the y direction, i.e., the

direction transversal to the stremwise; this means that

∂vT

∂y= 0 (6.63)

11Note that this frame of reference is the typical frame where the computational simulations and analyses areperformed.


3. If the airplane is small with respect to the wavelength characterisit of the gust, one has in

the BFR

vT (xA) = vT (xB/− U∞t) ≃ vT (t) (6.64)

4. Furthermore, suppose that the gust is vertical, i.e.,

vT = kw (6.65)

where w is the prescribed vertical component of the gust profile.

Considering the above assumptions, let us deal with the modeling of the aeroelastic gust re-

sponse; specifically, the Lagrange equations of motion in the s−Laplace domain, assuming a

discretization with a finite dimension equal to N and assuming also that the unsteady aerody-

namic load be aeroelastically due to the elastic deformation, can be written as

s2I q + Ω2 q = qDE(p;M∞) q (6.66)

where qT = q1, q2, ..., qN is the Laplace transform of the state-space vector built with the

lagrangian variable considered for the discretized problem, I is the identity matrix (i.e., suppose

that the discretized modes have been normalized assuming unit modal masses), Ω2 is the diagonal

matrix composed by the N squares of the angular natural frequencies on the diagonal, qD =12ρU

2∞ is the dynamical pressure, and E(p) is the GAF matrix12 associated to the N modes

assumed for the aeroelastic analysis: up to now, the aerodynamic input due to the presence of

the gust has not yet included in the analysis. Nevertheless, the matrix E can be decomposed as

(see Sec. 5.1)

E = E4 E3 E2 E1 (6.67)

or, in term of matrix components13

eiqD

= E4ij Cpj Cpj = E3jk φk φk = E2klχl χl = E1lm qm (6.68)

where the introduced quantities are the Laplace transform of the generalized forces (ei), of the

pressure coefficients (Cpj), of the velocity potential (φk), and of the normalwash (χl := ∂φl/∂n)

respectively.

12In the following the dependencies of the matrices by dimensional or undimensional variables s and p will beomitted for the sake of clarity.

13The convention of sum for all the repeated indeces has been assumed.

148

Now, if one consider a zero-th order panel discretization on the body surface for the Eq. 6.60,

together with Eq. 6.65, one has in the Laplace domain14

χl =M∑

m=1

E1lmqm − k · nl w (6.75)

where the boundary condition matrix E1 has been introduced neglecting, as usual for the dynamic

problem, the steady aerodynamic contribution. Thus, indicating with fχ the global boundary

condition terms (Eq. 6.75) collecting the elastic motion contributio together with the input gust

14The presented gust modeling assumes that the same gust disturbance affect all the points of the airplainsurface (see hypothesis 3). A more general approach can be also considered as in the following. Let us considerthe explicit structure of the discrete operator E1 apperaing in the last of the Eqs. 6.68, i.e., (see Sec. 5.1.1)

χl = E1lm qm = [sψm · nl − U∞i ·∆nm]x=xlqm (6.69)

where

∆nm =

g1 ×∂ψm

∂ξ2+

∂ψm

∂ξ1× g2

|g1 × g2|(6.70)

whereas ξ1 and ξ2 are two arbitrary material co-ordinates defined on the body surface, gα := ∂x/∂ξα (α = 1, 2)are two local covariant vectors tangent to the body surface in the reference (undeformed) configuration, and ψm

the m-th assumed mode shape function. Note that the Eq. 6.60 in the Laplace domain can be rewritten, usingthe Eq. 6.69, as

χlgust = χl + wl = [sψm · nl − U∞i ·∆nm]x=xlqm + wl (6.71)

where χl represents the normalwash contribution on the l − th panel generated by the elastic motion, whereas

wl := L[−vT · n|x=xl

]= wk · n|x=xl

(6.72)

is the normalwash contribution on the l − th given by the gust. Thus, one can observe that, within the assumedassumptions, the contribution of wl to the boundary conditions is equivalent to that of a plunge mode in theevaluation of the GAF matrix, i.e., it is of type ψgust := k (vertical translation) together with an associatedlagrangian variable

qgust :=w

s(6.73)

which is not an unknown but is the forcing term of the problem. This issue can be easily verified substituting inthe Eq. 6.69 the proposed definitions for ψgust and qgust: then, one can directly obtain χl = k · nl w =: wl. If

one indicates with E(gust) the coulmn of the GAF matrix E associated to a boundary condition mode like ψgust

one obtains as final Lagrange equation for the gust response

s2Iq + Ω2q = qD[E q+ E(gust)qgust

]= qD

[E q+

E(gust)

sw

](6.74)

Finally, It is worth to point out that the presented model for the gust input could be further improved keeping toconsider the parallelism of the use of the rigid mode in the aeroelastic analysis: indeed, in the presented theorythe four introduced hypotheses yield for the gust modeling an aerodyamic description identical to that inducedby a rigid body vertical mode. Nevertheless, the hypothesis number 3. could be assumed less restrictive assumingthat the function vT could depend also upon the longitudinal variable x. In this way, the induced unsteadyaerodynamics should be as given by a plunge rigid mode (ψgust1

= k) but also by a rigid pitch mode around thex0 axis and of type ψgust2

= (x− x0)k. Then, one could exthend this gust modeling introducing more and moremodes in the description without complicating the bae-available unsteady aerodynamic model.


contribution, one has

s2Iq + Ω2q = qDE4 E3 E2 fχ (6.76)

or (s2I + Ω2 − qDE

)q = qDE4 E3 E2 E1g w (6.77)

where the vector E1g is given by

E1gl:= −k · nl (6.78)

Therefore, the input/output gust problem is described by

q = Hg w (6.79)

where H the (column) transfer matrix of the aeroelastic system relative to the posed gust response

problem defined as

Hg(s;U∞, ρ∞,M∞) := qD(s2I+ Ω2 − qDE

)−1E4 E3 E2 E1g (6.80)

where all the functional and parametric dependencies of such a matrix have been pointed out.

Starting from Eq. 6.80, the aeroelastic system can be recast in first order form with state-space

matrix and input matrix dependent of the Laplace variable s, or

sI x = A(s) x+ B(s) w (6.81)

with xT = qT, sqT and

A(s) =

[0 I

−Ω2 + qDE(s) 0

]B(s) =

0

qDE4E3E2E1g

(6.82)

This confirms the integral-differential nature of the linear aeroelastic model and, in particular,

the fact that the dynamic input is not istantaneous at all.

In Figures 6.12-6.14 the magnitude in dB of some components of the aeroelastic frequency

response function vector are depicted as function of the frequencies in Hz: this example is

relative to a response analysis performed on the BFF configuration whose dynamic and stability

analysis has been introduced in the Section 6.1.2 (see Figs. 6.1-6.4. The lagrange variable

used for the response analysis are five; the corresponding flight condition are ϱ∞ = 1.117kg/m3,

M∞ = 0, and U∞ = 15m/s. The stability analysis for the same configuration has been performed

in Sec. 6.1.2. Note that the aeroelastic coupling is apparent in the depicted frequency response

curves: indeed, if the aerodynamics were not present, each curve would exhibit an isolated peak

relative to its associated and uncoupled natural frequency of vibration.

150

Figure 6.12: First and second components of the aeroelastic frequency response function H(ω)

Figure 6.13: Third and fourth components of the aeroelastic frequency response function H(ω)

Figure 6.14: Fifth component of the aeroelastic frequency response function H(ω)


Aeroelastic modeling description for the output dynamic loads

If one indicates with αn generic loads each induced by the assumption of the several mode shape

functions (ψ(n)) used in the aeroelastic analysis as assumed displacement (so-called modal loads),

then, because of the system linearity, the load due to a generic aeroelastic system response l(t)

(e.g., the bending moment in a structure point, the local stress, etc...) will be given on the base

of the actual state-space q(t), or

l(t) =N∑

n=1

αnqn(t) (6.83)

This Equation is one of the possible output relation for the aeroelastic system that can give the

aeroelastic dynamic loads: specifically, if deterministic input type are taken into account in the

dynamyc response analysis (for historycal reason, deterministic input is often denoted as discrete

input by aeroelasticians, see Ref. [10]), typically a so-called one-minus-cosine shape function

for different and suitable frequencies is considered for the w(t) input profile. Specifically, this

input function is recommended by the international requirement either considering a fixed value

for the input frequency (see FAR prescriptions, e.g., in Ref. [10] or a set of different frequencies

for several flight conditions). An enrichment of such a description can be achieved considering a

stochastic description of the gust input (storically known as continuous gust) as sum of contribu-

tions of continuously variable harmonics in term of power spectral density of the gust input (see

e.g., JAR prescription again). This issue will be considered in the following subsection. Figure

6.15 shows the time history of the load factor (namely, the vertical acceleration of the center

of mass or the acceleration associated to the vertical rigid body motion variable) of the BFF

configuration under a 1 -cos deterministic gust with a gust length of 12.5 chords (as prescribed

by JAR regulation) flying at U∞ = 15m/s; a comparison of the responses obtained including

and not including the effect of flexibility is presented. Figure 6.16 shows the time history of the

vertical rigid-body displacement for the same case study.

Continuous turbolence gust

Besides the deterministic model (e.g., the one minus cosine gust model, Ref. [10]), the stochastic

description is also employed for the gust input modeling: this is the case when only the energetic

content of the signal is known for the several harmonic components, i.e., when the Power Spectral

Density (PSD) Sw(ω) of the time signal w(t) is known as data input (see App. A).

Thus, suppose that HL(ω) is the global FRF vector relating the gust input w and the vector

l including, e.g., several dynamic loads: this FRF vector can be obtained composing Eq. 6.79

with Eq. 6.83. The relationship between the PSD function Sw(ω) of the input and the matrix

152

Figure 6.15: Time history of load factor due to a 1 − cos gust: comparison between rigid andelastic aircraft configuration

Figure 6.16: Time history of the vertical rigid-body displacement (q1(t) modal variable): com-parison between rigid and elastic aircraft configuration


of the PSD SL(ω) of the output is given by

SL(ω) = H∗L(ω) Sw(ω) HT

L(ω) (6.84)

where H∗L is the vector complex conjugate of HL. Typically the international regulation prescribe

function as representative of the power spectral density of the atmosferic turbolence in order

to certify the performance of the flexible airplane. The spectrum proposed by Von Karman for

the power spectral density of the vertical gust velocity is that considered by the regulation ACJ

25.341(B), which is an interpretative material to the JAR 25.341(B): the proposed spectrum

(Ref. [10]) is given in this case by

Sw(f) = σ2w

2L

U∞

[1 +

8

3

(1.339

2πL

U∞f

)2]

[1 +

(1.339

2πL

U∞f

)2]11/6 (6.85)

where f = ω/2π is the frequency measured in Hertz, σw is the root mean square (RMS) of the

vertical gust velocity, U∞ is the air speed, and L is a parameter indicating the turbolence scale.15

Figure 6.17 depicts typcal curves given by this kind of spectra. The previous formula is valid if all

the parameters are expressed in the british system, i.e., using feet for length and feet per second

for velocity. Thus, if one expresses the RMS σ2li

of a generic aeroelastic output component

li on the base of the PSD Sli of such an output, in the hypotheses of stationary and ergodic

input random process with mean value equal to zero and considering also the input/output

relationship shown by Eq. 6.84, one has

σli =

√∫ +∞

−∞H∗

Li(f)Sw(f)HLi(f)df = σw

√√√√√√√√√√∫ +∞

−∞|HLi(f)|

2 2L

U∞

[1 +

8

3

(1.339

2πL

U∞f

)2]

[1 +

(1.339

2πL

U∞f

)2]11/6df

or,

σ2qi = A σ2

w with A :=

∫ +∞

−∞|Hi(f)|2

2L

U∞

[1 +

8

3

(1.339

2πL

U∞f

)2]

[1 +

(1.339

2πL

U∞f

)2]11/6 df

15Note that a similar turbolence PSD model is that proposed by Dryden, i.e.,

Sw(f) = σ2w

2L

U∞

[1 + 3

(2πL

U∞f)2]

[1 +

(2πL

U∞f)2]2 (6.86)

Note that the above formula is an approximation of Eq. 6.85, but it has the advantage to represent the moduleof the transfer function of a linear system as all the exponents appearing on Eq. 6.86 are all integer numbers.

154

Figure 6.17: von Karman spectra (from Ref. [10]) for different turbolence scale length L andfor a fixed gust velocity rms σw = 1fps.

Note that the unic coefficient A is the result of the aeroelastic response analysis for continuum

gust and represents the ratio between the RMS of the incremental load and the RMS of the

random gust.

One of the most relevant employment of the previous result in the aeroelastic design is the

statistic evaluation of the number of crossing of a given level qi of the generic output signal qi,

per unit time, with positive slope. If the random process qi(t) is stationary and ergodic, and

with mean value equal to zero, the number N(qi) of crossing of the given level is (Rice equation,

see, e.g., [10])

N(qi) = N(0) e−1

2

(qiσqi

)2

(6.87)

where the number of zero crossing per unit time with positive slope is given in term of the gyro

radious of the function Sqi(f), i.e.,

N(0) =

√√√√√√√∫ ∞

0f2Sqi(f)df∫ ∞

0Sqi(f)df

(6.88)

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