Paper
Int’l J. of Aeronautical & Space Sci. 14(4), 310–323 (2013)
DOI:10.5139/IJASS.2013.14.4.310
Static Aeroelastic Response of Wing-Structures Accounting for In-Plane
Cross-Section Deformation
Alberto Varello* and Alessandro Lamberti**
Department of Mechanical and Aerospace Engineering, Politecnico di Torino, 10129 Torino, Italy
Erasmo Carrera***
Department of Mechanical and Aerospace Engineering, Politecnico di Torino, 10129 Torino, Italy
Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Abstract
In this paper, the aeroelastic static response of flexible wings with arbitrary cross-section geometry via a coupled CUFXFLR5 approach is presented. Refined structural one-dimensional (1D) models, with a variable order of expansion for the
displacement field, are developed on the basis of the Carrera Unified Formulation (CUF), taking into account cross-sectional
deformability. A three-dimensional (3D) Panel Method is employed for the aerodynamic analysis, providing more accuracy
with respect to the Vortex Lattice Method (VLM). A straight wing with an airfoil cross-section is modeled as a clamped beam,
by means of the finite element method (FEM). Numerical results present the variation of wing aerodynamic parameters,
and the equilibrium aeroelastic response is evaluated in terms of displacements and in-plane cross-section deformation.
Aeroelastic coupled analyses are based on an iterative procedure, as well as a linear coupling approach for different free
stream velocities. A convergent trend of displacements and aerodynamic coefficients is achieved as the structural model
accuracy increases. Comparisons with 3D finite element solutions prove that an accurate description of the in-plane crosssection deformation is provided by the proposed 1D CUF model, through a significant reduction in computational cost.
Key words: aeroelasticity, higher-order 1D finite elements, Carrera Unified Formulation, in-plane cross-section deformation
1. Introduction
The in-depth understanding of aeroelastic effects on
deformable lifting bodies (LBs), due to steady and unsteady
aerodynamic loadings, is a typical challenging issue for the
current design of aerospace vehicles [1]. Furthermore, with
the forthcoming employment of composite materials in nextgeneration aircraft configurations, such as High-Altitude
Long Endurance aircraft (HALE) [2], and strut-braced wings
[3], accurate evaluation of the aeroelastic response becomes
even more crucial [4].
Recently, special attention has been directed to the
profitable exploitation of the aeroelastic phenomena
comprehension , by studying the concept of morphing wings,
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: October 21, 2013 Revised : December 6, 2013 Accepted: December 12, 2013
Copyright ⓒ The Korean Society for Aeronautical & Space Sciences
which are able to adapt and optimize their shape depending
on the specific flight conditions and mission profiles [5, 6].
The smart wing is very flexible and could allow a number of
advantages, such as drag reduction and aeroelastic vibrations
suppression by means of adaptive control [7] and different
solutions, such as compliant structures [8], bi-stable laminate
composites [9], piezoelectric [10] and shape memory alloy
actuation [11].
In order to develop aeroelastic tools that are able to work
in any regime and with any LB geometry, the literature from
the last decades has been widely influenced by research
devoted to build reliable methods, to couple computational
fluid dynamics (CFD) or classical aerodynamic methods with
the finite element method (FEM) for structural modeling [12].
* Research assistant, corresponding author, e-mail: [email protected]
**Ph.D. student
***Professor of Aerospace Structures and Aeroelasticity
310
http://ijass.org pISSN: 2093-274x eISSN: 2093-2480
Alberto Varello Static Aeroelastic Response of Wing-Structures Accounting for In-Plane Cross-Section Deformation
the several composite rotor blades applications, the work
done by Jeon and Lee [29] concerning the steady equilibrium
deflections, via a large deflection type beam theory with
small strains, is worth mentioning. An example of the use of a
refined beam theory for aeroelastic analysis can be found in
[30], where the static and dynamic responses of a helicopter
rotor blade are evaluated by means of a YF/VABS model.
Higher-order 1D models with generalized displacement
variables, based on the Carrera Unified Formulation, have
recently been proposed by Carrera and co-authors, for the
static and dynamic analysis of isotropic and composite
structures [31]. The CUF is a hierarchical formulation,
which considers the order of the model as a free-parameter
of the analysis. In other words, models of any order can be
obtained, with no need for ad hoc formulations, by exploiting
a systematic procedure. Structural 1D CUF models were used
to analyze the structural response of isotropic aircraft wings,
under aerodynamic loads computed through the Vortex
Lattice Method (VLM), in [32]. The aeroelastic CUF-VLM
coupling was preliminarily formulated in [33] for isotropic flat
plates, and then extended to instability divergence detection
and the evaluation of composite material lay-up effects on
the aeroelastic response of moderate and high-aspect ratio
wing configurations, in [34]. Flutter analyses of composite
lifting surfaces were also presented in [35], by coupling the
CUF approach with the Doublet Lattice Method.
The present work couples a refined one-dimensional finite
element model based on CUF to an aerodynamic 3D Panel
Method, implemented in the software XFLR5. Two potential
methods are here compared: the VLM and the 3D Panel
Method. The aeroelastic static response of a straight wing
is computed through a coupled iterative procedure, and a
linear coupling approach. Particular attention is drawn to
the in-plane deformation of the wing airfoil cross-sections,
as well as the aeroelastic influence of free stream velocity.
Valuable examples are in the review articles by Dowell and
Hall [13], and Henshaw et al. [14]. Reduced approaches, for
instance panel methods, are widely used for the classical
aerodynamics of wings under some limitations [15],
allowing a sizeable reduction in terms of computational
cost [16, 17]. The assumption of undeformable airfoil-crosssections [18] is typically not proper for recent configurations,
since the weight reduction makes wings more flexible and
highly-deformable. Hence, two-dimensional (2D) plate/
shell and three-dimensional (3D) solid methods are usually
employed for the structural modeling, instead of classical
one-dimensional (1D) theories, such as the Euler-Bernoulli,
Timoshenko, or Vlasov theories [19].
With the advent of smart wings, detailed structural
and aeroelastic models are even more essential to fully
exploit the non-classical effects in wing design, due to the
properties characterizing advanced composite materials,
such as anisotropy, heterogeneity and transverse shear
flexibility. Beam-like components can be analyzed by means
of refined one-dimensional (1D) formulations, which have
the main advantage of a lower computational cost required
compared with 2D and 3D models. A detailed review of the
recent development of refined beam models can be found
in [20]. El Fatmi [21] improved the displacement field over
the beam cross-section, by introducing a warping function,
to refine the description of normal and shear stress of the
beam. Generalized beam theories (GBT) originated with
Schardt’s work [22] and improved classical theories, by
using a piecewise beam description of thin-walled sections
[23]. An asymptotic type expansion, in conjunction with
variational methods, was proposed by Berdichevsky et al.
[24], where a commendable review of prior works on beam
theory development was given. An alternative approach to
formulating refined beam theories, based on asymptotic
variational methods (VABS), has led to an extensive
contribution in the last few years [25].
A considerable amount of research activity devoted to
aeroelastic analysis and optimization was undertaken in
the last decades, by using reduced 1D models. A review
was carried out by Patil [26], who investigated the variation
of aeroelastic critical speeds with the composite ply layup of box beams, via the unsteady Theodorsen’s theory. A
thin-walled anisotropic beam model in-corporating nonclassical effects was introduced by Librescu and Song [27]
to analyze the sub-critical static aeroelastic response, and
the divergence instability of swept-forward wing structures.
Qin and Librescu [28] developed an aeroelastic model
to investigate the influence of the directionality property
of composite materials, and non-classical effects on the
aeroelastic instability of thin-walled aircraft wings. Among
2. Numerical models: refined 1D CUF model
and panel methods
2.1 Variable kinematic 1D CUF FE model
For the sake of completeness, some details about the
formulation of CUF finite elements are here retrieved from
previous works [32, 34]. A structure with axial length L and
cross-section Ω is discretized through a mesh of NEL 1D finite
elements. A cartesian coordinate system is defined with axes
x and z parallel to the cross-section, whereas y represents
the longitudinal coordinate. According to the displacementbased framework of CUF [31], the displacement field is
311
http://ijass.org
�
�
�
�
� ��� �
����
������������
��
��
�����������
��� �
� �������
�� �� ���
with respect to the length �, the inplane
(subscript
andinout-of-plane
�� is�)evaluated
, �(subscript
���)iscross-section
calculated
theofcase
of
where where
�� is evaluated
���in
, ����
� �and
is calculated
in �� . inIn �the
case
small
� ���and
� . In
stress and strain vectors in Eq. 3 are
related
to respect
the
displacement
via inplane
linear
differential
with
to
the length
the
(subscript
�)out-of-plane
and out-of-plane
(subsc
with
respect
to the
length
�, vector
the �,
inplane
(subscript
�) andmatrix
(subscript
�)
operators �� , �� and material stiffness
, ���
, �Eq.
as
follows:
�� ,in
�� related
stress
and���
strain
vectors
Eq.
3 are
related
the displacement
via diffe
linea
stress matrices
and
strain
vectors
in
3�are
to the to
displacement
vector vector
via linear
�, ��
� ���
��
�� � �� � operators
� and
operators
�
���material
material
stiffness
matrices
, ���
stiffness
matrices
��� , ����
����
���
��, ,(5)
��, , ����
�� ,as�follows:
�� as follo
� �and
�� � ��� �� � ��� ��
�� � �� �
�� ��������
�� ��� ��
�� � ����
�� � ������ �� �
� �
��
Using Eq.
5, of
Eq.functions
3 can be rewritten
in terms of virtual nodal displacements:
assumed to be an expansion of a certain
class
displacements:
�� ��������
�� ��� ��
�� � ����
�� � ��� �� �� �
� �
��
Fτ, which depend on the cross-section coordinates x
�
�
����
�5,��Eq.
� ��
��
(6) displacements:
��
Using
Eq.
3 can
be rewritten
in of
terms
of virtual
nodal
Using
35,can
rewritten
in terms
virtual
nodal(6)displacements:
�� � Eq.
�� �be
and z. Introducing the shape functions Ni and the nodal
� � �, � , ��
� � �, � , ��
(1)
�, �� � �� �x, z� �� ��� ���
(1)
���,displacement
�, �� � �� �x, z�vector
�� ��� �,�� the displacement vector ���,
with
�
�
�
�
����
����
���
��
��
���
���,
��
�
, ��
2�/2
and
� � �, � , �� � ���� �where,
2�/2 the 3 x 3 and 3 x 1 fundamental
���
������
�� �� ���
����
��nuclei
���
the ͵ ൈ ͵ and ͵ ൈ ͳ fundamental nuclei ࡷ௜௝ఛ௦ and ࡲ
components ux, uy, and uz where
becomes:
5 ఛ௜ are introduced. From Eq. 6 the goare introduced. From Eq. 6, the governing equation of motion
whereterm
contains
degrees
to the � �� elewhere ��� contains the degrees of freedom of the � �� expansion
corresponding
to the of� ��freedom
ele- of the � �� expansion term corresponding
���
� derived
, �� the through
�� �,
verning
equation
athe
finite
element
assembly
procedure:
can
be
derived
through
aൈ
finite
element
assembly
procedure:
the ͵ofൈmotion
͵ and can
͵ ൈbe
ͳ fundamental
where
nuclei
ࡷ
͵ ௜௝ఛ௦
ൈ
͵and
and
͵ఛ௜
are
ͳ fundamental
introduced.
From
nucleiEq.
ࡷ
6௜௝ఛ௦
the
andgoࡲఛ௜ are introduced. From E
(1)ࡲ
���, �, �� � �� �x, z� �� ���
���where
5
5
� � �, � (1)
, �� � ���� � 2�/2
ment͵node.
The compact
expression
ment node. The� compact
expression in Eq.
1 is the
based
repeated
subscripts
௜௝ఛ௦ in �Eq. 1 is based on Einstein's notation: repeated subscripts �
where
͵equation
ൈon͵ Einstein's
and
ൈ ͳnotation:
fundamental
nuclei
are introduced.
Fromthrough
Eq. 6 the
go- element
(7)
� �, � , �
ࡷ through
ࢗ ൌࡷࡲ
ఛ௜motion
�
verning
of
motion
can
be derived
verning
equation
aand
finiteࡲ
ofelement
assembly
can be derived
procedure:
a (7)
finite
assembly procedure:
�� ��� ���
� �of�,the
�(1)
,�����
where ���� contains
the �
degrees
of2�/2
freedom
expansion
term
corresponding
to
the � �� ele�
�,
�
�
�
�,
,
�
�
,
�
�
�
�
�,
�
�
�,
,
�
�
,
�
��
�
�,
�
,
����
�
�
�
�
�
�x,
���
(1)
���,and
�, ��
�
�
z�
�
�
௜௝ఛ௦ (1) Taylor's polynomials
�x,�
���
���
�x,Taylor's
���
���,
���,
�,����,�summation.
�����
������
and
�the
indicate
summation.
of
the Eq.
xItand
z variables
are employed
� indicate
Multivariate
polynomials
of
xcan
and
zderived
variables
are
��x,�z�
���z�
(1) nodal
(1)
���,
�,���
�,
� ��
�
z�
��z�
�
����,
��
��
� �x,
�is
��
���,
where
the
͵the
and
ൈstiffness
ͳ�
fundamental
nuclei
ࡷ
are
introduced.
From
6ࢗ the
go-is(7)
��
�
,,͵
�
verning
equation
of
through
a(1)
assembly
procedure:
�and
�
�,ࡲ
�ఛ௜equivalent
,�
����
��
�,
���ൈ�
,�͵
�motion
����
2�/2
where
ࡷ
structural
matrix
ࡲ,Multivariate
isemployed
the
of
forces.
��
��
��
��
��,
�,
�
����
��be
��,
� and
�
�
�
�2�/2
�
�,
,2�/2
��
����
�
����
2�/2
�
2�/2
is�vector
the
structural
stiffness
matrix,
and
the
where
ࡷ
ࢗfinite
ൌ
ࡲelement
ࡷshould
ൌ ࡲbe
��x,
� z�
�,
��
, ����, ��
��
���
�x,
(1)
���,
�, ��
� ��The
���,
(1)
���,
�,is
��
�(1)
����
z�
�,�� ���
�����
����
�
�,
�
,�
��
�
�
�,
,
�
��
�x,
���
���
ment
node.
compact
expression
in
Eq.
1
based
on
Einstein's
notation:
repeated
subscripts
�
௜௝ఛ௦
(1)
�z�
�
z�
�
�
�
௜௝ఛ௦
��
��
�
�
�
�,
�
,
�
����
�
2�/2
�
�
�,
�
,
�
����
�
2�/2
�of freedom
�
� �� ���,
��
�x,
���
of��
the
�� �
expansion
corresponding
the
�͵ ൈ
�, ��
��where
�
where
͵ eleൈfreedom
͵fundamental
and
͵ൈ
ͳcross-section
fundamental
nuclei
and�ࡲis
areFrom
introduced.
From
thewhich
go-that
where
ൈ
͵tois
and
ͳ
nuclei
ࡷ ��
and
ࡲ(1)
introduced.
the
go- Eq.
�term
��
��
ఛ௜ defined
���
��
�functions
�,
� �contains
�,
, ��
���
, the
����
�͵ ��
����
2�/2
� the
2�/2
(1)
���,
�,
��z�
�
vector
of
equivalent
nodal
forces.
It
should
be6 noted
the
of
the
τbe
-th
ఛ௜ࡷ
� �x, z�
��
��
and
asEq.
the6procedure:
expansion
is anofree parameter
here
as
functions
�are
and
��degrees
defined
the
expansion
order,
which
a͵
free
here�asthe
cross-section
�equation
�
�,
�
,of
� of
����
� to
2�/2
௜௝ఛ௦
verning
motion
can
derived
through
a�͵��
finite
element
assembly
��
��
��ࢗparameter
��
��
��
��
��
�
�,
�
, expansion
�
����
�on
2�/2
ࡷ
ࡲis
(7)
where �where
degrees
of
of
�of
expansion
term
corresponding
the
�to
ele� as
�
�,isof
,term
�of
theterm
͵ is
ൈ
and
ͳthe
fundamental
nuclei
and
ࡲ
areshould
introduced.
From Eq. nodal
6 the forces.
gothat
no
assumptions
the
expansion
order
have
been
made
so
far.
Therefore,
itand
isorder,
possible
tovector
where
��� �
contains
the�x,degrees
the
degrees
of� the
freedom
the
of� �the
�expansion
corresponding
corresponding
the
to
�the
ele�ൌ
elewhere
where
�freedom
�of
�� contains
contains
contains
the
degrees
thenoted
degrees
of
freedom
of
freedom
the
the
expansion
�where
expansion
term
corresponding
corresponding
tovector
theto �the
�stiffness
ele- ࡷ
��term
ఛ௜ is
where
ࡷ
the
structural
stiffness
matrix
where
and
ࡷ
ࡲൈ
is
the
structural
of eleequivalent
matrix
nodal
forces.
ࡲ
It
the
be of equivalent
����contains
��
�� �freedom
���
(1)
���,
�,
�
�
z�
�
assumptions
on the
expansion
orderto
have
so
expansion
corresponding
to
the
i-th
element
node.
The
��
��
�� ele��far been made.
�
� term
��
and ��
�onindicate
summation.
Multivariate
Taylor's
polynomials
of freedom
the
x2�/2
and
zthrough
variables
are employed
��
where
�
contains
the
of
freedom
of
the
�
expansion
term
corresponding
to
the
�
where
�
contains
the
degrees
of
of
�
expansion
term
corresponding
the
�
ele�
�
�,
�
,
�
����
�
�� degrees
��
��
ession
in
Eq.
1
is
based
Einstein's
notation:
repeated
subscripts
�
verning
equation
of
motion
can
be
derived
a
finite
element
assembly
procedure:
��
��
verning
equation
of
motion
can
be
derived
through
a
finite
element
assembly
procedure:
��
��
ees
degrees
of freedom
of
of
the
of the
�the degrees
expansion
� expansion
term term
corresponding
tonodes
theto �the
ele�� �of
ele�nodal
�nuclei
are
formulated
in(7)
the
present work and
the
formulation.
number
ofof
nodes
�
�� � where
��Elements
are
formulated
in
the
work
and
of
the
formulation.
Elements
number
ofis
where
���freedom
contains
ofexpression
freedom
ofiscorresponding
the
expansion
term
corresponding
to
� present
ele� the
�
ࡷ
ࢗ
ൌrepeated
ࡷterm
is�
the
stiffness
matrix
and
ࡲwith
is
the
vector
equivalent
forces.
Itit should
be
here
���node.
contains
the
degrees
of ment
freedom
ofwith
the
�Eq.
expansion
corresponding
to without
the
�the
eleTherefore,
itno
is
possible
tothrough
obtain
models
compact
in
based
on
Einstein’s
notation:
ment
The
compact
expression
in
Eq.
1in
based
Einstein's
notation:
subscripts
�of
obtain
higher-order
1D
models
changing
the
formal
expression
of
components
as
verning
equation
motion
can
be
derived
ahigher-order
finite
element
assembly
ment
ment
node.
node.
The
The
compact
compact
expression
expression
inEq.
1Eq.
is1on
1based
is
based
on
Einstein's
on
Einstein's
notation:
repeated
repeated
subscripts
subscripts
�ࡲ
�assumptions
ment
node.
node.
The
compact
The
compact
expression
expression
inthat
Eq.
in
1Eq.
is
based
1notation:
isrepeated
based
on
Einstein's
on
Einstein's
notation:
notation:
repeated
subscripts
subscripts
�so
�expansion
noted
nostructural
assumptions
on
the
expansion
noted
order
that
have
been
made
on
the
far.
Therefore,
order
is 1D
have
possible
beentoprocedure:
made so far. Therefore, it i
��
��
and
�
is
defined
as
the
expansion
order,
which
is
a
free
parameter
here
as
cross-section
functions
�
ment
The
based
on
Einstein's
notation:
repeated
� Einstein'sthe
�1 iis
ment
node.
The
compact
expression
in
Eq.
1 subscripts
is
based
notation:
repeated
�(7) nuclei
where
�Eq.
contains
the
degrees
ofxfreedom
of
the
�repeated
expansion
term
corresponding
toࡲthe
�ࡷ
ultivariate
Taylor's
polynomials
of
the
and
zinnotation:
variables
are
employed
ࢗeleൌ on
ࡲchanging
without
formal
expression
of the
repeated
subscripts
τEq.
and
indicate
summation.
Multivariate
ࡷ
ࢗൌ
(7)subscripts
��node.
ct
pression
expression
in
in
1Eq.
iscompact
1based
iscompact
based
on
Einstein's
onexpression
Einstein's
notation:
repeated
subscripts
subscripts
�
�
named
��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
of degrees
named
��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
of
degrees
ment
node.
The
expression
in
Eq.
1
is
based
on
Einstein's
notation:
repeated
subscripts
�
where
ࡷof isthe
the
structural
stiffness
matrix
and
is
the
vector
ofemployed
equivalent
nodal
should
be
noted
that
no
assumptions
the
order
have
been
made
so
far.ࡷ
Therefore,
it models
isItcomponents
possible
to asexpression
ent
The
expression
inMultivariate
1summation.
is based
on
Einstein's
notation:
repeated
subscripts
�the
well
as
classical
beam
models
as
Euler-Bernoulli's
and
Timoshenko's.
Higher-order
ࢗofൌthe
ࡲforces.
(7) co
andnode.
� indicate
summation.
Multivariate
Taylor's
polynomials
xof
and
zpolynomials
are
employed
and
and
� compact
indicate
� indicate
summation.
summation.
Multivariate
Taylor's
Taylor's
polynomials
polynomials
the
ofare
xvariables
the
and
xon
and
zsuch
variables
z expansion
variables
are
employed
are
employed
and
�and
indicate
� Eq.
indicate
summation.
Multivariate
Multivariate
Taylor's
Taylor's
polynomials
of
the
of
xcomponents,
and
xࡲzand
variables
z the
variables
are
are
employed
obtain
higher-order
1D
models
without
obtain
changing
higher-order
formal
1D
expression
models
without
nuclei
changing
theproformal
of the nuclei
as
well
as
classical
beam
models,
such
as
Taylor’s
polynomials
of
the
x
and
z
variables
employed
�xMultivariate
�and
are
formulated
in employed
the
present
work
and
of compact
the
formulation.
Elements
with
number
ofis
nodes
�the
and
�
indicate
summation.
Multivariate
Taylor's
polynomials
of
z
variables
are
�
and
�
indicate
summation.
Taylor's
polynomials
of
the
x
and
z
variables
are
employed
ment
node.
The
expression
in
Eq.
1
based
on
Einstein's
notation:
repeated
subscripts
�
and
�
is
defined
as
the
expansion
order,
which
is
a
free
parameter
�
where
ࡷ
the
structural
stiffness
matrix
and
ࡲ
is
the
vector
of
equivalent
nodal
forces.
It
should
be
where
ࡷ
is
the
structural
stiffness
matrix
and
ࡲ
is
the
vector
of
equivalent
nodal
forces.
It
should
be
� Multivariate
ion.
Multivariate
Taylor's
Taylor's
polynomials
polynomials
ofcross-section
the
of xthe
andxthe
zand
variables
z variables
are
employed
are
employed
freedom
(DOFs)
used
through
the
proposed
approach
of freedom
(DOFs)
used
through
proposed
approach
is:
indicate
summation.
Multivariate
polynomials
of
the
xiszof
and
z models
variables
are
employed
Euler-Bernoulli’s
and
Timoshenko’s.
Higher-order
models
here
as
functions
Fand
,�
and
N
as
the
noted
no
assumptions
the
expansion
order
have
made
so
far.ofis:
Therefore,
itcomponents
is possible
to
obtain
higher-order
1D
without
changing
formal
expression
asand
accurate
description
of
shear
mechanics,
the
in-plane
and
out-of-plane
cross-section
deτthe
dhere
�and
indicate
summation.
Multivariate
polynomials
ofthat
xclassical
and
variables
are
employed
where
ࡷ
the
structural
stiffness
matrix
and
ࡲtheisnuclei
theEuler-Bernoulli's
vector
of models
equivalent
forces. It Higher-order
should be
and
�� and
isTaylor's
asvide
the
expansion
order,
which
is
athe
free
parameter
as�cross-section
functions
�as
and
�functions
is�defined
is
defined
the
as
expansion
the
expansion
order,
order,
which
which
is
aisfree
is
aEuler-Bernoulli's
free
parameter
parameter
here here
as
cross-section
as cross-section
functions
�
��defined
�functions
�
and
is
�
defined
isdefined
defined
ason
the
as
expansion
the
expansion
order,
order,
which
which
isbeen
a free
isand
a parameter
free
parameter
here
here
cross-section
asTaylor's
cross-section
functions
�an
well
as
beam
models
such
as
well
asthe
classical
beam
Timoshenko's.
models
such
Higher-order
as
pro-nodal
Timoshenko's.
� as
�
named
��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
of
degrees
provide
an
accurate
description
of
the
shear
mechanics,
expansion
order,
which
is
a
free
parameter
of
the
formulation.
and
�
is
defined
as
the
expansion
order,
which
is
a
free
parameter
here
as
cross-section
functions
�
and
�
is
defined
as
the
expansion
order,
which
is
a
free
parameter
here
as
cross-section
functions
�
and
�
indicate
summation.
Multivariate
Taylor's
polynomials
of
the
x
and
z
variables
are
employed
noted
that
no
assumptions
on
the
expansion
order
have
been
made
so
far.
Therefore,
it
is
possible
to
�
�
are
formulated
in
the
present
work
and
ith
number
of
nodes
�
�
noted
that
nowhich
assumptions
on
the expansion� order have been made so far.��
Therefore,
is possible to
�as the
and
� is�defined
is defined
asexpansion
the� expansion
order,
which
isobtain
athe
free
isasaexpansion
parameter
free
parameter
nctions
ns here
�� and
�� cross-section
��
�expansion
���� �it
2�
�
����
�Poisson's
2�
andisorder,
�
defined
as
order,
which
issuch
a parameter
free
parameter
as
functions
higher-order
1D
models
changing
formal
expression
of
the
nuclei
components
as
well
classical
beam
models
Euler-Bernoulli's
Timoshenko's.
Higher-order
models
�number
formation,
effect
the
spatial
directions
and
the
torsional
mechanics
in
more
detail
than
and
�
defined
expansion
order,
which
isalong
anoted
free
re
cross-section
�with
that
no
assumptions
on
thepresent
order
have
been
made
soprofar.in-plane
Therefore,
is possible to
�nodes
� �with
are
in
the
present
work
and
of as
the
formulation.
Elements
ofisnodes
�nodes
(2)
(2)
� of
vide
accurate
description
of
the
shear
vide
mechanics,
an
accurate
the
description
in-plane
and
of
out-of-plane
the
mechanics,
cross-section
the
deanditout-of-plane
cross
�
�
�
are
formulated
are
formulated
in��without
the
in
present
the
present
work
work
and
and
of the
of formulation.
the functions
formulation.
Elements
Elements
with
with
number
number
of
�formulated
�ofwith
�
are
�as
formulated
are
formulated
inthe
the
in
present
the
workwork
and
and
of
the
formulation.
the
formulation.
Elements
Elements
number
number
of�formulated
nodes
of
nodes
�
� ��
����
�
�and
���
�
���
����
� as
�
���
the
in-plane
and
out-of-plane
cross-section
deformation,
Elements
with
number
ofthe
nodes
N
=4
are
in
the
�Nan
�
� �
��shear
��
2
2
of
freedom
(DOFs)
used
through
the
proposed
approach
is:
�
�
are
formulated
in
the
present
work
and
of
the
formulation.
Elements
with
number
of
nodes
�
�
�
are
formulated
in
the
present
work
and
of
the
formulation.
Elements
with
number
of
nodes
�
obtain
higher-order
1D
models
without
changing
the
formal
expression
of
the
nuclei
components
as
and
�
is
defined
as
the
expansion
order,
which
is
a
free
parameter
here
as
cross-section
functions
�
polynomials
as
shape
functions
[19].
The
number
of
degrees
obtain
higher-order
1D
models
without
changing
the
formal
expression
of
the
nuclei
components
as
�
�
�formulated
�
�
�
are
�
are
formulated
in
the
in
present
the
present
work
work
and
and
ments
sagrange
with
with
number
number
of
nodes
of
nodes
�
�
Poisson’s
effect
along
the
spatial
directions,
and
the
present
work,
and
named
B4,
using
third-order
Lagrange
�
�with number of nodesclassical
�functions
are formulated
in
the
present
work
anddirections
the��,
formulation.
Elements
�
as
classical
beam
models
such
as
Euler-Bernoulli's
and
Timoshenko's.
models
provide
an
accurate
description
of
the
shear
mechanics,
the
in-plane
and
out-of-plane
cross-section
de-of
models
do.
Thanks
to
the
CUF,
the
present
hierarchical
approach
is Higher-order
invariant
with
respect
to
� �
obtain
higher-order
1D
models
without
thespatial
formal
expression
thetorsional
nuclei components
�
are
formulated
in the
present
work
and
theofformulation.
Elements
with
number
ofLagrange
nodes
��polynomials
named
using
third-order
Lagrange
polynomials
as�well
shape
[19].
The
number
of
degrees
formation,
Poisson's
effect
along
the
spatial
formation,
Poisson's
and
the
effect
torsional
along
mechanics
the
indirections
more
detail
and
than
the
mechanics inasmor
named
named
��,
��,
usingusing
third-order
third-order
polynomials
asLagrange
shape
aspolynomials
shape
functions
functions
[19].
[19].
The
The
number
number
of
degrees
of
degrees
named
named
��,Lagrange
using
��,
using
third-order
third-order
Lagrange
polynomials
as
shape
as
shape
functions
functions
[19].
[19].
The
number
The
of
degrees
ofchanging
degrees
The
variational
statement
employed
is
thenumber
Principle
of
Virtual
Displacements:
The
variational
statement
employed
isfunctions
the Principle
of
Virtual
Displacements:
��
torsional
mechanics,
in
more
detail
than
classical
models
polynomials
as
shape
[19].
The
number
of
degrees
�
����
�
2�
named
��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
of
degrees
named
��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
of
degrees
well
as
classical
beam
models
such
as
Euler-Bernoulli's
and
Timoshenko's.
Higher-order
models
pro�degrees
are
formulated
in
the
present and
workTimoshenko's.
and
the formulation.
Elements
withfunctions
number
of
nodes
theof
proposed
approach
is: shape
well[19].
as classical
beam
models
such as
Euler-Bernoulli's
Higher-order
models pro� �of
(2) and out-of-plane
rhorder
Lagrange
Lagrange
polynomials
polynomials
as
as shape
functions
[19].
The
The
number
number
of degrees
�field
��the
����
�
��
���
��
vide
an accurate
description
of
shear
mechanics,
the
in-plane
de��,
using
third-order
Lagrange
polynomials
as
shape
functions
[19].
The
number
ofCUF,
degrees
formation,
Poisson's
effect
along
the
spatial
directions
and
the
torsional
mechanics
incross-section
more
detail
than
the
order
of
the
displacement
expansion.
More
details
are
not
reported
but
can
be
found
in to
well
as
classical
beam
models
such
as
Euler-Bernoulli's
and
Timoshenko's.
Higher-order
pro- wi
med
��,ofusing
third-order
Lagrange
polynomials
asused
shape
functions
[19].
The
number
of
degrees
of named
freedom
(DOFs)
used
through
the
proposed
approach
is:
2
do.
Thanks
to
the
CUF,
the
present
hierarchical
approach
is approachmodels
of
freedom
(DOFs)
used
through
proposed
approach
is:
classical
models
do.
Thanks
to
the
classical
the
present
models
hierarchical
do.
Thanks
approach
to here,
the is
CUF,
invariant
the
present
with
respect
hierarchical
is invariant
freedom
of freedom
(DOFs)
(DOFs)
used
used
through
through
the(DOFs)
proposed
the
proposed
approach
approach
is:
is:
of
freedom
of
freedom
(DOFs)
used
through
through
thethe
proposed
the
proposed
approach
approach
is:
is:
�
�
�
�
�
�
��
vide
anshape
accurate
description
of
the
shear
mechanics,
the (3)
in-plane
and
out-of-plane
cross-section
de- field
of
freedom
(DOFs)
used
through
the
proposed
approach
is:� ��
of �freedom
(DOFs)
used
through
the�
proposed
approach
is:
���,
����
� 2�
named
using
third-order
Lagrange
polynomials
as
functions
[19].
The
number
of
degrees
vide
an
accurate
description
of
the
shear
mechanics,
the
in-plane
and
out-of-plane
cross-section
de(3)
invariant
with
respect
to
the
order
of
the
displacement
�
�
��
�
���
�
�
��
�
�
��
�
��
�
��
�
��
�
���
�
��
�
�
��
�
��
ough
d���
through
the
proposed
the
proposed
approach
approach
is:
is:
���
� the
� � mechanics
���in more
� �the
� of
� Poisson's
���
(2)ofofthe
effect
along
the
spatial
directions
and
torsional
detail
than
of�freedom
(DOFs)
used
through
the���proposed
approach
is:
� proposed
��
� (DOFs)
���
classical
do.
Thanks
to2�the
CUF,
the
present
hierarchical
approach
is invariant
withcan
respect
to
work
[31].
The
variational
statement
is
the
Virtual
Displacements:
vide
an
accurate
description
of� displacement
the
shear
the but
in-plane
and
out-of-plane
cross-section
de-can
�� �approach
��the
����
�
2�formation,
freedom
is:
thePrinciple
field
expansion.
the
order
of
the
details
are
notmechanics,
reported
field
expansion.
here,
More
be
details
found
in
are not reported
here, but
��
��
�� ���
�
����
�
����
�
2�order
�models
2�
����
�displacement
����
� 2�
�
�More
�employed
2used through����
(2)
(2)
���
�
��
�����
� ����
expansion.
More
details
are
not
reported
here,
but
can
be
(2)
(2)
(2)
(2)
�
��
�
��
�
�
�
�
���
���
��
�
��
�
��
����
����
�
�
�
�
���
���
��
��
��
��
��
��
����
�
2� Poisson's
�and
����
2� andmechanics
formation,
effect
along
the spatial
directions
the torsional
mechanics
in
more detail than
Poisson's
effect
along the
spatial
directions
the �
torsional
in
more
detail
than
of freedom
(DOFs)
theformation,
proposed
approach
is:
2 �
2
2
2
2
�� ���
����
� ����
�used
2�� through
2�
(2)
(2)
�
����
2�A�order
classical
models
do.
to�
the
CUF,
the
present
hierarchical
approach
iswork
invariant
with
respect
�[31].
��
�
�energy
���
�and
��
�the
���
the
the
displacement
field
expansion.
More
details
are
reported
here,
but
can
found to
in
2.2
numerical
approach
for
wing
aerodynamic
analysis
��
Poisson's
effect
along
the
spatial
directions
and
the be
torsional
mechanics
in more
��
found
in
the
work
ofnot
��
���
����
� 2� �
(2)
(2)Thanks
is
internal
strain
energy
�[31].
is the
of
external
loadings
variationally
con-detail than
where
����formation,
is ���
theDisplacements:
internal
and
isof
work
of ����
external
loadings
variationally
����
the
of
work
ofcon[31].
�����
��
��
����
�
�where
�
���
oyed
is����
the�Principle
of
Virtual
��� work
(2)
��
��
2 � ����
�
���
�the
����
��
�strain
(2)2 the
��
(3) ���
�
��
����
�
�
���
�
��
�
���
�
��
�
�
��
�
��
�
��
�
2
2
��
���
�
�
�
�
���
2
The variational
statement
employed
is
the
Principle
ofemployed
Virtual
Displacements:
classical
models
do.
Thanks
to
the
CUF,
the
present
hierarchical
approach
is
invariant
with
respect
to
��
�
����
�
2�
classical
do.
Thanks
to
the
CUF,
the
present
hierarchical
approach
is
invariant
with
respect
to
The The
variational
variational
statement
statement
employed
employed
is
ismodels
Principle
the
Principle
of
Virtual
of
Virtual
Displacements:
Displacements:
2 the
The
variational
statement
employed
is
the
Principle
of
The
variational
The
variational
statement
statement
employed
is
the
is
Principle
the
Principle
of
Virtual
of
Virtual
Displacements:
Displacements:
�
(2)
the
the
displacement
field
expansion.
More
details
are
not
reported
here,
but can
found
in concentrated
the order
work
[31].
�Displacements:
��
���� method
� �is theand
���
2.2.1
Preliminaries
models
do.
Thanks
tohere
the
CUF,
the
present
hierarchical
approach
is invariant with
��
sistent
present
method
and
derived
theaerodynamic
case
of be
a analysis
generic
load respect to
sistent with
the present
derived
for
the
case with
ofclassical
afor
generic
concentrated
load
2.2
Aof
numerical
approach
wing
aerodynamic
2.2
AVirtual
numerical
analysis
approach
forfor
wing
The
statement
employed
Principle
of
Virtual
The
variational
statement
employed
isthe
the
Principle
of
Displacements:
2 here
Virtual
Displacements:
mployed
nt employed
is thevariational
isPrinciple
the Principle
of
Virtual
of
Virtual
Displacements:
Displacements:
�the Principle
2.2 A
numerical
approach
for
wing
aerodynamic
The
employed
is
of
Virtual
Displacements:
(3)
�e variational
�����variational
�� � statement
���� �statement
�
��
�
��
�
��
�
employed
theinternal
Principle
of
Virtual
Displacements:
thedisplacement
order
of�the field
displacement
field
expansion.
More
details
arehere,
not reported
here,
but in
can be found in
�where
the
order
of
the
expansion.
More
details
are
not
reported
but
can
be
found
is�isthe
strain
energy
and
is
the
work
of
external
loadings
variationally
con�������
�
���
�the
� � �� � ��
�
� numerical
� typically
�analysis
�
�of���
� ��
the
work
[31].
(3)
�of��
2.2
A
for
�
�������
The
evaluation
aerodynamic
loads
be
carried
through
CFDpoint
code��
which
������ �
���
���ison
�
��
�
��
�
order
the
expansion.
More
details
are
notsolves
reported here, but can be found in
(3)
(3)
(3) a(3)
�
����
���
�����
��
�
���
�2.2.1
�
��
�of
�
��
��
�,�wing
�load
�
��
��
���
�
���
�
�approach
��
�
�
�����
��
���
�
��
�
��
�
�
�displacement
��
�Preliminaries
analysis
Preliminaries
2.2.1
���
��
��,can
acting
on
the
load
application
� � ���statement
acting
the
arbitrary
����
,���
���aerodynamic
which
does
notarbitrary
nec-fieldout
����
��
���
�of
�Virtual
� ���
�application
���
���
���
���
�
��
�
��
�point
��
��
���
The variational
employed
the��
Principle
Displacements:
�
� , �� , �� �, which does not nec�� ����
��
��
�
�
� �
�
� �
� �
�
� � �� ���
�
�
(3)
�
�
��
�
�
��
�
��
�
��
�
��
�
��
�
���
�
�
��
�
�
��(3)� ����� �load
��� �
�work
��of
�
�
�
�
(3)�a generic
the
[31].
���
�method
� work
� hereofderived
��� �for
��� the case �
� � concentrated
the
[31].
(3)
(3)
�
�
�
�
��n���
�
���
���
�
�
�
��
�
�
��
�
��
�
�
�
��
��
�
��
�
��
�
�
��
�
sistent
with
the
present
and
of
� �is �the
�
���� ��loadings
(3)
����
� ��� �variationally
����
��
������
�� ���
�
energy �and� �����
of
external
conA�
numerical
approach
for
wing
aerodynamic
���
� �for
� the
2.2.1
Preliminaries
� ����
example
either
Navier-Stokes
equations
orThe
Euler
equations
This
analysis
hascarried
��������
�work
���
�2.2
��
��
work
of(3)
[31].
� ����
� �� � ��
���
The
evaluation
of
aerodynamic
loads
can
beanalysis
evaluation
typically
carried
of numerically.
aerodynamic
out through
loads
akind
CFD
canofbe
code
typically
which
solves out through a CFD code w
�
2.2.1
Preliminaries
essarily
lies
along
the
1D
finite
element
mesh,
unlike
standard
liesisalong
the
1D finite
element
mesh,
unlike
standard
1D
FE
models.
�
stands
for
the
virtual
��strain
isessarily
the
strain
energy
and
�internal
is�and
the
work
of
external
loadings
variationally
conwhere
��where
� internal
internal
the
strain
energy
energy
and
�
�
is
the
is
work
the
work
of
external
of
external
loadings
loadings
variationally
variationally
con����internal
�is
��� where
���
is
the
is
the
internal
strain
strain
energy
energy
and
�
and
�
is
the
is
work
the
work
of
external
of
external
loadings
loadings
variationally
variationally
concon- 1D FE models. � stands for the virtual
where
where
�
�
�
���the
���
���
���
���
���
���
(3)con�
��
�
���
�
�
��
�
�
��
�
��
�
��
�
�
2.2
A
numerical
approach
for
wing
aerodynamic
analysis
2.2
A
numerical
approach
for
wing
aerodynamic
analysis
���
�
�
�
�
���
�is�the
��internal
��case
�strain
�is the
acting
on
the
arbitrary
application
point
��
,variationally
��can
, ��assumptions
�,
which
does
not
nec�
���of
�some
����
and
�strain
the
work
of
external
loadings
conwhere
����
isload
the
internal
strain
energy
and
is
the
work
of
external
loadings
variationally
conwhere
�is
od strain
and
here
derived
for
the
aenergy
generic
concentrated
load
�
� L
2.2.1
Preliminaries
The
of
aerodynamic
loads
can
be
typically
���
The
evaluation
ofeither
aerodynamic
loads
befor
typically
carried
out
through
a analysis
CFD
code
which
solves
internal
energy,
and
Lcost
isbut
the
work
where
���
high
computational
under
itfor
iseither
possible
to
employ
simplified
approaches.
2.2
A
numerical
approach
wing
aerodynamic
int external
ext Navier-Stokes
nal
train
energy
energy
and
�the
�is
is
work
the
work
of
of
external
loadings
variationally
variationally
conconfor
example
equations
example
orevaluation
Euler
equations
Navier-Stokes
numerically.
equations
This
kind
or
of
Euler
analysis
equations
has numerically. This kind of
���internal
���the
isinternal
strain
energy
andderived
�is���loadings
isafor
the
work
of external
loadings
variationally
conwhere
�is���and
becomes:
variation.
By
using
Eq.
1,
��
becomes:
variation.
using
Eq.
1,and
��
the
strain
energy
�
the
work
ofand
external
loadings
variationally
conhere
����with
sistent
the
present
method
and
here
the
case
ofthe
aderived
generic
concentrated
load
���
���
���
sistent
sistent
with
with
theBy
the
present
present
method
method
and
and
here
here
derived
derived
for
the
for
case
case
of
a
of
generic
a
generic
concentrated
concentrated
load
load
sistent
sistent
with
with
the
present
the
present
method
method
and
here
here
derived
for
the
for
case
the
case
of
a
of
generic
a
generic
concentrated
concentrated
load
load
carried out through a CFD code, which solves, for example,
of external 2.2.1
loadings
variationally
consistent with the
2.2.1 Preliminaries
Preliminaries
sistent
with
the
present
method
and
here
derived
for
the
case
of
a 1D
generic
concentrated
load
sistent
with
theunlike
present
method
and
here
for
the
of assumed
a generic
concentrated
loadsolves
essarily
liesthe
along
the
1D
finite
element
mesh,
standard
FE
models.
�work,
stands
for
thecase
virtual
isapplication
the
internal
strain
energy
and
�generic
isdoes
the
work
of
external
loadings
variationally
conwhere
����
ofconsidered
aerodynamic
loads
canderived
be
typically
carried
out
through
a This
CFD
code
which
themethod
arbitrary
load
point
��
,�
�,
not
necfor
example
either
Navier-Stokes
equations
or
Euler
equations
numerically.
kind
of
analysis
has
InThe
the
wing
cases
in
the
present
the
flow
field
is
to
be
steady
and
the
fluid
���
�
�
�
2.2.1
Preliminaries
nt
ethod
and
and
here
here
derived
derived
for
for
case
the
case
of
a, �of
generic
a which
concentrated
load
load
aevaluation
computational
cost
but
under
some
aload
high
assumptions
computational
cost
but
to�
under
employ
some
simplified
assumptions
it is possible to employ
simplified
� equations
�or
�high
� it is possible
� concentrated
�present
with
the
and
here
derived
for
the
case
of
a, ����generic
concentrated
either
Navier-Stokes
Euler approaches.
equations
present
method,
and
here
derived
for
the
case
of
awhich
generic
� method
�the
��arbitrary
�for
(4)
(4)
��
�
��
�
�
��
�
�
�
��
�
��
��
�
�
��
�
�
�
�
��
�
tent
with
the
present
method
and
here
derived
the
case
of
a
generic
concentrated
load
� sistent
� ��
�
�
acting
on
arbitrary
load
application
point
��
,
�
�,
does
not
nec���
�
�
��
���
�
�
��
��
��
��
���
��
�
�
�
�
acting
�
acting
on
the
on
arbitrary
the
load
load
application
application
point
point
��
,
��
�
,
,
�
�
�,
,
�
which
�,
which
does
does
not
necnot
nec��
�
�
�
�
�
��
�
�
��
�
�
�
�
�
acting
�
acting
on
the
on
arbitrary
the
arbitrary
load
load
application
application
point
point
��
,
�
��
,
�
,
�
�,
,
which
�
�,
which
does
does
not
necnot
nec� ���
�
�
�� ��� ��� ��
� �be
� �
� � cancarried
�� ���The
��evaluation
��
� typically
��
� �through
�
� �
�
The
evaluation
of
aerodynamic
loads
be
carried
out
through
a
CFD
code
which
solves
of
aerodynamic
loads
can
typically
out
a
CFD
code
which
solves
�
numerically.
This
kind
of
has
acarried
high
computational
concentrated
load
, cases
acting
on
�
�arbitrary
���unlike
�
�
acting
onpoint
the
arbitrary
application
��acost
, ��but
, The
�the
�,the
which
does
not
nec�
�
��
��the
�not
��point
acting
on
the
arbitrary
load
application
point
��
�to
,small
�assumed
which
does
not
necbecomes:
By
using
Eq.
��
for
example
either
Navier-Stokes
equations
or
Euler
equations
numerically.
This
ofwork,
analysis
hasfluid
sistent
with
the
present
method
and1,
here
derived
for
the
case
of
generic
concentrated
load
a �which
high
computational
some
assumptions
itcases
isflow
possible
employ
simplified
approaches.
viscosity
decisive
viscous
effects
can
be
confined
into
aanalysis
region
(boundary
layers
�variation.
�� �application
�since
�under
�
� ,can
�
� �,
evaluation
of
aerodynamic
loads
be
typically
out
through
a CFD
code which
solves
�load
� load
element
mesh,
standard
1D
FE
models.
the
virtual
� does
�is
� necting
on
the
onarbitrary
the
point
��
��
,�load
�, ���
�
�,
, application
�load
which
does
not
not
necIn
wing
considered
in
the
present
In
the
work,
wing
the
considered
field
is
in
the
present
tokind
be
steady
and
the
the
flow
field
is assumed
to be
steady a
�, �
��
�stands
�
� �, for
�
�where
��application
acting
on
the
arbitrary
point
��in
, ����,
which
does
not
nec�� �
������acting
��
�., �
� �,
where
�
is
evaluated
in
��
,
�
�
and
�
is
calculated
in
�
.
In
the
case
of employ
small displacements
is
evaluated
in
��
,
�
�
and
�
is
calculated
In
the
case
of
small
displacements
�
����� ��
�
on
the
arbitrary
load
application
point
��
,
�
,
which
does
not
nec� ��1D
�
�
�
�
�
�
�
�
�
�
�
�
�
essarily
lies
along
the
finite
element
mesh,
unlike
standard
1D
FE
models.
�
stands
for
the
virtual
�
�
cost,
but
under
some
assumptions
it
is
possible
to
arbitrary
load
application
point
(xelement
yeither
which
does
not
essarily
essarily
lies along
lies along
the
1D
the
finite
1D
finite
element
element
mesh,
mesh,
unlike
unlike
standard
FE
1Dmodels.
FE
models.
�Euler
stands
� equations
stands
for
the
for
virtual
the�equations
virtual
essarily
essarily
lies
along
lies
along
the
1D
the
finite
1D
finite
element
mesh,
mesh,
unlike
unlike
standard
standard
1D
FE
1D
models.
FE
models.
stands
�
stands
for
the
for
virtual
the
virtual
for
example
Navier-Stokes
equations
or
Euler
numerically.
This
kind
of
analysis
has
P,standard
P, z1D
P),
for
example
either
Navier-Stokes
equations
or
numerically.
This
kind
of
analysis
has
�
�
� under
�
aInhigh
computational
some
assumptions
itnot
is
to
employ
simplified
approaches.
��
���� �lies
acting
theelement
arbitrary
load
application
point
��
,cost
�fluid
, �but
which
does
not
nec(4)
the
wing
cases
considered
in
the
present
work,
the
flow
field
is the
assumed
toviscous
beis
steady
and
thesince
fluid
��
�unlike
��
�
�
��
�1D
�
�
�
�
and
wake
regions).
can
be
thus
considered
ascan
inviscid
and
flow
field
irrotational,
for
either
Navier-Stokes
equations
or
equations
numerically.
This into
kindaof
analysis
has(bou
essarily
along
1D on
finite
mesh,
standard
FE
models.
���example
stands
for
the
virtual
lies
along
the
1D
finite
element
mesh,
unlike
standard
1D
FE
models.
�the
stands
for
the
virtual
�� �
�� � the
�
���
����,
���
�virtual
�The
��
becomes:
viscosity
is
not
decisive
since
the
viscous
viscosity
effects
is
bepossible
decisive
confined
since
into
aEuler
small
region
effects
(boundary
can
be layers
confined
small
region
simplified
approaches.
In
the
cases
considered
in the
�itefinite
necessarily
lie
along
the
finite
element
mesh,
unlike
D
element
element
mesh,
mesh,
unlike
unlike
standard
standard
1D
FE
1D
models.
FE
models.
�essarily
stands
�1D
stands
for
the
for
virtual
the
essarily
lies
along
the
1D
finite
element
mesh,
unlike
standard
1D
FE
models.
�
stands
forvirtual
the
virtual
with
respect
to(subscript
the
length
the inplane
(subscript
�) wing
and out-of-plane
(subscript
�) cross-section
with
respect
to
the
length
�,
the
inplane
(subscript
�)
and
out-of-plane
�) �,
cross-section
sarily
lies
along
the
1D
finite
element
mesh,
unlike
standard
1D
FE
models.
�
stands
for
the
becomes:
variation.
Byvariation.
usingByEq.
1, using
��
becomes:
becomes:
variation.
using
By
Eq.
Eq.
��
1,By
��
���1,
becomes:
variation.
variation.
using
By using
Eq. 1,Eq.��
1,
��becomes:
a
high
computational
cost
but
under
some
assumptions
it
is
possible
to
employ
simplified
approaches.
a
high
computational
cost
but
under
some
assumptions
it
is
possible
to
employ
simplified
approaches.
���
���
���
���
present
work,
the
flow
field
is
assumed
to beand
steady,
and the
standard
1D FEin
models.
δ stands
for
virtual
variation.
By
In
the
cases
considered
in
the
present
work,
the
flow
field
issome
assumed
tobe
bethus
steady
thelayers
fluid
viscosity
isthe
not
decisive
since
the
viscous
effects
be
confined
into
aassumptions
small
region
�
curl
velocity
vector
‫ݕ‬ǡcomputational
‫ݖ‬ሻ
equal
tocan
zero:
�Eq.
is
��� ,variation.
�unlike
and
�wing
isthe
calculated
inThe
���.fluid
In
the
case
of
small
displacements
aࢂሺ‫ݔ‬ǡ
high
cost
but
it(boundary
isis possible
to inviscid
employ
approaches.
variation.
By
using
1,���evaluated
��
becomes:
By
Eq.
1,
��
essarily
1D
finite
mesh,
standard
1D
FE
models.
stands
for
the
virtual
� ��
� �the
�ofusing
and
wake
regions).
can
be isthus
and
considered
wake
regions).
asunder
inviscid
The
and
fluid
the
canflow
field
considered
irrotational,
as
since simplified
and the flow
field is irrota
(4)
�����
� � lies
�����along
�where
��
���element
��� becomes:
���
����
��
�the
becomes:
�1,
��� becomes:
stress
and
strain
vectors
in
Eq.
3
are
related
to
the
displacement
vector
via
linear
differential
matrix
becomes:
variation.
By
using
Eq.
1,
��
stress
and
strain
vectors
in
Eq.
3
are
related
to
the
displacement
vector
via
linear
differential
matrix
�
�
�
fluid
viscosity
is
not
decisive
since
the
viscous
effects
can
�
�
�
�
using
Eq.
1,
δL
becomes:
���
�
�
�
�
�
�
�
�
becomes:
riation. By using Eq. 1, �����
(4)
ext
�����
���
��
�
�
��
�
�
�
�
��
�
In
the
wing
cases
considered
in
the
present
work,
the
flow
field
is
assumed
to
be
steady
and
the
fluid
(4)
(4)
(4)
(4)
��
�
��
�
�
��
�
�
��
�
��
�
�
�
�
�
�
��
�
��
�
�
�
�
��
��
��
�
��
�
�
��
�
��
�
�
��
�
�
�
�
�
�
��
�
�
��
�
�
��
��
In
the
wing
cases
considered
in
the
present
work,
the
flow
field
is
assumed
to
be
steady
and
the
fluid
��� ���
�� �
��
������ �����
�� ��
�� � ��
��� � �
�� ���� ��
�and wake
� �)The
viscosity
is�
not
decisive
the
effects
be
confined
into
a small
region
(boundary
layers
� ‫ ׏‬confined
� the
regions).
fluid
be
thus
considered
as
inviscid
and
the
flow
field
is
irrotational,
sincewake
ൈconsidered
ࢂcan
ൌof
૙
(8)
with�Eq.
respect
to
length
�,
the��
inplane
and
out-of-plane
�)
cross-section
Incan
the�viscous
wing
in
present
work,
the
flow
field
istoassumed
to be steady and the fluid
be
into
region
(boundary
layers
and
(4)
��
�case
���of
� small
�
�� the
��(subscript
�
��
�(4)
becomes:
variation.
1,
��since
�
��
�
�(subscript
��
�is
�
�
��
�a��small
�the
�
curl
of
the
velocity
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡcases
‫ݖ‬ሻ
curl
equal
tothe
zero:
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ ‫ݖ‬ሻ
is equal
zero:
�calculated
and���
������
is�By
. ���
In
the
���
��� vector
��
� (4)
��displacements
��the
��
� ���
��
�
(4)
�����
�
�
��using
���� ��
�� �
����,in
�
�
������
��
�
��
�
��
���� ��
�stiffness
� ��
operators
��as
,displacements
�follows:
material
matrices
��� , ��� , ��� , ��� as follows:
operators
and
material
matrices
����In
,the
�����case
,�����
, small
���
��
(4)can stiffness
��
��and
��
����
��
���viscosity
�
�in�
(4)
� and
���
���
�is��
��
��in
(4)
��
�
�
��
�
�
�
�
where����where
evaluated
��
, �in
����
�
is
calculated
�
.
of
is
not
decisive
since
the
viscous
effects
be
confined
into
a
small
region
(boundary
layers
���
�
�
��
��
��
viscosity
is
not
decisive
since
the
viscous
effects
can
be
confined
into
a
small
region
(boundary
layers
�� is��evaluated
isin
evaluated
��
in
,
��
�
,
�
and
�
and
�
is
�
calculated
calculated
in
�
in
.
In
�
.
the
In
case
the
case
of
small
of
small
displacements
displacements
where
where
�
� iswhere
��
���
�
�
is
evaluated
is
evaluated
��
in
,
�
��
�
,
and
�
�
�
and
is
�
calculated
is
calculated
in
�
in
.
In
�
the
.
In
case
the
case
of
small
of
small
displacements
displacements
regions). as
The
fluid can
be thus
considered
as inviscid,
and
�
� � �� �
� � ��� �
��
��
�
and
wake
regions).
fluid
be
thus
considered
inviscid
and
the
flow
field
isdisplacements
irrotational,
since
�are
��velocity
the
curl
ofto
the
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ�displacements
iscalculated
equal
�and
is
not
decisive
since
the
viscous
effects
be
confined
a small
stress
and out-of-plane
strain
in
Eq.
3is
related
the
displacement
vector
via
linear
differential
In
this
case
the
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
can
be(4)
considered
asInthe
acan
potential
function
��is�iscalculated
evaluated
in
��. vectors
, ��.the
�In(subscript
�
calculated
in
the
of
small
inThe
��
, ��viscosity
�can
and
�
in
��૙
.matrix
thegradient
case
ofofsmall
‫׏‬toൈzero:
ࢂൌ
‫ ׏‬ൈ ࢂinto
ൌ߶:
૙
(8) region (boundary layers
������
�
��
�
�
��
�
�
�is
�evaluated
��
�
�)calculated
and
�)
�In
�where
� .��In
��displacements
�case
� is
� cross-section
��
��
�he
in� ,where
�inplane
��
� , where
�and
� (subscript
and
�
is
in
in
�
case
the
case
of
small
of
small
displacements
the
flow
field
is
irrotational,
since
the
curl
of
the
velocity
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
��
�
�
�
�
�
�
�
�
�
�
�
�
is
evaluated
in
��
,
�
�
and
�
is
calculated
in
�
.
In
the
case
of
small
displacements
�flow field
�� � is irrotational,
�� �
��N
��
�(subscript
�� in
� displacements
�and
�
isrespect
evaluated
in
(x�)
, and
zthe
)The
calculated
y�)(subscript
. out-of-plane
Inthus
Finplane
�τ�,
�to
here
is �evaluated
in
��towhere
,length
��the
��with
and
�
calculated
. the
In
the
case
of
small
with �respect
to the
length
�,
(subscript
out-of-plane
�)
cross-section
and
wake
regions).
The
fluid
can
inviscid
andcross-section
theirrotational,
since
Pin
P�
i is
Pbe
wake
regions).
fluid
can
be
thus
considered
asout-of-plane
inviscid
and�as
the
flow
field
is
since
� with
�with
�tois
�and
with
respect
respect
to the
the
length
the
�,
inplane
the
inplane
(subscript
(subscript
�)
and
�)
and
out-of-plane
out-of-plane
(subscript
�)considered
cross-section
�)� cross-section
respect
the
length
the
length
�,
�,
inplane
inplane
(subscript
(subscript
�)
and
and
(subscript
(subscript
�) cross-section
�)
(5)
(5)
the curl
of
the
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ ‫ݖ‬ሻ
isvector
equal
to
zero:
�
�
���
��of
�
�cross-section
�� beand
�
�
�
�the
��to
�
��of
‫�׏‬
ൈThe
ࢂ�be
ൌ
૙
(8)flow
�
�
�‫ݖ‬ሻ
�via
��linear
� is(subscript
�the
(x,
y,the
z)can
is equal
to
zero:
and
regions).
fluid
be thus
considered
asa‫ݖ‬ሻ
inviscid
the
field
irrotational,
since
operators
�case
, vector
�
and
material
stiffness
matrices
�the
,the
, wake
���
as
follows:
‫߶׏‬
ൌ
ࢂ
(9)
�
��can
�and
��
�,velocity
��
�vector
�
�
�
�
��
��
��
with
respect
to
the
length
�,
inplane
�)
out-of-plane
(subscript
�)
cross-section
with
respect
to
the
length
�,
inplane
(subscript
�)
and
out-of-plane
(subscript
�)
In
this
case
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
In
can
this
case
considered
velocity
as
the
vector
gradient
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
potential
considered
function
߶:as
theisgradient
of a potential
the
of
small
displacements
with
respect
the
length
where
�
is
evaluated
in
��
,
�
�
and
�
calculated
in
�
.
In
case
small
displacements
3
are
related
to
the
displacement
differential
matrix
� (subscript
� �
� (subscript
�
�,
gththe
�,
inplane
the
inplane
(subscript
�)
and
�)
out-of-plane
and
out-of-plane
(subscript
�)
cross-section
�)
cross-section
with
respect
tovectors
the length
�,inplane
the
inplane
(subscript
�)
and
out-of-plane
(subscript
�)
cross-section
th
respect
to
the
length
�,
the
(subscript
�)
and
out-of-plane
(subscript
�)
cross-section
stress
and
strain
in
Eq.
3
are
related
to
the
displacement
vector
via
linear
differential
matrix
the
curl
of
the
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
is
equal
to
zero:
the
curl
ofvectors
the
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
is equal
todifferential
zero:
stressstress
and and
strainstrain
vectors
vectors
in
Eq.
in
3Eq.
are
3 related
are
related
the
displacement
the
via
linear
via n)
linear
differential
stress
stress
and
strain
and
strain
vectors
intovelocity
Eq.
into
3Eq.
are
3displacement
related
are related
tovector
the
tovector
displacement
the
displacement
vector
vector
viamatrix
linear
viamatrix
linear
differential
differential
matrix
matrix
L,
the
in-plane
(subscript
p)
and
out-of-plane
(subscript
‫ ׏‬be
ൈ
ࢂ
ൌ in
૙ terms
(8)
In
this
case
the
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
can
considered
asofthe
gradient
ofto
a potential
function
curl
of
the
velocity
vector
‫ݕ‬ǡvirtual
‫ݖ‬ሻ
equal
zero:
Using
5,
Eq.
3differential
can
rewritten
nodal
displacements:
Using
Eq.
rewritten
in�
terms
of�)
virtual
nodal
��velocity
�displacements:
�in��
�wing
�
�
�
the
analysis
of
aEq.
or��
an
airfoil
under
these
conjectures
can
belinear
performed
by potential
�
�Eq.
��)
stress
and
strain
vectors
in
Eq.
3 are
related
toHence,
the
displacement
vector
via
linear
matrix
stress
and
strain
vectors
3 the
are
related
tobe
the
displacement
vector
viais
differential
matrix
�
��
‫߶׏‬
ൌ ࢂ ࢂሺ‫ݔ‬ǡ
‫ ߶׏‬ൌ meࢂ ߶: (9)(8)
stiffness
matrices
�the
,5,
�Eq.
,3�can
, be�
as�via
follows:
with
respect
to
length
�,
inplane
(subscript
and
out-of-plane
(subscript
cross-section
��
��
��the
��
Eq.
sl in
3
Eq.
are
3
related
are
related
to
the
to
displacement
the
displacement
vector
vector
linear
via
linear
differential
differential
matrix
matrix
(5)
cross-section
stress
and
strain
vectors
in
Eq.
3
are
related
stress
and
strain
vectors
in 3Eq.
3related
are related
to
the
displacement
vector
via
linear
differential
matrix
ess
and operators
strain
vectors
in
Eq.
are
to
the
displacement
vector
via
linear
differential
matrix
operators
�operators
,
�
and
material
stiffness
matrices
�
,
�
,
�
,
as
follows:
�
�
�
�
�
�
�
ൈ
ࢂfollows:
ൌ
(8)
�
�material
�
‫׏‬
ൈ
ࢂ
(8)
���, operators
�
and
and material
matrices
�stiffness
�
, �matrices
,����
�
,matrices
,���
�
as
�
��� , �
��
��
operators
�� ,stiffness
���stiffness
��matrices
material
stiffness
�, ��
,��
�follows:
, , follows:
�
, �
as��
as૙follows:
��as
��
��ൌ
�, and
�and
� material
�� ,���
����
����
����
��
��
��, ,૙��
��
��‫׏‬
�
�
����
����
In this
case
the
velocity
vector
ࢂሺ‫ݔ‬ǡ��
‫ݕ‬ǡ ‫ݖ‬ሻ
can
be
considered
asan
the
gradient
of
adefined
potential
function
߶:
‫߶׏‬
ൌ�
ࢂ
(9)
‫���׏‬orbe
ൈ
ࢂ
ൌ
૙
(8) by p
The
describing
flow
field
around
object
can
abycombina(x,
y, z)as
can
be
considered
In�
this
case,
the
���
�
��
�an
��
�thods.
�
��
�
��
(6)performed
(6)
the
displacement
vector
via
differential
matrix
operators
��� ,�
�����to
and
material
stiffness
matrices
��Hence,
,linear
�
,function
�via
follows:
��a �
��
��
�� material
operators
�
, �potential
�
,���
,velocity
�of
asvector
follows:
�� the
the
analysis
astiffness
wing
ormatrices
anthe
airfoil
Hence,
under
these
analysis
conjectures
wing
can
be
performed
airfoil
under
potential
these
conjectures
mecan be
stress
and
strain
vectors
Eq.
3,�,�are
related
to
displacement
vector
linear
differential
matrix
�
���in
�
��
�
��
��
��
��ofas
�
�,��and
�� , the
��
��
��
� � stiffness
erial
material
stiffness
matrices
matrices
�
,
�
,
,
�
�
�
,
�
as
follows:
as
follows:
��material
��
�� ��
�� ��
��matrices
��
operators
�
, and
�Using
and
stiffness
�
,
�
,
�
,
�
as
follows:
(5)
�
�
��
��
��
��
erators
�
,
�
material
stiffness
matrices
�
,
�
,
�
,
�
as
follows:
�
�
�
�
�
�
�
In
this
case
the
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
can
be
considered
as
the
gradient
of
a
potential
function
߶:
�
�
�
Eq.
5,
Eq.
3
can
be
rewritten
in
terms
of
virtual
nodal
displacements:
In
this
case
the
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
can
be
considered
as
the
gradient
of
a
potential
function
߶:
�
� � � ���
��
��
��
��
as
the
gradient
of
a
potential
function
:
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
��
�
�
�
�
�
�
�
�
�
�
�
�
�
operators
,
,
and
material
stiffness
matrices
,
,
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
��
�
��
��
�
�� �
�
�
��
�
��
�
��
�
��
�
n
pp
pn
� �� �� p�
�
�
��
�
�
�� �
‫ ߶׏‬these
ൌ
ࢂor
(9)
Hence,
the
of
aas
an
airfoil
under
conjectures
can(5)
bedescribing
performed
bybody
potential
meof
singularities
such
aswing
doublets,
vortices,
sources
uniform
flux
over
thecan
external
surface.
(5)
In or
this
case
the(5)
velocity
vector
ࢂሺ‫ݔ‬ǡ
‫ݕ‬ǡ
‫ݖ‬ሻ
can
be
considered
as the
of an
a potential
function
߶: as
�
�analysis
�
������
�(5)
�(5)
� around
�
function
describing
thods.
the
The
potential
be defined
the
flow
as
field
agradient
combinaaround
object can
be defined
�
������matrices
�����tion
�
�thods.
�
�
operators �� , �
and
���
,�
,The
,���
���
follows:
���
�
�
�����material
��field
�
�� �function
� an object
�
����
�
�
�������
�potential
�
��
�
��
��
�
��
�
�
���
���� ��
������
�������
����flow
��
��
��
�
�
�
�
�
�
��� �
��
�������
�,�
��
���stiffness
�
��
,�
as
follows:
�
��
��
���
��
�
��
���
�
�
�
�����
���
� np
��
��
��
��� �
nn
� ����� �
�
�
�
�
�
�
(5)
(5)
�
�
�
�
����
�
��
�
��
�
� �� �
�� � ���
�������
�
��5��
�
5
�
�
�
‫߶׏‬
ൌ
ࢂ
(9)
(9)
(5)
(5)
�
‫߶׏‬
ൌ
ࢂ
(9)
�
�
��
�
��
(6)
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
���
��� ��� �
��analysis
��
(5)
tten
in��
terms
of virtual��nodal
displacements:
� these
�� conjectures
�
�� � can be performed by potential me���� ���� �
����
������
�
�
��
(5)
Hence,
the
a such
wingas
ordoublets,
an airfoil
under
� ��
��
��� �
thods.
The
function
describing
the
flow
field
around
ansuch
object
can‫߶׏‬
beൌdefined
as asources
combina��
��
����
�
���
��
��potential
���
��� of
ࢂvortices,
(9) b
����
���
tion
singularities
vortices,
tion
of
singularities
or uniform
asflux
doublets,
over
the
external
body
surface.
or uniform flux over the external
��
��
�
�
��
�
�of
�
�terms
�
�
�5,
�Eq.
�
Using
Eq.
5,Using
Eq.
can
be
rewritten
in
terms
virtual
nodal
displacements:
�3be
��
�virtual
�displacements:
�of
��
�nodal
��
� of virtual
�3
�5,
Using
Eq.3 5,
Eq.
Eq.
Eq.
can
3���
be
can
rewritten
be
rewritten
terms
in
terms
ofrewritten
virtual
nodal
6 sources
Using
Using
Eq.
Eq.5,in3of
Eq.
can
can
be
rewritten
in
terms
in
ofdisplacements:
virtual
nodalnodal
displacements:
displacements:
(5)
(5)
Hence,
the
analysis
of
a
wing
or
an
airfoil
under
these
conjectures
can
be
performed
by
potential
me� Using
���� Eq. 5, Eq. �3 can� be�rewritten
Hence,
the
analysis
of
a
wing
or
an
airfoil
under
these
conjectures
can
be
performed
by
potential
meHence,
the
analysis
of
a
wing
or
an
airfoil
under
these
� �5,nodal
�� potential
�
���be
��function
in termsUsing
of �virtual
Eq.
Eq.
3displacements:
can
rewritten
in termsvortices,
ofthe
virtual
nodal around
displacements:
�displacements:
�of
�of��virtual
��� nodal
����in�terms
�
(6)
�displacements:
��
�� virtual
��be
thods.
The
describing
flow
anairfoil
object
can the
bethese
defined
asbody
a combinabe
written
rewritten
in terms
nodal
tion
of��
singularities
such
as�doublets,
sources
or uniform
flux over
external
surface.
Hence,
the analysis
offield
a wing
or an
under
conjectures
can be performed by potential me� terms
�
can
rewritten
in
of virtual
nodal
displacements:
����
��
�
� � ����
����
ingUsing
Eq. 5,Eq.
Eq.5,3Eq.
can3be
rewritten
in ��
terms
virtual
5����������
�nodal
��
�
(6)
�����
�
���
��
������
��
�
���
���
������ ���
��
(6) 6(6)
conjectures
can be performed
The
(6) (6)by potential methods.
�� �of��
�� displacements:
6
��
��
�� �
����
�� ����
�� ��
�����
�� ��
�The����
� potential
thods.
The
function
describing
the
flow
field
an
object
can
be
defined
as
a
combina� field
� around
����
thods.
potential
function
describing
the
flow
around
an
object
can
be
defined
as
a
combina���
�
��singularities
�in��terms
��
�potential
��(6)
���or uniform
���� � vortices,
Using
Eq.
in Eq.
terms
of virtual
� 5, Eq.
�
(6)
���� 3 can�beUsing
�� �sources
��3 �
�� displacements:
�� function
tion
of
such
as
doublets,
flux
over
the
external
body
surface.
Eq.
5,
can
be
rewritten
of
virtual
nodal
�
�nodal
����
�����
���
����� ��
�rewritten
����� ��
describing
the
flow
field
around
an
object
(6)
(6)
thods.
The
potential
function
describing
the
flow
field
around
an
object
can
be defined as a combina���� ��
��
�
�� ���� �
����
�� �
� �� ���� ����
���
���� ��
(6) (6) 6
�� ����� ��
5
tion
of
singularities
such
as
doublets,
vortices,
sources
or
uniform
flux
over
the
external
body
surface.
� of ����
�
tion
singularities
such
as
doublets,
vortices,
sources
or
uniform
flux
over
the
external
body
surface.
5��� � ��
���� �
5 �� �
5 ��
5
5 tion of singularities(6)
such
6 as doublets, vortices, sources or uniform flux over the external body surface.
Int’l J. of Aeronautical & Space Sci. 14(4), 310–323 (2013)
5
5
5
5
5
DOI:10.5139/IJASS.2013.14.4.310
5
5
6
312
6
6
Alberto Varello Static Aeroelastic Response of Wing-Structures Accounting for In-Plane Cross-Section Deformation
after a trial and error process, the best results can be
can be defined as a combination of singularities, such as
obtained by using just the Dirichlet BC type [38]. The 3D
doublets, vortices, sources, or uniform flux over the external
Panel Method employs a low-order panel method. The LLT
body surface. According to the detailed exposition in [36],
e detailed exposition in [36], the equation to be used to compute the solution of the
approach is not able to evaluate the pressure coefficients on
the equation to be used to compute the solution of the
blem is the Laplace’s equation:
the wing surface, but only the lifting loads along the lifting
aerodynamic problem is Laplace’s equation:
According to the detailed exposition in [36], the equation to be used to compute the solution of the
line. The VLM is able to analyze the pressure coefficients,
(10) to(10)
According
be used to compute the solution of the
‫׏‬ଶ Ԅ ൌ Ͳto the detailed exposition in [36], the equation
but only on the reference surface, which is defined as
aerodynamic problem is the Laplace’s equation:
the mean surface between the upper and the lower wing
problem
is
the
Laplace’s
equation:
on describes a potential flowaerodynamic
field
only
if
the
compressibility
effects
can
be
neglected,
Laplace’s equation describes a potential flow
field only
‫׏‬ଶ Ԅ ൌofͲthe
he detailed exposition in [36],
the
equation
to
be
used
to
compute
the
solution
surfaces. The 3D Panel (10)
Method is able to calculate the
if the compressibility effects can be neglected,
as occurs
ଶ
just
3D
Panel
Method
employs
low-order
panelthe
method.
results presented afterwards where the free stream velocity
is using
rather the
low.
Otherwise
(10)
Ԅ ൌ Ͳ BC type [38]. The
‫׏‬Dirichlet
pressureeffects
coefficients
on bothathe
upper and
lowerThe
wing
forLaplace's
the results
presented
afterwards,
the free
equation
describes
a potentialwhere
flow field
only stream
if the compressibility
can be neglected,
oblem is the Laplace’s equation:
surfaces.
Therefore,
this
method
offers
the
most
realistic
velocity
is
rather
low.
Otherwise,
some
corrections,
e.g.
the
LLT approach
is not
able
to evaluate
the pressure
coefficients
on the wing surface, but only the lifting
, e.g. Prandtl-Glauert transformation,
are necessary
as explained
in flow
[37]. field
The
assumpLaplace's equation
describes
a potential
only
if the
compressibility
effects
can be neglected,
as‫׏‬occurs
results presented afterwards where (10)
the free stream velocity
is rather
low.
Otherwise field.
ଶ
description
of the
aerodynamic
Ԅ ൌ Ͳfor thetransformation,
Prandtl-Glauert
are necessary, as explained
loads
along
thewhere
liftingthe
line.
The
VLMvelocity
is able to
the pressure
coefficients but only on the referuced lead to an integral-differential
which expresses
the
potential
function
in
as occursequation
for the results
presented
afterwards
free
stream
is analyze
rather low.
Otherwise
in [37]. The
assumptions
here introduced
lead to an integrale.g. Prandtl-Glauert
transformation,
are necessary as explained in [37]. The assumpion describes a potential flowsome
field corrections,
only if the compressibility
effects
can be neglected,
differential
equation,
which For
expresses
the
function
ence
which
isare
defined
as theasmean
surface
between
the upper and the lower wing surfaces.
t of the fluid domain as a some
combination
of singularities.
thesurface
sake
ofpotential
brevity,
this
corrections,
e.g. Prandtl-Glauert
transformation,
necessary
explained
in [37].
The assumptions
here
introduced
lead
to
an
integral-differential
equation
the potential static
function in
3. Aeroelastic
response analysis via
in
an
arbitrary
point
of
the
fluid
domain
as
a
combination
he results presented afterwards where the free stream velocity is rather low. Otherwise which expresses
The
3D
Panel
Method
is
able
to
calculate
the
pressure
coefficients
on
both the upper and the lower
eported here but more details
can
be
found
in
[36].
Among
the
potential
methods,
the
tions
here
introduced
lead
to
an
integral-differential
equation
which
expresses
the
potential
function
in
the CUF-XFLR5 approach
ofan
singularities.
Forofthe
ofdomain
brevity,
this
equation is not
arbitrary
thesake
a combination
ns, e.g. Prandtl-Glauert transformation,
arepoint
necessary
asfluid
explained
inas
[37].
The assump-of singularities. For the sake of brevity, this
reported
here,
more
details
cansurfaces.
be
found
in [36].
Among
wing
Therefore,
method offers
description
of the aerodynamic field.
an be formulated followingan
a arbitrary
low-order
orbut
a of
highapproach.
(firstpoint
the order
fluid
domain
as Low-order
a combination
ofthis
singularities.
Forthe
themost
sakerealistic
of brevity,
this
equation
is
not
reported
here
but
more
details
can
be
found
the potential
the static response of the wing
the
potential
methods,
the
panel
methods
can
be
formulated
In this
work themethods,
aeroelastic
duced lead to an integral-differential equation which expresses the potential function in in [36]. Among
3.more
Aeroelastic
static
analysis
CUF-XFLR5
approach
hod employs triangular or quadrilateral
panels
having
values
of can
singularities
equation
reported
here
details
be response
found
[36].
Among
the potential
methods,
the
followingis anot
low-order,
or constant
abut
high-order
approach.
The in
lowisviacomputed
an(firstiterative procedure, based on a
panel methods
can
be formulated
following
abrevity,
low-order
or a high- order
approach. through
Low-order
int of the fluid domain as aorder
combination
of
singularities.
For
the
sake
of
this
(first-order) panel method employs triangular or
coupled
CUF-XFLR5
method.
Hence,
the aerodynamic
In following
thisusework
aeroelastic
of the wing
is computed
an iterative
procedure
Hess and Smith approach. panel
Higher-order
methods
instead
higher
than
first-or astatic
methodspanel
can be
formulated
athe
low-order
high-response
order approach.
Low-order
(first- through
order)
panel
method
employs
triangular
or
quadrilateral
panels
having
constant
values
of
singularities
quadrilateral
panels
having
constant
values
of
singularities’
analysis
is
performed
through
the
potential
methods
reported here but more details can be found in [36]. Among the potential methods, the
based
a quadrilateral
coupled CUF-XFLR5
method.
Hence,
the XFLR5,
aerodynamic
analysis is performed
through
the
. paraboloidal panels) and aorder)
varying
singularity
strength
over
eachon
panel.
panel
method
employs
triangular
or
constant
values
of singularities
strength,
such
as the
Hess and
Smith
approach.
The panels
higher-having
available
in
as previously
mentioned;
whereas,
strength
suchor
asaHess
Smith
approach.
Higher-order
instead use higher than firstcan be formulated following
a low-order
high-and
order
approach.
Low-order
(first-panel methods
order
panel methods
instead
use higher
than first-order
variable kinematic 1D CUF models provide the structural
methods
available
in XFLR5,
previously
mentioned,
whereas variable kinematic 1D CUF
implementation of aerodynamic
potential
strength
such asmethods
Hess and Smithpotential
approach.
Higher-order
panel
methodsasinstead
use higher
than firstpanels
(e.g. paraboloidal
panels)
andand
a varying
singularity
order
panels
(e.g.
paraboloidal
panels)
a
varying
singularity
strength
over
each panel.with a variable expansion order N.
wing
deformation
ethod employs triangular or quadrilateral panels having constant values of singularities
strength
over
each
panel.
models
provide
the structural
wing
deformation
with
a variable expansion order N.
ware developed by Andre order
Deperrois.
It(e.g.
performs
viscouspanels)
and inviscid
aerodynamic
panels
paraboloidal
and
a varying
singularity
strength
over each
panel.
2.2.2 XFLR5:panel
an implementation
of aerodynamic
potential
s Hess and Smith approach. Higher-order
methods instead
use higher than
first- methods 3.1 Iterative procedure
3.1
Iterative
procedure
ils and wings using three potential
methods:
the Lifting Line
Theory
(LLT),
the
VLM methods
2.2.2 XFLR5:
an implementation
of
aerodynamic
potential
is
a software
developed
It performs viscous and inviscid aerodynamic
XFLR5:
an implementation
ofAndre
aerodynamic
potential
g. paraboloidal panels) and2.2.2 a XFLR5
varying
singularity
strength
over by
each
panel.Deperrois.
Figureprocess,
1 showswhich
in detail
aeroelastic
iterative
Figure
1
shows
in
detail
the
aeroelastic
iterative
startsthe
with
the evaluation
of theprocess,
presl Method. The LLT methodXFLR5
derives
from
the
Prandtl's
wing
theory
and
considers
the
is
a
software
developed
by
Andre
Deperrois.
It
performs
viscous
and inviscid aerodynamic
methods
analysis
on
airfoils
and
wings
using
three
potential
methods:
the
Lifting
Line
Theory
(LLT),
the
VLM
which
starts
with
the
evaluation
of
the
pressure
coefficients
n implementation of aerodynamic potential methods
XFLR5
a software
by
Andre
Deperrois.
sure three
coefficients
for
the undeformed
wing
configuration.
Thethe
further
steps to be repeated
for each
ite-to
distribution of vortices. Theanalysis
VLM
considers
a wing
asdeveloped
an using
infinitely
thin
lifting
surface
onisairfoils
and wings
potential
methods:
theItLifting
Line
(LLT),
VLM
for
theTheory
undeformed
wing
configuration.
The further
steps
and the 3DItPanel
Method.
The and
LLTinviscid
method aerodynamic
derives from the Prandtl's wing theory and considers the
ftware developed by Andreperforms
Deperrois.
performs
viscous
and viscous
inviscid
aerodynamic
analysis on
be repeated for each iteration are:
ration
are:requires
n of vortices placed over and
a wing
reference
surface. This
method
thefrom
non-the Prandtl's wing theory and considers the
the 3D
Panel Method.
The LLT
method
derives
airfoils
and
wings,distribution
using three
potential
methods:
the Lifting
1.infinitely
post-processing
wing
as
a
linear
of
vortices.
The
VLM
considers
a
wing
as
an
thin liftingcalculation
surface of the aerodynamic forces;
oils and wings using three potential methods: the Lifting Line Theory (LLT), the VLM
Line
Theory
(LLT),
the
VLM,
and
the
3D
Panel
Method.
1.
post-processing
calculation
of
the
aerodynamic
forces;
ition on the reference surface
as
a
boundary
condition.
Hence,
the
normal
component
2.
structural
analysis
wing as a linear distribution of vortices. The VLM considers a wing as an infinitely thin lifting surfaceof the wing, subject to the
via
a distribution
of vortices
placed
over
wingtheory
reference
method requires the nonThe
LLT
method
from
Prandtl’s
el Method. The LLT method
derives
from the derives
Prandtl's
wing
theory
andawing
considers
the andsurface. This aerodynamic
forces previously computed;
2.
structural
analysis
of
the
wing
subject
to the requires
aerodynamic
forces previously computed;
panel
normal
vector
is equal
to The
elocity ࢂ௜ on the generic i-th
via
aaerodynamic
distribution
of vortices
placed
over a࢔wing
reference
surface. This method
the non௜ of
considers
thecondition
wing
aswith
a linear
distribution
vortices.
3.
newthe calculation
of the aerodynamic pressure
penetration
on
the
reference
surface
as
a
boundary
condition.
Hence,
normal
component
r distribution of vortices. The VLM considers a wing as an infinitely thin lifting surface
VLM
considers
a
wing
as
an
infinitely
thin
lifting
surface,
coefficients
for
the
new
configuration;
3. surface
new calculation
of thecondition.
aerodynamic
pressure
coefficients
for thedeformed
new deformed
configuration;
penetration condition on the reference
as a boundary
Hence,
the normal
component
onplaced
the
generic
i-th
aerodynamic
normal
vector ࢔௜ is evaluation
equal to
induced
velocity
ࢂ௜This
athe
distribution
of vortices
over
a wing
on of vortices placed overvia
aofwing
reference
surface.
method
requires
the reference
non- panel with 4.
post-processing
of the wing deformation
4. post-processing
evaluation
of thenormal
wingand
deformation
cross-section
distortion.
i-th aerodynamic
panel with
vector
࢔௜ isand
equal
to
of
induced
velocityrequires
ࢂ௜ on the
ൌ Ͳ method
(11)
ࢂthe
surface.
thegeneric
non-penetration
condition
௜ ȉ࢔
௜ This
cross-section
distortion.
zero:
dition on the reference surface
as areference
boundarysurface
condition.
the normal
component
on the
as aHence,
boundary
condition.
Hence, the
displacements
evaluated
in specific
displacements
specific sections
distributedare
regularly
along
the wingsections
span.
n this method can be foundzero:
in [15]. The 3D Panel Method Structural
schematizes
the wing sur-are evaluated in Structural
normal
component
of
the
induced
velocity
on
the
generic
ȉ
࢔
ൌ
Ͳ
(11)
ࢂ
௜
௜
distributed
regularly
along
the
wing
span.
The
cross-section
velocity ࢂ௜ on the generic i-th aerodynamic panel with normal vector ࢔௜ is equal to
The
cross-section
as the in-plane
displacement,
a quantity
which expresses
i-th
aerodynamic
panel
withand
normal
vector
is Ͳequal
todefineddistortion
tion of doublets and sources.
The
strength of the
doublets
sources
isࢂcalculated
to s is
(11)thei.e.
௜ ȉ ࢔distortion
௜ ൌ
s is defined
as
in-plane
displacement,
i.e. a
Further details on this method can be found in [15]. The 3D Panel Method schematizes the wing surzero:
quantity
thatand
expresses
the in-plane
difference
between
the
thebe
in-plane
difference
between
theMethod
deformed
shape
thewing
“undeformed”
shape
of the airfoil
crossriate boundary conditions (BCs),
be of
Dirichletor
Neumann-type.
AcFurtherwhich
detailsmay
on this
method
can
found
in
[15]. The
3D Panel
schematizes
the
surface
as
a
distribution
of
doublets
and
sources.
The
strength
of
the
doublets
and
sources
is
calculated
to
deformed
shape
and
the
“undeformed”
shape
of
the
airfoil
(11) (11)
ࢂ௜ ȉ ࢔௜ ൌ Ͳ
section:
eator of the program, afterface
a trial
error process,
the best
results
canThe
be obtained
as aand
distribution
of doublets
and
sources.
strength of the doublets
and sources is calculated to
cross-section:
theThe
appropriate
boundary
conditions (BCs),
which
may be of Dirichlet- or Neumann-type. Acon this method can be found meet
in
[15].
3D
Panel
Method
schematizes
the
wing
surFurther details on this method can be found in [15].
meet the appropriate boundary conditions (BCs), which may be of Dirichlet- or Neumann-type.
Acଶ
(12)
(12)
‫ ݏ‬ൌ ටο‫ݑ‬௫ଶ ൅ ο‫ݑ‬
The
3D Panel
the
wing
as a
cording
to theMethod
creator
ofschematizes
the program,
afteris
acalculated
trialsurface
and error
can
௭ be obtained
ution of doublets and sources. The7 strength
of the doublets
and sources
to process, the best results
distribution
doublets
sources.
strength
of process,
the
cording
to theof
creator
of theand
program,
afterThe
a trial
and error
the best results can be obtained
ο‫ݑ‬appropriate
are
the cartesian where
components
of the△u
relative
displacement vector ο࢛ along the
whereorto
ο‫ݑ‬
priate boundary conditionsdoublets
(BCs), which
may be of
Neumann-type.
Ac△ux and
and sources
is Dirichletcalculated
meet
௫ andthe
௭
z are the cartesian components of the
7
� � �,
�, ��the
� �chord
�� ��� ���
relative
displacement
vector
△ ���,
along
boundary conditions (BCs), which may be of 7Dirichlet- or
� �x, z� direction
� � �,
chordresults
direction
the transversal direction z, respectively, between the deformed cross-section
and
creator of the program, after a trial and error process, the best
can xbeand
obtained
x and the transversal direction z, respectively, between
Neumann-type. According to the creator of the program,
where ��� contains the degrees of freedom of the � �� expansion t
7
8
313
ment node. The compact expression in Eq. 1 is based on Einstein
http://ijass.org
and � indicate summation. Multivariate Taylor's polynomials of th
here as cross-section functions �� and � is defined as the expansi
of the formulation. Elements with number of nodes �� � � are f
the average
distortion for the generic�i-th iteration,
respectively.
The average
distortion
�
��
� cross-section
� � �,
� , �� � ���� � 2�/2
, and �� � are the lifting coefficient, the moment coefficient, and
where ���� is equal to 10�� , ��� , ��
the average
distortion
the
generic
i-th
iteration,
respectively.
average
distortion
��is is
defined
inEq.
Eq.18.
18.AA
linear
approach
isadopted
adopted
usual
classical
aeroelasticity:forfo
��for
defined
inis
linear
approach
isThe
asasusual
ininclassical
aeroelasticity:
�� iscross-section
defined in Eq.
18. A linear
approach
adopted
as
usual
classical
for
each
itera��� contains
the degrees
of in
freedom
ofaeroelasticity:
the � �� expansion
term
corresponding to th
the average cross-section distortion for the generic i-thwhere
iteration,
respectively.
The average
distortion
�� is defined
18. A linearloads
approach
is
as deformed
usual
in classical
aeroelasticity:
each
iterationthe
theadopted
aerodynamic
loads
computed
deformed
wing
configuration
appliedto
tion
aerodynamic
loads
computed
forforthethedeformed
wing
areareapplied
tion in
theEq.
aerodynamic
computed
for
wing
configuration
areforapplied
toconfiguration
the undeThethe
compact
expression
in Eq.
1 is based
on Einstein's
notation: repeated su
�� is defined in Eq. 18. A linear approach is adopted asment
usualnode.
in classical
aeroelasticity:
for each
iteration the formed
aerodynamic
loads computed
for
the
deformed
wing
configuration
are
applied
to
the undeformed
configuration,
without
changing
the
structural
stiffness
matrix��ofofEq.
Eq.7.7.
formed
configuration,
without
changing
the
structural
stiffness
matrix
configuration,
without
changing
thesummation.
structural
stiffness
matrix
�
of Eq. 7.
andwing
� indicate
Multivariate
tion the aerodynamic loads computed for the deformed
configuration
are applied
to the Taylor's
unde- polynomials of the x and z variables are
formed
configuration,
without
the
structuralloads
stiffness
matrix � of Eq. 7.
3.2
Aerodynamic
loadscomputation
computation
3.2
Aerodynamic
Int’l J. of Aeronautical & Space Sci. 14(4),
310–323
3.2(2013)
Aerodynamic
loadschanging
computation
here as cross-section
�� and � is defined as the expansion order, which is a free
formed configuration, without changing the structural stiffness
matrix � offunctions
Eq. 7.
The
aerodynamic
load
computed
XFLR5
isa distributed
a distributed
and
thiswork
workit itis ism
3.2 Aerodynamic
loads
computation
The aerodynamic
load
computed pressure
bybyXFLR5
and
The aerodynamic load computed
by XFLR5 is
a distributed
andisin
this
work itpressure
ispressure
modeled
asininthis
of the formulation. Elements with number of nodes �� � � are formulated in the present
3.2 Aerodynamic loads computation
the base
section.
Given model,
athe
structural
model,
the
base
section
corresponds
tobase
undeformed
crossand
in this
work
it is to
modeled
as work
distributed
forces.as panel is evaluated as:
deformed
cross-section
andThe
base
Given
base
section.
Given
athe
structural
model,
the
base
section
corresponds
the in
undeformed
ction.
Given
a structural
the section.
base
section
corresponds
tothe
the
undeformed
crossthe
base
Given
athe
structural
model,
thesection.
section
corresponds
to the
crossaerodynamic
computed
bypressure,
XFLR5
aundeformed
distributed
pressure
and
this
itcrossisaerodynamic
modeled
distributed
forces.
The
generic
force
acting
j-th
aerodynamic
distributed
forces.
generic
force
acting
ononthe
j-th
panel is evaluated as:
distributedload
forces.
The generic
force is
acting
onThe
the
j-th
aerodynamic
panel
is the
evaluated
as:
The
generic
force
onLagrange
the
aerodynamic
is
a structural model,
the base section
corresponds
to the
named
��, using
polynomials
as panel
shape functions
[19]. The number
The aerodynamic
load computed
by XFLR5
is a distributed
pressure
andthird-order
inacting
this work
it is j-th
modeled
as
section
shifted
and
rotated
in
such
a
way
that
its
leading
edge
and
trailing
edge
points
corresponds
to
and
in
way
thatacting
its edge
leading
and
trailingtoedge
points
corresponds
ed and rotated in such a way
that itsshifted
leading
edge
andsection
trailing
edge
corresponds
toa and
section
and
rotated
in
suchshifted
aand
waypoints
thatrotated
its
leading
edge
trailing
distributed
forces.
Thesuch
force
onpoints
theedge
j-th
aerodynamic
panel
is evaluated
as:� �to
11
�
evaluated
as:
undeformed
cross-section,
shifted
rotated
in
such
ageneric
way
�
1corresponds
� � ��
� · approach
·��·� �·���
�
�· ��· �·���· �
(15)
� �
of freedom �(DOFs)
used
is:
� · is
�
· �through
· the
�� proposed
distributed forces. The generic force acting on the j-th aerodynamic
evaluated
� panel
� · �� as:
2 2a �
that
its
and
trailing
edge
points
correspond
toa structural
2 a�structural
the leading
edge edge
and trailing
edge
points
ofedge
the
deformed
cross-section
obtained
by
such
the
leading
edge
and
trailing
edge
points
of theobtained
deformed
obtained by such
structural
edge
and trailing
points
theleading
deformed
cross-section
obtained
by
such
a structural
theof
leading
edge
and
trailing
edge
points
of
the
deformed
cross-section
such
1 by cross-section
�
�
(15)
(15)
�� � · �� · �� · �� · ��
the leading edge and trailing edge points, respectively, of1the
�� � ���� � 2�
�
�� � · �� · ��� · �� · ��2
model.
���� � � (15)
����� � ��
model.
model.
deformed cross-section obtained by such a structural model.
2
2
where
thefree
freestream
stream
velocity
and
isthe
the
airof
density.
areao
velocity,
and
is
the
where
where
����is isthe
the
stream
velocity
and
� �is�isthe
airair
density.
����is isthethearea
The
iterative
process
in
Fig.
1
is
stopped
once
the
where
�
is
the
free
stream
velocity
and
�
is
the
air
density.
�
area
the j-th
�
�
,
the
moThe
iterative
process
in
Fig.
1
is
stopped
once
the
convergences
of
the
lifting
coefficient
�
The
process
in Fig.
1 ismostopped
the
convergences
theemployed
lifting
��� ,�the
�of once
e process in Fig. 1 is stoppedThe
once
the convergences
of
the
coefficient
�convergences
mo- coefficient
iterative
process in Fig.
1 iterative
islifting
stopped
once the
theThe
lifting
coefficient
�of
variational
statement
is the Principle
ofmoVirtual Displacements:
� , the
� , the
density. Aj is the area of the j-th aerodynamic panel, which
convergences of the lifting coefficient
moment
where ��CLis, the
the free
stream velocity
and �� panel
is thewhich
air density.
�� is coefficient
the area of�� �therefers
j-th to. According to XFLR
� thepressure
aerodynamic
pressure
According to XFLR5
aerodynamic
which �the
coefficient
�� XFLR5
�refers to.
cross-section
distortion
ofisthe
wing
sections
are
achieved
simultaneously.
ment�coefficient
�� , and the
the
pressure
coefficient
refers
to
refers
According
to
XFLR5
notation,
aerodynamic
panel
which
the
pressure
coefficient
,
and
the
cross-section
distortion
of
thepanel
wing
achieved
simultaneously.
ment
coefficient
�
,
and
the
cross-section
distortion
of
the
wing
sections
are
achieved
simultaneously.
cient
coefficient
C
,
and
the
cross-section
distortion
of
the
wing
�
,
and
the
cross-section
distortion
of
the
wing
sections
are
achieved
simultaneously.
ment
coefficient
�
where
�
the
free
stream
velocity
and
�
is
the
air
density.
�� sections
is the�are
area
of to.
theAccording
j-th
�
M
�
�
�
�
�
� ���
���� �� �when
����� � ��� �
����� � are
� �� � ���positive
notation,
normal� �vectors
considered
sections are achieved simultaneously.
The description
�� is negative
refersThe
to.�positive
According
to
XFLR5
notation,
aerodynamic
panel
whichofcontrols
the
pressure
coefficient
�considered
�
andtheir
their
verseis isouter-poin
outer-po
normal
vectors
are
positive
when
Theof
description
of a similar
iterative
process
can
be
found
also
in
[12].
The
convergence
are
is
negative
and
verse
normal
vectors
are
considered
when
�
The
description
of
a
similar
iterative
process
can
be
found
also
in
[12].
convergence
controls
are
�
�
tion
a similar iterative
process
can
be
found
also
in
[12].
The
convergence
controls
are
The description aerodynamic
of a similar iterative
process
canpressure
bevectors
found are
alsoconsidered
in [12].
convergence
controls
� is outer-pointing.
negative
their
verse
Each
normal
�� istheir
ispositive
negative,
and
verseandisnotation,
outer-pointing.
Each
refers
to. when
According
to are
XFLR5
panel
which
the
��The
a similar iterative
process can
also
be found
in [12].coefficient
The
the
internal
straintheir
energy
and
�
is본문
the work
of external
loadings
variatio
where
���� �is�� is
aerodynamic
force
transferred
from
the
내
수식
(15)
밑에
문장.
thus:
negative
and
verse
isaerodynamic
outer-pointing.
Each
are considered positive
when
��� aerodynamic
convergence
controls are thus:
thus: normal vectors
aerodynamic
force
transferred
from
the
aerodynamic
model
thestructural
structural
model
aerodynamic
force
is isis
transferred
from
model
totothe
model
folf
thus:
�
aerodynamic
force is
from
the
aerodynamic
model
to the
the
structural
model
following
the
is negative
and
their
verse
is
outer-pointing.
Each
normal vectors are considered
positive when
��transferred
model
to
the
structural
model,
following
the
approach
밑에
문장.
본문내내수식
수식(15)
(15)
밑에
문장.
�
�
��� 본문
sistent
with
the� ���
present
method
and
here
derived
for the case
of
�
���
�described
�
��� ���
����� ���
� �
� transferred
���
force��
is��
thefor
aerodynamic
to2.1
the
structural
model
following
the
approach
in
section
2.1for
forthethe
generic
concentrated
load�.
�. a generic concent
���
�
� from
� model
��described
� � 수식
approach
section
generic
concentrated
load
� ��내
�밑에
��
�문장.
���� � �����
본문
(15)
�in� section
�
� in
�����
����� ���aerodynamic
described
in�generic
section
2.1
for
the
generic
concentrated
load
approach
described
2.1
the
concentrated
load
�.
�����
� �� (13)
(13)
�
�
�
����
(13)
�
����
�
�
����
(13)
�
����
�
�
����
�
� ���� �from the �aerodynamic
� ���� model to the� structural (13)
aerodynamic
force
is transferred
model following the
�
�
���
��
�
��
���
��
�approach
���� 2.1 for� .the
� ��
��� �� actingload
on the
point �� , � , �� �, which doe
�
�� ���concentrated
described in section
generic
�. arbitrary load application
9 each iteration, �the�three-dimension
9 For
9 the three-dimensional
For
each
iteration,
deformed
approach
described
in
section
2.1
for
the
generic
concentrated
load
�.
�
���
For
each
iteration,
the
three-dimensional
deformed
configuration of the wing
�
���
�
���
(14)
� �� �
���(14)
(14)
(14) element
��� � � �� ��� � ��� � �� � � ����
�
��iteration,
�
���each
essarily lies
along
the
1D
finite
mesh,
unlike
standard
1D
FE models. � stands for
For
each
iteration,
the
three-dimensional
deformed
configuration
ofthe
the
wing
isbuilt
built
using
11
airfoils
For
the
three-dimensional
deformed
configuration
ofofusing
the
wing
is
using
11the
airfoils
For
each
iteration,
the
three-dimensional
deformed
configuration
wing
isbuilt
using
11
airfoils
(14)
configuration
of
the
wing
is
built
11
airfoils
along
�
����
9
�
����
�
�
����
along the half-wing span at a distance o
�
�� � is built using
For each��iteration,
the�� three-dimensional deformed
11
airfoils
�� � configuration of the
along
the
half-wing
span
at
a
distance
of
0.5
m
from
9 wing
half-wing span, at a distance of 0.5 m from each other. The first each other. The first sect
becomes:
along
the
half-wing
span
adistance
distance
of
0.5
mfrom
from
each
other.
The
first
section
lies
atthe
the
wing
root.
variation.
By
using
Eq.
1,
��
along
the
half-wing
span
atataatadistance
ofof0.5
m
each
other.
first
section
lies
atatthe
wing
root.
along
the
half-wing
span
0.5
mfrom
each
other.
The
first
section
lies
wing
root.
��� The
�
�
��
�
�
�
��
�
�
The
, the
�����,toll
�� ,is
and
��span
are
the
lifting
coefficient,
the
moment
coefficient,
where
is��
equal
�moment
��
�coefficient,
-4
�lifting
��and
are
coefficient,
the
moment
andwing is discretized through a lattic
is
equal
toare
10
lies
at
the
wingisroot.
root.
The
wing
iscoefficient,
discretized
, ���to
, �10
,where
and
are
the
lifting
coefficient,
and
is
equal����
to 10
, �������
are
the,lifting
10
�, �
� , and
,the
�of�
,0.5
and
the
coefficient,
the the
moment
coefficient,
and
where
����
is equal
equal
toto
along
half-wing
at10
awhere
distance
m ��from
each
other.
The
firstsection
section
lieslifting
atThe
the
wing
�
wing
discretized
through
a lattice ofthrough
quadrilateral aerodynamic panels.
�
��
�Let
� number
�panels.
be
the
The
wing
isdiscretized
discretized
through
alattice
lattice
ofquadrilateral
quadrilateral
aerodynamic
panels.
aa lattice
ofofquadrilateral
aerodynamic
Let
be
the
The
wing
isisdiscretized
through
quadrilateral
aerodynamic
Let
be
the
number
The
wing
through
alattice
aerodynamic
Let
�
the
number
��
�panels.
� ��
�����
�����
�be
�
��
���
coefficient,
the
coefficient,
and
the
average
cross��� � ��panels.
��
the average cross-section
distortion
the moment
generic
i-th
iteration,
respectively.
average
�iteration,
of��
the
cross-section
distortion
fordistortion
the
generic
i-th
respectively.
The average ��
distortion
�panels along the chord line and let
cross-section
distortion for
generic
i-th
iteration,
respectively.
The
average The
distortion
thewing
average
cross-section
distortion
the
generic
i-th
iteration,
respectively.
The
average
distortion
The
isfor
discretized
through
aaverage
latticeforof
quadrilateral
aerodynamic
panels.
Let
�
be
the
number
��
be the
the number of panels along th
of panels
along
chordline,
line and
and let ��� be
number �of��panels
along
thethe
chord
section distortion for the generic i-th iteration, respectively.
the
number
along
the
half-wing
span.
panels
along
the
chord
line
and
let
�is��
where
evaluated
in ��
, �panels
� and
�� the
isthe
calculated
inspan.
��When
. When
InWhen
the case of small disp
be
the
number
ofof
panels
along
half-wing
span.
ofofof
panels
along
the
chord
line
and
letlet
����
bebe
the
number
along
half-wing
panels
along
the
chord
line
and
�
�of
�panels
number
of��
panels
the
half-wing
span.
When
the
VLM
��in isEq.
defined
Eq. approach
18. A linear
approach
isEq.
adopted
as
in classical
aeroelasticity:
for
each
�in in
distortion
is defined
defined
Eq.
18.
linear
�� usual
is
Eq.
18.
A
linear
approach
isiteraadopted
as
usual
in
classical
aeroelasticity:
for each
iterathe
VLM
is employed the total numbe
18. A in
linear
isis average
adopted
usual
classical
aeroelasticity:
for
each
itera��panels
defined
inthe
18.inAline
linear
approach
is adopted
asAusual
in classical
aeroelasticity:
foralong
each iterabe
the
number
of
panels
along
the half-wing
ofThe
alongas
chord
and
let
���
the span.
VLMWhen
is employed the total number of aerodynamic
panels ��� is equa
��AP
is
employed,
the
number
aerodynamic
panels
is� � . For (subscript
�out-of-plane
��
approach is adopted as usual in classical
aeroelasticity:
for
with
respect
thetotal
length
�,panels
theof
�) cro
is(subscript
equal
���
�and
�N
the
VLM
isemployed
employed
the
total
number
ofto
aerodynamic
�
isisequal
tototo
� ���)
�
. �For
the
the
VLM
isare
employed
the
total
number
ofofaerodynamic
panels
�inplane
equal
�
�
.
For
the
the
VLM
is
the
total
number
aerodynamic
panels
�
��
��
��
��
��
��
��
��
tion
the
aerodynamic
loads
computed
for
the
deformed
wing
configuration
applied
to
the
undetion
the aerodynamic
loads
computed
for the deformed
wing
are
applied
to the Nunde�configuration
�
odynamic
for
the
deformed
wing
configuration
are
to the
unde3D
Panel be
Method ��� must
be
tion
the
loads
computed
forapplied
the
deformed
wing
configuration
are�applied
to. For
the unde� calcula
the
3D
Panel
Method,
must
͵
ൈ ͵ and loads
͵ ൈ ͳcomputed
fundamental
nuclei
ࡷ௜௝ఛ௦the
and
ࡲ
are
introduced.
From
Eq.
6 the
goequal to
���
�
the
the
VLM
isaerodynamic
employed
the
number
of
aerodynamic
panels
�
each
iteration,
aerodynamic
loads
computed
for
the�
AP
ఛ௜total
�� is equal
��
the 3D Panel Method �
must be calculated as � � ��� � ��� � � �
��� to the�displacement
�
� � �as
��are
� � � �thevector
�, where
stress
andbestrain
vectors
in�Eq.
3�
related
calculated
as
the
term
the
3D
Panel
Method
�be
must
calculated
���
��� the
�
�term
�term
����,����
where
term
3D
Panel
�
calculated
as
���
where
3D
Panel
Method
���must
must
be
calculated
as� ��
�
���
��of�
,� where
the
�
��� isisthe
the via linear differen
deformed
wing
configuration
areconfiguration,
applied
to
the
undeformed
��
formed configuration,
without
changing
the
structural
stiffness
matrix
�
of�Method
Eq.
7.stiffness
��
��
��
��
��
��
�����
formed
without
changing
the
structural
stiffness
matrix
��
Eq.
7.
iguration,
without
changing
the
structural
stiffness
matrix
�
of
Eq.
7.
�
�
formed
configuration,
without
changing
the
structural
matrix
�
of
Eq.
7.
number
of panels along the wing span
�
�
ation of motion can be derived
through
a finite
assembly
procedure:
�
3D Panel
Method
���element
must be
calculated
as � ��� ��� � � ��� , where
the
term
�
�
�
is
the
�
�� � � along
is the number
of
thenumber
wing span,
on the
upper
���
� �panels
of panels
along
the wing span on the upper an
configuration, without changing the structural
��
�� is the
� stiffness
operators
�the
, upper
�upper
and
material
stiffness
matrices
�
, In�
, ���of
, the
�the
� panels
�the
�the
��In
��
�� as follows:
number
of
onon
and
lower
surfaces
of
wing.
addition,
the
number
ofcomputation
thewing
wing
span
and
lower
surfaces
ofthe
the
wing.
addition,
� �loads
���
� panels
��� along
isalong
the the
number
ofspan
panels
along
wing
span
on
the
upper
and
lower
surfaces
3.2 Aerodynamic
loads computation
�
3.2
Aerodynamic
amic
loads computation
3.2
Aerodynamic
loads
computation
and
lower
surfaces
of
the
wing.
In
addition,
the
term
term
�
�
is the number of panel
matrixof
7.
ࡷ panels
ࢗofൌEq.
ࡲ along
(7) surfaces of the wing. In addition, the
��
number
the wing span on the upper and lower
�
wing. In addition, the term � � ���
is�the
number
placed on the t
���
�� sake
�sake
�of
�panels
��
is
the
ofonpanels
on
thethetiptip�lateral
cross�tip
�placed
�lateral
� number
�lateral
��of
�
�panels
��
term
�
iswork
number
ofand
panels
placed
on
the
lateral
cross-sections.
For
the
term
��
�
isthe
theitby
number
of
panels
placed
the
tip
cross-sections.
For
the
of
The aerodynamic
load
computed
by
XFLR5 isload
apressure
distributed
pressure
and
in
this
is
modeled
as
wing.
In
addition,
the
term
�
�
�
is
the
number
of
placed
on
cross-sections.
The
aerodynamic
load
computed
XFLR5
is
a
distributed
pressure
and
in
this
work
it
is
modeled
as
��
��
namic
load
computed
by
XFLR5
is
a
distributed
and
in
this
work
it
is
modeled
as
��
The
aerodynamic
computed
by
XFLR5
is
a
distributed
pressure
in
this
work
it
is
modeled
as
convenience,
only half-wing is analyze
� is the vector of equivalent nodal forces. It should be
s the structural stiffness matrix
and
ࡲ
��half-wing
� ��� �� �
�convenience,
�� �
��of
term
� ���
is the number
panels placed on the tip lateral cross-sections.
For
sake
of
sections. For
thethe
sake
of
convenience,
only
is ��� �since
�
3.2 Aerodynamic
loadsofcomputation
For
the
sake
only the
half-wing
is analyzed
the aerodynami
convenience,
only
half-wing
is
analyzed
since
the
aerodynamic
loads
are
considered
to
be
symmetric
convenience,
only
half-wing
is
analyzed
sinceisas:
the
aerodynamic
loads
are
considered
to
be
symmetric
distributed
forces. force
The generic
force
acting
onThe
the j-th
aerodynamic
panel
is
evaluated
as:
For
the
sake
of
convenience,
only
half-wing
analyzed
since
the
aerodynamic
loads
are
considered
to
distributed
forces.
The
generic
force
actingpanel
on
the
j-th aerodynamic
panel
is evaluated
as:
analyzed,
since
the
aerodynamic
loads
are
considered
to
be
orces.
The
generic
acting
on
the
j-th
aerodynamic
panel
is
evaluated
as:
distributed
forces.
generic
force
acting
on
the
j-th
aerodynamic
is
evaluated
with
respect
to
the
wing
root.
no assumptions on the expansion
order
have
been
made
so
far.
Therefore,
it
is
possible
to
convenience, only half-wing is analyzed since the aerodynamic loads areUsing
considered
Eq. 5, to
Eq.
3 symmetric
can be rewritten
in terms
of wing
virtualroot.
nodal displacements:
bebesymmetric
to the
The aerodynamic load computed by XFLR5 is a distributed
symmetric with
respect towith
the respect
wing root.
1
with
respect
wing
root.
with
respecttotothe
the
wing
root.to1the wing�root. �
be1
symmetric
with
respect
�
1
�expression
��formal
· ·��
· ��� · �� · �of
�
�
(15)
����
er-order 1D models without
changing
as� �� � (15)
· � · � · � · ��
� the
�root.
� the
· � to
·�the
�
·�
�� �
(15)
with
respect
wing
� · �components
· � � · �(15)
·�
�� nuclei
��� � ��
�� ���
results���� �
2 � � �4. Numerical
2 � � 2� �
2 � � � �
4. Numerical results
sical beam models such as Euler-Bernoulli's and Timoshenko's. Higher-order models pro4.1 Aerodynamic assessment
4.1 Aerodynamic assessment
5
urate
description
of
the
shear
mechanics,
the
in-plane
and
out-of-plane
cross-section
de�� isstream
the free
stream
and
�� density.
is
the air
density.
��� isof
the
area
ofdensity.
theandj-th
where
is the
the
free
stream
velocity
�
is
the
air
density.
�
is
the
area
of
the
j-th and 3D Panel Method, which
Firstly
an
aerodynamic
assessment
of VLM
iswhere
the free
velocity
andvelocity
the free
air
���
area
the
j-th
�
�
where
��� isis the
stream
velocity
and
is
the
air
�
is
the
area
of
the
j-th
� is
�
�
Firstly an aerodynamic assessment of VLM and 3D Panel Method, which are able to evaluate the
Poisson's effect along the spatial directions and the torsional
mechanics
in
more
detail
than
�
� pressure coefficients on the wing surface, is performed analyzing the effects o
�
� XFLR5
refers
to.panel
to
notation,
which aerodynamic
the
pressure panel
��the
refers to.notation,
According to XFLR5 notation,
aerodynamic
which notation,
the
pressure
coefficient �to
to.
According
toAccording
XFLR5
caerodynamic
panel whichpanel
the pressure
coefficient
�coefficient
which
pressure
coefficient
�
� refers
� refers
pressure
coefficients
on to.
the According
wing surface,� isXFLR5
performed analyzing the effects of two typical geometricodels do. Thanks to the CUF, the present hierarchical approach is invariant with respect to
�
al parameters: the airfoil thickness and the camber line. A straight wing is co
�
�
�
is negative
andistheir
verse
outer-pointing.
Each
normal
vectors arepositive
considered
positive
when
and line.
theirEach
verse
is outer-pointing.
Each the wing span
normal
vectors
are
considered
positive
�� is
� and
their positive
verse
outer-pointing.
Each
ors
are considered
when
�vectors
and when
their
verse
isnegative
outer-pointing.
normal
are �
considered
when
�� isis negative
� is negative
al
parameters:
the
airfoil
thickness
and
the
camber
A
straight
wing is considered:
the displacement field expansion. More details are not reported here, but can be found in
is 10 m and the airfoil chord is 1 m long as drawn in Fig.2a, where the righ
force isfrom
transferred
from the
aerodynamic
to the
the
structural
following
the1 m long
force
ismtransferred
from
the
model
to the
structural
model
the is depicted.
caerodynamic
force is transferred
the
aerodynamic
model
toaerodynamic
themodel
structural
model
following
the
is
10
and model
themodel
airfoil
is
as drawn
in Fig.2a,
thefollowing
right half-wing
aerodynamic
force
is transferred
from
aerodynamic
tochord
theaerodynamic
structural
model
following
the where
[31].
This wing configuration is also used in the following structural and aeroelasti
approach
section
2.1 forconcentrated
the genericinconcentrated
load
approach
described
section
2.1 forload
theisgeneric
concentrated
load �.structural and aeroelastic analyses. The effect of
scribed
in described
section 2.1infor
theapproach
generic
load
�.2.1 for
This in
wing
configuration
also used
in the following
described
section
the�.
generic
concentrated
�.
rical approach for wing aerodynamic analysis
the camber line on the aerodynamic field is evaluated using NACA 2415, 341
9
the camber
9
9 line on the aerodynamic9 field is evaluated using NACA 2415, 3415 and 4415 airfoils. The
minaries
analysis of the influence of the airfoil thickness is then carried out using the
analysis of the influence of the airfoil thickness is then carried out using the symmetric NACA 0005,
ion of aerodynamic loads can be typically carried out through a CFD code which solves
0010 and 0015 airfoils. The number of aerodynamic panels is chosen as a c
0010 and 0015 airfoils. The number of aerodynamic panels is chosen as a compromise between the
e either Navier-Stokes equations or Euler equations numerically. This kind of analysis has
limit number of panels that can be used in XFLR5 (= 5,000) [38] and the num
limit number of panels that can be used in XFLR5 (= 5,000) [38] and the number of panels required in
putational cost but under some assumptions it is possible to employ simplified approaches.
order to achieve convergence in the aerodynamic results. In the following
Fig. 1. Aeroelastic iterative
procedure,
with
controllers
on the
aerodynamic
coefficients
andthe
wing
defor- analyses, the choice of
order
to on
achieve
convergence
in
results. In
following
Fig. 1. Aeroelastic iterative procedure, with controllers
aerodynamic
coefficients
andaerodynamic
wing deformation.
cases considered in the present work, the flow field is assumed to be steady and the fluid
�
�
��� � �� and ��� � �� remains the same.
�
mation.
�
� �� and ��� � �� remains the same.
���
not decisive since the viscous effects can be confined into a small region
(boundary layers
For the present assessment analysis the free stream velocity is assumed to be
DOI:10.5139/IJASS.2013.14.4.310
the present assessment
egions). The fluid can be thus
considered as inviscid and the flow fieldFor
is irrotational,
since 314analysis the free stream velocity is assumed to be �� � �� ��� such that
the compressibility effects can be neglected. The air density is assumed to b
the compressibility effects can be neglected. The air density is assumed to be � � � ����� ��⁄�� .
he velocity vector ࢂሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ is equal to zero:
1 1 of attack  of the wing is equal to 3 deg. In all the following ana
The angle
1 of attack  of the wing is equal to 3 deg. In all the following analyses the air density �
The angle
�
‫׏‬ൈࢂൌ૙
(8)
the velocity vector ࢂሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ can be considered as the gradient of a potential function ߶:
10
tion the aerodynamic loads computed for the deformed wing configuration
�� is defined
inconfiguration,
Eq. 18. A linear
approach
is adoptedstructural
as usual stiffness
in classical
aeroelasticity
without
changing
matrix
of Eqa
theformed
camber
line on
on the
the aerodynamic
aerodynamic
field isisthe
evaluated using
using NACA
NACA
2415,�3415
3415
the
camber
line
field
evaluated
2415,
an
formed configuration, without changing the structural stiffness matrix � of E
tion the3.2
aerodynamic
loads
computed
for the deformed wing configuration are appli
Aerodynamic
loads
computation
analysis
of the
the influence
influence
of
the airfoil
airfoil thickness
thickness isis then
then carried
carried out
out using
using the
the sym
sym
analysis
of
of
the
3.2 Aerodynamic loads computation
formedThe
configuration,
without
changing the
structural
matrix � of and
Eq. 7.
aerodynamic
loadThe
computed
byof
XFLR5
is stiffness
a distributed
this
0010 and
and
0015 airfoils.
airfoils.
The
number
of
aerodynamic
panels isispressure
chosen as
as aaincomp
com
0010
0015
number
aerodynamic
panels
chosen
The aerodynamic load computed by XFLR5 is a distributed pressure and in th
3.2 Aerodynamic
loads computation
distributed
The
generic
force
on the(=
aerodynamic
panel
is eva
limit
number forces.
of panels
panels
that
can be
be
usedacting
inDeformation
XFLR5
(=j-th
5,000)
[38] and
and the
the number
number
number
of
that
can
used
in
XFLR5
5,000)
[38]
Alberto Varello Static Aeroelastic Response of Wing-Structureslimit
Accounting
for
In-Plane
Cross-Section
distributed forces. The generic force acting on the j-th aerodynamic panel is ev
The aerodynamic load computed by XFLR5 is a distributed
pressure �and in this work
1
order to
to achieve
achieve convergence
convergence in
in the
the aerodynamic
aerodynamic
In
the following
following ana
ana
order
·results.
��� · ��In
· �the
�� � · �� results.
�
21
�
�
· �� · �� · �� panel
· �� is evaluated
�� �
distributed
forces. The �generic
force acting on the
j-th aerodynamic
�
4. Numerical results
�� � �� and
choice of �
thesame.
same. 2
and �
���
� ��
�� remains
remains the
same.
���
�� and
remains
�� �
�� �
For the present assessment analysis, the free1stream velocity
�
�
· ��stream
· ��� · �
�� the
� · �� is
For the
present assessment
assessment analysis
analysis
the2free
free
stream
velocity
is assumed
assumed to
to be ��
For
velocity
4.1 Aerodynamic assessment
is assumed
tothe
bepresent
suchstream
that the
compressibility
where
�� =50m/s,
is the free
velocity
and ��
is the air
density. be�� ��is
where
�� isThe
theairfree
stream
velocity toand
� is the air density. �� i
effects can
becompressibility
neglected.
density
isneglected.
assumed
beair
the
compressibility
effects
can be
be
neglected.
The
air�density
density
assumed to
to be
be ��
the
effects
can
The
isis assumed
�
Firstly, an aerodynamic assessment of the VLM and the
3
aerodynamic
panel
which
the
pressure
coefficient
=1.225kg/m . The angle of attack α of the wing is equal to 3 �� refers to. According
�
3D Panel Method, which are able to evaluate the pressure
Accordi
aerodynamic
panel
The
angle
of attack
attack
ofwhich
the
wing
equal�to
to coefficient
deg.
Inair
all�density.
therefers
following
analyse
where
��
is the
free
stream
velocity
and
the
�to.
the ar
�
angle
of
 of
the
wing
isispressure
equal
33isdeg.
all
the
following
deg. In
allThe
the
following
analyses,
the
airthe
density
and
theIn
�
� is analyse
�
coefficients on the wing surface, is performed analyzing
normal vectors are considered positive when �� is negative and their verse i
angle
of
attack
α
will
be
invariable
parameters.
The
results
and
angle
of airfoil
attack
 will
invariable
parameters.
The
results
focus
on
variation
ofnegative
the
� � is
andparameters:
the the
angle
ofthe
attack
 will
be be
invariable
parameters.
The
results
focus
on
the the
variation
the
and their to
verse
normal
vectors
are
considered
positive
when
the effects of two typical geometrical
�refers
to. According
X
panel
which
the
pressure
coefficient
���of
focusaerodynamic
on theaerodynamic
variation
of force
the spanwise
local from
lifting
coefficient
is
transferred
the
aerodynamic
model to the structura
thickness and the camber line. A straight wing is considered:
10
10
wing
span
defined
spanwise
local
lifting
coefficient
�� along
thethe
wing
span,
defined
along
the
wing
span
defined
as:isas:transferred from the
spanwise
local
lifting
coefficient
�� along
�
aerodynamic
force
to theisstructu
negative and model
their verse
outer
normal vectors
are considered
positive when �� is aerodynamic
the wing span is 10 m, and the airfoil chord is 1 m long, as
approach described in section 2.1 for the generic
concentrated load �.
����
drawn in Fig. 2a, where the right half-wing is depicted. This
����described in section 2.1 for the generic concentrated load �.
�� ���
�� ���
� � 1 approach
aerodynamic
is transferred from the aerodynamic(16)
model
(16)to the structural mode
1 � � ��force
�
wing configuration is also used in the following structural
9(16)
� ����
� ����
�� �� �
�2�2
����
� ����
2 2� �� �
9 load �.
and aeroelastic analyses. The effect of the camber line on
approach described in section 2.1 for the generic concentrated
the aerodynamic field is evaluated,where
using
NACA
2415,
3415
where
����
and
����
are
the
chord
and
the
Lift
Force
generated
by
the
pressure
acting
on
the
where,
c(y)
and
L(y)
are
the
chord
and
the
Lift
Force,
���� and ���� are the chord and the Lift Force generated by the pressure acting on the pa-pa9 panels
and 4415 airfoils. The analysis of the influence of the airfoil
respectively, generated by the pressure acting on the
and
the can
angle
of

invariable
parameters. The
nels
with
span-length
2 ����
placed
at the
y coordinate.
More
details
can
be attack
found
in will
[32].
As
a first
nels
with
span-length
2
����
placed
at
the
y
coordinate.
More
details
be
found
in
[32].
Asbe
a first
thickness is then carried out, using the symmetric NACA
with span-length 2e(y), placed at the y coordinate. More
0005, 0010 and 0015 airfoils. The result,
number
aerodynamic
can
behalf-wing)
found is
inreported
[32].
As in
a first
result,
the
trend
along the wing span define
spanwise
lifting
coefficient
y axis
(right
is reported
inlocal
Fig.
3a.
This
analysis
is��carried
result,
the
trend
�� along
the the
y details
axis
(right
half-wing)
Fig.
3a.
This
analysis
isofcarried
theof
trend
of of
�� along
panels is chosen as a compromise between the limit number
along(b)the
y
axis
(right
half-wing)
is
reported
in
Fig.
3a.
This
NACA 2415 airfoil cross-section, with variable thickness and 2 cells
����
out
considering
the
variation
ofanalysis
the
airfoil
thickness.
As
expected,
VLM
is not
toairfoil
take
into
out(=
considering
theand
variation
airfoil
thickness.
As
expected,
the the
VLM
is not
ableable
take
into
ac- acof panels that can be used in XFLR5
5,000)
[38],
the of the
is carried
out,
considering
the
variation
oftothe
�� ���
�
1
� � � � � � 2 � ����
number of panels required, in order tocount
achievevariation
convergence
thickness.
Asitexpected,
the VLM ispressures
not ableontothetake
into
of airfoil
thickness,
since
computes
aerodynamic
wing
reference2 � �
count the the
variation of airfoil
since
it computes
aerodynamic
pressures on the
wing
reference
Fig.thickness,
2. Geometrical
configuration
of the straight
wing.
(a) Plan view of the straight half-wing
in the aerodynamic results. In the following analyses, the
account the variation of airfoil thickness, since it computes
where
����
andthe
����
arethethe
and the Lift Force gene
surface,
underestimates
respect
to the
Panel
Method.
contrary,
the
3D
Panel
surface,
andand
underestimates
�� �with
respect
to the
3D 3D
Panel
Method.
On On
the
contrary,
3Dchord
Panel
� with
nels
span-length
2increases,
����
placed
the y coordinate. More
Method
is able
to evaluate
change
of the
lifting
coefficient
aswith
the
airfoil
thickness
increases,
asatcan
Method
is able
to evaluate
the the
change
of the
lifting
coefficient
as
the
airfoil
thickness
as can
result, the trend of �� along the y axis (right half-wing) is repo
be seen
in Fig.
be seen
in Fig.
3a. 3a.
out considering
the camber
variation
of changes.
the airfoil
as the
Figure
reports
trend
of the
spanwise
local
lifting
coefficient
camber
lineline
changes.
It Itthickness. As expect
Figure
3b 3b
reports
the the
trend
of the
spanwise
local
lifting
coefficient
�� �as
� the
count
the
variation
airfoil
thickness,
since it computes aerod
is evident
both
aerodynamic
methods
to analyze
influence
of of
the
camber
line.
Comis evident
thatthat
both
aerodynamic
methods
are are
ableable
to analyze
the the
influence
of the
camber
line.
Com-
Fig. 2. Geometrical configuration of the straight wing.
surface,
and
underestimates
�and
with
respect
� thus
paring
Figs.
it should
to be
noted
spanwise
local
lifting
coefficient,
thus
the to the 3D Panel M
paring
Figs.
3a 3a
andand
3b 3b
it should
to be
noted
thatthat
thethe
spanwise
local
lifting
coefficient,
and
the
Method is able to evaluate the change of the lifting coefficient
aerodynamic
pressures,
is(b)
affected
more
by
camber
change
the
airfoil
thickness
change.
aerodynamic
pressures,
is affected
more
byairfoil
the the
camber
lineline
change
thanthan
the
airfoil
thickness
(a) Plan view of the straight
half-wing
NACA
2415
cross-section,
with
variable
thickness
and
2 cells change.
29
be seen
in
Fig.realistic
3a.
It can
be
concluded
Panel
Method
is able
to provide
a more
realistic
evaluation
of the
It can
be
concluded
thatthat
the the
3D 3D
Panel
Method
is able
to provide
a more
evaluation
of the
Fig. 2. Geometrical configuration of the straight
wing.
(a) Effect
Figure
3b
reports
the trend
of the
spanwise
pressure
distribution
wing
VLM.
Moreover,
the
Panel
Method
affords
pressure local lifting coeffic
pressure
distribution
on on
the the
wing
thanthan
the the
VLM.
Moreover,
the
3D 3D
Panel
Method
affords
pressure
evident
that
both
aerodynamic
methods
loads
actual
wing
surface,
which
fundamental
an
accurate
study
of the
actual
wing
loads
on on
the the
actual
wing
surface,
which
are are
fundamental
for for
anis accurate
study
of the
actual
wing
de-de- are able to analyze
Figs.
3a reference
andreference
3b it surface
should
to for
be
noted that the spanwis
airfoil
distortion,
in lieu
of loads
applied
a fictitious
wing
surface
as for
formation andand
airfoil
distortion,
in lieu
of loads
applied
on on
a paring
fictitious
wing
as
(a) Plan view of the straightformation
half-wing
pressures,
is aerodynamic
affected
more by the camber line ch
VLM
case.
These
reasons
make
Panel
Method
the
recommended
classical
aerodynamic
the the
VLM
case.
These
reasons
make
the the
3D 3D
Panel
Method
theaerodynamic
recommended
classical
following
aeroelastic
wing
analyses.
tooltool
for for
the the
following
aeroelastic
wing
analyses.
It can be concluded that the 3D Panel Method is able to pro
Structural
assessment
4.24.2
Structural
assessment
pressure distribution on the wing than the VLM. Moreover,
loads1Don1D
the
actual
wing
surface,
which
are fundamental for a
In order
to validate
results
given
proposed
higher-order
CUF
approach
a comparison
In order
to validate
the the
results
given
by by
the the
proposed
higher-order
CUF
approach
a comparison
of of
formation
and
airfoil
distortion,
in lieu
static
structural
wing
response
is here
performed
with
MSC
Nastran.
Only
right
half-wing
the the
static
structural
wing
response
is here
performed
with
MSC
Nastran.
Only
the the
right
half-wing
of of loads applied on a
the VLM
case. These
reasons
the 3D Panel Method th
straight
configuration
introduced
in the
previous
aerodynamic
assessment
Fig.
is consithe the
straight
configuration
introduced
in the
previous
aerodynamic
assessment
(see(see
Fig.
2a)2a)
ismake
consiof the airfoil thickness
(a) Effect of the airfoil thickness
(a) Effect of the airfoil thickness
(b) Effect of the airfoil camber line
tool
for the
aeroelastic
wing
analyses.
29
dered
here
due
toPressure
loads
and
structural
symmetry.
A the
clamped
boundary
condition
is
taken
into
account
dered
here
due
toPressure
and
structural
symmetry.
A the
clamped
boundary
condition
is
taken
into
account
Table
1.
distribution
on the
wing
along
spanwise
direction
forfollowing
the the
structural
assessment.
Table
1.loads
distribution
on
the
wing
along
spanwise
direction
for
structural
assessment.
Fig. 3. Effects of the (a) airfoil thickness and (b) camber line on the spanwise local lifting coefficient Cl of the straight wing, along the y axis. Com�
4.2
Structural assessment
� �
⁄3,�
��������,
�����
����������
� ���� �
�� ���������.
�� �
30
⁄� ,�o��. ���
� ������,
� ��������
���
�
� ���������.
, ����
α=3
=50m/s,
=1.225kg/m
parison of the VLM and the 3D Panel Method.
� �11
11
In order to validate the results given by the proposed higher-or
315
y [m]
y [m]
the static structural wing
response is here performed with MS
������
http://ijass.org
��� ���
the straight configuration introduced in the previous aerodyna
����
� ��������
����
� ����1.001.00
����
� ��������
����
� ����0.750.75
dered here due to loads and structural symmetry. A clamped b
����
� ��������
����
� ����0.500.50
����
� ��������
����
� ����0.250.25
11
Int’l J. of Aeronautical & Space Sci. 14(4), 310–323 (2013)
aerodynamic pressures on the wing reference surface, and
underestimates Cl with respect to the 3D Panel Method. In
contrast, the 3D Panel Method is able to evaluate the change
of the lifting coefficient as the airfoil thickness increases, as
can be seen in Fig. 3a.
Figure 3b reports the trend of the spanwise local lifting
coefficient Cl as the camber line changes. It is evident that
both aerodynamic methods are able to analyze the influence
of the camber line. Comparing Figs. 3a and 3b, it should be
noted that the spanwise local lifting coefficient, and thus the
aerodynamic pressures, are affected more by the camber line
change than the airfoil thickness change. It can be concluded
that the 3D Panel Method is able to provide a more realistic
evaluation of the pressure distribution on the wing than
the VLM. Moreover, the 3D Panel Method affords pressure
loads on the actual wing surface, which are fundamental
for an accurate study of the actual wing deformation and
airfoil distortion, in lieu of loads applied on a fictitious
wing reference surface, as for the VLM case. These reasons
make the 3D Panel Method the recommended classical
aerodynamic tool for the following aeroelastic wing analyses.
root cross-section (at y=0), whereas the tip cross-section is
free. The cross-section employed is a 2415 NACA airfoil, with
constant thickness equal to 2 mm. A spar with a thickness
equal to 2 mm is inserted along the spanwise direction,
at 25% of the chord. The isotropic material adopted is
aluminum: Young’s modulus E=69GPa, and Poisson’s ratio
v=0.33.
Due to the small thickness and the well-known aspect
ratio restrictions typical of solid elements, this wing is
modeled in MSC Nastran by 214,500 solid Hex8 elements
and 426,852 nodes, corresponding to 1,280,556 degrees of
freedom (DOFs). The same structure is analyzed through
CUF models with a variable expansion order up to N=14, and
discretized through a 1D mesh of 10 B4 finite elements (31
nodes). The number of DOFs depends on N, as expressed in
Eq. 2; for instance, with 10 B4 elements and N=14, the DOFs
are 11,160. However, an analysis of the present structure is
also carried out through a Nastran shell FE model, but it is
not reported herein, for the sake of brevity. Nonetheless,
the error obtained between 1D CUF and shell results is
comparable with the error obtained between 1D CUF and
Table
1. Pressure
distribution
on on
the the
wing
along
the the
spanwis
Table
1. Pressure
distribution
wing
along
span
Table 1. Pressure distribution
on the
wing along
spanwise
direction
for thedirecstructural assessment.
Tableon1. Pressure
distribution
on the
the
wing along
spanwise
Table 1. Pressure distribution
the
wing along
the spanwise
direction
for thethe
structural
assessment.
�
� �
⁄�⁄��
�� ������,
�����
�� ��������
� ���� ���
�� �
, �����
� ������,
� ��������
��� ���
��
, ����
=50m/s,
=1.225kg/
tion, for the structural assessment.
�
4.2 Structural assessment
�
�
⁄��� ,� �����
In order to validate the results given by�the
proposed
�������,
� �� ���,
� � � � ������ ��.
� � �����
⁄�m�3,,��
��� � ���������
�� �
� ���������.
����
� � �
� ��
� � �
higher-order 1D CUF approach, a comparison of the static
y [m]
������
structural wing response is here performed, with MSC
1.00
����
�
�
�
����
Nastran. Only the right half-wing of the straight configuration
y [m]
������
(b) Effect of the airfoil camber line
���� � � � ���� 0.75
introduced in the previous aerodynamic assessment (see Fig.
Fig. 3. Effects of the (a) airfoil thickness, and (b) camber line, on the spanwise local lifting coefficient
���� � � ����
� ����
� �1.00
� ���� 0.50
2a) is considered here, due to loads and structural symmetry.
����
�
�
� ���� 0.25
of the
straight wing,
along thecondition
axis. Comparison
VLM
and the for
3D the
Panel Method.
A clamped
boundary
is takenofinto
account
���� � � � ���� 0.75
,
,
.
���� � � � ����
y [m]
y [m]
���
����
� ��������
����
� ����1.00
����
� ��������
����
� ����0.75
����
� ��������
����
� ����0.50
0.50
���� ����
� ��������
� ����0.25
for ����
different
Table 2. Convergent values of lifting coefficient ������ and moment coefficient ��
���� � � � ����
0.25
��
structural models. �� = 30 m/s, ���� = 0.4637, ��
= - 0.1629.
������
Model
N=1
N=4
N=8
N=9
N = 10
N = 12
N = 14
(a) Airfoil upper surface
(a) Airfoil upper surface
���� �� ��
����
��
0.4643
0.4641
0.4667
0.4877
0.4953
0.5034
0.5090
�����
- 0.1633
- 0.1634
- 0.1659
- 0.1823
- 0.1886
- 0.1950
- 0.1994
2
2
3
6
8
9
10
DOFs
279
1,395
4,185
5,115
6,138
8,463
11,160
(b) Airfoil lower surface
(b) Airfoil
lower surface
Table 3. Convergence of the average distortion
�� [mm]
in the iterative aeroelastic analysis for different
Fig. 4. Percent error obtained by different 1D CUF models in the computation of the distortion along
Fig. 4. Percent error obtained by different 1D CUF modelsstructural
in the computation
of the cross-section
distortion along
and
at
m/s.(b) lower surfaces,
models.
at the
� �airfoil
� the
m.(a)
�upper
�=
the airfoilAirfoil
(a) upper,
and (b) lower surfaces,
at
wing
tip30
cross-section
m). Structural asthe wing tip cross-section (y=5m). Structural assessment: static
wing response
to a variable pressure
distribution.
Reference (solution:
Nastran solid.
sessment: static wing response to a variable pressure distribution. Reference solution: Nastran solid.
Model
DOI:10.5139/IJASS.2013.14.4.310
31
N=1
N=4
N=8
N=9
N = 10
1
0.0402
0.0135
0.1729
1.1441
1.4177
2 316
0.0403
0.0136
0.1816
1.4721
1.9198
Iteration
3
4
5
6
7
8
9
10
11
12
0.0403
0.0136
0.1820
1.5624
2.1159
0.1821
1.5868
2.1930
0.1821
1.5934
2.2234
1.5951
2.2353
1.5956
2.2400
1.5958
2.2419
2.2426
2.2429
-
-
(a) Relative lifting coefficient
(b) Moment coefficient
0,556 degrees of freedom (DOFs). The same structure is analyzed through CUF models with a
expansion order up to N=14 and discretized through a 1D mesh of 10 B4 finite elements (31
Thiswith
section
focuses
on the
results
regarding the
equilibrium
aeroelastic
The number of DOFs depends on N as the
expressed
in Eq. 2;
for instance,
10 B4
elements
base section.
Given
a structural
model,
the
base
section
corresponds
to the
undeformed
cross- response of a wing exposed to
a free
stream
ܸஶ edge
ൌ ͵Ͳ݉Ȁ‫ݏ‬
via theedge
iterative
procedure. This analysis aims at
14 the DOFs are 11,160. However, ansection
analysis
of the
carried
also
shifted
andpresent
rotated structure
in such
a is
way
thatvelocity
itsout
leading
and trailing
pointsCUF-XFLR5
corresponds to
Alberto
Varello
Static
Aeroelastic
Response
ofdeformed
Wing-Structures
Accounting
In-Plane
Cross-Section
Deformation
evaluating
the
influence
expansion
order for
Nby
on
the aeroelastic
behavior
of the structure, as the
a Nastran shell FE model, but it is not reported
herein
forand
thetrailing
sake
ofedge
brevity.
Nonetheless,
theof CUF
the
leading
edge
points
of the
cross-section
obtained
such
a structural
accurate
description
the cross-section distortion depends on N. The same material and straight wing
tained between 1D CUF and shell results
is comparable with the error
obtained
betweenof1D
model.
assessment are employed here, see Fig. 2a. In this case, the
solid results.
configuration
as
those
considered
in the
theis
previous
assessment
employed
Fig. 2a. In this
thedepicted
moThe
iterative
process
in
Fig.
1
is
stopped
once
the
convergences
of
lifting
coefficient
�� , are
d solid results.
cross-section
the NACA
2415 airfoil,
inhere,
Fig. see
2b. The
A variable pressure distribution step-like along the
spar thickness
t3 is constant
spanwise direction is applied to the upper
and
lower wing is the
equal
2 mm;
case
2415 airfoil
depictedand
in Fig.
2b.toThe
spar whereas,
thickness the
‫ݐ‬ଷ is constant
the cross-section
distortion
of theNACA
wing sections
are achieved
simultaneously.
ment
coefficient
�� , and
ble pressure distribution step-like along the
spanwise
direction
is applied
tothe
thecross-section
upper
and lowsurfaces, in order to simulate a real pressure distribution, see
skin thickness of upper and lower surfaces varies gradually,
equal
tostructural
2can
mmbewhereas
the2 skin
thickness
of
upper
lower
surfaces
from 2
The structural
description
of see
a similar
iterative
process
found
also
in
[12].
convergence
controls
Table
1. The
static
response
of the
wing
is evaluated
mm
(t1 The
in Fig.
2b)
to 1 and
mm
(t2 inare
Fig. 2b), varies
in thegradually
zone
surfaces in order to simulate
a real
pressure
distribution,
Table
1.and
The
static
re- from
in terms of the distortion s at the tip cross-section. For the
between 40% and 45% of the chord. This particular choice is
mm (‫ݐ‬ଵ in the
Fig.upper
2b) toand
1 mm (‫ݐ‬ଶ in Fig. 2b) in the zone between the 40% and the 45% of the chord.
thus:
of the wing is evaluated inupper
termsand
of the
distortion
s atFigs.
the tip
lower
surfaces,
4a cross-section.
and 4b show For
the percent
coherent with the purpose of studying a highly-deformable
�
���
�coherent
��� with the
This
particular
choice
is
purpose of studying a high-deformable nonclassical crosserror
e
obtained
by
computing
the
distortion
through
1D
cross-section.
�� � through ��
� ��
�
urfaces, Figs. 4a and 4b show the percent error e obtained computing��the
1D�nonclassical
� �distortion
(13)
� ���� as�
� ����
�
�The 1D structural mesh consists of 10 B4 elements. For the
CUF models and the Nastran solid model, which
is
taken
��
��
section.
odels and Nastran solid model,
which is taken as reference:
reference:
sake of brevity, a convergent study on the number of mesh
� consists
��� � � �� ���
elements
is B4
notelements.
reportedFor
here.
In fact,
the choice
of 10 B4study on the
The
1D
structural
mesh
of 10
the sake
of(14)
brevity,
a convergent
‫ݏ‬ே௔௦௧௥௔௡ െ ‫ݏ‬ଵ஽ ஼௎ி
� ����
�
݁ ൌ ͳͲͲ
��
(17)
(17) elements yields a good evaluation of displacements for all the
‫ݏ‬ே௔௦௧௥௔௡
number of mesh elements is not reported here. In fact, the choice of 10 B4 elements yields a good
points of the structure, as detailed in [32, 34], where a similar
�
, and �� � are the lifting coefficient, the moment coefficient, and
where ���� is equal to 10�� , ��� , ��
cted in Figs. 4a and 4b, the proposed
1D
FEs
provide
a
convergent
solution
by
gradually
apTable
1.
Pressure
distribution
onpoints
the
the spanwise
direction
thewhere
structural
assessment.
structural
case
in wing
terms
of
wing
configuration
andfor
applied
As depicted in Figs. 4a and 4b, the proposed
1Ddisplacements
FEs
evaluation of
for all the
of thealong
structure,
as detailed
in [32,
34],
a similar
the averagesolution,
cross-section
distortion for
the generic i-th iteration,
respectively.
The average
distortion
aerodynamic
loads was
studied
via the present structural
provide a convergent
by gradually
approaching
�
ng the Nastran solid results as the expansion order increases from 8structural
to 14, according
to the
⁄�� , ����
� ��in���,
� of
�
�����
��
� � � aerodynamic
� ������ ��.loads was studied via the
case
terms
wing
configuration
and�
applied
�
model,
and
successfully
assessed
the Nastran solid results, as the expansion order��increases
� � � with a commercial FE solid
�� is defined in Eq. 18. A linear approach is adopted as usual in classical aeroelasticity: for each iteramodel.
from
8 to[39].
14, according
conclusions
made
in
ions made in previous CUF
works
For N=14 to
thethe
maximum
percent
error
isprevious
about model
3% forand
present
structural
successfully assessed
FE solid model.
y [m] with a commercial
������
Theconfiguration
aeroelastic analysis
is now
CUF works [39].
N=14, the maximum
percent
is
tion For
the aerodynamic
loads computed
for error
the deformed
wing
are applied
to thecarried
unde- out following the
����
��
���� 1.00
er surface and about 2.7%about
for the
For the wing
configuration
considered,
theisiterative
The
analysis
now carried
out following
the�iterative
coupled
procedureinCUF-XFLR5
decoupled
procedure
CUF-XFLR5
described
Fig.
3%lower
for thesurface.
upper surface,
and about
2.7%aeroelastic
for the
lower
formed configuration, without changing the structural stiffness matrix � of ����
Eq. 7.� � � ���� 0.75
1, and varying N. The convergence process on the lifting
surface. For the wing configuration considered, the choice of
of N=14 seems hence to be accurate enough to detect the cross-section
distortion
an acscribed
in Fig.with
1 and
varying N. The convergence
on the
lifting and moment coefficients is
0.50
���� �process
� � ����
and moment coefficients is drawn in Fig. 5a, by means
N=14 seems hence
to
be
accurate
enough
to
detect
the
cross3.2 Aerodynamic loads computation
���� � � � ���� 0.25
drawn
inrespect
Fig. 5a
means
a dimensionlessparameter
parameter ‫ܥ‬௅ Ȁ‫ܥ‬௅௖௢௡௩ , and
e error with respect to Nastran
3Ddistortion
results andwith
withan
a remarkable
terms
oftoby
DOFs
of aofdimensionless
section
acceptable reduction
error
within
andininFig.
Fig.5b,5b,respectively.
The
aerodynamic
load
computed
by
XFLR5
is
a
distributed
pressure
and
in
this
work
it
is
modeled
as
the Nastran 3D results, and with a remarkable
reduction in
Table
1. Pressure
on the wing
the final
convergentvalues
value of
of the
lifting
coefficient
which
differentcoefficient
for
each
expansion
or-different
‫ܥ‬௅௖௢௡௩ isTable
91% reduction, 11,160 vs. 1,280,556).
����distribution
andismoment
��
for
2. Convergent
lifting
coefficient
������
terms of DOFsdistributed
(about a 91%
reduction,
11,160
vs.
1,280,556).
����
Table
2. Convergent
values
oflifting
lifting
coefficient
������
andfor
moment
coefficient
��
for different
forces. The generic force
acting
on
the j-thvalues
aerodynamic
panel
is evaluated
as:
and
moment
Table
2. Convergent
values
of
coefficient
����
����
����
����
Table
2. 2.
Convergent
of
lifting
coefficient
�
and
moment
coefficient
�
different
Table
Convergent
values
of
lifting
coefficient
�
and
moment
coefficient
�
for
different
�
�
�
����
���� �
௖௢௡௩ models.
�in
������,
��� � ��������⁄�� , ����
�� =30
Table 2. Convergent values of lifting
coefficientas�well
andthe
moment
coefficient���moment
different
coefficient
, for different
m/s,
der employed
as
final convergent
coefficient
‫ܥ‬ெ
, as reported
Table 2.
oelastic coupling
�
�
�� structural
��
structural
models. ��
= 30
m/s,
�� �=��0.4637,
����� == 0.4637,
- 0.1629.
structural
models.
==300.1629.
��
= - 0.1629.
1models.
��=
�
�����
structural
models.
����
= ·30
m/s,
0.4637,
-=m/s,
0.1629.
�
structural
=�30
m/s,
����==0.4637,
= 0.4637,���
��
- 0.1629.
4.3 Aeroelastic coupling
(15)
��
��� · �
�
·
�
·
�
�
�
�
�
�
structural
models.
�
=
30
m/s,
�
=
0.4637,
�
=
0.1629.
�
�
12
Hence,
a different
2 � choice of N influences the structural response of the wing to the aerodynamic loads
���� � �
����
This section focuses on the results regarding the
Model
DOFs
������
��
����� ����� � ����
�� ��
y [m]
����
and consequently affects
also
the
aerodynamic
analysis,
due
to
coupling.
The
higher
Model
�
��
DOFs
����
��� ��
�����
��the aeroelastic
����
����
����
����
����
�
�
DOFs
Model
�
�
Model
�
�
DOFs
�
�
�
equilibrium aeroelastic responseModel
of a wing� ����
exposed�to
�
�
���� �N
����
����
����a
� ��
0.4643
0.1633
2
279
=
1
DOFs
�����
�
�
(b)
Airfoil
lower
����
௖௢௡௩
free stream velocity
via the
iterative
CUF-XFLR5
where �� =30m/s,
is the free
stream
andorder
�� is
the air
density.
�
isN =
the1 appears
area
of the j-th
- 0.1634
2‫ ܥ‬௖௢௡௩ (‫ܥ‬ெ
1,395
Nsurface
=the
40.4643
0.4643
- 0.1633
2 ) and the279
initial
thevelocity
expansion
employed
more 0.4641
difference
0.4643
-� 0.1633
2 2 between
279
NN
==
11
- 0.1633
279௅
0.4643
- 0.1633
2 N = 8 279 0.4667
N=1
- 0.1659
3
4,185
procedure. This analysis aims at evaluating
the influence
of
����
0.4641
- 0.1634
2
N
= 4 For
௜௡
௜௡ CUF models
Fig.aerodynamic
4. Percent error
obtained
different
in9to.
theAccording
computation
the
distortion
- 0.1634
2 2 cases
1,395
N�
=��=
4 refers
value
‫ܥ‬
(‫ܥ‬1D
evaluated
for
undeformed
wing.
the
presented
in this
work, the1,395
number
0.4641
- 0.1634
1,395
N
4 Nthe
to XFLR5
notation,
which
theby
pressure
0.4877
-of
0.1823
6 along
5,115
=0.4641
௅ coefficient
ெ ) of
the CUF expansion
order panel
N on the
aeroelastic
behavior
0.4641
- 0.1634
2
1,395
N=4
����
0.4953
8
6,138
N = 10
0.4667
- 0.1659
3
4,185
N = 8 - 0.1886
the structure,
as
the accurate
description
ofofthe
cross-section
0.4667
- 0.1659
3 m).
4,185
N
=wing
8 8 tip
0.4667
- 0.1659
3 coefficient
4,185 isasN
=
� the
iterations
to
achieve
the
convergence
of
the
lifting
the
same
as
that
one
rethe normal
airfoil
(a) upper,
and
surfaces,
cross-section
(
Structural
0.4667
-required
0.1659
3
4,185
N = (b)
8 lower
is
negative
and
their
verse
is
outer-pointing.
Each
vectors
are considered
positive
when
�at
0.5034
- 0.1950
9
8,463
N = 12
�
����
distortion depends on N. The same material and straight
0.4877
- 0.1823
6
5,115
N=9
- 0.1823
66
5,115
NN
==
9 9 N =0.4877
0.4877 0.5090
- 0.1823 - 0.1994
5,115
10
11,160
14
quired
to achieve
the convergence
of 5,115
the
moment coefficient.
It can be solid.
seen that the increase of N cor0.4877
- 0.1823
6
N=9
wing configuration
as
those
considered
in
the
previous
sessment:
static
wing
response
to
a
variable
pressure
distribution.
Reference
solution:
Nastran
aerodynamic force is transferred from the aerodynamic model to the structural
following
the
0.4953
- 0.1886
8
6,138
N = model
10
NN
==
1010
0.4953
0.4953
- 0.1886
- 0.1886
88
0.4953 to -an
0.1886
8
6,138
N = 10
responds
increasing
number
of�.
iterations
ܰ௜௧௘௥
approach described in section 2.1
for the generic
concentrated
load
N = 12
0.5034
- 0.1950
N = 12
N = 14
௖௢௡௩஼ಽ ஼ಾ
6,138
6,138
required
to achieve the
convergence
of
- 0.1950
9
8,463
0.5034
99
8,463
NN
==
1212
0.5034
- 0.1950
8,463
9
8,463
0.5090
- 0.1994
10
11,160 for different
N =13
14
Table 3. Convergence
the
average -distortion
��10
[mm]
in 11,160
the
iterative aeroelastic
analysis
==
1414 of 0.5090
9 NN
0.5090 - 0.1994
0.1994
10
11,160
0.5090
- 0.1994
10
11,160
0.5034
- 0.1950
structural models. Airfoil cross-section at � � � m. �� = 30 m/s.
Model
Iteration
1
2
3
4
5
6
7
8
9
10
N = 1 0.0402 0.0403 0.0403
N = 4 0.0135 0.0136 0.0136
(a) Relative liftingNcoefficient
Moment coefficient
= 8 0.1729 0.1816 0.1820 0.1821(b) 0.1821
(a) Relative lifting coefficient
(b) Moment coefficient
N = 9 1.1441 1.4721 1.5624 1.5868 1.5934 1.5951 1.5956 1.5958
Table 1. Pressure distribution on the wing along the spanwise direction for the structural assessment.
N = 10 in 1.4177
1.9198
2.1159analysis,
2.1930
2.2234 models
2.2353 with
2.2400
2.2419
2.2426
2.2429
Fig. 5. Convergence of lifting and moment coefficients
the iterative
aeroelastic
for structural
different
accuracy.
Aerody�
1.6738
2.6774 2.7340 2.7600 2.7719 2.7774 2.7799 2.7811
⁄�� , 2.2852
� ������, ��N�=�12
��������
�� ��� � ���������.
���� � 2.5542
namic method: 3D Panel. �� =30m/s.
�
Fig. 5. Convergence of lifting Nand
in the 2.9456
iterative3.0250
aeroelastic
struc- 3.0990 3.1014
= 14moment
1.7925 coefficients
2.4670 2.7867
3.0646analysis,
3.0844 for
3.0941
“-“ : convergence achieved with a tolerance ���� � ���� .
tural models with different accuracy. Aerodynamic method: 3D Panel.
y [m]
������
317
���� � � � ����
1.00
���� � � � ����
0.50
���� � � � ����
32
���� � � � ����
0.75
0.25
.
2
http://ijass.org
11
12
2.7816
3.1027
2.7818
3.1033
3
mm (‫ݐ‬ଵ in Fig. 2b) to 1 mm (‫ݐ‬ଶ in Fig. 2b) in the zone between the 40% and the 45% of the chord.
number of mesh elements is not reported here. In fact, the choice of 10 B4 elements yields
a good
evaluation
of displacements for all the points of the structure, as detailed in [32, 34], where a similar
hpoints
the purpose
of
studying
a
high-deformable
nonclassical
crossof the structure, as detailed in [32,
34], where
structural
casea similar
in terms of wing configuration
and
applied
aerodynamic
loads
was
This particular choice
is coherent
with
thestudied
purposevia
of the
studying a high-deformable nonclassical crossevaluation of displacements for all the points of the structure, as detailed in [32, 34], where a similar
structural case in terms of wing configuration and applied aerodynamic loads was studied via the
iguration and applied aerodynamic loadspresent
was studied
viamodel
the and successfully assessed with a commercial FE solid model.
structural
section.
coefficients.
be clearly explained afterwards
as a consequence
thetendency will be clearly explain
aerodynamic
coefficients. of
This
structural case in terms of wing configuration and aerodynamic
applied aerodynamic
loadsThis
wastendency
studied will
via the
present
structural model and successfully assessed with a commercial FE solid model.
B4
elements.
For
the
sake
of
brevity,
a
convergent
study
on
the
ly assessed with a commercial FE solid model.
The aeroelastic analysis is now carried out following
the
iterative
coupled
procedure
CUF-XFLR5
deThe 1D
structural
mesh consists
of 10enriches
B4introduction
elements.
For
the sakefield.
of brevity,
convergent
study on the
introduction
of higher-order
terms
in the
model formulation
which
the displacement
of higher-order
terms ina the
model formulation
wh
present structural model and successfully assessed with
a commercial
FE solid model.
Thea aeroelastic
analysis is now carried out following the iterative coupled procedure CUF-XFLR5 deorted
here.
In
fact,
the
choice
of
10
B4
elements
yields
good
out following the iterative Int’l
coupled
procedure
CUF-XFLR5
described& in
Fig.Sci.
1 and
N.(2013)
The convergence
the lifting
and reported
moment here.
coefficients
isthe choice of 10 B4 elements yields a good
J. of Aeronautical
Space
14(4),varying
310–323
number��process
of
meshon
elements
is
In fact,
average cross-section
distortion
is now
introduced
in not
order to evaluate
the
aeroelastic
deformaAn average
cross-section
distortion �� is now introduced in ord
The aeroelastic analysis is now carried out following An
the iterative
coupled procedure
CUF-XFLR5
described
in Fig. 1 and varying N. The convergence process on the lifting and moment coefficients is
econvergence
points of theprocess
structure,
as
detailed
in
[32,
34],
where
a
similar
௖௢௡௩
on the lifting and moment
coefficients
is
and all
in Fig.
5b, respectively.
drawn in Fig. 5a by means of a dimensionless
parameter
‫ܥ‬௅ Ȁ‫ܥ‬௅
evaluation
displacements
the points
thecross-section
structure,
as shape
detailed
in [32,
34], where
a similar
tion of on
thethe
cross-section
along
the of
wing
span.
Givenfor
an airfoil
the average
distor-along
tion ofofthe
the wing
span. Given
an a
scribed in Fig. 1 and varying N. The convergence process
lifting andshape
moment
coefficients
is
௖௢௡௩cross-section,
and in
Fig.Given
5b, respectively.
drawn
in Fig. 5a by means of a dimensionless
parameter
‫ܥ‬௅ Ȁ‫ܥ‬௅the
௖௢௡௩
section
shape
along
wing
span.
an
airfoil
crossfiguration
and
applied
aerodynamic
loads
was
studied
via
the
௖௢௡௩
and
in
Fig.
5b,
respectively.
mensionless parameter ‫ܥ‬௅ Ȁ‫ܥ‬
௅
respectively.
isisthe
valueofofthe
thelifting
lifting
the final
final convergent
convergent௖௢௡௩
value
coefficient
which
different
for each expansion
or‫ܥ‬௅
structural
case in
termsis of
wing configuration
and applied
aerodynamic loads was studied via the
tion �� is‫ܥ‬defined
tion �� is
section, the average distortion
is defined
definedas:
as:
Ȁ‫ܥ‬௅ as:and in Fig. 5b, respectively.
drawn in Fig. 5a by means of a dimensionless parameter
coefficient,
which
is different
expansion value
orderof the lifting coefficient which is different for each expansion orthe௅each
final convergent
‫ܥ‬௅௖௢௡௩ isfor
ully
assessed
with
a
commercial
FE
solid
model.
the lifting coefficient which is different for
expansion
or-as the final convergent moment coefficient ‫ ܥ‬௖௢௡௩ , as reported in Table 2.
der each
employed
as well
present structural model
ெ and successfully assessed with a commercial FE solid model.
employed,
as the coefficient
final convergent
coefficient
� � � ��
value as
of well
the lifting
which moment
is different
for each expansion or- � � � ��
‫ܥ‬௅௖௢௡௩ is the final convergent
௖௢௡௩
aerodynamic
coefficients.
This tendency
clearly
explained ��afterwards
as a co
der
employed
as
well
as
the
final
convergent
moment
coefficient
‫ܥ‬ெ
, as reported
in Tablewill
2. be (18)
(18)
��
�
d
out
following
the
iterative
coupled
procedure
CUF-XFLR5
de�
௖௢௡௩
ergent moment coefficient ‫ܥ‬ெ
, as
ininTable
2.2. choice of N influences theThe
as reported
reported
� the
��is wing
Hence,
aTable
different
structural
response
of
to
the
aerodynamic
loads
� �� deaeroelastic
analysis
now
carried
out
following
the
iterative
coupled
procedure
CUF-XFLR5
௖௢௡௩
Fig.
6.
Spanwise
distribution
of
the
average
distortion
of
the
airfoil
cross-sections,
for
different
der employed as well as theHence,
final convergent
moment
coefficient
‫ܥ‬ெ , asthe
reported in Table 2.
a different
choice
of N
influences
introduction
in the modelloads
formulation which enriches the displa
Hence,
aisdifferent
choice ofstructural
N influences the structural
responseofofhigher-order
the wing to terms
the aerodynamic
convergence
process
on the
lifting
moment
coefficients
es
the structural
response
of the
wingand
to the
aerodynamic
loads
where
�
is
the
curvilinear
coordinate
the
surface
and
�� ishigher
the distortion
the and moment
and
consequently
affects
also
the
aerodynamic
analysis,
due
toexternal
aeroelastic
coupling.
The
where
curvilinear
coordinate
along thecoefficients
external airfoil
structural
models.
. The
where
lthe
is
theairfoil
curvilinear
coordinate
along
external
scribed
inalong
Fig.
1 and
varying
N.
convergence
process
on the
theoflifting
is
response of the wing to the aerodynamic loads,
and
Hence, a different choice of
N influences the structural response of the wing to the aerodynamic loadsAn average cross-section distortion �� is now introduced in order to evaluate the aero
௖௢௡௩
and
consequently
affects
also
the
aerodynamic
analysis,
due
to
the
aeroelastic
coupling.
The
higher
Ȁ‫ܥ‬
and
in
Fig.
5b,
respectively.
mensionless
parameter
‫ܥ‬
airfoil
surface,
and
s
is
the
distortion
of
the
single
point
of
௅
௅
consequently
also
affects
the
aerodynamic
analysis,
due
to
௖௢௡௩
௖௢௡௩
odynamic analysis, due to the aeroelasticthe
coupling.
Thesingle
higheremployed
௖௢௡௩
point of the
airfoil
in Eq.
12.a(‫ܥ‬
Itdimensionless
that
� isexternal
positive
) and the
initial
expansion
order
theexternal
more difference
‫ܥ‬௅ of
single
point
of the
in Eq. 12.
and surface
in Fig. defined
5b, respectively.
drawn surface
inappears
Fig. defined
5abetween
by means
parameter
‫ܥ‬a Ȁ‫ܥ‬
ெis noteworthy
௅ airfoil
external airfoil surface defined
in Eq. 12. It is௅ noteworthy
and consequently affectsthe
also
the aerodynamic
analysis,
due to the
aeroelastic
coupling.
The the
higher
aeroelastic
coupling.
The
higher
the
expansion
order
௖௢௡௩
௖௢௡௩
tion
of
the
cross-section
shape
along
the
wing
span.
Given
an
airfoil
cross-section,
th
(‫ܥ‬
)
and
the
initial
the
expansion
order
employed
the
more
difference
appears
between
‫ܥ‬
f thedifference
lifting coefficient
is different
for
each
expansion
or௖௢௡௩
௅
ெ
(‫ܥ‬
)‫ܥ‬and
ore
appears which
between
‫ܥ‬௅௖௢௡௩ the
௜௡ the
௜௡quantity
that
s convergent
is a ��presented
positive
quantity,
and
a null
for
the
ெ more
andfor
a null
value
for௖௢௡௩
the
average
meansvalue
no
Figure
6avalue
plots
the
trend
of average
quantity
and
null value
for
the
�� means
value
(‫ܥ‬ெ
)initial
evaluated
thebetween
undeformed
wing.
the
in distortion.
this
work,
the
number
employed,
appears
isFor
thedistortion
finalcases
of the
lifting
coefficient
which
isaverage
different
fordistortion
each expansion
or- no
‫ܥ‬௅௖௢௡௩
௅ difference
௖௢௡௩
(‫ܥ‬
) and the initialtion �� is defined as:
the expansion order employed
appears between
‫ܥ‬௅
௜௡
௖௢௡௩ the more difference
distortion
distortion.
Figure
6 plots
the trend of
value
) evaluated
forெ
thethe
undeformed
wing. For means
the casesnopresented
in this
work,
the number
vergent moment coefficient ( ‫ܥ‬ெ
,) asand
reported
in Tablevalue
2. ‫ܥ‬௅௜௡ (‫ܥ‬ெ
the initial
evaluated
for
eformed wing. For the cases presented in of
thisiterations
work, therequired
number
௖௢௡௩
the average
distortion
along der
the employed
wing
span
showing
which
are
the
most
in-plane
deformed
the
average
distortion
along
the
spaninshowing
to
achieve
the
convergence
of
the
lifting
coefficient
is
the
same
as
that
one
reas
well
as
the
final
convergent
moment
coefficient
‫ܥ‬airfoil
, aswing
reported
Table 2. which a
ெ which
the average distortion along the wing span, showing
௜௡
undeformed
wing. Forwing.
the cases
in thisinwork,
the the number
value ‫ܥ‬௅௜௡ (‫ܥ‬ெ
) evaluated
for the undeformed
For thepresented
cases presented
this work,
� � � ��
of iterations
of
the
lifting
coefficient
is
the
same
as
that
one
reces the structural response of the wing to the aerodynamic
loads required to achieve the convergence
arespanwise
the
most
in-plane
deformed
cross-sections
in equilibrium response. A r
onvergence of the lifting coefficient
is the
same
that
one
reTable
1.achieve
Pressure
distribution
onofthe
along
the
spanwise
direction
the
assessment.
Table
1. cross-sections
Pressure
distribution
on
the
wing
along
the
for
the
structural
�� of
�of
number of
iterations
required
to
achieve
convergence
in the
the
static
aeroelastic
response.
A
remarkable
variation
in
the
trend
thewing
cross-sections
in
the
static
aeroelastic
quired
toas
the
convergence
thewing
moment
coefficient.
Itchoice
can
bedirection
seen
that
thestructural
increase
of airfoil
Nassessment.
corHence,
aequilibrium
different
of
N for
influences
the
structural
response
the
� �� to the aerodynamic loads
of iterations required to of
achieve
the
convergence
of
the
lifting
coefficient
is
the
same
as
that
one
rethe
static
aeroelastic
equilibrium
response.
A
remarkable
lifting coefficient
is higher
thetosame
asthe
that
required of
to
quired
achieve
convergence
coefficient. It can be seen that the increase of N corodynamic analysis, due to thethe
aeroelastic
coupling. The
� the
� moment
஼ಾ
�accuracy
�
he moment coefficient. It can be seen thatresponds
the
ofaverage
N �cor-��distortion
ಽ�
coefficients.
This
will
be clearlyof
explain
the
structural
model
chosen.
Models
with
antendency
⁄��
average
distortion
appears
depending
oncoupling.
the accuracy
the str
��increase
�����
��appears
��⁄iterations
�
��
�
��.
, ��depending
����,
��
���,
�number
�����
�௖௢௡௩஼
���the
�required
������
��.
, Itand
��
consequently
affects
the
aerodynamic
analysis,
due
tothe
the
aeroelastic
the
convergence
of
to
an
increasing
of
�moment
���
��on
variation
inofalso
the
trend
ofaerodynamic
the
average
distortion
appears,
������
���
�������
achieve
the
convergence
of
the
coefficient.
can
�ܰ௜௧௘௥
�to
isachieve
the
curvilinear
coordinate
along
external
airfoil
surface The
and higher
� is the
quired to achieve the convergence௖௢௡௩
of the ௖௢௡௩
moment coefficient. It can be seen that the increase of N cor-where
௖௢௡௩஼ಽ ஼ಾ
(‫ܥ‬
)
and
the
initial
more difference
appears
between
‫ܥ‬
aerodynamic
coefficients.
This
tendency
will
be
clearly
explained
afterwards
as
a
consequence
of
the
required
to
achieve
the
convergence
of
responds
to
an
increasing
number
of
iterations
ܰ
௖௢௡௩஼ ஼
depending
accuracy
oftothe
model secchosen.
௅that theெincrease of N corresponds to the increasing
௜௧௘௥ aton
be seen
௖௢௡௩
introduction
of
higher-order
in the
formulation
expansion
reveal that the
section
� the
� �the
m appears
be structural
the
most
distorted
to achieve
the convergence
of order higher than 9the
iterations ܰ௜௧௘௥ ಽ ಾ required
expansion
order
higher
thanterms
9 reveal
thatmodel
the section
at initial
� �whi
�
the
order
employed
more
difference
appears
between
‫ܥ‬௅௖௢௡௩
ெ It )isand
y [m]
���
y [m]expansion
���
௖௢௡௩஼ ஼
13
single
of the
external
surface
Eq. (‫ܥ‬
12.
noteworthy
tha
���
���point
Models
with
an expansion
orderairfoil
higher
than 9defined
revealinthat
required totoachieve
the
convergence
of
responds to an increasing
numberofofiterations
iterations ܰ௜௧௘௥ ಽ ಾ introduction
number
required
achieve
the
of
higher-order
terms
in
the
model
formulation
which
enriches
the
displacement
field.
deformed wing. For the cases presented in this work, the number
13
� � ��‫�����ܥ‬
1.00 at y=4
����
An
average
cross-section
��inisthis
nowwork,
introduced
in orde
௜௡� ����
௜௡1.00
tion. coefficients. This ����
tion.
the
section
m appears
to be
the
most
distorted
value
undeformed
For
thedistortion
casesdistortion
presented
the number
convergence of aerodynamic
tendency
13
௅ (‫ܥ‬ெ ) evaluated
quantity for
andthe
a null
value forwing.
the average
�� means
no distortion.
Figure 6
0.75
����
�
�
�
����
0.75
���� distortion
� � � ����
13
An
average
cross-section
��
is
now
introduced
in
order
to
evaluate
the
aeroelastic
deformasection.
onvergence of the lifting coefficient
is the same
as that afterwards,
one rewill be clearly
explained
as a consequence
tion
of
cross-section
shape3along
span.
Given
an
For this cross-section,
Table
3�presents
numerical
values of
in the Table
iterative
For distortion
thisthecross-section,
presents
the
numerical
values
0.500.50
��������
� �� �����
�the
����
of
iterations
required
toaverage
achieve
theaverage
convergence
of
the�� lifting
is the
the wing
same
that
one
re- aio
distortion
along 3the
wing
span
showing
which
are
theasmost
in-plane
For the
this
cross-section,
Table
presents
thecoefficient
numerical
of the introduction of higher-order terms in the model
0.25
����
�
�
�
����
0.25
����shape
���
����the wing span. Given an airfoil cross-section, the average distoralong
he moment coefficient. It can be seen that the increase of N cor-tion of the cross-section
tion
�� isindefined
as:aerodynamic
values
ofconvergence
average
distortion
the
iterative
aeroelastic
aeroelastic
analysis forfield.
different
structural
theories.
As
occurred
for
convergence
of
aeroelastic
analysis
for
different
structural
theories.
AsNoccurre
formulation, which enriches
the displacement
quired
to achieve
the
thethe
moment
coefficient.
It can
beresponse.
seen
thatAthe
increase of
cor- i
cross-sections
inofthe
static
aeroelastic
equilibrium
remarkable
variation
௖௢௡௩஼ಽ ஼ಾ
analysis,
for
different
structural
theories.
As
occurred
for
tion �� isisdefined
as:
An to
average
distortion
now introduced
required
achievecross-section
the convergence
of
iterations ܰ௜௧௘௥
���� �
����
�
஼
௖௢௡௩஼
coefficients, the number of iterations �����
required to achieve thecoefficients,
convergence
of ಽnumber
��ಾincreases
as
� � � �� to achie
the
of iterations
�����
required
��������ܰ
����
����
required
to of
achieve
the�� convergence
of
responds��to
an
increasing
number
ofappears
iterations
convergence
of
aerodynamic
the
number
moment
coefficient
��
forcoefficients,
different
Table
2. aeroelastic
Convergent
values
of lifting
coefficient
and
moment
coefficient
��
for different
Table
2. Convergent
values
of of
lifting
coefficient
�the
average
distortion
depending
on the
accuracy
the structural
� model chosen
in order to evaluate
the
deformation
the
cross௜௧௘௥
� and
� ��
� � � ��
13
N, and consequently DOFs,
increases.��In ��fact, increasing
N, the structural
model
(18)In fact, increasing the e
N, order
and consequently
DOFs,
increases.
�� � the expansion
expansion �order
that the section at � � � m appears to be the m
structural
models.
�� =��
30=m/s,
���� �=���0.4637,
�� �=�- 0.1629.
structural
models.
30 m/s,
= 0.4637,
= - 0.1629.
13
�� higher than 9 reveal
where � is the curvilinear coordinate along the external airfoil
becomes in general more deformable approaching the real structural becomes
behavior.inIt general
means that
commorea deformable
approaching the real stru
tion.the external airfoil surface and � is the distortion of the
where � is the ����
curvilinear
coordinate
along
��������
�� ���� ��
��������
����
single point of the external airfoil surface defined in Eq. 12.
Model
DOFs
�� �� ����������
Model �� ��
DOFs
plete three-dimensional displacement field as well as local effects are evaluated
with an increasing
acplete three-dimensional
displacement
field as well as local effec
For
this
3 presents the
values of average distortion
single
the
external
airfoil
surface
in279
Eq.279
12. Table
It is noteworthy
thatnumerical
� is a positive
0.4643
- 0.1633
2 defined
N = N1 point
0.4643
- 0.1633
2 cross-section,
= 1 of
quantity and a null value for the average distortion �� means no
curacy, especially
structures
see Figs.
4a and 4b.
the with high-deformable cross-se
especially
for Since
structures
- 0.1634
2 2cross-sections,
1,395
N = N4 = 4for0.4641
0.4641 with
-high-deformable
0.1634
1,395curacy,
analysis
structural
occurred
for the convergence
quantity
a null value -for
the averageaeroelastic
distortion
�� 4,185
meansfor
no different
distortion.
Figure 6theories.
plots theAs
trend
of
3 3
N = N8 =and
0.4667 0.1659
- 0.1659
4,185
8 0.4667
the average distortion along the wing span showing which ar
model accuracy
the structural
deformation
therefore
more sensitive
to the
variations
accuracy
increases,
theofstructural deformation is therefo
����
�
0.4877
- 0.1823
6 is
5,115
N = N9 =increases,
0.4877along
-the
0.1823
6
5,115model
9 distortion
coefficients,
the
number
�����
required airfoil
to achieve the convergence o
the average
wing span
showing
which
are oftheiterations
most in-plane
deformed
cross-sections
in
the
static
aeroelastic
equilibrium response. A re
0.4953
- 0.1886
8 8
6,138
N
=
10
0.4953
0.1886
6,138
N
=
10
aerodynamic loads, which are different for each iteration following the
convergentloads,
trend which
in Figs.
aerodynamic
are5adifferent for each iteration follo
N,
and
consequently
DOFs,
increases.
In
fact,
increasing
the
0.5034
0.1950
9
8,463
N
=
12
0.5034
- 0.1950 equilibrium
9 response.
8,463
N = 12 in the
cross-sections
static aeroelastic
A remarkable variation in the trend of the expansion order N, the
average distortion appears depending
on the accuracy of the stru
����
�
���� �
0.5090
0.1994
10
11,160
N
=
14
0.5090
0.1994
10
11,160
N
=
14
highlight
that, given
an expan- �����
and 5b, leading to an increasing ����� . Numerical results in Table 3and
. Numerical results in
5b, leading
to an increasing
becomes
in
general
more
deformable
approaching
the
real
average distortion appears depending on the accuracy of the structural model chosen. Models with anstructural behavior. It m
expansion order higher than 9 reveal that the section at � � � m
sion order, a higher number of iterations is necessary to achieve convergence
on astructural
distortion
sion order,
higher number
of iterations is necessary to achiev
three-dimensional
as well
as localseceffects are evaluated with
expansion order higher than 9 reveal thatplete
the section
at � � � mdisplacement
appears to befield
the most
distorted
tion.
Table
1. Pressure
distribution
on the see
wingFigs.
along4a
thea
Table
3. Convergence
of the
distortion
�� [mm]
inon
the
aeroelastic
analysis
for
different
1. Pressure
distribution
wing
along
the
spanwise
direction
for
structural
assessment.
Table
3. Convergence
of average
the Table
average
distortion
�� [mm]
intheiterative
the
iterative
aeroelastic
analysis
forthe
different
curacy,
especially
for structures
with
high-deformable
cross-sections,
tion.
34
Forcross-section
this cross-section,
3 presents
the numerical values o
Fig. 7. Deformation
at y=4m,Table
computed
for
Fig.
Fig.6. 6.Spanwise
Spanwisedistribution
distributionofofthe
theaverage
average distortion
distortion ‫ݏ‬ҧ of
of the
the airfoil
airfoil cross-sections
for differentof the airfoil
�
�
�
⁄�� , �sensitive
� ������, ���
��������more
�� =30m/s.
⁄model
structural
models
with
� ������,
�
�
� ��values
�
���������.
�� =30m/s.
����
cross-sections,
for
different
structural
models.
m/s.
structural
models.
cross-section
at �atTable
���
�
accuracy
increases,
theaccuracy.
structural
deformation
is �therefore
to��
��� � � ��
30� ,m/s.
structural
models.
Airfoil
cross-section
����m.
m.
� � different
� =�30
�=
For
this
cross-section,
3���������
presents
the
of average
distortion
�� in the iterative
33 Airfoil
�
14 numerical
14
structural models. ܸஶ ൌ ͵Ͳ݉Ȁ‫ݏ‬.
aeroelastic
analysis
for
different
structural
theories.
As
occurred
Table 1. Pressure distribution on the wing along the spanwise direction for the structural assessment.
Table
3. C3.onvergence
in the
iterative
aeroelastic
analysis,
different
structural
models.
Airfoil
at following the convergent
Table
Convergenceofofthe
theaverage
average distortion
distortion
‫ݏ‬ҧ [mm]
the for
iterative
aeroelastic
analysis
forfor
different
aerodynamic
loads,
which
different
for cross-section
each
iteration
aeroelastic
analysis
different
structural
theories.
As
occurred
forare
the
convergence
of aerodynamic
���� �
�
coefficients, the number of iterations �����
required to achie
y=4m. �� =30m/s.
� ������, ��� � ��������⁄�� , ���� � �� ��� � ���������.
����
�
y [m]
structural models. Airfoil cross-section at ‫ ݕ‬ൌ Ͷ m. ܸஶ = 30� m/s.
y
[m]
���
����
� leading to an increasing
resultsasin Table 3 highlight
that,
and 5b,
����� . Numerical
coefficients, the number of iterations �����
required to achieve���the convergence
of �� increases
N,
and
consequently
DOFs,
increases.
In
fact,
increasing
the ex
Iteration
Iteration
���� � �on� str
���
����
� � � ���� number
1.00
Model
Model
iterations
is 12
necessary
achieve convergence
N,
fact,
increasing
order
structural
1 1
2 2
3 and
4 4
5 DOFs,
6 increases.
7 In7sion
8order,
9 the
10 10 of 11
13 13tomodel
3 consequently
5
6
8 a higher
9 expansion
11 N,12the
becomes
in general
more
deformable
approaching the real stru
yIteration
[m]
������
Model
N=1
N=4
N=8
N=9
N = 10
N = 12
���� � � � ���
����
N=N
1 = 10.0402
0.0403
0.0403
- - - - -� � �- ����
- - - - 0.0402
0.0403
0.0403 - - 0.75
becomes
in general more
deformable
approaching the real structural
behavior.
It- means -that a com2N
3 0.0135
4 0.0136
50.0136
6
810
N=
4 = 40.0135
0.0136
-����7- � � -� ����
-9 - - 11- - 12 - - 13plete
- three-dimensional
- 0.0136
-displacement
field as well as
local effec
1.00
���� � � � ���
���� � � � ���� 0.50
=N
8 = 0.0403
0.1816
0.1821
0.1821
-- -- - - - - - - - - - - - - 80.1729
0.1729
0.1820
0.1821
0.1821
0.0402 N
0.0403
- 0.1816
-0.1820
plete three-dimensional
displacement
field as well as local effects are evaluated with an increasing ac����
�1.5934
� �1.5934
����1.5951
0.75
cross-se
N=N
9 = 91.1441
1.4721
1.5624
1.5868
1.5956
1.5958
- - especially
- - for
- structures
-14 -with high-deformable
1.1441
1.4721
1.5624
1.5868
1.5951
1.5956
1.5958
- curacy,
���� �
� � ���
����
� � �- ����
0.25
0.0135 0.0136 0.0136
N=N
10= 10
1.4177
1.9198
2.1159
2.1930
2.2234
2.2353
2.2400
2.2419
2.2426
2.2429
1.4177
1.9198
2.1159
2.1930
2.2234
2.2353
2.2400
2.2419
2.2426
2.2429
curacy, especially
for����
structures
see Figs. 4a and 4b. Since the
���� � � �
0.50 with high-deformable cross-sections,
model accuracy
increases,
the structural deformation is therefo
=N
12= 12
1.6738
2.2852
2.5542
2.7719
2.7774
2.7799
2.7816
2.7818
1.6738
2.2852
2.5542
2.6774
2.7340
2.7600
2.7811
2.7816
2.7818 - 0.1729 N0.1816
0.1820
0.1821
0.1821
- 2.6774
- 2.7340
- 2.7600
- 2.7719
- 2.7774
- 2.7799
-2.7811
����
�3.0250
�increases,
�3.0250
����3.0646
0.25
model
accuracy
the
structural
deformation
is3.0990
therefore
more
sensitive
to
the3.1035
variations
N=N
14= 14
1.7925
2.4670
2.7867
2.9456
3.0844
3.0941
3.0990
3.1014
3.1027
3.1033
1.7925
2.4670
2.7867
2.9456
3.0646
3.0844
3.0941
3.1014
3.1027
3.1033
3.1035 of
1.1441 1.4721 1.5624 1.5868 1.5934 1.5951 1.5956�� 1.5958
- aerodynamic loads, which are different for each iteration follow
“-“ : “-“
convergence
achieved
withwith
a tolerance
���� ����
� ��� ��
. �� .
: convergence
achieved
a tolerance
1
1.4177
1.9198
2.1159
2.1930
loads,2.2419
which2.2426
are different
2.2234aerodynamic
2.2353 2.2400
2.2429
1.6738
2.2852
2.5542
2.6774
2.7340and2.7600
2.7719to 2.7774
2.7799
5b, leading
an increasing
N = 14 DOI:10.5139/IJASS.2013.14.4.310
1.7925 2.4670 2.7867 2.9456 3.0250
3.0646
3.0844
3.0941
for each
iteration
following
convergent trend in Figs.
5a�
����
- and 5b, the
. Numerical results in T
leading to an increasing �����
���� � 2.7816 2.7818
�2.7811
���� . Numerical results in Table 3 highlight that, given an expansion order, a higher number of iterations is necessary to achiev
3.0990
318 3.1014
3.1027
3.1033
3.1035
sion order,
33 a higher number of iterations is necessary to achieve convergence on structural distortion
“-“ : convergence achieved with a tolerance ‫ ݈݈݋ݐ‬ൌ ͳͲିସ .
2
2
14
14
tion distortion
�� is defined
as: depending on the accuracy of the structural model chosen. Models with an
average
appears
��� m appears to be the most distorted secexpansion order higher than 9 reveal that the section at� �� ��
(18)
�� �
aerodynamic coefficients. This tendency
will be clearly explained afterwards as a consequence of the
� ��
tion.
introduction
of higher-order
terms
in theairfoil
modelsurface
formulation
the of
displacement
field.
where � is the curvilinear
coordinate
along the
external
and �which
is theenriches
distortion
the
For this cross-section, Table 3 presentsAlberto
the numerical
average distortion
the iterative
Varello values
Static of
Aeroelastic
Response��ofinWing-Structures
Accounting for In-Plane Cross-Section Deformation
Anexternal
average airfoil
cross-section
in orderthat
to evaluate
the aeroelastic deformasingle point of the
surfacedistortion
defined in�� is
Eq.now
12.introduced
It is noteworthy
� is a positive
aeroelastic analysis for different structural theories. As occurred for the convergence of aerodynamic
tion
of the
shape along
the wing
GivenFigure
an airfoil
cross-section,
quantity and a null
value
forcross-section
the average
distortion
�� means
nospan.
distortion.
6 plots
the
the trend
same.ofthe average distor���� � required to achieve the convergence
coefficients, the numberofofiterations
iterations �����
required to achieve the convergence of �� increases as
For N>8, the displacement field becomes accurate enough
Fig.tion
7.of
Deformation
of
airfoil
cross-section
at
m, computed
for structural
modelsIn
withdeformed
dif��along
isincreases
defined
as:
as N,
andshowing
consequently
DOFs,
increase.
the average distortion
thethewing
span
which
are
the
most
in-plane
airfoil
relevantly take into account a cross-section distortion for
N, and consequently DOFs, increases. In fact, increasing the expansion order N, the structuraltomodel
fact,
increasing
the
expansion
order
N,
the
structural
model
ferent accuracy.
.
cross-sections in the static aeroelastic equilibrium response. A remarkable
trendairfoil
of thecase considered, as can also be seen in Fig. 7. As
� � � �� variation in the the
becomes,
in general,
more
the that a com(18)
�� �approaching
becomes in general more
deformable
approaching
the deformable,
real structural
behavior. It means
previously
explained, given a structural model, the distortion
� ��
real structural
behavior.
This means
a complete
average distortion appears
depending
on the accuracy
of thethat
structural
modelthreechosen. Models with an
is computed
by comparing the deformed cross-section to the
plete three-dimensional dimensional
displacement field
as well as local
are evaluated
with
an increasing
acdisplacement
field,effects
as well
as localairfoil
effects,
where � is the curvilinear
coordinate
along
the external
surface and � is the distortion of the
corresponding
base section. For the sake of simplicity, only
expansion order higher
than
9
reveal
that
the
section
at
�
�
�
m
appears
to
be
the
most
distorted
secare evaluated
with increasing
accuracy,see
especially
for 4b. Since the
curacy, especially for structures
with high-deformable
cross-sections,
Figs. 4a and
the
base
section
N=1 is plotted in Fig. 7.
single
point of
the highly-deformable
external airfoil surface
defined in Eq.
It is noteworthy that � isfor
a positive
structures
with
cross-sections,
see12.
Figs.
tion.
As
expected,
low-order
models provide a correct
model accuracy increases,
the 4b.
structural
deformation
is therefore
more sensitive
to the variations of
4a and
Since the
model accuracy
increases,
the structural
quantity and a null value for the average distortion �� means no distortion.
Figure
6
plots
the
trend
of
evaluation of the bending and torsional structural
For this cross-section,
Table 3 presents
the numerical
values of
distortion
deformation
is therefore
more sensitive
to average
the variations
of �� in the iterative
aerodynamic loads, which are different for each iteration following the convergent trend in Figs.
5a but not an exhaustive description of the in-plane
behavior,
the
average distortion
wing
span showing
are the most in-plane deformed airfoil
aerodynamic
loads,along
whichtheare
different
for eachwhich
iteration,
aeroelastic analysis for different
structural theories. As occurred for the convergence of aerodynamic
deformation. This conclusion is confirmed by Fig. 8, where
���� �
. Numericaltrend
resultsininFigs.
Table5a3 and
highlight
that, given an expanand 5b, leading to an increasing
following�the
5b, leading
���� convergent
cross-sections in the static
aeroelastic
equilibrium
response.
A
remarkable
variation
the trend ofs the
the
airfoilindistortion
computed by variable kinematic
���� � . The numerical results in Table 3
to an ofincreasing
coefficients, the number
iterations �����
required to achieve the convergence of �� increases as
models is depicted
thecomputed
upper surface,
at y=4m. The
sion order, a higher number Fig.
of iterations
is necessary
to achieve
convergence
on structural
8. Distortion
of the
airfoil
upper
ofof thedistortion
cross-section
at along m,
for different
highlight
that,
given
an depending
expansion
order,
a highersurface
number
average
distortion
appears
on
the accuracy
of
the structural
model chosen. Models with an
maximum
N, and consequently DOFs, increases. In fact, increasing the expansion order N, the structural
model distortion value is reached in the part of the
iterations is necessary to achieve convergence on structural
to the
expansion
order higher
than 9 reveal that the section
� � � m appearscross-section
to be the most next
distorted
sec-trailing edge, since the stiffening
structural
. atcoefficients
than models.
convergence
on real
aerodynamic
becomes in generaldistortion
more deformable
approaching the
structural behavior.
It means that a comeffect
due
to
the
spar
at
25% of the chord limits the cross௖௢௡௩஼ ஼
௖௢௡௩௦
thetolerance
toleranceememployed is
aerodynamic coefficientstion.
(ܰ௜௧௘௥
൐ ܰ௜௧௘௥ ಽ ಾ ),, although
although the
section acdistortion. Nonetheless, the chordwise position of
plete three-dimensional displacement field as 14
well as local effects are evaluated with an increasing
the
maximum
distortion
For this cross-section, Table 3 presents the numerical values of average distortion �� in the
iterativepoints on the airfoil upper and
curacy, especially for structures with high-deformable cross-sections, see Figs. 4a and 4b.
Sincesurfaces
the
lower
changes, depending on the accuracy of the
ement field becomes accurate
enough
to relevantly
take structural
into account
a crossaeroelastic
analysis
for different
theories.
As occurred for the structural
convergencemodel,
of aerodynamic
see Table 4. As a consequence, it is worth
model accuracy increases, the structural deformation is therefore more sensitive to the variations
of
pointing out that the increase of N is relevant, not only for
he airfoil case considered,coefficients,
as can be seen
in Fig.
7. As previously
the also
number
of iterations
� ���� �explained,
required to achieve the convergence of �� increases as
aerodynamic loads, which are different for each iteration����
following the convergent trend an
in Figs.
5a detection of distortion values, but also of the
accurate
del the distortion is computed
by
comparing
the
deformed
cross-section
to
the
N, and consequently
increases. In fact, increasing the expansion order
N, the
structural model
accurate
shape-type
deformation.
����DOFs,
�
. Numerical results in Table 3 highlight that, given an expanand 5b, leading to an increasing �����
In
general,
improvements
of the structural theory
ction. For the sake of simplicity,
the basemore
section
for N=1 is
plotted in Fig.
becomesonly
in general
deformable
approaching
the real structural behavior. It means that a comyield
more realistic deformations of the wing, until a
sion order, a higher number of iterations is necessary to achieve convergence on structural
distortion
good convergence
is achieved
for N=14, according to the
plete three-dimensional displacement field as well as local effects are evaluated
with an increasing
acconclusions made for Figs. 4a and 4b in the structural
er models provide a correct
evaluation
of the
bending and
torsional
structural cross-sections, see Figs. 4a and 4b. Since the
curacy,
especially
for structures
with
high-deformable
assessment. In other words, the difference between the
results obtained
through
xhaustive description of the
in-plane
deformation.
is confirmedis therefore more sensitive
14
model
accuracy
increases,This
the conclusion
structural
deformation
to the variations
of the generic (N-1)-th and N-th
expansion orders decreases and becomes minimal for N=14.
Table following
1. Pressure distribution
on thetrend
wing in
along
the 5a
spanwise direction for the structural assessment.
irfoil distortion ‫ ݏ‬computed
by variable
kinematic
is depicted
along
the
aerodynamic
loads,
which models
are different
for each
iteration
the For
convergent
this reason,
it Figs.
is possible
to consider the fourteenth35 surface of the cross-section at
Fig. 8. Distortion of the airfoil upper
�
� sufficiently �
order
model
accurate
to describe the aeroelastic
����
�
⁄
� ������,
���3 �
��������that,
� ,given
�� �� � ���������.
�� =30m/s.
���� �
y=4m,
for the
different
structural
models. results
Ͷ m. The maximum distortion
iscomputed
reached
in
part
the. Numerical
cross-section
in Table
highlight
an expanand 5b,value
leading
to an increasing
� of
�
����
ge since the stiffening effect
to the
spar atnumber
25% of of
theiterations
chord limits
the cross-to achieve convergence on structural distortion
siondue
order,
a higher
is necessary
netheless, the chordwise position of the maximum distortion points on the airfoil
y [m]
ces changes depending on the accuracy of the structural model, see Table 4. As
���� � � � ����
1.00
���� � � � ����
0.50
14
worth pointing out that the increase of N is relevant not only for an accurate de-
alues but also of the accurate shape-type deformation.
ents of the structural theory yield more realistic deformations of the wing until a
������
���� � � � ����
0.75
���� � � � ����
0.25
achieved for N=14, according to the conclusions made for Figs. 4a and 4b in the
In other words, the difference between the results obtained through the generic
ansion orders decreases and becomes minimal for N = 14. For this reason it is
(a) Relative lifting coefficient
(b) Relative moment coefficient
he fourteenth-order model sufficiently accurate to describe the aeroelastic beha-
ere considered.
ty influence
Relative
liftingcoefficients
coefficient
moment
coefficient
Fig. 9. Convergence (a)
of lifting
and moment
in the iterative aeroelastic analysis,(b)
for Relative
different free
stream velocities.
Structural model: N
=14. Aerodynamic method: 3D Panel.
Fig. 9. Convergence of lifting and moment coefficients in the iterative aeroelastic analysis, for differ-
stream
velocities.
model:
establishing the influence of theent
freefree
stream
velocity
on the wingStructural
distortion. The
mployed for this analysis is the same as the that one used in the previous study.
ious conclusion, the structural model considered is N = 14. The free stream ve-
e 25, 30, and 35 m/s. As in the previous analysis, the aerodynamic convergence
N = 14.
319Aerodynamic method: 3D Panel.
http://ijass.org
Int’l J. of Aeronautical & Space Sci. 14(4), 310–323 (2013)
the structural model considered is N=14. The free stream
velocities considered are 25, 30, and 35 m/s. As in the previous
analysis, the aerodynamic convergence process is presented
4.4 Free stream velocity influence
process is presentedthrough
throughthe
thedimensionless
dimensionless parameter
parameter ‫ ܥ‬Ȁ‫ ܥ‬௖௢௡௩ , as
as illustrated
illustrated in Fig. 9a. The
process is presented through the dimensionless parameter ‫ܥ‬௅ Ȁ‫ܥ‬௅௖௢௡௩ , as illustrated௅ in ௅Fig. 9a. The
This analysis aims at establishing the influence of the free
in Fig. 9a. The convergence of the moment coefficient is also
௖௢௡௩
.
convergence of the moment coefficient is also shown in Fig. 9b through
the parameter ‫ܥ‬ெ Ȁ‫ܥ‬ெ
௖௢௡௩
stream velocity on the wingconvergence
distortion. The
wing
configuration
.
of the
moment
coefficient is also
shown
Fig.
throughthe
theparameter
parameter ‫ܥ‬ெ Ȁ‫ܥ‬ெ
shown
in in
Fig.
9b,9b
through
employed for this analysis is the same as that used in the ௖௢௡௩
௖௢௡௩
In this
case, ‫ܥ‬௅
and ‫ܥ‬ெ
represent the final convergent values of the lifting and moment coef௖௢௡௩
In this
‫ܥ‬௅௖௢௡௩ and conclusion,
‫ܥ‬ெ
represent the final
convergent values of the lifting and moment coefprevious study. According
to case,
the previous
behavior of the structure here considered.
ficients for a given ܸ . As occurred for the previous aeroelastic analysis, the trends do not show any
ficients for a given ܸஶ . As occurred for the ஶprevious aeroelastic analysis, the trends do not show any
numerical problems such as oscillations. From Figs. 9a and 9b it is important to note that the number
numerical problems such as oscillations. From Figs. 9a and 9b it is important to note that the number
௖௢௡௩஼ ஼
ಽ ಾ
required to achieve the aerodynamic convergence increases as ܸஶ increases,
௖௢௡௩஼ ஼of iterations ܰ௜௧௘௥
of iterations ܰ௜௧௘௥ ಽ ಾ required
௖௢௡௩ to achieve the aerodynamic convergence increases as ܸஶ increases,
Table 4. Convergent average distortion ‫ݏ‬ҧ
[mm] of the cross-section at ‫ = ݕ‬4 m for different
and the final convergent values are much different from the initial values, as summarized in Table 4.
and the final convergent values are much different from the initial௎ௌ
values, as summarized
in Table 4.
௅ௌ
structural models. Values and chordwise position of the maximum distortions ‫ݏ‬௠௔௫
[mm] and ‫ݏ‬௠௔௫
The reason of this behavior is easily explained by the fact that an increasing free stream velocity
The reason of this behavior is easily explained by the fact that an increasing free stream velocity
[mm] on airfoil upper and lower surfaces. ܸஶ = 30 m/s.
means increasing aerodynamic loads, and consequently higher structural deformations, and lastly a
means increasing aerodynamic loads, and consequently higher structural deformations, and lastly a
more relevant coupling effect on the aeroelastic response of the wing. In fact, an increasing airfoil dismore
coupling effect
response
In fact, anDOFs
increasing airfoil dis௖௢௡௩௦
௎ௌ on the aeroelastic
௅ௌ of the wing.
‫ݔ‬௦೘ೌೣ
‫ݔ‬௦೘ೌೣ
ೆೄ Ȁܿ
ಽೄ Ȁܿ
ܰ௜௧௘௥
‫ݏ‬ҧ ௖௢௡௩relevant
‫ݏ‬௠௔௫
‫ݏ‬௠௔௫
tortion
the most
cross-section0.24
at ‫ ݕ‬ൌ Ͷ m is
obtained with ܸஶ according to numerical
0.0403 for the 3most deformed
0.0718for
0.33deformed
279
N=1
to numerical
tortion
cross-section
at ‫ ݕ‬ൌ 0.0439
Ͷ m is obtained
with ܸஶ according
௖௢௡௩
0.0136
3
0.0103‫ݏ‬ҧ
0.33Fig.
0.0251 of theatairfoil
0.23
1,395
NTable
= 4 4. Convergent
average
distortion
[mm]
of
the
cross-section
‫ݕ‬
=
4
m
for
different
11. D
istortion
upper
and
lower
surfaces
of the crossresults in Table 5 and airfoil deformed profiles in Fig. 10. Also for velocity
values different from
results
in Table
airfoil
deformed0.74
profiles
in
Fig.
10.y=4m,
Also
forlower
velocity
values
different
0.1821
5 5atand
0.5267
0.4797
N =of8 the airfoil
11. Distortion
of
the airfoil
upper 0.75
and
surfaces
of the cross-section
at fromvelocim, computed
section
at
computed
for4,185
different
free stream
Fig. 10. Deformation
cross-section
y=4m,
computed Fig.
௎ௌ
௅ௌ
௖௢௡௩
models.
Values
position
ofConvergent
the maximum
distortions
[mm]achieve
Table
4.
average
distortion
‫ݏ‬ҧ‫ݏ‬௠௔௫
[mm]
ofand
the ‫ݏ‬cross-section
at ‫ݕ‬
4 m for different
1.5958
8and chordwise
4.6073
0.74
4.1253
0.75
5,115
Nstructural
= cross-section
9 stream
Fig. 10. Deformation
of the airfoil
at
m,Structural
computed
for
different
free
ve௠௔௫
Structural
model:
N=14.
for different
free
velocities.
model:
N=14.
͵Ͳ݉Ȁ‫ݏ‬,
astream
higher
number
ofstream
iterations
necessary
convergence
on=structural
distortion
for different
freeties.
velocities.is
Structural
model:to
N = 14.
͵Ͳ݉Ȁ‫ݏ‬,
a higher
number
of iterations0.73
is necessary5.1626
to achieve convergence
on
structural distortion
2.2429
10
6.9936
0.79
6,138
N
=
10
௖௢௡௩ model: N = 14.
௎ௌ
௅ௌ
locities. ‫ݏ‬ҧStructural
Convergent average
average distortion
distortion
‫ݏ‬ҧ௖௢௡௩
[mm] of
of the
the
cross-section
at
‫ݕ‬
=
4
m
for
different
Convergent
[mm]
cross-section
at
‫ݕ‬
=
4
m
for
different
structural
models. Values and chordwise position of the maximum
distortions
‫ݏ‬௠௔௫ [mm] and ‫ݏ‬௠௔௫
on airfoil
upper
and[mm]
lowerofsurfaces.
ܸ
m/s.
௖௢௡௩஼
ஶ = 300.73
௖௢௡௩௦
ಽ ஼ಾ positions
the
cross-section
at
y=4m,
forfor
different
structural
Values
and
chordwise
Table
2.7818
9.5341
5.7456
0.82models.
8,463൐
N[mm]
=average
12
Table4. 4.Convergent
Convergent
average distortion
distortion
‫ݏ‬ҧ ௖௢௡௩12
[mm] of
the
cross-section
at
‫ ݕ‬aerodynamic
=4m
different
ܰ௜௧௘௥
), see Table 5.
than
convergence
on
lifting
coefficient
௖௢௡௩஼
௖௢௡௩௦
ಽ ஼ಾ(ܰ௜௧௘௥
), see Table 5.
than convergence
on aerodynamic
lifting coefficient (ܰ௜௧௘௥ ൐ ܰ௜௧௘௥
௎ௌ
௅ௌ
௎ௌ
௅ௌ
models. Values
Values and
and chordwise
chordwise position
positionofof
ofthe
themaximum
maximum
distortions
[mm]
and ‫ݏݏ‬௠௔௫
models.
the
maximum
distortions
and
3.1035‫ݏݏ‬௠௔௫
[mm], [mm]
on theonairfoil
upper
lower
surfaces.
13 and
10.7482
0.73
6.0178
0.82
11,160
N = 14 distortions
airfoil
upperand
and
lower
surfaces.
ܸஶ=30m/s.
= 3038m/s.
௠௔௫ [mm]
௠௔௫
௎ௌ
௅ௌ
structural models. Values and chordwise position of the maximum distortions ‫ݏ‬௠௔௫
[mm] and ‫ݏ‬௠௔௫
The
limitation
of
distortion
close
to
the
airfoil
leading
edge due to the spar is enhanced for ܸஶ ൌ
The37 ௖௢௡௩
limitation of
distortion close
to the airfoil leading ௅ௌ
edge due to the spar is enhanced for ܸஶ ൌ
airfoil
upper and
and lower
lower surfaces.
surfaces. ܸܸஶ
= 30
30 m/s.
m/s.
irfoil upper
ஶ=
௖௢௡௩௦
௎ௌ
‫ݔ‬௦೘ೌೣ
‫ݔ‬௦೘ೌೣ
ೆೄ Ȁܿ
ಽೄ Ȁܿ
DOFs
‫ݏ‬ҧ
‫ݏ‬௠௔௫
‫ݏ‬௠௔௫
[mm] on airfoil upperModel
and lower surfaces.
ܸஶ = ܰ
30௜௧௘௥
m/s.
௖௢௡௩௦
௎ௌ upper ‫ݔ‬and
௅ௌ
͵ͷ݉Ȁ‫ݏ‬. The
trends of‫ݏ‬ҧ ௖௢௡௩
distortion
on the airfoil
lower surfaces,
which
‫ ݔ‬ಽೄ are
Ȁܿ indicated
ೆೄ Ȁܿ
ܰ௜௧௘௥
Model
DOFsas US
‫ݏ‬௠௔௫
‫ݏ‬௠௔௫
௦೘ೌೣ
͵ͷ݉Ȁ‫ݏ‬.
The
trends
of
distortion
on
the
airfoil
upper
and
lower
surfaces,
which
are
as US௦೘ೌೣ
0.0403
3
0.0718
0.33
0.0439
0.24
279indicated
N=1
௖௢௡௩௦
௖௢௡௩௦
௖௢௡௩
௎ௌ
௎ௌ
௅ௌ
Ȁܿ N =‫ݏݏ‬௠௔௫
Ȁܿ
‫ݔݔ‬௦௦೘ೌೣ
‫ݔݔ‬௦௦೘ೌೣ
ೆೄ Ȁܿ
ಽೄ Ȁܿ
0.0136
ೆೄ
ಽೄ
3DOFs
0.33
0.0251
1,395 0.0439
4௅ௌ
ܰ௜௧௘௥
DOFsand 0.0103
ܰ
‫ݏ‬ҧ‫ݏ‬ҧ௖௢௡௩
‫ݏݏ‬௠௔௫
௠௔௫
௠௔௫
௜௧௘௥
೘ೌೣ ௖௢௡௩
೘ೌೣ
LS respectively,
are
depicted
at0.23
‫ ݕ‬ൌ Ͷ for
velocities. 0.24
It is important
3in Fig. 11
0.0718
0.33different
279to note
0.0403
௖௢௡௩௦
௎ௌ
௅ௌ N = 1
‫ ݔ‬ೆೄare
Ȁܿ depicted
ಽೄ atȀܿ‫ ݕ‬ൌ ͶDOFs
‫ݏ‬ҧ
Model
‫ݏ‬௠௔௫
LS‫ݏ‬௠௔௫
respectively,
in Fig. ‫ݔ‬11
for different velocities.
It is4,185
important to note
௦೘ೌೣ
5௦೘ೌೣ
0.5267
0.74
0.4797
0.75
N =ܰ8௜௧௘௥ and0.1821
Model
0.0403
0.0403
33
0.0136
0.0136
33
0.1821
0.1821
55
1.5958
1.5958
88
2.2429
2.2429
10
10
2.7818
2.7818
12
12
3.1035
3.1035
13
13
0.0718
0.33
0.0439
0.24
279 that 4.6073
0.0718
0.33
0.0439
0.24
3
0.0103
0.33 5,115
0.0251
1,395
N = 4 of
0.0136 and 4.1253
1.5958
8 279
0.74
0.75
N=
9
deformations
upper
lower
surfaces
remarkably
differ also
because 0.23
of a different
aerody3 that deformations
0.0718
0.33
0.0439
0.24 remarkably
279 differ also because of a different aerodyN=1
0.0403
of
upper and
lower surfaces
2.2429
10
6.9936
0.73
5.1626
0.79
6,138
N
=
10
0.0103
0.33
0.0251
0.23
1,395
0.0103
0.33
0.0251
0.23
1,395
5
0.5267
0.74
0.4797
0.75
4,185
N = ௖௢௡௩
8
0.1821
௖௢௡௩
,0.73
moment
coefficient
‫ܥ‬ெ
, not
andonly
average
disTableN 5.
Convergent
values of 12
lifting
coefficient
‫ܥ‬௅ distribution.
pressure
Table
6 shows
that0.82
the8,463
maximum
distortion values on the airfoil
3 namic
0.0103
0.33namic
0.0251
0.23
1,395
N=4
0.0136
2.7818
9.5341
5.7456
= 12
pressure distribution.
Table
6 shows
that not only
the maximum
distortion values
on the airfoil
0.5267
0.74
0.4797
0.75
4,185 10.7482N = 9 0.73
0.5267
0.74
0.75
4,185
8 ௖௢௡௩
0.7411,160
0.75
5,115
1.5958
௖௢௡௩4.6073
௖௢௡௩
௖௢௡௩4.1253
௖௢௡௩ Table
3.1035
13
6.0178
0.82coefficient
N =0.4797
14 distortion
Table
5. 5.
Convergent
Convergent
of‫ݕ‬of
lifting
coefficient
, moment
coefficient
‫ܥ‬ெ
‫ܥ‬ெ
, and
, and
average
average
disdisTable
average
‫ݏ‬ҧof
[mm]
of
the
cross-section
=
4 mcoefficient
for
different
௎ௌvalues
௅ௌ
௅‫ܥ‬௅ , moment
5 [mm]0.5267
0.74
0.75
4,185‫ܥ‬௅ௌ
N =4.8Convergent
0.1821
cross-section
at ‫ݕ‬0.4797
ൌ
Ͷvalues
mat௎ௌ
forlifting
different
free
stream
velocities
ܸ [m/s].
tortion
‫ݏ‬ҧ ௖௢௡௩
௎ௌ
௎ௌ
upper (‫ݏ‬௠௔௫ଵ ,௅ௌ‫ݏ‬௠௔௫ଶ )௅ௌand lower (‫ݏ‬௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ ) surfaces
ஶ changes as ܸஶ varies, but also their cor-
) and lower (‫ݏ‬௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ ) surfaces changes as ܸஶ varies, but also their cor(‫ݏ‬
௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ
4.6073
0.74
4.1253upper 0.75
0.75
5,115
4.6073
0.74
4.1253
5,115
10
6.9936
5.1626ܸ ܸ [m/s].
0.79
6,138
Nof=ofcross-section
10cross-section
௎ௌ 2.2429at at‫ ݕ‬ൌ
௅ௌͶm mforfor
[mm]
Ͷ௠௔௫
different
differentfree
free0.73
stream
streamvelocities
velocities
tortion
‫ݏ‬ҧ ௖௢௡௩
‫ݏ‬ҧ ௖௢௡௩[mm]
structural
models.
Values and 8chordwise4.6073
position௜௡
oftortion
the
maximum
distortions
‫ݏ‬௠௔௫
[mm] and‫ݕ‬5,115
‫ݏ‬ൌ
ஶ ஶ [m/s].
0.74
4.1253
0.75
N
=
9
1.5958
௜௡
௖௢௡௩
௖௢௡௩ and average
௖௢௡௩
௖௢௡௩
Table
CConvergent
onvergent
values
ofoflifting
coefficient
distortion
cross-section
y=4m,
Table5. Convergent
values
lifting
moment
coefficient
andaverage
average
disTable
5.5.
values
lifting
coefficient
, ,moment
moment
coefficient
4.‫ܥܥ‬Convergent
, ,and
average
disdistortion
‫ݏ‬ҧ ௖௢௡௩ [mm]
[mm]
the cross-section
at ‫ ݕ‬for
= 4 m models
for different
responding
chordwise
This
aspect highlights
theofofimportance
ofathigher-order
espeStructural
model:
N coefficient
=coefficient
14. ‫ܥ‬௅ ‫ܥܥ‬௅=௅ 0.4637,
‫ܥ‬ெ
=Table
- 0.1629.
ெெ positions.
This
highlights
the importance
models
6.9936
0.73
5.1626responding
0.79 chordwise
6,138positions. N
6.9936
0.73
5.1626
0.79
6,138
9.5341 of higher-order
0.73
5.7456 espe- 0.82
8,463
= 12aspect
2.7818
௜௡ ௜௡12
Structural
Structural
model:
model:
NN
= N=14.
=
14.14.‫ܥ‬௅௜௡
‫ܥ‬௅௜௡=0.79
0.4637,
= 0.4637,‫ܥ‬ெ
‫ܥ‬6,138
= -=0.1629.
- 0.1629.
[m/s].
Structural
model:
=0.4637,
=-0.1629.
different
free
stream
velocities
[mm]
upper
and
lower
surfaces.
ܸ
=
30
m/s.
ெ
ஶ
10
6.9936
0.73
5.1626
N =on10airfoil
2.2429
௖௢௡௩
௎ௌ
௅ௌ
[mm] ofof cross-section
cross-section atat ‫ݕݕ‬ൌൌͶͶ mm for
forcially
different
free
streammodels.
velocities
[m/s].
tortion ‫ݏ‬ҧ‫ݏ‬ҧ௖௢௡௩
[mm]
different
stream
velocities
ܸܸ
[m/s].
tortion
structural
Values
and
chordwise
position of the
maximum
‫ݏ‬௠௔௫
[mm]
and ‫ݏ‬௠௔௫
ஶஶ
for free
an
accurate
evaluation
of13
in-plane
cross-section
distortion
of distortions
high-deformable
structures.
9.5341
0.73
5.7456cially for
0.82an accurate
8,463
9.5341
0.73
5.7456
0.82
8,463
10.7482
0.73
6.0178
0.82
11,160
N in-plane
= 14
3.1035
evaluation of
cross-section
distortion
of high-deformable
structures.
12௜௡
9.5341௜௡௜௡
0.73
5.7456
0.82
8,463
N = 12
2.7818
஼ಽ ஼upper
௖௢௡௩௦ܸஶ = 30 m/s.
௖௢௡௩
ಾ
on௖௢௡௩
airfoil
and
lower surfaces.
Structuralmodel:
model:NN==14.
14. ‫ܥܥ‬௅௜௡
0.4637,
0.1629. ‫ ܥ‬௖௢௡௩ [mm]ܰ
Structural
‫== ܥܥ‬
--0.1629.
௅ ==0.4637,
ܰ
‫ݏ‬ҧ ௖௢௡௩
ܸஶ௎ௌ
5. Conclusions
௖௢௡௩
௖௢௡௩
஼ ஼ ಽ ஼ಾ
ெ ௅ௌ
௜௧௘௥
௖௢௡௩௦
௖௢௡௩௦
10.7482
0.73
6.0178
0.82 ெெ ‫ܥ‬௅ ‫ ݔ‬11,160
11,160
௖௢௡௩௦
10.7482
0.73
0.82
௖௢௡௩
௖௢௡௩
௖௢௡௩
௖௢௡௩
௜௧௘௥
‫ݔ‬
ೆೄ Ȁܿ
ಽೄ Ȁܿ
ܰ௜௧௘௥
‫ܥ‬
‫ܥ‬
‫ܥ‬
‫ܥ‬
‫ݏ‬ҧ ௖௢௡௩
‫ݏ‬ҧ ௖௢௡௩ ܰ௜௧௘௥
ܸ
ܸ
ܰ6.0178
‫ݏ‬ҧ ௖௢௡௩
Model
DOFs
ܰܰ ಽ ಾ
‫ݏ‬
‫ݏ‬௠௔௫
5.
Conclusions
ஶ
ஶ
௦
௦
௅
௅
ெ
ெ
௠௔௫
௜௧௘௥
೘ೌೣ
೘ೌೣ
13
10.7482
0.73
6.0178
0.82
11,160 ௜௧௘௥௜௧௘௥
N = 14
3.1035
25
N=1
0.4879
-0.1827
7
1.7269
9
௎ௌ
௅ௌ on the௅ௌbasis of Carrera Unified Formulation
Variable kinematic 1D finite elements ௎ௌ
were formulated
and kinematic
chordwise
positions
of25the maximum
distortions
‫ݏ‬௠௔௫ଵ
, 1.7269
‫ݏ‬௠௔௫ଵ
, ‫ݏ‬௠௔௫ଶ
௖௢௡௩
௖௢௡௩
25
0.4879
-0.1827
-0.1827
7 ,13
7‫ݏܥ‬ெ
9 dis9 Formulation
௠௔௫ଶ
3 Values
0.0718
0.33
0.0439
0.24
279
0.0403Table
30
0.5090
10
3.1035
Variable
1D
finite
on
the
basis
of Carrera
Unified
௖௢௡௩௦
Table 6.
5.
Convergent
values
of
lifting
coefficient
‫ܥ‬were
,formulated
moment
coefficient
,1.7269
and
average
௖௢௡௩
஼஼ಾ
௖௢௡௩
஼஼ಽ-0.1994
௖௢௡௩
௎ௌ
௅ௌ
௖௢௡௩௦
௖௢௡௩
௖௢௡௩
௖௢௡௩
௖௢௡௩
௖௢௡௩ 0.4879
௖௢௡௩
ಽ
ಾ elements
௅ ௖௢௡௩௦
ܸܸ
ஶஶ
‫ܥܥ‬௅௅
35
‫ܥܥ‬ெெ
ܰܰ
‫ݏ‬ҧ‫ݏ‬ҧ Modelܰܰ௜௧௘௥
௜௧௘௥ ‫ݏ‬ҧ
௜௧௘௥
0.5608 ௜௧௘௥
-0.2394
௖௢௡௩
18
ܰ௜௧௘௥
6.3296
‫ݏ‬௠௔௫
22
‫ݔ‬௦೘ೌೣ
ೆೄ Ȁܿ
‫ݏ‬௠௔௫
‫ݔ‬௦೘ೌೣ
ಽೄ Ȁܿ
DOFs
Table
4. upper
Convergent
average
distortion
‫ݏ‬ҧ 0.5090
[mm]
the
cross-section
at
=method
43.1035
m
forfree
different
(CUF)
and
coupled
to anofaerodynamic
implemented
in XFLR5. The aeroelastic stat3030of
0.5090
-0.1994
-0.1994
10
3.1035
13aeroelastic
13
0.0103
0.33
0.0251
0.23
1,395
[mm]3‫ݏ‬ҧon
airfoil
and
lower
surfaces
cross-section
at
‫ݕ‬free
= 43Dm10panel
for‫ ݕ‬different
stream
௖௢௡௩
(CUF)
and
coupled
to an
aerodynamic
method
implemented
in XFLR5.
of
cross-section
at 1.7269
‫ݕ‬
ൌNthe
Ͷ=3D
for
different
velocities
ܸThe
tortion
ஶ [m/s].
3 stream
0.0718
0.33
0.0439 stat- 0.24
279
1m panel
25
0.4879 [mm]
-0.1827
1.7269
25
0.4879
-0.1827
77
99 0.0403
௎ௌ
௅ௌ
models. Values
chordwise
position
of thewing
maximum
distortions
[mm]
and ‫ݏ‬௠௔௫
3535 of
0.5608
0.5608
-0.2394
-0.2394
1818 ‫ݏ‬௠௔௫ 6.3296
6.3296
22
22
5 structural
0.5267
0.74 and
0.4797
0.75
4,185
N=8
0.1821
ic
response
a
straight
with
a
high-deformable
airfoil
cross-section
was
computed
through
a
௎ௌ
௎ௌ
௅ௌ
௅ௌ
Structural
model:
Nwith
=௜௡14.
velocities
ܸpositions
Table
6. Values
and
chordwise
positions
ofof14.
the
distortions
[mm]
on the airfoil
upper
and0.0251
lower
surfaces
of
Table
6. Values
and
chordwise
theamaximum
distortions
‫ݏ‬13
, ‫ݏ‬௠௔௫ଵ ,airfoil
‫ݏ‬௠௔௫ଶ
௜௡
ஶ [m/s].
ic
response
of
straight
high-deformable
cross-section
was0.33
computed
through
a 0.23
௠௔௫ଵ
௠௔௫ଶ
3
0.0103
1,395
N‫ݏ‬a=
4 , 13
0.0136
30
0.5090
-0.1994
10wing
3.1035
30
0.5090
3.1035
Structural
model:
N-0.1994
=
‫ܥ‬maximum
0.4637,
‫ܥ‬
௅ = 10
ெ = - 0.1629.
at
for
different
free
stream
Structural
model: N=14.
[mm] 4.6073
on airfoil
upper
and
lowervelocities
surfaces.
ܸஶ =[m/s].
30
m/s.
8 y=4m,
0.74
4.1253
0.75
N = 9 the cross-section
1.5958
coupled
iterative
for5,115
an increasing
structural accuracy
and0.4797
for different0.75
free stream
veloci[mm] on airfoil upper
lower surfaces
of the cross-section
= 4Nm=for
different
free
stream
5 ve- and
0.5267
0.74free stream
4,185
8procedure
0.1821
35 and0.5608
0.5608
-0.2394
18 at
6.3296
22structural
35
18
22
coupled-0.2394
iterative
procedure
for‫ݕ‬6.3296
an
increasing
accuracy
for different
veloci10
6.9936
0.73
5.1626
0.79
6,138
N = 10
2.2429
௎ௌ N = 14.‫ ݔ‬ೆೄ Ȁܿ
௎ௌ
௅ௌ
௅ௌ
model:
locities ܸஶ [m/s]. Structural
16
‫ݔ‬N ೆೄ
‫ݔ‬௦ಽೄ Ȁܿ 4.1253
‫ݔ‬௖௢௡௩
Ȁܿ4.6073
ಽೄ
8௦೘ೌೣభ
0.74
0.75
5,115
= 9 Ȁܿ௖௢௡௩
1.5958
ܸஶ
஼ಾ
‫ݏ‬௠௔௫ଵ
‫ݏ‬஼௠௔௫ଵ
‫ݏ‬௠௔௫ଶ
‫ݏ‬௠௔௫ଶ
௖௢௡௩௦
௖௢௡௩
ಽ16
௦೘ೌೣభ
௖௢௡௩௦
೘ೌೣమ
௎ௌ ௦೘ೌೣమ ܰ ‫ ݔ‬ೆೄ Ȁܿ
ܰ‫ݔ‬௜௧௘௥
‫ܥ‬௅௖௢௡௩ܰ௜௧௘௥
‫ܥ‬ெ
‫ݏ‬ҧ‫ ݏ‬௅ௌ
ಽೄ Ȁܿ
Model ܸஶ ‫ݏ‬ҧ ௖௢௡௩
DOFs
‫ݏ‬௠௔௫
௜௧௘௥
௦೘ೌೣ
௦೘ೌೣ
12
9.5341
0.73
5.7456
0.82
8,463 ௠௔௫
N = 12
2.7818
25
6.089225
0.72
0.29
10 0.83 6.9936
0.79
6,138
N=
10
2.2429
0.4879 0.8670
-0.1827
73.0324 1.7269
91.6285 0.730.46 5.1626
3
0.0718
0.33
0.0439
0.24
279
N
=
1
0.0403
30
10.7482
0.73
1.5540
0.30
6.0178
0.82
2.6232
0.45
10.7482
0.73
0.82 ‫ݏ‬10
11,160
N = 14
3.1035
௎ௌ
௅ௌ6.0178
௎ௌ
௅ௌ
Ȁܿ
‫ݔ‬
Ȁܿ
Ȁܿ
‫ݔ‬
Ȁܿ
‫ݔ‬௦13
‫ݔ‬
ೆೄ
ಽೄ
ೆೄ
ಽೄ
30
0.5090
-0.1994
3.1035
13
ܸஶ
‫ݏ‬௠௔௫ଵ
‫ݏ‬
‫ݏ‬
௦೘ೌೣమ
௦೘ೌೣమ
௠௔௫ଵ
௠௔௫ଶ
௠௔௫ଶ
12 0.81 9.5341
0.82
8,463
N௦೘ೌೣభ
=
12
2.7818
35 ೘ೌೣభ 21.2323
0.74
0.31
13.9437 6.3296
4.5026 0.730.43 5.7456
0.5608 33.1618
-0.2394
18
22
0.0103
0.33
0.0251
0.23
1,395
N = 4 35 0.0136
25
6.0892
0.72
0.8670
0.29
3.0324
1.6285
0.46
13
10.7482
0.73
6.0178
0.82
11,160
N 0.83
= 14
3.1035
5
0.5267
0.74
0.4797
0.75
4,185
N=8
0.1821
30
10.7482
0.73
1.5540
0.30
6.0178
0.82
2.6232
0.45
8
4.6073
0.74
4.1253
0.75
5,115
N=9
1.5958
DOI:10.5139/IJASS.2013.14.4.310
320 4.5026
35
21.2323
0.74
3.1618
0.31
13.9437
0.81
0.43
10
6.9936
0.73
5.1626
0.79
6,138
N = 10
2.2429
N=4
0.0136
N = 12
2.7818
12
3
9.5341
0.73
5.7456
0.82
8,463
N = 14
3.1035
13
10.7482
0.73
6.0178
0.82
11,160
௖௢௡௩
௖௢௡௩
௖௢௡௩
process
processisAlberto
ispresented
presented
through
through
the
dimensionless
dimensionless
parameter
parameter
‫ܥ‬௅‫ܥ‬Ȁ‫ܥ‬
, asillustrated
illustratedinfor
inFig.
Fig.9a.
The
The, as illustrated
௅ Ȁ‫ܥ‬
௅ dimensionless
௅ , as
process
is Response
presented
through
the
‫ܥ‬௅9a.
Ȁ‫ܥ‬
in Fig. 9a. The
Varello
Staticthe
Aeroelastic
of
Deformation
௅ Cross-Section
process is presented through
the dimensionless
parameter
‫ܥ‬௅ Ȁ‫ܥ‬௅௖௢௡௩
,Wing-Structures
as illustrated
in Accounting
Fig. 9a.parameter
The In-Plane
௖௢௡௩
௖௢௡௩
௖௢௡௩
. . , as illustrated
convergence
convergence
of of
thethe
moment
moment
coefficient
coefficient
is
also
also
shown
shown
in in
Fig.
Fig.
9b
9b
through
through
the
parameter
parameter
‫ܥ‬ெ‫ܥ‬Ȁ‫ܥ‬
ெ
ெȀ‫ܥ‬
ெ௖௢௡௩
.
convergence
moment
coefficient
is‫ܥ‬the
also
shown
9b
the parameter
‫ܥ‬ெ Ȁ‫ܥ‬
௖௢௡௩
process
isisinpresented
through
the
dimensionless
parameter
‫ܥ‬௅Ȁ‫ܥ‬
in Fig.
9a.ெ
The
௅ through
. in Fig.
the moment‫ ܥ‬coefficient
isillustrated
also
shown
Fig.of
9bthe
through
the
parameter
௖௢௡௩
ெ Ȁ‫ܥ‬ெ
esented through the convergence
dimensionlessof parameter
Ȁ‫ܥ‬
,
as
in
Fig.
9a.
The
௅
௅
௖௢௡௩
(VLM).
As
far
as
the
use
of
1D
higher-order
models
is
process is presented through the dimensionless ௖௢௡௩
parameter
‫ܥ‬௅௖௢௡௩
Ȁ‫ܥ‬௅ , as illustrated
in Fig. 9a.
The
௖௢௡௩
௖௢௡௩
௖௢௡௩
௖௢௡௩
௖௢௡௩
In
this
this
case,
case,‫ܥ‬௖௢௡௩
and
and‫ܥ‬ெ
‫ܥ‬ெconvergence
represent
the
final
final
convergent
convergent
values
values
of
thethe
lifting
lifting
and
and
moment
moment
coefcoefIn In
this
case,
and
represent
final
௅‫ܥ‬௅
Inrepresent
this
case,
‫ܥ‬the
andofcoefficient
‫ܥ‬the
represent
the
finalin
convergent
values
ofparameter
the lifting ‫ܥ‬
andȀ‫ܥ‬
moment
. coefofthe
the
isof
also
shown
Fig.
9b
through
the
௅ moment
ெ lifting
ெ
In thisis case,
‫ܥ‬௅௖௢௡௩ inand
‫ܥ‬9b
represent
the final
convergent
values
and
moment
coef௖௢௡௩
concerned,
the following
main
conclusions
can beெdrawn:
ெ through
Ȁ‫ܥ‬
.
of the moment coefficient
also
shown
Fig.
the
parameter
‫ܥ‬
ெ
ெ
௖௢௡௩
convergent
values
of
the
lifting
and
moment
coefficients
convergence of the moment coefficient is also shown in Fig. 9b through the parameter ‫ܥ‬ெ Ȁ‫ܥ‬ெ .
1.
The
introduction
of
higher-order
terms
௖௢௡௩
௖௢௡௩
. As
occurred
occurred
forfor
the
the
previous
previous
aeroelastic
analysis,
analysis,
the
the
trends
trends
doaeroelastic
do
notnot
show
show
any
anythe lifting
ficients
ficients
forfor
a given
a givenܸஶܸ.ஶAs
.ெ
As occurred
for
previous
analysis,
the trends
do in
not the
show
ficients
for
ܸ
In
this
case,
‫ܥ‬the
and
‫ܥ‬aeroelastic
represent
thethe
final
convergent
values
of
and moment
coef-any
ஶ
௅a given
forforathrough
given
respectively.
Asthe
occurred
for
previous
௖௢௡௩
As
occurred
previous
aeroelastic
analysis,
the
do
not
show
any
given
ܸஶ ., dimensionless
௖௢௡௩ is ficients
presented
the
parameter
‫ܥ‬௅ Ȁ‫ܥ‬
, as
illustrated
in trends
Fig. 9a.
The
‫ܥ‬௅௖௢௡௩ andprocess
‫ܥ‬ெ
the afinal
convergent
values offor
the
lifting
and
moment
coef௅
௖௢௡௩represent
௖௢௡௩
displacement
field
is
even
more
important
for
In this case, ‫ܥ‬௅
and ‫ܥ‬ெ
represent
theproblems
final
convergent
values
of theFrom
lifting
and9a
moment
coefaeroelastic
analysis,
the
trends
not show
any
numerical
numerical
numerical
problems
such
such
asdo
as
oscillations.
oscillations.
From
Figs.
Figs.
9a
and
and
9b
it is
itfor
is
important
important
to to
note
note
that
thethe
number
number
numerical
problems
as9boscillations.
From
9athat
and
9b
it is
important
note
thatshow
the number
. such
As
occurred
the
previous
aeroelastic
analysis,
the
do not
any
ficients
for9aa and
given
ܸ
the
analysis,
due
to trends
thetofluid-structure
௖௢௡௩
numerical
problems
such as oscillations.
From
Figs.
9b
itஶparameter
is
important
that
the Figs.
number
. aeroelastic
convergence
of
the
moment
coefficient
is
also
shown
in
Fig.
9b
through
the
‫ܥ‬ெtoȀ‫ܥ‬note
the previoussuch
aeroelastic
analysis, the
trends
do 9a
not and
show9b,
anyit is
given ܸஶ . As occurred for problems,
as oscillations.
From
Figs.
ெ
aeroelastic
analysis,
the
trends
do
not
show
any
ficients for a given ܸஶ . As occurred for the previous
஼
஼
௖௢௡௩஼
௖௢௡௩஼
coupling.
ಽ ಾ
ಽ ಾ
௖௢௡௩஼
ಽ ஼ಾ
required
required
to
achieve
achieve
the
aerodynamic
aerodynamic
convergence
convergence
increases
increases
asܸஶ9b
ܸஶincreases,
of௖௢௡௩஼
of
iterations
iterations
ܰ௜௧௘௥
஼ಾ ܰ
important
to
note
that
thetonumber
ofoftoiterations
numerical
problems
such
as oscillations.
Fromܸ
Figs.
9a as
and
itincreases,
is
important toincreases
note thatasthe
required
to achieve
the
aerodynamic
convergence
ܸஶnumber
increases,
iterations
ܰthe
௜௧௘௥
ಽ
௖௢௡௩
௖௢௡௩
௜௧௘௥
achieve
aerodynamic
convergence
increases,
of ‫ܥ‬
iterations
In as
thisoscillations.
case,
andܰ
‫ܥ‬
theimportant
final
convergent
values
the
lifting andincreases
moment
coef௜௧௘௥
blems such
Figs.
and required
9b it is
tothe
note
that
the of
number
2.Inas
theஶcase
that the wing is rather flexible, the in-plane
௅ From
ெ 9a represent
numerical problems such asrequired
oscillations.
From Figs.
9a and 9b it isconvergence
important to note
that the number
to achieve
the aerodynamic
increases
஼ಾfrom
௖௢௡௩஼ಽfrom
and
and
thethe
final
final
convergent
convergent
values
values
are
much
much
different
different
the
the
initial
initial
values,
values,
as
as
summarized
summarized
inconvergence
in
Table
Table
4.
4. aincreases
cross-section
deformation
has
great
impact
and
the
final
convergent
values
are
much
different
from
the
initial
values,
as summarized
inthe
Table 4.
required
to
achieve
the
aerodynamic
as ܸஶ on
increases,
of are
iterations
ܰthe
௖௢௡௩஼ ஼
௜௧௘௥
and
theasfinal
convergent
values
are
much
different
from
initial
values,
as not
summarized
increases,
and
the
final
convergent
values
are
much
.
As
occurred
for
the
previous
aeroelastic
analysis,
the
trends do
show anyin Table 4.
for
a
given
ܸ
ஶ
required
to
achieve
the
aerodynamic
convergence
increases
as
ܸ
increases,
ܰ௜௧௘௥ ಽ ಾficients
஼
௖௢௡௩஼ಽ ಾ
ஶ
alteration
of
the
aerodynamic
loadings.
required
to from
achieve
aerodynamic
convergence
increases
as ܸ4.
increases,
of iterations ܰ௜௧௘௥
ஶfact
different
thethe
values,
as
summarized
in this
Table
The
Thereason
reason
ofinitial
ofthis
thisbehavior
behavior
is
iseasily
easily
explained
explained
bybythevalues
the
factthat
ananincreasing
increasing
free
freestream
stream
velocity
velocity
The
reason
of
isthat
easily
explained
by
that
an
increasing
free stream
velocity
and
the
convergent
are
different
from
the
initial
values,
as summarized
in Table
4.
The reason
of as
thisoscillations.
behavior isFrom
easily
explained
byfinal
the
fact
that behavior
an
increasing
free
stream
velocity
3.
Th
e higher
thethe
freefact
stream
velocity,
the more
marked
numerical
problems
such
Figs.
9a
and
9b
it
is
important
to
note
thatmuch
the
number
convergent values are muchThe
different
from
the
initial
values,
as summarized
in by
Table
4.fact
reason
for
this
behavior
is
easily
explained
the
and the final convergent values are
much
different
from
the
initial
values,
as
summarized
in
Table
4.
the
distortion
effect.
means
meansincreasing
increasingaerodynamic
aerodynamic
loads,
loads,
and
and
consequently
higher
higher
structural
structural
deformations,
deformations,
and
andlastly
lastly
a a
means
increasing
aerodynamic
loads,
andin-plane
consequently
deformations,
lastly a
The
reason
ofconsequently
this
behavior
is
easily
explained
byin
the
fact
that
anstructural
increasing
free stream and
velocity
௖௢௡௩
௖௢௡௩஼
that
free
stream
means
increasing
ಽ ஼an
ಾ increasing
means
increasing
loads,
andvelocity
consequently
higher
structural
deformations,
and
lastly
a 9a.higher
process
is aerodynamic
presented
through
the
dimensionless
‫ܥ‬௅ Ȁ‫ܥ‬
,ஶAs
asincreases,
illustrated
Fig.
The
required
to achieve
aerodynamic
increases
of iterations
ܰ௜௧௘௥
௅as ܸ
f this behavior
is easily
explained
by
the
fact
that
anthe
increasing
free convergence
streamparameter
velocity
far
as
the
present
hierarchical
one-dimensional
The reason of this behavior
is easily
explained
by
the
fact
that
an
increasing
free
stream
velocity
aerodynamic
loads,
and
consequently
higher
structural
more
more
relevant
relevant
coupling
coupling
effect
effect
onon
the
the
aeroelastic
aeroelastic
response
response
of of
theloads,
the
wing.
wing.
In
In
fact,
fact,
anan
increasing
increasing
airfoil
airfoil
disdis-Indeformations,
more
relevant
coupling
effect
on
aeroelastic
response
of
the
wing.
fact, an increasing
airfoiladismeans
increasing
aerodynamic
and
consequently
higher
structural
and lastly
௖௢௡௩
approach
is concerned,
the
more relevant
coupling
effect
ondifferent
the coefficient
aeroelastic
response
of the
In
fact,
anthe
increasing
dis. results point out that:
convergence
the
moment
is
also
shown
in wing.
Fig.
9b
through
the
parameter
‫ܥ‬ெ Ȁ‫ܥ‬
ெ
and the final
values of
are
much
the
initial
values,
asasummarized
in
Table
4. airfoil
deformations,
and
lastly,
a morefrom
relevant
coupling
effect
on
sing aerodynamic
loads,convergent
and
consequently
higher
structural
deformations,
and
lastly
means increasing aerodynamic loads,
and
consequently
higher
structural
deformations,
and
lastly
a
a.
Th
eܸஶ
CUF
is an ideal
tool
to easily compare different
according
according
numerical
numerical
tortion
tortion
forfor
thethe
most
most
deformed
deformed
cross-section
cross-section
at
‫ݕ‬most
ൌ
‫ݕ‬ൌ
Ͷ effect
Ͷm m
is is
obtained
with
with
ܸஶ
accordingairfoil
to numerical
theat
deformed
cross-section
‫ݕ‬
ൌ Ͷ of
mtothe
isto
obtained
ܸஶincreasing
the aeroelastic
response
of௖௢௡௩
the wing.
fact,
increasing
more
coupling
onܸobtained
the
aeroelastic
response
wing.
In with
fact, an
dis௖௢௡௩
according
toat
numerical
tortion
the most
cross-section
at tortion
‫ݕ‬Inrelevant
ൌthe
Ͷ for
man
is
obtained
with
ஶ of
Inbehavior
this
case,isdeformed
‫ܥ‬easily
and
‫ܥ‬ெ
final
convergent
values
thevelocity
lifting
and
moment
coef-since the model accuracy is a
thisfor
byrepresent
the
fact
that
anairfoil
increasing
stream
higher-order
theories,
coupling The
effectreason
on theofaeroelastic
response
wing.
In fact,
an increasing
dis- free
௅of theexplained
more relevant coupling effect
on the
aeroelastic
response
of
the
wing.
In
fact,
an
increasing
airfoil
disairfoil
distortion
for
the
most
deformed
cross-section
at
y=4
results
resultsin inTable
Table5 5and
andairfoil
airfoil
deformed
deformed
profiles
profiles
in
Fig.
Fig.10.10.
Also
Also
forfor
velocity
velocity
different
different
from
from
results
in
Table
5 in
and
deformed
profiles
in
10.analysis.
Also
for velocity
valuestodifferent
from
free
parameter
of
the
numerical
tortion
forFig.
the
mostAlso
deformed
cross-section
at
‫ ݕ‬ൌvalues
Ͷvalues
mFig.
is
obtained
with
ܸஶ according
resultsmin
Table
5 and
airfoil
profiles
in
10.
forairfoil
velocity
values
different
from
isat
obtained
with
according
the
numerical
results
., As
occurred
for
the
previous
aeroelastic
analysis,
the
trends
show any
ficients
given
ܸஶdeformed
means cross-section
increasing
aerodynamic
and
consequently
higher
structural
deformations,
and
lastly
a do not
according
to numerical
e most deformed
‫ ݕ‬ൌfor
Ͷ amloads,
is obtained
with
ܸஶ to
according
to
numerical
tortion for the most deformed cross-section
at
‫ݕ‬
ൌ
Ͷ
m
is
obtained
with
ܸ
b.
The
in-plane
airfoil
cross-section
deformation
is
well͵Ͳ݉Ȁ‫ݏ‬,
͵Ͳ݉Ȁ‫ݏ‬,
higher
a higher
number
numberofprofiles
ofiterations
iterations
isஶis
necessary
tofor
toachieve
achieve
convergence
convergence
on
structural
distortion
distortion
in Table
5 and aairfoil
deformed
in Table
Fig.
10.
Also
͵Ͳ݉Ȁ‫ݏ‬,
anecessary
higher
number
of
iterations
is necessary
to
convergence
on structural
in
5 and
airfoil
deformed
profiles
inonstructural
Fig.
10.achieve
Also
for
velocity values
differentdistortion
from
͵Ͳ݉Ȁ‫ݏ‬,
a higher
number
ofsuch
iterations
is results
necessary
toFigs.
achieve
convergence
onairfoil
structural
described
by the
thenumber
proposed 1D structural model, in
numerical
problems
as oscillations.
From
and
it is important
todisnotedistortion
that
coupling
effect
on the
aeroelastic
response
of the
wing.
In9a
fact,
an9bincreasing
ble 5 and more
airfoilrelevant
deformed
profiles
in
Fig.
10.
Also
for
velocity
values
different
from
௖௢௡௩
velocity
values
different
from
30m/s,
a
higher
number
of
results in Table 5 and airfoil
deformed
profiles
in
Fig.
10.
Also
for
velocity
values
different
from
process is presented through the dimensionless
Ȁ‫ܥ‬௅ , as ௖௢௡௩௦
illustrated
in Fig.஼ 9a. The FE solution,
஼௅
௖௢௡௩஼
௖௢௡௩஼
௖௢௡௩௦
௖௢௡௩௦ parameter
ಽ ஼ಾ
ಽ‫ܥ‬
௖௢௡௩஼
three-dimensional
൐൐
ܰ௜௧௘௥
ܰ௜௧௘௥
),ಾagreement
see
), see
Table
Table
5.with
5. a൐
than
than
convergence
convergence
onon
aerodynamic
aerodynamic
lifting
lifting
coefficient
(ܰ
(ܰ
௖௢௡௩஼
͵Ͳ݉Ȁ‫ݏ‬,
a coefficient
higher
number
ofಽ ஼iterations
isgood
necessary
to (ܰ
achieve
convergence
structural
௜௧௘௥
௜௧௘௥
ܰ௜௧௘௥ ಽ ಾon
), see
Table 5.distortion
than
convergence
on
aerodynamic
lifting
coefficient
௖௢௡௩௦
ಾ
௖௢௡௩஼ಽ ஼to
௜௧௘௥
iterations
isonnecessary
convergence
on
structural
ಾ achieve
൐ܸ
ܰ
), see
Table
5.
convergence
aerodynamic
lifting
coefficient
(ܰ
according
to
numerical
tortion
for than
the most
deformed
cross-section
at
‫ݕ‬
ൌ
Ͷ
m
is
obtained
with
௜௧௘௥
required
to
achieve
the
aerodynamic
convergence
increases
as
ܸ
increases,
iterations
ܰachieve
gher number
of iterations
is of
necessary
to
convergence
on
structural
distortion
௜௧௘௥
ஶ
ஶ
௜௧௘௥
௖௢௡௩
and
with
a
remarkable
reduction
in
terms
of DOFs.
͵Ͳ݉Ȁ‫ݏ‬, a higher number distortion
of iterationsthan
is necessary
to
achieve
convergence
on
structural
distortion
convergence on
of the
moment
coefficient
is also shown in Fig. 9b through the parameter
‫ ܥ‬஼Ȁ‫ܥ‬ெ .
convergence
the
lifting
௖௢௡௩஼
௖௢௡௩௦ forfor
ಾ
The
Thelimitation
limitation
of ofdistortion
distortionclose
close
toaerodynamic
tothe
theairfoil
airfoil
leading
leading
edge
edge
due
due
tototothe
the
spar
sparis isenhanced
enhanced
ܸஶܸஶ
ൌtoಽெ
ൌthe
The
limitation
of
distortion
close
the
airfoil
leading
edge
due
spar
is
enhanced
for ܸஶ ൌ
൐
ܰ
),
see
Table
5.
than
convergence
on
aerodynamic
lifting
coefficient
(ܰ
c.A
convergent
trend
of
displacements
and
aerodynamic
௜௧௘௥
஼in
௖௢௡௩஼
௜௧௘௥
௖௢௡௩௦
The
limitation
of
distortion
to ಽthe
leading
edge
due to values
the spar
is enhanced
ಾ airfoil
results in Table
5coefficient
and the
airfoil
deformed
profiles
Fig.
10.
Also
for
velocity
different
from for ܸஶ ൌ
and
final
convergent
values
much
different
൐close
ܰ
),௖௢௡௩
Table
ence on aerodynamic
lifting
(ܰ
coefficient
, see
see
Table
5.
௖௢௡௩஼
௖௢௡௩௦ are
௖௢௡௩ from the initial values, as summarized in Table 4.
௜௧௘௥
ಽ ஼5.
ಾ
௜௧௘௥
In this
‫ܥ‬
and
‫ܥ‬
represent
the
final
convergent
values
of
the
lifting
and
moment
coef൐
ܰ
),
see
Table
5.
than convergence on aerodynamic lifting coefficient
(ܰcase,
coefficients
isindicated
achieved
as
the structural model
௅ ௜௧௘௥
ெ
௜௧௘௥
͵ͷ݉Ȁ‫ݏ‬.
͵ͷ݉Ȁ‫ݏ‬.
The
The
trends
of
of
distortion
distortion
on
on
the
the
airfoil
airfoil
upper
upper
and
and
lower
lower
surfaces,
surfaces,
which
which
areare
indicated
as
US
US spar
The
limitation
oftrends
distortion
close
to
the
airfoil
leading
͵ͷ݉Ȁ‫ݏ‬.
The
trends
of
distortion
on
airfoil
upper
and
lower
surfaces,
which
are indicated
Theachieve
limitation
of
distortion
close
to
the
airfoil
leading
edge
dueasto
the
is enhanced
for ܸஶas
ൌ US
Thereason
trends
of edge
distortion
the
airfoil
upper
and
lower
surfaces,
which
arethe
indicated
as US
͵Ͳ݉Ȁ‫ݏ‬,
higher
number
of iterations
is necessary
to
convergence
structural
distortion
The
of
this
behavior
is spar
easily
by
the
an
increasing
free stream
velocity
accuracy
increases.
This proves that the proposed 1D
n of distortion
closea ͵ͷ݉Ȁ‫ݏ‬.
to the airfoil
leading
due
toonthe
is explained
enhanced
for
ܸஶ fact
ൌ onthat
edge,
spar,
is enhanced
for to
trends
.=35m/s.
Asspar
occurred
for the previous
ficients
for
a given
ܸஶthe
The limitation of distortion
closedue
to to
thethe
airfoil
leading
edge
due
isThe
enhanced
for ܸஶ ൌaeroelastic analysis, the trends do not show any
and
and
LSLS
respectively,
respectively,
areare
depicted
depicted
in in
Fig.
Fig.
11
11
at at‫ݕ‬஼ ൌ
‫ݕ‬ൌ
Ͷdistortion
Ͷforfor
different
different
velocities.
It
is is
important
to
to
note
note
higher-order
approach
does
notwhich
introduce
additional
LS
respectively,
are
depicted
invelocities.
Fig.
11 upper
atIt ‫ݕ‬
Ͷimportant
for
different
velocities.
It isindicated
important
͵ͷ݉Ȁ‫ݏ‬.
The
trends
on the
airfoil
and
lower
surfaces,
are
as to
USnote
௖௢௡௩஼
௖௢௡௩௦
ಽ ಾof
andthe
LSofairfoil
respectively,
arethe
depicted
inupper
Fig.loads,
11
atand
‫ݕ‬ൌ
Ͷܰsurfaces,
for
different
velocities.
important
to ൌnote
distortion
on
airfoil
and
lower
means
increasing
aerodynamic
consequently
structural
and lastly a
൐indicated
),higher
see
Table
5. It isdeformations,
convergence
on
aerodynamic
lifting
coefficient
(ܰ
trends of than
distortion
on
upper
and
lower
surfaces,
which
are
as which
US
௜௧௘௥and
௜௧௘௥
numerical
problems
in
the
aeroelastic
analysis
of
numerical
problems
such
as
oscillations.
From
Figs.
9a
and
9b
it
is
important
to
note
that
the
number
͵ͷ݉Ȁ‫ݏ‬. The trends of distortion
onthat
the
airfoil
and
lower
surfaces,
which
areremarkably
indicated
as
US
are indicated
as USupper
and
LS,upper
respectively,
are
depicted
in differ
that
deformations
deformations
of of
upper
and
and
lower
lower
surfaces
surfaces
remarkably
differ
also
also
because
because
of
a different
afor
different
aerodyaerodythat
deformations
ofalso
upper
and
lower
surfaces
differ
also because
a different
aerodyand
LS
respectively,
are
depicted
in
Fig.
11
at with
‫ ݕ‬of
ൌremarkably
Ͷ
different
It isofimportant
to note
that deformations
of upper
andairfoil
loweron
surfaces
remarkably
differ
because
of
awings
different
arbitrary
cross-section
more
relevant
coupling
effect
the
aeroelastic
response
of
the
wing.
In
fact,
an
airfoil
dis- velocities.geometry.
limitation
ofFig.
distortion
close
to
the
leading
edge
due
to
the
spar
is
enhanced
for
ܸஶ increasing
ൌ aerodyctively, areThe
depicted
in Fig.
11
at
‫ݕ‬
ൌ
Ͷ
for
different
velocities.
It
is
important
to
note
௖௢௡௩஼
௖௢௡௩
11
at
y=4,
for
different
velocities.
It
is
ಽ஼
ಾ important to note
isTable
presented
through
theonly
dimensionless
parameter
‫ܥ‬௅values
Ȁ‫ܥ‬௅ on
,the
as
illustrated
in Fig. 9a.toThe
and LS respectively, are depicted namic
innamic
Fig.pressure
11
at of
‫ݕ‬distribution.
ൌdistribution.
Ͷ process
for different
It not
istonot
important
toaerodynamic
noted.A
required
achieve
the
convergence
increases
as ܸஶ
increases,
iterations
ܰTable
௜௧௘௥
pressure
6velocities.
shows
6 shows
that
that
only
the
the
maximum
maximum
distortion
distortion
values
on
the
airfoil
airfoil
higher
number
of
iterations
isdistortion
necessary
namic
pressure
distribution.
Table
6surfaces
shows
that
only differ
the
maximum
valuesachieve
onaerodythe airfoil
that
deformations
of
upper
and
lower
remarkably
also
because
of a different
that
ofTable
the 6upper
and
lower
namic
pressure
distribution.
shows
that
not
only
the
distortion
on
thenot
airfoil
to numerical
tortion
for
the most
deformed
cross-section
at surfaces,
‫ݕ‬surfaces
ൌ aerodyͶmaximum
mwhich
isalso
obtained
with values
ܸஶasaccording
͵ͷ݉Ȁ‫ݏ‬.
The
trends
ofdeformations
distortion
ondiffer
the
airfoil
upper
and
are
indicated
US
ions of upper
and lower
surfaces
remarkably
also because
of lower
a different
௖௢௡௩than for
convergence
on
structural
distortion
.
of௅ௌthe
moment
is also
shown
in Fig.
9b through
the parameter
‫ܥ‬ெ Ȁ‫ܥ‬4.
that deformations of upperremarkably
and lower surfaces
remarkably
differ
also
because
of
acoefficient
different
aerody௎ௌ௎ௌbecause
௎ௌthe
௎ௌ convergence
௅ௌ
௅ௌare
௅ௌ pressure
final
convergent
values
much
different
from
the
initial
values,
as
summarized
in Table
ெ
differ,
aerodynamic
௎ௌ
௎ௌ
௅ௌas
௅ௌ
,and
‫ݏ‬, ௠௔௫ଶ
‫ݏ‬௠௔௫ଶ
)of
and
)different
and
lower
lower
(‫ݏ‬௠௔௫ଵ
(‫ݏ‬
‫ݏ‬, ௠௔௫ଶ
‫ݏ‬distribution.
surfaces
) surfaces
changes
changes
ܸஶthat
ܸ
varies,
but
also
also
their
their
corcorupper
(‫ݏ‬௠௔௫ଵ
௠௔௫ଵ
௠௔௫ଶ
, ‫)ݏ‬௠௔௫ଶ
)asand
lower
(‫ݏ‬as
,ஶvaries,
‫ݏ‬௠௔௫ଶ
) but
surfaces
changes
ascoefficients.
ܸஶ varies,
butonalso
their cor(‫ݏ‬, ௠௔௫ଵ
௎ௌ upper
௎ௌ (‫ݏ‬௠௔௫ଵ
௅ௌ
௅ௌ upper
namic
pressure
Table
6 velocity
shows
not
only
maximum
distortion
values
the airfoil
௠௔௫ଵ
convergence
onthe
aerodynamic
,
‫ݏ‬
)
and
lower
(‫ݏ‬
,
‫ݏ‬
)
surfaces
changes
ܸ
varies,
but
also
their
corupper
(‫ݏ‬
results
in
Table
5
and
airfoil
deformed
profiles
in
Fig.
10.
Also
for
values
different
from
ஶ
௠௔௫ଵ
௠௔௫ଶ
௠௔௫ଵ
௠௔௫ଶ
and
LS
respectively,
are
depicted
in
Fig.
11
at
‫ݕ‬
ൌ
Ͷ
for
different
velocities.
It
is
important
to
note
e distribution. Table 6 shows
that not onlyTable
the maximum
on
airfoil
distributions.
6 showsdistortion
that notvalues
only
thethemaximum
௖௢௡௩
௖௢௡௩
namic pressure distribution. Table 6 shows that The
not only
the
maximum
distortion
values
on
the
airfoil
In
this
case,
‫ܥ‬
and
‫ܥ‬
represent
the
final
convergent
values
of
the
lifting
and
moment
coefreason
of
this
behavior
is
easily
explained
by
the
fact
that
an
increasing
free
stream
velocity
Theseofreasons
makemodels
the
future
use of the proposed CUF௅ ௎ௌ
ெ
responding
responding
chordwise
chordwise
positions.
positions.
This
This
aspect
aspect
highlights
the
the
importance
importance
of
higher-order
higher-order
models
espeespe௎ௌ
௅ௌ
௅ௌ
responding
positions.
This, aspect
the
importance
higher-order
espedistortion
on
theThis
airfoil
upper
and
, chordwise
‫ݏ‬highlights
and
lower
(‫ݏ‬௠௔௫ଵ
‫ݏ‬on
)highlights
surfaces
changes
as
ܸஶ of
varies,
but also models
their corupper
(‫ݏ‬necessary
௠௔௫ଵthe
௠௔௫ଶ
responding
chordwise
positions.
aspect
highlights
importance
higher-order
models
espe͵Ͳ݉Ȁ‫ݏ‬,
avalues
higher
number
ofremarkably
iterations
isdiffer
to )achieve
structural
distortion
௎ௌ
௅ௌ
௅ௌ
that lower
deformations
of
upper
and
lower
surfaces
also௠௔௫ଶ
because
of aofconvergence
different
XFLR5aerodyapproach
appear
promising for a versatile flight
, ‫ݏ‬௠௔௫ଶ
)௎ௌ
and
surfaces
௎ௌ (‫ݏ‬௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ )௅ௌ
௅ௌchanges as ܸஶ varies, but also their corlower cially
(‫ݏ‬cially
,
‫ݏ‬
)
surfaces
changes
as
ܸ
varies,
but
also
their
corupper (‫ݏ‬௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ ) andlower
surfaces
changes
as
varies,
but
.
As
occurred
for
the
previous
aeroelastic
analysis,
the
trends
do
not
show
any
ficients
for
a
given
ܸ
means
increasing
aerodynamic
loads,
and
consequently
higher
structural
deformations,
and
lastly
a
ஶ ஶcross-section
௠௔௫ଵ
௠௔௫ଶ
forfor
an
an
accurate
accurate
evaluation
evaluation
ofcially
of
in-plane
in-plane
cross-section
distortion
distortion
of
high-deformable
high-deformable
structures.
structures.
forchordwise
andistortion
accurate
evaluation
ofof
cross-section
distortion ofofhigh-deformable
structures.
optimization
tool. the
responding
positions.
This
aspect
highlights
importance
higher-order models
espe஼in-plane
௖௢௡௩஼
௖௢௡௩௦
ಾ
ciallydistribution.
for
antheir
accurate
evaluation
of
in-plane
cross-section
of
high-deformable
structures.
also
corresponding
chordwise
positions.
This
aspect
൐
ܰ௜௧௘௥ onಽ the
),airfoil
see
Table 5.
than
convergence
on aerodynamic
lifting
coefficient
(ܰ
namic pressure
Table
6 shows
that
not
only
the
maximum
distortion
values
hordwise positions.
This aspect
highlights
the
importance
of higher-order
models
espe௜௧௘௥
responding chordwise positions. This
aspect
highlights
the
importance
of
higher-order
models
espenumerical
problems
such
as
oscillations.
From
Figs.
9a
and
9b
it
is
important
to
note
that
the
number
more
relevant
coupling
effect
on
the
aeroelastic
response
of
the
wing.
In
fact,
an
increasing
airfoil
dis5. 5.
Conclusions
Conclusions
highlights
the importance of higher-order
models,
in
5. Conclusions
cially
for an accurate
evaluation
of in-plane cross-section distortion of high-deformable structures.
௎ௌ5. Conclusions
௎ௌ
௅ௌ
௅ௌ
, ‫ݏ‬௠௔௫ଶ
) and
lower distortion
(‫ݏ‬
, ‫ݏ‬of
) surfaces
changes
as ܸஶ varies,
buttoalso
upper (‫ݏ‬௠௔௫ଵ
The
limitation
ofaccurate
distortion
close
to the
airfoil
leading
edge due
the their
spar coris enhanced for ܸஶ ൌ
ccurate evaluation
of in-plane
cross-section
high-deformable
structures.
௠௔௫ଵ
௠௔௫ଶ
௖௢௡௩஼
particular
for ancross-section
evaluation
of
the
in-plane
crossಽ ஼ಾ
cially for an accurate evaluation
ofVariable
in-plane
distortion
of
high-deformable
structures.
according
to numerical
tortion
for
the
most
deformed
cross-section
at
‫ݕ‬
ൌbasis
Ͷbasis
mof
isof
obtained
with
ܸஶ
required
toon
achieve
the
aerodynamic
convergence
as ܸஶ increases,
of
iterations
ܰ௜௧௘௥were
Variable
kinematic
kinematic
1D
1D
finite
finite
elements
elements
were
formulated
formulated
on
thethe
Carrera
Carrera
Unified
Unified
Formulation
Variable
kinematic
1D
finite
elements
were
formulated
onFormulation
the increases
basis of Carrera
Unified Formulation
5.
Conclusions
References
Variable
kinematic
1D
finite
elements
were
formulated
on
the
basis
of
Carrera
Unified
Formulation
section
distortion
of
highly-deformable
structures.
responding chordwise
positions.
This aspect
highlights
importance
higher-order
models
espe-are indicated as US
͵ͷ݉Ȁ‫ݏ‬.
The trends
of distortion
on the airfoil
upperofand
lower surfaces,
which
ns
5. Conclusions
resultstoin
5 and
airfoil
deformed
profiles
in
Fig.in10.
Also
velocity
values
different from
and
the
final
convergent
values
are
much
different
from
thefor
initial
values,
as
summarized
in Table
4.
(CUF)
(CUF)
and
and
coupled
coupled
to
anTable
an
aerodynamic
aerodynamic
3D
3D
panel
panel
method
method
implemented
implemented
in
XFLR5.
XFLR5.
The
The
aeroelastic
statstat(CUF)
and
coupled
to
an
3D
panel
method
implemented
in XFLR5.
The Formulation
aeroelastic
statVariable
kinematic
1D
finite
elementsThe
were
formulated
onaeroelastic
the basis
of Carrera
Unified
(CUF)
and evaluation
coupled toof
anin-plane
aerodynamic
3D panel
method
implemented
inaerodynamic
XFLR5.
aeroelastic
[1] Fung,
Y.C.,statAnto Introduction
to the Theory of
an
accurate
cross-section
distortion
structures.
matic 1D cially
finite for
elements
were
on the are
basis
of Carrera
Unified
andformulated
LS respectively,
depicted
in Fig.
11 at Formulation
‫ݕ‬ofൌhigh-deformable
Ͷ for different velocities.
It is important
note
Variable kinematic 1D finite elements
were
formulated
on
the
basis
of
Carrera
Unified
Formulation
a higher
number
of
iterations
isairfoil
necessary
to achieve
convergence
onthrough
structural
The
reason
ofresponse
behavior
is easily
explained
the fact
that
an increasing
free
stream
velocity through a
ic ic
response
response
of͵Ͳ݉Ȁ‫ݏ‬,
of
a straight
a straight
wing
wing
with
with
athis
high-deformable
a high-deformable
airfoil
cross-section
cross-section
was
computed
computed
through
a2008.
a distortion
Aeroelasticity,
Dover
Publications,
ic
of a cross-section
wing
with
a by
high-deformable
was
(CUF)
and
coupled
tostraight
an aerodynamic
3D
panelwas
method
in
XFLR5.
Thecomputed
aeroelastic static response of a straight wing with a high-deformable
airfoil
was
computed
throughimplemented
a airfoil cross-section
5.aerodynamic
Conclusions
upled to an
3D
panel
method implemented
The aeroelastic
stat- differ also because
5.
Conclusions
that
deformations
of upper in
andXFLR5.
lower surfaces
remarkably
of
a
different
aerody[2]
Patil,
M.J.,
Hodges
D.H.,
and
Cesnik
C.E.S.,
“Nonlinear
஼
௖௢௡௩஼
௖௢௡௩௦
ಽ
ಾ
(CUF) and coupled to an aerodynamic
3D iterative
panel
method
implemented
in
XFLR5.
The
aeroelastic
stat൐
ܰhigher
),
see velociTable
5. for different
than
convergence
oncoupled
aerodynamic
lifting
coefficient
(ܰ
means
increasing
aerodynamic
loads,
and
structural
deformations,
and lastly
a
coupled
coupled
iterative
procedure
procedure
for
for
an
an
increasing
increasing
structural
accuracy
accuracy
and
forfor
different
different
free
stream
stream
veloci௜௧௘௥
௜௧௘௥free
iterative
procedure
for consequently
an
increasing
structural
accuracy
and
free stream
velociic
response
of
astructural
straight
wing
with
a and
high-deformable
airfoil
cross-section
was
computed
through
coupled
iterative
procedure
for
an
increasing
structural
accuracy
and
for
different
free
stream
velociaeroelasticity
and
flight
dynamics
of
high-altitude
long- a
kinematic
1D finite
elements
were formulated
on
thethat
basis
of
Carrera
Unified Formulation
f a straightVariable
wing with
a high-deformable
airfoil
cross-section
computed
through
a maximum
namic
pressure
distribution.
Table was
6 shows
not
only
the
distortion values on the airfoil
ic response of a straight wingVariable
with a high-deformable
airfoilrelevant
cross-section
was
computed
through
a response
16 the
more
coupling
effect
aeroelastic
of to
the
an increasing
airfoil
diskinematic
finite
elements
were
formulated
endurance
aircraft”
,wing.
Journal
of
Aircraft,
Vol.ܸஶ38,
1, 2001,
The1D
limitation
of
distortion
close
to16on
the
airfoil
edgestructural
due
the
sparInisfact,
enhanced
for
ൌ No.
16
coupled
procedure
for anleading
increasing
accuracy
and
for different
free
stream
veloci16iterative
௎ௌ
௎ௌ 3D and
௅ௌimplemented
௅ௌ
(CUF)
to
an
aerodynamic
panel
method
in
XFLR5.
The
aeroelastic
stative procedure
forand
an coupled
increasing
structural
accuracy
for
different
free
stream
veloci,
‫ݏ‬
)
lower
(‫ݏ‬
,
‫ݏ‬
)
surfaces
changes
as
ܸ
varies,
but
also
their
corupper
(‫ݏ‬
on
the
basis
of
the
Carrera
Unified
Formulation
(CUF)
and
pp.
88-94.
ஶ
௠௔௫ଵ
௠௔௫ଶ
௠௔௫ଵ
௠௔௫ଶ
coupled iterative procedure for an increasing structuraltortion
accuracy
different
freecross-section
stream velociaccordingastoUS
numerical
forand
theoffor
most
deformed
at and
‫ ݕ‬ൌlower
Ͷ m issurfaces,
with are
ܸஶ indicated
͵ͷ݉Ȁ‫ݏ‬. The trends
distortion
on the
airfoil upper
which
16obtained
coupled to
an aerodynamic 3D panel method, implemented
[3] Sulaeman,
E., Kapania, R., and Haftka, R.T., “Parametric
16
ic response of a straight
wing
with
a
high-deformable
airfoil
cross-section
was
computed
through
a
responding chordwise16positions. This aspect highlights the importance of higher-order models espein XFLR5. The aeroelastic
static
response
a straight
wing
studies
flutter
in avelocity
, Proceeding
in Table
and
airfoil
deformed
Fig.
10.speed
Also for
valueswing”
different
from of
and LS results
respectively,
are 5of
depicted
in Fig.
11 at ‫ݕ‬profiles
ൌ Ͷ forinof
different
velocities.
Itstrut-braced
is important
to
note
coupled iterativewith
procedure
for
an
increasing
structural
accuracy
and
for
different
free
stream
velocia
highly-deformable
airfoil
cross-section
was
computed
43rd AIAA/ASME/ASCE/AHS/ASC
Structures, Structural
cially for an accurate evaluation of in-plane cross-section distortion ofthe
high-deformable
structures.
͵Ͳ݉Ȁ‫ݏ‬,
a of
higher
of iterations
is necessarydiffer
achieve
convergence
on structural
distortion
thatiterative
deformations
uppernumber
and lower
surfaces remarkably
also
because
of a different
aerodythrough a coupled
procedure,
for
increasing
Dynamics,toand
Materials
Conference,
AIAA
Paper
2002-1487,
16
5. Conclusions
structural accuracy and for different free stream velocities.
Denver,
CO,
April
2002.
௖௢௡௩஼ಽ ஼ಾ
௖௢௡௩௦
namic pressure
distribution.
6 showslifting
that not
only the (ܰ
maximum
values
the5.airfoil
൐ distortion
ܰ௜௧௘௥
), see on
Table
than convergence
on Table
aerodynamic
coefficient
௜௧௘௥
AnVariable
aerodynamic
assessment
confirmed
that
the 3D Panel
Hodges,
D.H.,
and Pierce, G.A., Introduction to
kinematic
1D finite elements
were
formulated
on the basis of[4]
Carrera
Unified
Formulation
௎ௌ
௎ௌ
௅ௌ
௅ௌ
, ‫ݏ‬evaluation
anddistortion
lower
‫ݏ‬௠௔௫ଶ
surfaces
changes
ܸஶ tovaries,
but
also
their cor(‫ݏ‬The
Method provides a upper
more realistic
of the (‫ݏ‬
pressure
Structural
Dynamics
and
Aeroelasticity,
close, to
the )airfoil
leading
edgeasdue
the spar
is
enhanced
forCambridge
ܸஶ ൌ
௠௔௫ଵlimitation
௠௔௫ଶ ) of
௠௔௫ଵ
(CUF)
and
coupled
to
an
aerodynamic
3D
panel
method
implemented
in
XFLR5.
The
aeroelastic
statdistribution on the wing, than the Vortex Lattice Method
University Press, 2002.
responding
chordwise
positions.
This aspect
highlights
the importance
higher-order
espe- as US
͵ͷ݉Ȁ‫ݏ‬.
The trends
of distortion
on the
airfoil upper
and lowerof
surfaces,
whichmodels
are indicated
ic response of a straight wing with a high-deformable airfoil cross-section was computed through a
cially for
anLS
accurate
evaluation
in-planeincross-section
of high-deformable
structures.
and
respectively,
are of
depicted
Fig. 11 at ‫ ݕ‬distortion
ൌ Ͷ for different
velocities. It
is important to note
coupled iterative procedure for an increasing structural accuracy and for different free stream velocihttp://ijass.org
321
5. Conclusions
that deformations of upper and lower surfaces remarkably differ also because of a different aerody-
16
Variablenamic
kinematic
1D distribution.
finite elements
were
formulated
on only
the basis
of Carreradistortion
Unified values
Formulation
pressure
Table
6 shows
that not
the maximum
on the airfoil
௎ௌ to an௎ௌ
௅ௌ method
௅ௌ
(CUF) and
coupled
aerodynamic
3D(‫ݏ‬
panel
implemented in XFLR5. The aeroelastic stat, ‫ݏ‬௠௔௫ଶ
) and lower
upper
(‫ݏ‬௠௔௫ଵ
௠௔௫ଵ , ‫ݏ‬௠௔௫ଶ ) surfaces changes as ܸஶ varies, but also their cor-
ic response
of a straight
wing positions.
with a high-deformable
airfoil cross-section
wasofcomputed
through
a esperesponding
chordwise
This aspect highlights
the importance
higher-order
models
Int’l J. of Aeronautical & Space Sci. 14(4), 310–323 (2013)
Hou, G.J.W., “Efficient nonlinear static aeroelastic wing
analysis”, Computers & Fluids, Vol. 28, No. 4, 1999, pp. 615628.
[19] Bathe, K.J., Finite Element Procedures, Prentice Hall,
Upper Saddle River, New Jersey, 1996.
[20] Kapania, K., and Raciti, S, “Recent advances in
analysis of laminated beams and plates, part II: Vibrations
and wave propagation”, AIAA Journal, Vol. 27, No. 7, 1989, pp.
935-946.
[21] El Fatmi, R., “Non-uniform warping including the
effects of torsion and shear forces. Part I: A general beam
theory”, International Journal of Solids and Structures, Vol.
44, No. 18-19, 2007, pp. 5912-5929.
[22] Schardt, R., “Extension of the engineer’s theory
of bending to the analysis of folded plate structures”, Der
Stahlbau, Vol. 35, 1966, pp. 161-171.
[23] Silvestre, N., and Camotim, D., “Second-order
generalised beam theory for arbitrary orthotropic materials”,
Thin-Walled Structures, Vol. 40, No. 9, 2002, pp. 791-820.
[24] Berdichevsky, V.L., Armanios, E., and Badir, A.,
“Theory of anisotropic thin-walled closed-cross-section
beams”, Composites Engineering, Vol. 2, No. 5-7, 1992, pp.
411-432.
[25] Yu, W., Volovoi, V.V., Hodges, D.H., and Hong, X.,
“Validation of the variational asymptotic beam sectional
analysis (VABS)”, AIAA Journal, Vol. 40, No. 10, 2002, pp.
2105-2113.
[26] Patil, M.J., “Aeroelastic tailoring of composite box
beams”, Proceedings of the 35th Aerospace Sciences Meeting
and Exhibit, AIAA Paper 97-0015, Reno, NV, January 1997.
[27] Librescu, L., and Song., O., “On the static aeroelastic
tailoring of composite aircraft swept wings modelled as thinwalled beam structures”, Composites Engineering, Vol. 2, No.
5-7, 1992, pp. 497-512.
[28] Qin, Z., and Librescu, L., “Aeroelastic instability of
aircraft wings modelled as anisotropic composite thinwalled beams in incompressible flow”, Journal of Fluids and
Structures, Vol. 18, No. 1, 2003, pp. 43-61.
[29] Jeon, S.M., and Lee, I., “Aeroelastic response and
stability analysis of composite rotor blades in forward flight”,
Composites: Part B, Vol. 32, No. 1, 2001, pp. 249-257.
[30] Friedmann, P.P., Glaz, B., and Palacios, R., “A moderate
deflection composite helicopter rotor blade model with an
improved cross-sectional analysis”, International Journal of
Solids and Structures, Vol. 46, No. 10, 2009, pp. 2186-2200.
[31] Carrera, E., Giunta, G., and Petrolo, M., Beam
Structures: Classical and Advanced Theories, John Wiley &
Sons, 2011.
[32] Varello, A., Carrera, E., and Demasi, L., “Vortex lattice
method coupled with advanced one-dimensional structural
[5] Sofla, A.Y.N., Meguid, S.A., Tan, K.T., and Yeo, W.K.,
“Shape morphing of aircraft wing: status and challenges”,
Materials & Design, Vol. 31, No. 3, 2010, pp.1284-1292.
[6] Gamboa, P., Vale, J., Lau, F.J.P., and Suleman, A.,
“Optimization of a morphing wing based on coupled
aerodynamic and structural constraints”, AIAA Journal, Vol.
47, No. 9, 2009, pp. 2087-2104.
[7] Suleman, A., and Costa. A.P., “Adaptive control of
an aeroelastic flight vehicle using piezoelectric actuators”,
Computers & Structures, Vol. 82, No. 17-19, 2004, pp. 13031314.
[8] Kota, S., Hetrick, J.A., Osborn, R., Paul, D., Pendleton,
E., Flick, P., and Tilmann, C., “Design and application of
compliant mechanisms for morphing aircraft structures”,
Proceedings of the SPIE Vol. 5054, Smart Structures and
Materials 2003: Industrial and Commercial Applications of
Smart Structures Technologies, San Diego, CA, 12 August
2003.
[9] Diaconu, C.G., Weaver, P.M., and Mattioni, F.,
“Concepts for morphing airfoil sections using bi-stable
laminated composite structures”, Thin-Walled Structures,
Vol. 46, No. 6, 2008, pp. 689-701.
[10] Lim, S.M., Lee, S., Park, H.C., Yoon, K.J. and Goo, N.S.,
“Design and demonstration of a biomimetic wing section
using a lightweight piezo-composite actuator (LIPCA)”,
Smart Materials and Structures, Vol. 14, No. 4, 2005, pp. 496503.
[11] Elzey, D.M., Sofla, A.Y.N., and Wadley, H.N.G., “A
shape memory-based multifunctional structural actuator
panel”, International Journal of Solids and Structures, Vol. 42,
No. 7, 2005, pp. 1943-1955.
[12] Bhardwaj, M.K., “Aeroelastic analysis of modern
complex wings using ENSAERO and NASTRAN”, Technical
Report NASA 19980235582, 1995.
[13] Dowell, E.H., and Hall, K.C., “Modeling of fluidstructure interaction”, Annual Review of Fluid Mechanics,
Vol. 33, 2001, pp. 445-490.
[14] Henshaw, M.J. de C. et al., “Non-linear aeroelastic
prediction for aircraft applications”, Progress in Aerospace
Sciences, Vol. 43, No. 4-6, 2007, pp. 65-137.
[15] Katz, J., and Plotkin, A., Low-Speed Aerodynamics,
Cambridge University Press, 2001.
[16] Wang, Y., Wan, Z., and Yang, C., “Application of highorder panel method in static aeroelastic analysis of aircraft”,
Procedia Engineering, Vol. 31, 2012, pp. 136-144.
[17] Yang, C., Zhang, B.C., Wan, Z.Q., and Wang, Y.K., “A
method for static aeroelastic analysis based on the highorder panel method and modal method”, Science China
Technological Sciences, Vol. 54, No. 3, 2011, pp. 741-748.
[18] Newman, J.C. III, Newman, P.A., Taylor, A.C. III, and
DOI:10.5139/IJASS.2013.14.4.310
322
Alberto Varello Static Aeroelastic Response of Wing-Structures Accounting for In-Plane Cross-Section Deformation
doublet lattice method”, Composite Structures, Vol. 95, 2013,
pp. 539-546.
[36] Maskew, B., “Program VSAERO theory Document: a
computer program for calculating nonlinear aerodynamic
characteristics of arbitrary configurations”, NASA contractor
report CR-4023, 1987.
[37] Oosthuizen, P.H., and Carscallen, W.E., Compressible
fluid flow, McGraw-Hill, 1997.
[38] Deperrois, A., Guidelines for XFLR5 v6.03, 2011.
[39] Carrera, E., and Varello, A., “Dynamic response
of thin-walled structures by variable kinematic onedimensional models”, Journal of Sound and Vibration, Vol.
331, No. 24, 2012, pp. 5268-5282.
models”, Journal of Aeroelasticity and Structural Dynamics,
Vol. 2, No. 2, 2011, pp. 53-78.
[33] Varello, A., Demasi, L., Carrera, E., and Giunta, G., “An
improved beam formulation for aeroelastic applications”,
Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and Materials Conference,
AIAA Paper 2010-3032, Orlando, FL, 12-15, April 2010.
[34] Carrera, E., Varello, A., and Demasi, L., “A refined
structural model for static aeroelastic response and
divergence of metallic and composite wings”, CEAS
Aeronautical Journal, Vol. 4, No. 2, 2013, pp. 175-189.
[35] Petrolo, M., “Flutter analysis of composite lifting
surfaces by the 1D Carrera Unified Formulation and the
323
http://ijass.org