In e na ional Fo um on Ae oelas ici y and S uc u al Dynamics
IFASD 2024
17-21 June 2024, The Hague, The Ne he lands
FLIGHT DYNAMIC-AEROELASTIC RESPONSE OF A HIGHLY
FLEXIBLE AIRCRAFT WITH DISTRIBUTED PROPELLERS
Albe o Gallego Pozo1and Rauno Ca alla o1
1Uni e si y Ca los III o Mad id
Leganes, Mad id, Spain
auno.ca [email p o ec ed]
Keywo ds: Ae oelas ici y; highly- lexible Wings; Dis ibu ed Elec ic P opulsion; geome ic
nonlinea i ies; ligh dynamic-ae oelas ic coupling; whi l lu e .
1 INTRODUCTION
1.1 Ae oelas ic S abili y o Highly-Flexible Wings wi h Dis ibu ed P opelle s: S a e o
he A
Wi hin he eme gen elec ic ai c a ma ke , Dis ibu ed Elec ic P opulsion (DEP) is a p omis-
ing concep . I p o ides bene i s in e ms o ae odynamics and p opulsi e e iciency, noise
educ ion, and ehicle con ol [1]. Some o he new ai c a concep s applying DEP ha e high
aspec a io wings o imp o ed e iciency. The ligh s uc u e, in conjunc ion wi h he la ge
p opelle s and mo o s mass and ine ia, esul s in an enhanced s uc u al lexibili y o he wing,
which can lead o an in e ac ion be ween elas ic de o ma ions and igid body ligh dynamics
ha a o s ae oelas ic ins abili ies. This in e ac ion may in u n be in ensi ied by he gy oscopic
e ec s induced by he p opelle s o a ion, and he p opelle s ae odynamics.
In e o [2], he ae oelas ici y o a wing clamped a he oo ea u ing dis ibu ed p opelle s
is analyzed, and i is shown ha he angula momen um o a la ge wing ip p opelle s has an
impo an e ec on he ae oelas ici y o he wing. Con ibu ion [3] explains ha p opelle whi l
modes ha e an impac on he s abili y o he wing, a e p oposing a amewo k o s udy he
ae oelas ic s abili y o a wing clamped a he oo ea u ing dis ibu ed p opelle s and linea
beams o accoun o he s uc u e. E o [4] assessed he in luence o whi l lu e on he
design o he DEP Ai c a X-57 h ough mul ibody dynamic analyses on a semi-span clamped
model.
Wi h he push owa ds mo e e icien ai c a , s uc u al geome ic nonlinea i ies a e p og es-
si ely becoming mo e ele an and need o be included in he design. These nonlinea i ies a e
gene ally a consequence o e y la ge de lec ions bu can also be d i en ea ly, a non-la ge
de lec ions, by pa icula a chi ec u es, such as in Joined Wings cases [5]. Among he con-
sequences o hese nonlinea i ies a he ae oelas ic le el a e he non-conse a i e p edic ion o
s a ic and dynamic ins abili ies and he ela i e change o mechanisms d i ing hem [6]. The el-
e ance o his phenomenon is exempli ied by he Pazy wing es case [7], a benchma k model
o geome ically nonlinea ae oelas ic s udies in ol ing la ge de lec ions in low-speed low,
and by he as body o li e a u e abou i .
A ho ough s udy o ae oelas ic ins abili ies should conside possible coupling wi h ligh dy-
namics esponses. A phenomenon such as Body F eedom Flu e [8] is an example d i en by
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such in e ac ion. On he o he hand, e en i no ae oelas ic ins abili y occu s, such coupling can
impac lying quali ies. The o e all s abili y p oblem should be o mula ed conside ing a lying
lexible body [9, 10].
In he cu en s a e o he a , he e a e s udies on he ae oelas ic assessmen o dis ibu ed elec-
ic p opulsion and on ligh dynamic-ae oelas ic coupling. Howe e , no esea ch has ocused
on highly lexible wings (and hei ela i e nonlinea i ies) ea u ing dis ibu ed p opulsion while
accoun ing o he concu en coupling wi h ligh dynamics.
1.2 Challenges and Con ibu ions
The INDIGO (In eg a ion and Digi al Demons a ion o Low-emission Ai c a Technologies
and Ai po Ope a ions) p ojec [11, 12], inanced wi hin he Ho izon Eu ope p og amme, e-
uni es academia, esea ch cen es and ai po s o iden i y he ma gins o imp o emen in ai po
Local Ai Quali y and Noise (LAQN) esul ing om he in oduc ion o a new non-con en ional
mid- ange ai c a . The no el ai ame ea u es dis ibu ed p opulsion based on hyb id elec-
ic/sus ainable and con en ional uel powe ain and la ge aspec - a io wing capable o ly qui-
e ly and in ze o- o-low-emission mode (i.e. elec ic and SAF) a low al i udes nea ai po s and
eso s o con en ional a ia ion uel only when equi ed, e.g., a highe al i udes o o echa ge
ba e ies du ing c uise.
The concu en in eg a ion o se e al p opelle s along a highly lexible wings inc eases he
complexi y o ae oelas ic esponse o he ollowing easons:
• Ha ing ypically lowe na u al equencies, he ae oelas ic modes ha e mo e oom o
in e ac ion wi h he ligh -dynamic ones.
• Wi h he wing being mo e lexible, he coupling wi h he classic whi l- lu e phenomenon
is possibly igh e ; he same holds o he coupling wi h he ligh dynamics, whe e, a he
p opelle le el, i is jus a displacemen and o a ion o he moun poin .
• Wi h he wing being mo e lexible, non-negligible di e ences a ise when assuming a non-
de o med e e ence condi ion o he eal one in ligh . This is ue bo h a he ae odynamic
le el and he s uc u al one (geome ic nonlinea i ies).
This s udy p esen s he de i a ion o he sys em o equa ions, inco po a ing all ele an coupled
physics, and i s implemen a ion in o a digi al ool. The so wa e’s capabili ies a e subsequen ly
demons a ed using a syn he ic baseline model o an ai c a wi h a highly lexible wing and
DEP. Ae oelas ic s abili y analyses a e conduc ed using app oaches o inc easing complexi y
ega ding he couplings and physical phenomena conside ed. The esul s a e analyzed and dis-
cussed, emphasizing he di e ences in ae oelas ic beha io when u ilizing models ha include
speci ic couplings and physical e ec s.
2 THE COUPLED FLIGHT DYNAMIC-AEROELASTIC STABILITY MODEL
This sec ion p esen s he heo e ical amewo k behind he coupled ligh dynamic-ae oelas ic
s abili y module. Fi s ly, he pe u ba ion equa ions o mo ion a e de i ed o a gene al equi-
lib ium condi ion. Then, he non-linea im module o ob ain he equilib ium condi ion is ex-
plained. Modal analysis is discussed o se e as app op ia e basis o he analysis o he (small)
displacemen s ela i e o he non-linea equilib ium. The ae odynamic models o he uns eady
ae odynamics o bo h li ing su aces and p opelle s a e hen desc ibed. Finally, he comple e
sys em o equa ions go e ning he ligh dynamic-ae oelas ic s abili y o he ai c a is de i ed
and a solu ion p ocedu e is sugges ed.
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2.1 The Fligh Dynamic-Ae oelas ic S abili y Equa ion
The S abili y Equa ion is de i ed s a ing om Lag ange’s equa ion:
d
d (∂T
∂˙η)−∂T
∂η +∂U
∂η =∂(δW))
∂(δη)(1)
The o mula ion is o ien ed owa ds he in eg a ion o linea ini e elemen models, which a e
he basis in ae oelas ici y. The nex assump ions a e made o he de i a ion o he heo e ical
amewo k:
•Assump ion 1. The ai c a is disc e ized in o ini e elemen s using a lumped mass ap-
p oach. Thus, he ai c a is disc e ized in o poin s (k) wi h a mass (mk) and an ine ia
enso (Jk). Each o a ing mass (p opelle ) is hen modelled as a poin wi h a mass and
an ine ia enso .
•Assump ion 2. Local ansla ional and o a ional elas ic displacemen s wi h espec o
an equilib ium condi ion a e small; pe u ba ion heo y is used and linea elas ic heo y
applies.
•Assump ion 3. Elas ic displacemen s a e desc ibed using an o hogonal mode shapes
basis ob ained om a ee- ee modal analysis o he ai c a .
2.1.1 Kinema ics: posi ion and eloci y o a gene ic poin o he ai c a
Two e e ence ames a e de ined: an ine ial ame (ΣI) a ached o he Ea h’s su ace and a
body ame (ΣB) a ached o he ai c a . Fo a gene ic poin o he ai c a :
R=R0+ 0+u(2)
whe e:
•R0≡posi ion o he o igin o ΣB
• 0≡posi ion o a di e en ial o mass in he e e ence ai c a ’s con igu a ion
•u≡ a ia ion o he posi ion o he di e en ial o mass wi h espec o he e e ence
con igu a ion due o de o ma ions
The nex assump ion is made: Assump ion 4.d 0
d B
= 0
Le us di e en ia e be ween any gene ic poin o he ai c a (deno ed as wi h he sub-index k),
all he poin s bu he p opelle poin s (sub-index i) and he poin s associa ed o each p opelle
(sub-index p). The linea eloci ies o poin s iand pa e exp essed in he same o m, bu hei
o a ional eloci ies exp essions di e :
dRk
d I
= +ωB,I×( 0k+uk) + ˙
uk= + ˜ωB,I( 0k+uk) + ˙
uk
Ωi=ωB,I+˙φi
Ωp=ωB,I+˙φp+ωPH
(3)
whe e:
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•˙
() = d
d ()B
•ωB,Iis he angula eloci y o e e ence ame ΣBwi h espec o ΣI. The adop ed s an-
da d is ha o he Tay -B yan Angles:
ωB,IB
=
p
q
=L
˙
ϕ
˙
θ
˙
ψ
;L=
1 0 −sinθ
0cosϕ cosθsinϕ
0−sinϕ cosθcosϕ
(4)
•˙φiis he angula eloci y o poin idue o de o ma ions; ˙φiis de ined acco ding o he
Tay -B yan angles (each angle de ines a o a ional de o ma ion angle a poin i). These
angles de ine a new e e ence ame ΣAiwi h espec o ΣB.
˙φiAi
=Li
ϕi
θi
ψi
;Li=
1 0 −sinθi
0cosϕicosθisinϕi
0−sinϕicosθicosϕi
;q o i=
ϕi
θi
ψi
(5)
•˙φpis he angula eloci y o poin pdue o de o ma ions and is de ined, again, acco ding
o he Tay -B yan angles (each angle de ines a o a ional de o ma ion angle a poin p).
These angles de ine a new e e ence ame ΣApwi h espec o ΣB
• The symbol ”˜” abo e a a iable deno es he skew-symme ic ma ix co esponding o a
ec o ( o c oss-p oduc s), such ha :
˜ω=
0−ωzωy
ωz0−ωx
−ωyωx0
(6)
•ωP,H is he angula eloci y ec o o he o a ing mass a poin p.
2.1.2 Kine ic ene gy
The kine ic ene gy o he ai c a can be exp essed as he sum be ween he ansla ional kine ic
ene gy and he o a ional kine ic ene gy (T=EkinT+EkinR).
EkinT=1
2X
i
˙
RT
i˙
Rimi+1
2X
p
˙
RT
p˙
Rpmp=1
2m T −X
k
mk T(˜ 0k+˜uk)ωB,I+X
k
mk T˙uk−
−1
2X
k
mkωB,IT(˜ 0k+ ˜uk)(˜ 0k+ ˜uk)ωB,I+X
k
mkωT
B,I (˜ 0k+ ˜uk)˙uk+1
2X
k
mk˙uT
k˙uk(7)
EkinR=1
2X
i
ΩiTJiΩi+1
2X
p
ΩpTJpΩp=1
2X
k
ωT
B,IJkωB,I+X
k
ωB,ITJk˙φk+1
2X
k
˙φT
kJk˙φk+
+X
p
ωB,ITJpωPH +X
p
˙φT
pJpωPH +1
2X
p
ωPHTJpωPH (8)
whe e Jkis he ine ia enso o poin kand Gpis he gy oscopic ma ix o he p− h p opelle :
Gp=ωPH
0Ixz −Ixy
−Ixz 0Ix
Ixy −Ix0
p
(9)
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2.1.3 Simpli ica ion o he kine ic ene gy exp ession
Recalling assump ions 2and 3(small ampli ude ib a ion a ound he equilib ium s a e), dis-
placemen s can be w i en as a ( unca ed) supe posi ion o (linea ized abou he e e ence sys-
em) o s uc u al modes a ound he equilib ium s a e:
u= [Φ]ηE(10)
Mean axes [13] abou he non-linea equilib ium s a e a e used as body- e e ence ame, such
ha :
• O igin o he body ame (mean axes ame) is loca ed a he ins an aneous cen e o mass:
ZV
ρdV = 0 −→ X
k
mk( 0+ud) = 0(11)
• In e nal linea momen um ela i e o elas ic ansla ions is ze o:
ZV
˙udρdV = 0 −→ X
k
mk˙udk=0(12)
• In e nal linea ized angula momen um ela i e o elas ic de o ma ions is ze o:
ZV
˜ 0˙udρdV = 0 −→ X
k
mk˜ 0˙udk+X
k
Jk˙q o dk=0(13)
The p e ious wo mean axes cons ain s a e au oma ically sa is ied since elas ic displacemen s
a e app oxima ed using he modal shapes o he ee- ee ai c a .
ZV
˙udρdV =
nE
X
i=1
dηi
d ZV
ϕiρdV = 0
ZV
˜ 0˙udρdV =
nE
X
i=1
dηi
d ZV
˜ 0ϕiρdV = 0
(14)
Two u he hypo hesis which a e common when analyzing he ligh -dynamic-ae oelas ic s a-
bili y using mean axes a e [14] [15] [16]:
• Pe u ba ion de o ma ions and de o ma ions a es a e collinea :
ZV
˜ud˙ud= 0 −→ X
k
mkωT
B,I (˜ 0k+ ˜uk)˙uk+X
k
ωB,IT[BRAk]JkLk˙q o k= 0 (15)
• The change in ine ia due o pe u ba ion de o ma ions is negligible:
1
2X
k
ωB,IT(−mk(˜ 0k+ ˜uk)(˜ 0k+ ˜uk)+[BRAk]Jk[AkRB])ωB,I≈
≈1
2X
k
ωB,IT(−mk˜ 0k˜ 0k+Jk)ωB,I=1
2ωB,ITJ0ωB,I(16)
A e hese simpli ica ions, he kine ic ene gy exp ession eads:
T=1
2m T +1
2ωB,ITJ0ωB,I+1
2˙ηT
EMEE ˙ηE+1
2X
p
ωPHTJpωPH+
+X
p
ωB,ITJpωPH −X
p
q o p
T[Gp]ωB,I+X
p
˙qT
o pJpωPH +X
p
˙qT
o p[Gp]q o p
(17)
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2.1.4 Po en ial ene gy
The po en ial ene gy o he ai c a is he sum o he g a i a ional po en ial ene gy Ugand he
elas ic s ain ene gy Ue.
Ug=−ZV ol
(g·R)dm =−g·ROBm
Uel =1
2ZV ol X
ijkl
CijklεijεkldV =1
2ηET[KEE]ηE
(18)
2.1.5 Equa ions o mo ion
The gene alized coo dina es ha will be used a e he posi ion o he o igin o he mean axes
ame, he o ien a ion angles o he mean axes ame ela i e he ine ial one and he elas ic
modal coo dina es desc ibing displacemen s ela i e o he equilib ium. Applying Lag ange
equa ions esul s in he ollowing sys em o equa ions:
m˙ +ωB,I0 + ˜ωB,I 0=
∂FB
∂p0
δp
J0˙ωB,I+ ˜ωB,I0J0ωB,I+ ˜ωB,IJ0ωB,I0+X
p
[Gp]ωB,I+X
p
[Gp]˙q o p+
+˜ωB,I0X
p
[Gp]q o p=
∂MB
∂p0
δp
MEE ¨ηE+KEEηE+X
p
[Φ o p]T[Gp][Φ o p]˙ηE+X
p
[Φ o p]T[Gp]ωB,I=∂δW
∂(δηE)
(19)
whe e is now he pe u ba ion eloci y ec o , 0 he e e ence eloci y ec o , ωB,I he
pe u ba ion angula speed o he mean axis ame, ωB,I0 he e e ence angula speed o he
mean axis ame, and δpa ec o con aining all he pe u ba ions ha a ec he o ces and
momen s.
Fo comple eness, he kinema ic equa ions need o be added o he p e ious sys em o equa ions.
The ine ial eloci y o he ai c a is de ined by he a e o displacemen o i s cen e o mass:
V= o+ =d o
d I
+d
d I
= ( ˙
XEiI+˙
YEjI+˙
ZEkI) + ( ˙xEiI+ ˙yEjI+ ˙zEkI) =
= (UiB+VjB+WkB)+(uiB+ jB+wkB)
(20)
whe e he sub-index Ideno es ine ial e e ence ame and B e e s o he mean axes ame.
The o ien a ion o he mean axes ame wi h espec o he ine ial one is de ined by he Eule
angles (Φ = ϕ0+ϕ,Θ = Θ0+θ,Ψ = ψ0+ψ), acco ding o he Tay -B yan Fo malism.
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2.2 Re e ence Condi ion: T im
2.2.1 The ae odynamic p oblem
The i e a i e na u e o he non-linea lexible im equi es he use o a as ae odynamic ool o
e alua e he o ces on he de lec ed con igu a ions. Mo eo e , dis ibu ing o a y de ices ahead
o he wing esul s in complex ae odynamic in e ac ions be ween he wake shed by he blades
and he downs eam su aces. The selec ed comp omise is o calcula e he ae odynamics o he
ai c a wi h DUST, a mid- ideli y ool de eloped o p o ide as and eliable ae odynamic sim-
ula ions o Ve ical Take-O and Landing (VTOL) ai c a con igu a ions. The ool has been
shown o p o ide eliable and as p edic ions o he ae odynamic pe o mance o uncon en-
ional VTOL ai c a [17–22].
The ma hema ical o mula ion behind his so wa e elies on he Helmhol z’s decomposi ion
o he eloci y ield, which allows o ecas he ae odynamic p oblem as a combina ion o a
bounda y alue p oblem o he po en ial pa o he eloci y and a mixed panels- o ex pa i-
cles model o he ee o ici y ield in he low. The li ing su aces a e modelled wi h su ace
panels, whe eas p opelle s a e modelled using li ing lines. Vo ici y is shed om he ailing
edge o bo h li ing lines and li ing su aces and is hen con ec ed acco ding o he local e-
loci y, e ec i ely conside ing he swi l impa ed by he p opelle s on he low impinging on he
wing.
The eade is e e ed o [23] o mo e de ails abou he de i a ion.
2.2.2 The s uc u al p oblem
The s uc u al in-house sol e (pyBeam) [24] is based on a 6 DoF geome ically non-linea
beam o mula ion. The Eule -Be noulli beam kinema ic assump ion is conside ed. The equa-
ion go e ning he displacemen s o he s uc u e in i s disc e ized Fini e Elemen (FE) o m
is:
G(us) = s− in (us) = 0(21)
whe e us, sand in a e, espec i ely, he nodal gene alized displacemen s, he ex e nal and
in e nal load o ces ec o .
The p e ious equa ion is sol ed by a New on-Raphson me hod:
Kus=−G(us)(22)
whe e K=∂G(us)
∂usis he Jacobian/ angen ma ix.
2.2.3 Splines and mesh de o ma ion me hods
Ae odynamic and s uc u al g ids a e gene ally non-coinciden . A Mo ing Leas Squa es Algo-
i hm is used o compu e he spline ma ix ha ela es s uc u al and ae odynamic coo dina es,
displacemen s and o ces [24]. Le xs∈RNsbe he coo dina es o he s uc u al nodes, and
xa∈RNabe he coo dina es o he ae odynamic mo ing g id, hen i is possible o de ine a
spline ma ix S=S(us,ua), such ha :
ua=Sus
s=ST a
(23)
whe e ua ep esen s he displacemen s o he ae odynamic g id, us he displacemen s o he
s uc u al g id, s he o ces and momen s on he s uc u al nodes, and a he o ces and momen s
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on he ae odynamic nodes. The same p ocedu e can be used o ans e displacemen s and o ces
among o he ele an poin s o he g id, such as elemen cen e s.
Wi hin he p esen o mula ion, a e sol ing o s uc u al displacemen s, bo h he ae odynamic
and s uc u al g ids can be upda ed by simply summing he displacemen s o he ae odynamic
and s uc u al g id coo dina es o he p e ious i e a ion i−1:
xai=xai−1+u∗
a−→ Da= 0
xsi=xsi−1+u∗
s−→ Ds= 0 (24)
whe e a elaxa ion pa ame e αcan be applied o he bounda y displacemen s o ensu e s abili y
o he me hod:
u∗
a=αui
a+ (1 −α)ui−1
a
u∗
s=αui
s+ (1 −α)ui−1
s
(25)
2.2.4 The im p oblem: Fluid-S uc u e In e ac ion (FSI)
The equilib ium condi ion unde conside a ion is s eady-le el ligh . The equa ions ha go e n
his equilib ium a e:
F(xa,c) = Lcosα −W
Mycg =0;c=α
δe(26)
whe e αand δea e he Angle o A ack (AoA) and ele a o (o any o he con ol su ace)
de lec ion, espec i ely, and Mycg is he pi ching momen wi h espec o he cen e o g a i y.
The equilib ium along he longi udinal di ec ion ( h us equa ion) has been emo ed, since in
a i s app oxima ion i is independen om he o he wo. Gi en he h us ha each p opelle
needs o p oduce, he necessa y p opelle ’s collec i e pi ch is i s calcula ed using Blade Ele-
men Theo y ins ead o he Vo ex Pa icle Me hod o accele a e nume ical compu a ions, while
e aining simila accu acy.
Gi en an ae odynamic g id, he p e ious equa ion can be sol ed using a Good-B oyden’s
me hod [25], which alls wi hin he class o quasi-New on me hods. Due o he quasi linea
ela ion be ween ae odynamic esponse, his me hod apidly con e ges. The whole solu ion
p ocedu e o ind he non-linea im is desc ibed in he nex image.
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Figu e 1: Non-linea im wo k low.
2.3 Modal Analysis
A e con e gence o he im p ocedu e, he de lec ed shape is ob ained. F om he s uc u al
shape, i is possible o ob ain he upda ed mass and s i ness ma ices. A modal analysis is hen
pe o med o ob ain he upda ed modes. Such shapes ep esen he modal basis o be used when
o mula ing and esol ing he pe u ba ion ae oelas ic equa ion.
2.4 Enhanced DLM o Po en ial Uns eady Ae odynamics
The Double La ice Me hod (DLM) is a as ly used me hod o compu e he uns eady ae ody-
namics o li ing su aces [26]. Howe e , in i s o iginal o mula ion, i only calcula es loads due
o local pi ching and plunging o he li ing su aces, neglec ing con ibu ions due o in-plane
mo ion and in-plane o ces. This app oxima ion p o ides good esul s on con en ional wings.
Howe e , i may ail o p o ide accu a e esul s on con igu a ions whe e in-plane loads and he
co esponding momen s a e impo an , like T- ails [27]. The in-plane o ces can be ele an
also when s udying he ee- ee ai c a and impo an in e ac ions be ween la e al-di ec ional
ligh -dynamic and ae oelas ic modes. In a high aspec - a io wing, in-plane loads may cause
non-negligible in-plane bending due o he highe lexibili y. Fu he mo e, he gy oscopic mo-
ion o p opelle s ends o couple he yaw and pi ch mo ion o he p opelle ’s axis (whi l mode).
When he o o ’s mass is su icien ly la ge compa ed o he wing’s s i ness, his can lead o a
coupling be ween in-plane bending, ou -o -plane bending, and o sion. Hence, he e may be a
need o e ain in-plane loads and mo ion in high aspec - a io wings wi h dis ibu ed p opelle s.
In he p esence o la ge de lec ions he app oxima ion o conside ing he ae odynamic o ces
abou he unde o med con igu a ion may lead o inco ec p edic ion. The e ec s o s eady ou -
o -plane bending and in-plane bending can be simply modelled by app oxima ing he de o med
li ing su ace wi h se e al wing segmen s wi h di e en sweep and dihed al. Howe e , he
inclusion o angle o a ack, wis and cambe is mo e complex, due o he es ic ion ha he
eloci y goes in he x−di ec ion in he de i a ion o he p essu e po en ial equa ion. The e o e
a classic DLM canno model hese e ec s and modi ica ions a e equi ed.
To his aim, he gene al o m o he non-pene a ion bounda y condi ion (e alua ed a he Con-
ol Poin - CP - o each ae odynamic box j) needs o be de i ed [28]:
Vj·nj= 0 −→ (U∞+u0j+u1jeiω −iωhjeiω )·(n0j+ eiω ×n0j)=0 (27)
9
IFASD-2024-182
whe e xR=u w p q xEyEzEϕ θ ψT
To include he uns eady ae odynamics in he s a e-space sys em, lag s a es a e de ined:
ik
ik +βjV∞0
ˆceηR
ηE=xlagj(6+nE)×1−→ ˙xlagj=L1L206×nE
0nE×12 InE×nExR
˙ηE−βjV∞0
ˆcxlagj(51)
The new s a e ec o is:
x=
˙xR
˙ηE
ηE
xlagj
(52)
and he s a e-space sys em ( o one lag s a e) is now:
Ano lag 02(6+nE)×(6+nE)
0(6+nE)×2(6+nE)I6+nE
˙xR
¨ηE
˙ηE
˙xlag1
=
=
Bno lag
q∞0A2+1R
06×(6+nE)
q∞0A2+1E
0nE×(6+nE)
L1L206×nE06×nE
0nE×12 InE×nE0nE
−β1V∞0
ˆcI6+nE
xR
˙ηE
ηE
xlag1
(53)
This sys em can be ecas ed in o a classical eigen alue-eigen ec o analysis:
(sI −A−1B)ξ=0(54)
whe e he ec o o s a e-space ampli udes ξcon ains he ampli udes o igid body s a es, ae oe-
las ic s a es and a i icial ae odynamic s a es. Modes acking is equi ed o iden i y he ligh
dynamic-ae oelas ic eigen alues and eigen ec o s. The acking mechanism is based on he
i e a i e co ela ion o he eigen ec o s o successi e speeds.
3 APPLICATION TO A DEP AIRCRAFT WITH A HIGH ASPECT RATIO WINGS
In his sec ion, a syn he ic es case is p oposed o demons a e he capabili ies o he p esen ed
amewo k.
3.1 Baseline ai c a
The es model consis s o a modi ied e sion o he NASA X-57, see Figu e 7. The ae odynamic
and s uc u al models a e based on a high aspec a io wing wi h sweep angle, a igid ho izon al
ailplane and a igid e ical ailplane. The wing, HTP and VTP a e igidly connec ed o he
cen e o mass o he ai c a . The s uc u al FE model is a classic s ick model, i.e., he s uc u e
is desc ibed by beams and igid connec ions, as shown in Figu e 8. The wing’s in e nal s uc u e
16
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Figu e 7: NASA X57.
is a single-cell aluminium wingbox. An added dis ibu ed mass is conside ed o accoun o
non-s uc u al mass.
Figu e 8: S uc u al g id.
Figu e 9: P opelle ’s modelling in s uc-
u al g id.
The es case p esen s 2 la ge ip p opelle s and 4 smalle p opelle s dis ibu ed along he wing.
The p opelle s a e coun e - o a ing, wi h he p opelle s on he y−posi i e side o a ing coun-
e clockwise and hose on he o he side clockwise. The main p ope ies o he es ai c a a e
Figu e 10: Ae odynamic g id o im p ocedu e.
collec ed in Tables 1, 2, 3.
17
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Figu e 11: DLM ae odynamic g id.
Table 1: Wing, HTP and VTP geome y
Wing HTP VTP
span [m] 9.639 2.518 1.254
oo cho d [m] 0.756 0.783 1.773
ip cho d [m] 0.334 0.783 0.699
Leading edge sweep [º] 9.879 0 48.469
Su ace [m2]5.255 1.972 1.550
AR [-] 17.681 3.212 -
Table 2: Ai c a mass p ope ies
Pa ame e Value
ai c a mass [kg] 1174.82
wing mass (no including p opelle s) [kg] 144.70
o al p opelle s mass [kg] 124.66
es o ai c a mass [kg] 925.46
Xcg [m] 0.6821
Ycg [m] 0
Zcg [m] -0.5382
Ixx [kg·m2] 4386.923
Iyy [kg·m2] 6399.972
Izz [kg·m2] -115.952
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Table 3: P opelle s p ope ies.
C uise p opelle High-li p opelle
Diame e [m] 1.024 0.387
oo cho d [m] 0.100 0.050
Numbe o blades 3 5
Ai oil MH117 MH114
Powe [kW] 43 14.4
Ro a ional speed [ ad/s] 2250 4548
Th us [N] 578.98 220 (0 in c uise)
Inclina ion angle wi h espec o wing [º] 0 0
o a ing mass [kg] 8.12 2.00
non- o a ing mass [kg] 36.77 5.44
Pi ch s i ness [Nm] 12398.42 -
Yaw s i ness [Nm] 14751.24 -
3.2 Tes Cases
The ee- ee s abili y analysis o he baseline ai c a is pe o med acco ding o he cases
lis ed in Table 4. The column P opelle indica es i gy oscopic and ae odynamic e ec s o
Table 4: Tes cases.
Case P opelle s Rigid-elas ic coupling Re e ence condi ion DLM Re e ence ae odynamic shape
Case 1 No No Unde o med Basic Unde o med
Case 2 Yes No Unde o med Basic Unde o med
Case 3 Yes Yes Unde o med Basic Unde o med
Case 4 Yes No De o med Basic Unde o med
Case 5 Yes Yes De o med Basic Unde o med
Case 6 Yes Yes De o med Enhanced Unde o med
Case 7 Yes Yes De o med Enhanced De o med
he p opelle s a e included. The column Rigid-elas ic coupling e e s o including he ligh dy-
namics/ae oelas ic modes coupling. The column Re e ence condi ion speci ies i he s uc u al
p ope ies a e e alua ed on he unde o med con igu a ion, o on he de lec ed immed one. The
column DLM e e s o he employmen o he classic o enhanced e sions o he DLM. And,
he las column, Re e ence ae odynamic shape indica es whe he he unde o med shape o he
de lec ed one is used o he e alua ion o he ae odynamic o ces.
Table 5: C uise condi ions
V∞[m/s]h [m] ρ[kg/m3]
77.17 2438.4 0.9629
3.3 Resul s
In his sec ion, he esul s o he Fligh Dynamic-Ae oelas ic S abili y analyses pe o med o
all cases lis ed in Table 4 a e p esen ed, and a b ie discussion is p o ided.
3.3.1 No mal modes
Tables 6 and 7 summa ize he esul s o he modal analysis on he ai c a when conside ing i s
unde lec ed (jig-shape) and de lec ed (in- ligh ) shapes. The p opelle s a e no o a ing, o he -
19
IFASD-2024-182
wise, he modal analyses will esul s in ou -o -phase mode shapes.
Table 6: Na u al modes o he unloaded ai c a (in i s
jig-shape con igu a ion)
Mode F equency [Hz]
1S. Fi s Bending 1.2973
1A. Fi s Bending 2.1451
2S. Fi s To sion 5.238
2A. Fi s To sion 5.2453
3S. In-plane Bending 5.7043
4S. Second Bending 6.618
3A. Second Bending 6.792
4A. In-plane Bending 7.1448
5S. Tip p opelle yaw 8.9689
5A. Tip p opelle yaw 9.0702
Table 7: Na u al modes o he ai c a in i s in- ligh de-
lec ed shape
Mode F equency [Hz]
1S. Fi s Bending 1.2918
1A. Fi s Bending 2.1814
2S. Fi s To sion 4.9998
2A. Fi s To sion 5.1424
3S. In-plane Bending 5.7137
3A. Second Bending 6.5152
4S. Second Bending 6.6025
4A. In-plane Bending 6.9997
5S. Tip p opelle yaw 8.95
5A. Tip p opelle yaw 9.0439
The le e s (S) and (A) e e o symme ic and an i-symme ic modes; mo eo e , he shapes
a e desc ibed highligh ing he mos ep esen a i e ea u es: in ac , modes 2S/A o 4S/A a e
ac ually a combina ion o ou -o -plane bending, in-plane bending, o sion and p opelle pi ch.
A compa ison be ween he wo cases (unde lec ed and de lec ed s uc u e) shows a gene al
sligh d op o he na u al equencies (wi h excep ion o mode 1A). A isual ep esen a ion
o he modes is p o ided in Sec ion 5.1. I is in e es ing o obse e ha in-plane and ou -
o plane mo ions a e uncoupled in he plana case (jig-shape), bu coupled in he de o med
wing. In addi ion, he p opelle pi ch mode, which has a equency o app oxima ely 8 Hz
when he moun ing wing sec ion is in ini ely igid, has a signi ican pa icipa ion in he second
bending and i s o sion o bo h jig-shape and ligh -shape cases. Howe e , while i has a
ele an pa icipa ion in he in-plane bending o he jig-shape case, i s pa icipa ion in he in-
plane bending o he in- ligh shape is negligible.
3.4 S abili y analyses
To suppo he discussion, se e al pic u es and ables ga he ing he ele an da a will be used.
Figu es 12 and 13 show he oo loci o he i s modes o Cases 1 and 2, espec i ely; oge he
wi h Table 8 hey will suppo he discussion on e ec s o wing-p opelle in e ac ion.
Figu es 14 is employed o highligh e ec s o ligh dynamic-ae oelas ic coupling in Cases 2 o
7.
Tables 9 and 10 epo he da a o ypical symme ic lu e occu ences. In pa icula , inspec ing
he oo -loci o all Cases, i is in e ed ha lu e cases can be ga he ed in wo g oups, acco d-
ing o he ins abili y equency. These wo g oups a e e e ed o wi h Type 1 (highe equency)
and Type 2 (lowe equency). Bo h Tables p o ide, o each lu e onse , he p ope ies o he
uns able ae oelas ic mode in e ms o pa icipa ion o he modal basis. Hence, he eal ampli-
ude in he shape should be weigh ed acco ding o he ampli ude o modal base (which is mass
no malized).
20
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Figu e 12: Case 1. Roo Locus
Figu e 13: Case 2. Roo Locus
Table 8: Modes a e in oduc ion o gy oscopic e ec s a V∞= 0 m/s
Modes 2S wi h Gy . (| |,∠)Mode 2A wi h Gy . (| |,∠)Mode 3S wi h Gy . (| |,∠)
2S 0.94∠0◦2A 0.95∠0◦2S 0.56∠98◦
3S 0.1∠86◦3A 0.04∠209◦3S 0.75∠0◦
4S 0.05∠185◦4A 0.05∠92◦4S 0.06∠89◦
5S 0.33∠91.04◦5A 0.32∠90◦5S 0.34∠177◦
Desc ip ion S Backwa d whi l Desc ip ion A Backwa d whi l Desc ip ion S Backwa d whi l
21
IFASD-2024-182
3.4.1 Gy oscopic e ec s
The equencies on he pu e imagina y axis, i.e., o no incoming low, con ain o Case 2 he
e ec s o he ine ial coupling wi h he o a ing p opelle s. Table 8 shows ha he modes 2S,
2A, and 3S o Case 2, i.e., including he gy oscopic e ec s, a e p e y di e en han he ones o
Case 1. Fo example, he new 2S mode, is now a supe posi ion o he o iginal modes 2S, 3S, 4S
and 5S, each wi h a di e en phase; his mode now ea u es a s ong backwa d whi l componen .
O e all, he inclusion o gy oscopic e ec s couple in-plane bending, p opelle pi ch, yaw, and
o sion, esul ing in e ec i e whi l modes o he p opelle . As a consequence, he equencies
o modes 2S and 3S la gely dec ease.
3.4.2 Wing-p opelle s ae oelas ic coupling
Inspec ion o he oo loci o Case 1 and 2 (Figu es 12 and 13) depic s a complex pic u e.
In Case 1, h ee ins abili y occu ences a e de ec ed o he i s 4 modes. Fo he symme ic
case, mode 2S and 3S become uns able, e en hough mode 2S shows a hump-mode lu e .
The equencies o he lu e a e p e y close, a ound 5.7 Hz, sugges ing possible complex
in e ac ions. Fo he an i-symme ic case, mode 2A is he one becoming uns able. Flu e
equency is also simila o he one o he symme ic case.
In Case 2, i.e., in oducing he wing-p opelle s coupling, he lu e pic u e is p e y di e en .
Bo h modes 2S and 3S, ea u ing backwa d-whi ls, become uns able, bu a p e y di e en
lu e equencies ( 4.2 and 5.9 Hz, app oxima ely).
Fo he 2S ins abili y, he lu e speed is educed o 10%. The lu e mechanism changes
om being a pu ely o sion-bending coupling o being a combina ion o o sion-bending and an
e ec i e backwa d whi l o he ip p opelle , including he p opelle yaw (see Table 10).
Fo he ins abili y o mode 3S, a simila lu e mechanism is obse ed. Howe e , he lu e
speed is la gely inc eased and in-plane mo ion becomes mo e ele an , (see Table 9).
Mode 2A, ea u ing a backwa d-whi l, also lu e s sligh ly o e 4 Hz.
3.4.3 Fligh dynamic and ae oelas ic coupling
A en ion is now d awn o he di e ences among Cases 2 o 7 conce ning he ligh dynamic
modes, i.e., he eigen ec o s ha esemble he classic ligh -dynamic modes o a igid ai c a .
Figu e 14 shows he damping and equency o he sho pe iod and he du ch oll modes o he
lexible ai c a .
22
IFASD-2024-182
Figu e 14: Damping and equency esponse o ligh dynamic modes o he lexible ai c a .
The equency e olu ion o hese modes is obse ed o be nea ly independen o he ideli y o
he app oach. Fo example, Case 2, which disca ds he e ec s o lexibili y on ligh dynamic
esponse, al eady p edic s he equency easonably well. This is no he case o he damping o
he sho -pe iod mode. S a ing om Case 3, in which he ligh dynamic/ae oelas ic coupling
is aken in o accoun , a dec ease in damping is obse ed. This is possibly a consequence o
he ypical coupling be ween he sho -pe iod mode and he i s symme ic bending mode.
Addi ionally, he con ibu ions in oduced by he EDLM also educe he damping o he sho -
pe iod mode (Case 5 s. Cases 6 7).
Wi h ega ds o he damping o he Du ch oll, he eade may no ice a dec ease a low speeds
o Case 7, leading o ins abili y. Fo he o he cases, damping is only ma ginally changing.
A possible explana ion is ha a de lec ed wing is equi alen o a wing wi h mo e dihed al,
esul ing in la ge oll damping. This, in u n, inc eases he di e ence in magni ude be ween
oll and yaw damping, he eby educing he o e all s abili y o he Du ch oll.
3.4.4 Flu e Type 1
F om his poin on, only he symme ic modes a e conside ed o he sake o cla i y and con-
ciseness. As men ioned abo e, om Case 2 o Case 7, all lu e occu ences ha e been no ed o
happen a ound wo equency anges. Type 1 includes all lu e occu ences wi h a equency
o abou 5.5-5.9 Hz. The mode ha loses s abili y is Mode 3S, which, a ze o wind speed, p i-
ma ily ea u es in-plane bending, o sion, and ip-p opelle yaw, esembling a backwa d-whi l
mode. Figu e 15 shows mode 3S damping and equency s speed.
F om inspec ion, i can be in e ed ha upda ing he s uc u al pa o he ae oelas ic sys em wi h
he eal s i ness and ine ial dis ibu ion (geome ic nonlinea i ies), i.e., om Case 4 onwa d,
has a signi ican impac on he lu e onse . The equency o he ae oelas ic mode inc eases
by app oxima ely 0.2 Hz o mo e, while damping is educed, leading o an ea lie onse o
lu e ( lu e speed d ops by abou 100 m/s). F om Case 4, he lu e mechanism also appea s
o change, wi h he uns able ae oelas ic mode showing an inc eased pa icipa ion o Mode 2S
23
IFASD-2024-182
( i s o sion) and an almos negligible pa icipa ion o Mode 4S (second bending). Mo eo e ,
looking a he phases o he 2S, 3S and 5S modes sugges s ha lu e mode is go e ned by an
e ec i e backwa d whi l mode o he p opelle ’s hub (no e he di e ences in he phases o 2S
and 5S be ween Cases 2 and 4). I should be no ed, howe e , ha he na u al modes e alua ed
on he de lec ed shape do no closely esemble hose e alua ed on he jig shape, see Sec ion 5.3.
The eade is e e ed o Sec ion 5.5 o he modes o de o med shape including gy oscopic
e ec s.
Fu he analyses a e ongoing o gain a deepe unde s anding o he esul s.
Figu e 15: Damping and equency o ae oelas ic mode 3S.
Table 9: Flu e ype 1. Symme ic case. Flu e mechanism, equency, and speed.
Flu e Type 1
Na u al Mode Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7
w0 0 1.38∠15◦0 0.05∠265◦0.07∠280◦0.05∠282◦
q0 0 0.18∠158◦0 0.04∠271◦0.04∠265◦0.03∠259◦
1S. 1s Bending (S) 0.10∠181◦0.25∠255◦0.26∠255◦0 0 0 0
2S. 1s To sion (S) 1.00∠0◦0.53∠124◦0.51∠127◦0.86∠35◦0.88∠35◦0.86∠35◦0.86∠34◦
3S. In-Plane Bending (S) 0.30∠8◦1.00∠0◦1.00∠0◦1.00∠0◦1.00∠0◦1.00∠0◦1.00∠0◦
4S. 2nd Bending (S) 0.43∠168◦0.93∠234◦1.00∠234◦0.06∠206◦0.06∠206◦0.03∠206◦0.03∠200◦
5S. P opelle yaw (S) 0 0.60∠169◦0.62∠169◦0.37∠221◦0.38∠221◦0.36∠221◦0.35∠220◦
Flu e Mechanism 2nd Bending - 1s o sion - In-plane
bending (S)
In-plane bending - 1s To sion -
P op.Yaw (S) (Equi alen o a back-
wa d whi l o ip p opelle )
Flu e F equency [Hz] 5.75 5.89 5.89 5.65 5.66 5.65 5.65
Flu e Speed [m/s] 92.75 152.25 158 51.74 50.75 59.25 62.25
3.4.5 Flu e Type 2
Type 2 includes all lu e occu ences wi h a equency o abou 4.2-4.3 Hz. The mode ha
loses s abili y is he 2S, which, a ze o wind speed, p ima ily ea u es o sion and ip-p opelle
yaw, esembling a back-whi l mode. Figu e 16 shows mode 2S damping and equency s
speed.
24
IFASD-2024-182
Focusing on he end o damping, h ee main jumps a e no iced. The i s one, om Case 3 o
4, is a s abilizing e ec induced by upda ing he s uc u al pa o he ae oelas ic sys em wi h
he eal s i ness and ine ial dis ibu ion (geome ic nonlinea i ies). The second jump, again
s abilizing is obse ed when including he ligh dynamic-ae oelas ic coupling ( om Case 4 o
5, bu also om Case 2 o 3). The hi d jump, p omo ing ins abili y, is he in oduc ion o he
ae odynamic e ec s ypically neglec ed in he classic DLM ( om Case 5 o Cases 6 and 7). The
lu e mechanism changes be ween Case 2 & 3 and Cases 6 & 7. Now, mode 1S ( i s bending)
and 4S (second bending) pa icipa e in he uns able ae oelas ic mode.
Figu e 16: Damping and equency o ae oelas ic mode 2S.
Table 10: Flu e ype 2, symme ic case. Flu e mechanism, equency, and speed.
Flu e Type 2
Na u al Mode Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7
w0 0 0.69∠296◦- - 2.64∠275◦2.12∠272◦
q0 0 0.08∠104◦- - 0.20∠135◦0.21∠133◦
1S. 1s Bending (S) 0.09∠181◦0.17∠185◦0.19∠186◦- - 0.92∠173◦0.79∠172◦
2S. 1s To sion (S) 1.00∠0◦1.00∠0◦1.00∠0◦- - 1.00∠0◦1.00∠0◦
3S. In-Plane Bending (S) 0.58∠159◦0.13∠91◦0.13∠92◦- - 0.33∠94◦0.31∠89◦
4S. 2nd Bending (S) 0.36∠169◦0.19∠184◦0.19∠185◦- - 0.85∠352◦0.71∠351◦
5S. P opelle yaw (S) 0 0.40∠93◦0.40∠93◦- - 0.69∠81◦0.65∠82◦
Flu e Mechanism 2nd Bending - 1s To -
sion - In-plane Bending
(S)
1s To sion - P op. Yaw
(S) (Equi alen o a
backwa d whi l o ip
p opelle )
- - Ou -o -plane Bending -
1s To sion
Flu e F equency [Hz] 5.67 4.24 4.24 - - 4.28 4.26
Flu e Speed [m/s] 88.75 81.75 86.25 - - 151.25 143.75
4 CONCLUSIONS
In his pape , he o mula ion o a amewo k o he ae oelas ic s abili y assessmen o highly-
lexible con igu a ions ea u ing dis ibu ed elec ic p opulsion is p esen ed. The app oach con-
side s he ai c a ee in he ai , hus e aining he ligh dynamic-ae oelas ic coupling. Ad-
di ionally, i akes in o accoun he ae oelas ic e ec s induced by he p opelle as well as he
e ec s o la ge de lec ions. To his aim, an ad-hoc enhanced e sion o he DLM has been de-
eloped and in eg a ed. Mo eo e , a im p ocedu e has been es ablished o ind he e e ence
25
IFASD-2024-182
Figu e 28: Mode 2S. Fi s Symme ic To sion
Figu e 29: Mode 3S. Fi s Symme ic In-Plane Bending
32
IFASD-2024-182
Figu e 30: Mode 4S. Second Symme ic Bending
Figu e 31: Mode 5S. Symme ic P opelle yaw
33
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5.4 An isymme ic modes
Figu e 32: Mode 1A. Fi s An isymme ic Bending
Figu e 33: Mode 2A. Fi s An isymme ic To sion
34
IFASD-2024-182
Figu e 34: Mode 3A. Second An isymme ic Bending
Figu e 35: Mode 4A. Fi s An isymme ic In-Plane Bending
35
IFASD-2024-182
Figu e 36: Mode 5A. An isymme ic P opelle yaw
5.5 Gy oscopic modes o he in- ligh ai c a
Table 11: Modes o he de o med ai c a a e in oduc ion o gy oscopic e ec s a V∞= 0 m/s
Na u al mode Gy mode 1 Na u al Mode Gy mode 2 Na u al Mode Gy mode 3
2S 0.88∠0◦2A 0.94∠0◦2S 0.64∠35◦
3S 0.28∠129◦3A 0.06∠272◦3S 0.72∠0◦
4S 0.08∠345◦4A 0.05∠318◦4S 0.08∠213◦
5S 0.38∠78◦5A 0.33∠87◦5S 0.27∠223◦
Desc ip ion S Backwa d whi l Desc ip ion A Backwa d whi l Desc ip ion S Backwa d whi l
F equency [Hz] 4.14 F equency [Hz] 4.19 F equency [Hz] 5.65
6 ACKNOWLEDGEMENTS
The ac i i ies desc ibed in his pape ha e been ca ied ou unde he p ojec INDIGO (In e-
g a ion and Digi al Demons a ion o Low-emission Ai c a Technologies and Ai po Ope a-
ions), coo dina ed by R. Ca alla o om Uni e sidad Ca los III de Mad id. INDIGO p ojec [11,
12] has ecei ed unding om he Eu opean Clima e, In as uc u e and En i onmen Execu i e
Agency (CINEA) unde he Ho izon Eu ope p og amme unde g an ag eemen No 101096055.
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