Aeroelastic analysis of an aircraft with a high aspect ratio wing for medium altitude and long endurance capability was attempted in this paper. In order to achieve such an objective, various structural models were adopted. The traditional approach has been based on a onedimensional EulerBernoulli beam model. The structural analysis results of the present beam model were compared with those by the threedimensional NASTRAN finite element model. In it, a taper ratio of 0.5 was applied; it was comprised of 21 ribs and 3 spars, and included two control surfaces. The relevant unsteady aerodynamic forces were obtained by using ZAERO, which is based on the doublet lattice method that considers flow compressibility. To obtain the unsteady aerodynamic force, the structural mode shapes and natural frequencies were transferred to ZAERO. Two types of unsteady aerodynamic forces were considered. The first was the unsteady aerodynamic forces which were based on the onedimensional beam shape; the other was based on the threedimensional FEM model shape. These two types of aerodynamic forces were compared, and applied to the foregoing flutter analysis. The ultimate goal of the present research is to analyze the possible interaction between the rigidbody degrees of freedom and the aeroelastic modes. This will be achieved after the development of a reliable nonlinear beam formulation that would validate the current results as well as enable a thorough investigation of the nonlinearity. Moreover, such analysis will allow for an examination of the abovementioned interaction between the flight dynamics and aeroelastic modes with the inclusion of the rigid body degrees of freedom.
1. Introduction
Over the past few years there has been a growing need and demand for mediumaltitude, longendurance aircrafts. Mediumaltitude, longendurance aircrafts, abbreviated in the present paper as MALE aircrafts, are designed for longrange and longendurance intelligence and surveillance purposes. For instance, MALE aircrafts are suited to execute a broad range of intelligence and surveillancerelated tasks, including monitoring environments for nuclear or chemical contamination, target acquisition, reconnaissance missions, aerial refueling, strategic missile defense, cellular telephone relaying, meteorology, weather forecasting, and numerous others. When compared to other aircraft, the use of a MALE aircraft affords great advantages in the execution of these and other related tasks. The many assets of a MALE aircraft are largely the product of its design; for instance, MALE air vehicles are generally smaller than most other manned air vehicles and accordingly cost less to operate, as fuel and hangar costs are much lower. Moreover, the expenditures that are related to the human factor are minimal. As an illustration, the interior of the unmanned aircraft is optimized since it does not require any seat or complex instruments’ panel given that these aircraft do not carry human operators. In summary, the everincreasing use of MALE aircraft for a wide array of purposes throughout the world has elevated the analysis of all aspects of this aircraft to an endeavor of utmost importance.
One of the most distinguishing features of MALE aircraft design is the incorporation of high aspect ratio wings. Similarly to innovations in fixed and rotary wing aeroelasticity, the study and novelties in high aspect ratio wings has increased dramatically in the past few decades. However, whereas the study of fixed and rotary wing aeroelasticity heralded the publication of several review articles and studies in the field
[1

2]
, the literature on MALE aircraft remains minimal. In other words, there is no open literature available on MALE aircraft with highaspect ratio wings. Hence, Van Schoor, et al., has analyzed the aeroelastic properties of highaltitude, lowendurance (HALE) unmanned air vehicles with high aspect ratio wings
[3]
. The authors endeavored to study and account for the aeroelastic characteristics and the control of these highly flexible aircraft. Previous investigations in this field employed linearized modes, including rigid body modes, to predict the stability of the aircraft under different flight conditions. Through these studies, it was concluded that unsteady aerodynamics, combined with the flexibility of the aircraft, provided a better correlation. Nevertheless, the importance of the geometrical nonlinearity in HALE aircrafts was highlighted later; see
[4
,
5
and
6]
. The abovementioned works have concluded that the linear analysis based on the undeformed shape may not be accurate in the case of HALE aircrafts with high aspect ratio wings. Furthermore, in reference
[7]
, investigations were carried out to explore the effects of the large payloadinduced deformations and the effects of the parameters of the fuselage and horizontal tails on the flight dynamic characteristics of a highly flexible aircraft. Another key point of this work to undertake here means to account for or to investigate in sensitivity of some aeroelastic characteristics upon altering configuration parameters. This increased sensitivity was identified to be the result of the strong coupling of the highly flexible structure and aerodynamics. Given these points, for highly flexible aircrafts such as MALE aircraft, the coupled effects of large deflection due to structural flexibility and flight dynamics (e.g., roll controllability) as well as other aeroelastic effects (e.g., gust response, flutter instability) must be properly accounted for by applying a nonlinear aeroelastic formulation.
In this paper, an aeroelastic analysis of the high aspect ratio wings of a MALE aircraft will be developed and validated. These aircrafts are able to effectively perform mediumaltitude, longendurance surveillance missions largely because of their structural design. Chiefly, the long and slender wings of these vehicles are able to maximize the lifttodrag ratio. At first, the crosssectional inertia and stiffness properties will be extracted using the Variational Asymptotic Beam Sectional Analysis (VABS)
[8]
. With this in mind, the free vibration analysis will be carried out and natural frequencies will be obtained. Hence, the aeroelastic response analysis will be performed with the first few modes. In the meantime, a MALE aircraft may also undergo large deformations under normal operating loads leading to a geometrically nonlinear behavior. In view of that, the nonlinearities will be considered in the next stage of this research. For instance, a nonlinear model would facilitate the study of other important aspects of a MALE aircraft such as the trim effect. In addition to that, the next phase will also comprise investigations on the effects and inherent consequences of the inevitable interaction between rigidbody flight dynamics and aeroelastic modes.
2. Analytical Approach
 2.1 Structural Modeling
The analysis that has been carried out examines and considers the properties of onedimensional beams. In order to do so, a structural model of an EulerBernoulli beam, which is a displacementbased beam, was created in order to investigate its various structural properties. The linear behaviors of these high aspect ratio wings and the behavior of these flexible wings under unsteady subsonic aerodynamic forces were investigated
[9]
. First, the degrees of freedom had to be defined. Next, mass and stiffness matrices were obtained from the minimum energy kinetic principle. The structural analysis results of the present beam model were validated with the threedimensional NASTRAN result.
Figure 1
shows the threedimensional finite element wing construction. It has two control surfaces. Structurally, the stiffness of the EulerBernoulli beam is the result of its composite materials.
3D FEM Wing Model
 2.2 Aerodynamic Modeling
This section was written exceptionally well. It was clear and concise.
The flutter matrix equation is derived in the Laplace domain in terms of the generalized mass matrix,
M
, the generalized stiffness matrix
K
, and the generalized aerodynamic force matrix
Q
and given through Eqs.
(1) and (2).
This can be rewritten as:
The KMethod
The basic equation for flutter analysis by the KMethod is:
It is obtained by replacing by
ik
Eq. (2), where
ig_{s}
is the added artificial complex structural damping that is proportional to the stiffness. Also, the dynamic pressure
q
_{∞}
can be written as:
Where
ρ
is the air density.
The KMethod equation can be obtained by substituting Eq. (4) into Eq. (3) and dividing the resultant equation by
ω
^{2}
Where
is the complex eigenvalue of Eq. (5).
To solve for these eigenvalues, unsteady aerodynamic computations must be conducted at several reduced frequencies. These reduced frequencies are defined here as the “reduced frequency list”.
Q(ik)
are generated at a given Mach number of interest for each reduced frequency. For a given air density
λ
's, the eigenvalue of Eq. (5) in terms of ’s are solved in the complete reduced frequency list. For n structural modes, there are n eigenvalue corresponding to n modes at each reduced frequency.
The flutter frequency
ω_{f}
, the airspeed
V_{f}
, and artificial damping
g_{s}
are given in Eq. (6):
The GMethod
The GMethod is a damping perturbation method It means that this method is based on damping perturbation. (Ed. note: This is unclear. Do you mean, ‘a method based on damping perturbation’? Please confirm). It includes a first order damping term in the flutter equation.
This first order term is rigorously derived from the Laplace domain aerodynamics. In the GMethod, it is assumed that a function of
Q(p)
=
Q(g+ik)
exists for
g
≥ 0,
g
˂0.
This premise is based on the fact that the Laplace transform of the time domain unsteady aerodynamics for divergent
g>0
and constant amplitude motions
g=0
is analytic. Due to this analytical continuation, can be extended along the imaginary axis for small
Q(p)
by means of a damping perturbation method:
The term
in Eq. (7) is not available in the KMethod. However, if
Q(p)
is analytic, it must satisfy the CauchyRiemann equations such that:
Combining Eqs. (8) and (9) leads to the following general condition:
Equation (10) is valid in the complete pdomain except along the negative real axis where discontinuity due to a branch cut in subsonic flow occurs.
Thus, the term
can be replaced by:
Since
Q(ik)
can be provided by the kdomain unsteady aerodynamic methods,
Q(ik)
can be computed from
Q(ik)
by a central differencing scheme.
Substituting Eq. (11) in Eq. (7) gives the approximated pdomain solution of in terms of
Q(p)
and for small :
Substituting Eq. (12) in Eq. (2) yields the GMethod equation:
 2.3 Wing Analysis
In order to analyze the main wing of the aircraft, mass and stiffness matrices were first obtained from the crosssectional analysis performed by VABS
[8]
. Afterwards, an evaluation of the structural model of the EulerBernoulli beam allowed obtaining the relevant mode shapes and natural frequencies of the beam. These properties, as well as the newlyobtained system matrices, were further analyzed and adapted in order to be properly implemented in ZAERO (See
Figure 2
). To analyze the aeroelastic stability of the onedimensional beam, .FRE files were built and used in ZAERO DLM. Similarly, the threedimensional NASTRAN model supplied the .F06 files that were utilized for the flutter analysis. The latter was considered as a reference to validate the onedimensional beam model.
Main wing analysis procedure
 2.4 Complete Aivrcraft Analysis
For complete aircraft configuration, the assemblage combination of multiple beams was considered. As for the main wing, both mass and stiffness properties were obtained throughout the dimensional reduction carried out by VABS; which constitutes a tool that enables an efficient highfidelity crosssectional analysis. Next, the free vibration analysis results for both main and tail wings were achieved from the onedimensional beam analysis. Later, this outcome was combined with the properties of the fuselage (See
Figure 3
). Upon compiling all the data, ZAERO was employed to perform the flutter analysis, the results of which were again compared with those obtained by the threedimensional finite element model.
Complete aircraft analysis procedure
 2.5 Crosssectional Analysis
VABS produced a modeling of the crosssection for the high aspect ratio wing. For instance, VABS implements a rigorous dimensional reduction from the three dimensional description to a onedimensional model
[8

9]
. The crosssection of a high aspect ratio wing is presented below (See
Figure 4
). The inertial and stiffness properties will be presented in the numerical results.
CrossSection at the wing tip and the root
3. Numerical Results
 3.1 Structural Analysis
Based on the main threedimensional NASTRAN model, a more detailed representation of the finite elements for each of the aircraft’s components is depicted. Further geometric and structural properties that are determined through the use of the NASTRAN model are placed into corresponding tables and figures. All onedimensional results have been compared to the corresponding threedimensional NASTRAN results, the latter of which is considered to be a more accurate representation of the actual geometrical and structural data of the aircraft.
The results of this study will start with the present crosssectional analysis using the active twist rotor (ATR) blade data
[11]
, and the aeroelastic analysis of the MALE UAV wing. In the threedimensional NASTRAN model, a taper ratio of 0.5 was applied. This is based on the MALE model that has
Crosssectional analysis results
Crosssectional analysis results
Natural frequencies of the single main wing
Natural frequencies of the single main wing
Natural frequencies of the single tail wing
Natural frequencies of the single tail wing
Comparison upon the natural frequencies of the single main
Comparison upon the natural frequencies of the single main
been used throughout this study, which is comprised of 21 ribs and 3 spars. For the structural analysis, approximately 1,000 grid points are applied.
Using the obtained stiffness and mass properties, the free vibration analysis is carried out and natural frequencies are acquired for both the ATR blade and the MALE UAV wing. The relevant natural frequencies are obtained from a displacementbased formulation (linear) and MSCNASTRAN model (threedimensional) for the ATR blade
[12]
.
Further analysis is then performed which aims to determine the correlation between the one dimensional beam model and the threedimensional finite element model results in terms of mode shapes and natural frequencies. To begin with, the stiffness and mass results obtained from the threedimensional NASTRAN model are used as inputs within VABS which implements a significant dimensional reduction. The details of the crosssectional geometry and material properties are then included as inputs, allowing the existing beam models to be considered as well.
This table description is very well written. It is quite evident at this point that the paper is of high written quality as few revisions have been needed.
In
Table 2
, the structural analysis results of the onedimensional beam are presented. In terms of natural frequencies, the onedimensional beam demonstrates unsatisfactory correlation with the threedimensional model illustrated by the significant relative difference in terms of the natural frequencies.
It was then concluded that the differences obtained in both mode shapes and natural frequencies might be due to the assumption made at the very beginning of this structural analysis. To clarify, the same mass distribution as the one used throughout the threedimensional finite elements model was also used for the onedimensional beam analysis. In other words, the implementation of these mass properties throughout VABS was attempted before performing the onedimensional beam analysis. Consequently, this was suspected to be the cause of the differences noted in the previous results. For that purpose, an iterative procedure has been introduced and used for a more relevant mass distribution. For instance, this iterative procedure was applied based on a new optimized mass density distribution. The results obtained through this new approach show increasingly similar solutions between the onedimensional beam and the threedimensional model. A comparison between the previous onedimensional model and the revised one is provided in
Table 4
. It can be seen that the relative difference between the new onedimensional model and the threedimensional finite elements model is diminishing.
The obtained results indicate that the effect regarding the number of ribs on the mode shapes, natural frequency
Mode shapes of the single main wing predicted by NASTRAN
Mode shapes of the single tail wing by NASTRAN
Crosssection of the present high aspect ratio wing
and the flutter speed are quite marginal. Therefore, it can be concluded that changing the number of ribs does not influence the natural frequencies or the flutter speed in any way. Likewise, changing the spars would influence the bending and torsion frequencies in addition to the flutter speed.
In summary, from a threedimensional elasticity description to a onedimensional continuum model, all the details regarding the crosssectional geometry and material properties are included as inputs to calculate both structural and inertial coefficients. Later, these properties can be directly imported into the onedimensional beam analysis to predict the global behavior of the wing. In fact, this imported data is necessary for predicting pointwise threedimensional distributions of the displacement, strain and stress over the crosssection by VABS (See
Figure 5
). Thereafter, the wing undergoes a mass tuning in each airfoil, leading to the stiffness and mass properties that have been used in the EulerBernoulli beam model. Moreover,
Table 4
indicated that the results obtained through the previous analysis of the onedimensional beam model do not closely match those obtained from the NASTRAN finite element model, which is the most accurate. It was concluded that the differences obtained in both mode shapes and natural frequencies might be due to the mass properties used at the very beginning. Therefore, the same mass distribution as the one that has been used to obtain the three dimensional model was adopted. It was then implemented using VABS before analyzing the onedimensional beam model, which would prove to reduce the disparities among the results of the two models.
The vibration mode shapes are obtained through a one dimensional analysis of the linear EulerBernoulli beam. Moreover, ribs undergo a mass tuning since both their mass and stiffness properties could not be modeled in VABS. The mode shapes are presented below (See
Figures 6
and
7
). Later, these mode shapes are compared to the threedimensional finite element data (See
Figure 8
). Through this comparison, the similarities and correspondence between the results in these two models becomes increasingly apparent.
Mode shapes of the single main wing (linear beam)
 3.2 Flutter Analysis
Upon obtaining the previous results, the aeroelastic analysis is performed using the first six modes of the EulerBernoulli beam. In order to facilitate the wing flutter analysis, a combination of the onedimensional beam and ZAERO double lattice method is used. The results are obtained by the KMethod and GMethod, both of which use the generalized mass and stiffness matrices that were acquired using MATLAB for the onedimensional beam, and then implemented into the .FRE files. The flutter result for the main wing is validated with the threedimensional finite element model aeroelastic analysis which is based on the NASTRAN .F06 generated files as shown in
Table 5
. The flutter speed for the main wing was determined to be 670 km/h when combining the linear beam and ZAERO double lattice method. This was later validated with the results produced by the threedimensional model’s flutter. The difference between the linear beam and the finite element model, in terms of the main wing flutter speed result, was found to be less than 8%. Besides, the dominant structural mode of the main wing flutter was found to be the first inplane bending mode in the threedimensional finite elements model. In terms of the frequencies, a difference of 56% was noticed for the main wing. By analyzing the plots corresponding to the main wing (See
Figure 9
), there was a mode whose imaginary root slightly went across the horizontal axis. It has been identified as being the 8th mode. This mode coincides with the aerodynamic lag mode added throughout ZAERO aeroelastic analysis, which has a zero frequency. Thus, it was anticipated that instability would occur at the corresponding flight speed.
In
Table 6
, the results pertaining to the tail wing are presented. The flutter speed for the tail wing was determined
Main Wing Flutter Prediction Results
Main Wing Flutter Prediction Results
Tail Wing Flutter Prediction Results
Tail Wing Flutter Prediction Results
to be 792 km/h when the linear beam and ZAERO double lattice method were combined. The difference between the linear beam and the finite element model, in terms of the tail wing flutter speed result, was found to be approximately 2%. The dominant structural mode of tail wing flutter was assimilated as the first outofplane bending mode in the threedimensional finite element prediction. Also, the relative difference between the linear beam and the threedimensional tail wing flutter speed result in terms of frequencies was lower than 30%. From the plots related to the tail wing (See
Figure 10
), one of the modes demonstrated abrupt behavior. It was identified as the 5
^{th}
mode, which is, related to the first aerodynamic lag mode. Moreover, it corresponds to zero frequency and thus, it is possible that the tail wing undergoes a single degree of freedom flutter.
As for the entire aircraft, the flutter result from the threedimensional finite elements model is presented in
Table 7
. It was observed that the flutter speed in this case is lower than the one predicted in both
Tables 5
and
6
. Also, significant frequency coalescence was found in this case, unlike the previous mainwing only configuration (See
Figure 11
). This corresponds to the 14
^{th}
structural mode. The frequency coalescence that occurred between the 13
^{th}
andv 14
^{th}
structural modes can be clearly noticed. The damping of the above mentioned modes goes over the horizontal axis, which corresponds to the flutter speed (See
Figure 11
(c)  (d).). The coalescence that occurs throughout the entire aircraft configuration could be the product of the behaviors of the combined aircraft components’ interaction. One important aspect that cannot be determined without this combination is the antisymmetric coupling that is caused by the pairing of both wings. Furthermore, results indicated that the instability in the mainwing only configuration corresponds to a single degree of freedom flutter, whereas significant frequency coalescence was found throughout the entire aircraft.
In summary, the flutter determined by the present linear beam approach shows similar and correlating flutter speed and frequency results with those determined by the NASTRAN model. However, the accuracy of the frequency versus flight speed, as well as the damping versus speed plots, appeared to remain dubious. It is particularly difficult in this instance to determine the precise reasons for the discrepancies and account for these inaccuracies. As such, the onedimensional beam that has been used is considered as a crucial step before proceeding further in this analysis. Likewise, aerodynamic predictions were not validated independently. Subsequently, any aerodynamic predictions will need to be extracted and validated before further flutter analysis is conducted. For the aforementioned reasons, the need for an advanced model that would provide an expansion upon the present, and comparatively simple beam model, arises. For this reason, a separate aerodynamic dissection should be provided along with the structural
Complete Aircraft Flutter Prediction Results
Complete Aircraft Flutter Prediction Results
Damping and Frequency plots by the 3D FEM and the 1D beam model for the main wing
Damping and Frequency plots by the 3D FEM model for the tail wing
Damping and Frequency plots by the 3D FEM for the complete aircraft
analysis. Hence, this would enable enhanced accuracy and a more sophisticated analysis of the high aspect ratio wing.
4. Improved Approach to Capture the Geometrical Nonlinearity
One of the chief goals of this paper remains to provide an analysis of the interaction between the full six rigidbody degrees of freedom and aeroelastic modes. This very issue is believed to be a core component in understanding and enhancing the safety of unmanned aircrafts. Several notable incidents in these types of aircraft, such as the HELIOS aircraft crash, only serve to illustrate the gravity and urgency of this issue
[13]
. After exploring the development of the linear beam, as well as its structural and aeroelastic analysis, all while using the threedimensional NASTRAN model as a reference; it is clear that the best way to understand this important facet of unmanned air vehicles is to use a nonlinear beam formulation along with the finitestate unsteady aerodynamics. This approach would afford a much more accurate and allencompassing analysis of the high aspect ratio wings. As it now stands, this method is commonly used for rotorcrafts. However, the first implementation of this method, described in the work developed by Peters and He
[14]
, dealt solely with fixed wings. Above all, using a combination of the nonlinear intrinsic beam and the finitestate unsteady aerodynamics, either for a fixed or rotary wing, would provide an accurate and thorough method for the analysis. On the other hand, another approach could also provide a thorough method to analyze the overall structure of these aircrafts. This method is based on the structural model for a highly flexible wing itself, which has also been largely used for rotorcrafts (especially for the hover case), in terms of the frequency domain. This method is an implementation of the nonlinear beam formulation without any simplifying ordering scheme. Of particular importance within this approach is the combination of finitestate dynamic inflow, unsteady aerodynamics, and the nonlinear intrinsic formulation. Another advantage of this method is that there is no degree of freedom reduction required to determine the nonlinearities, which is an indisputable advantage to this method. Therefore, this analysis could be both conducted in terms of either the frequency or time domain using any of these methods.
In conclusion, the theory presented is only applicable to a rotating blade, either cantilevered or simply supported in its present understanding. Consequently, this approach, one that is based on this structural model, will need to be modified in order to enable the analysis of a fixed wing with an incoming free stream, which will represent a different air flow pattern when compared to rotary wings. Moreover, the finitestate unsteady aerodynamics model, in its current form, also possesses its own inherent limitations, since it is based on an ideal fluid equation and neglects viscosity and compressibility. Therefore, it would be more accurate to consider a possible extension and reinforcement of this approach by gradually including and accounting for these effects. In brief, the use of the nonlinear model would provide a much more accurate and precise result. In addition, an important feature and future objective of the current paper will be to perform the analysis and simulation of the flight dynamic response considering the six rigidbody degrees of freedom. By adopting this approach, the flutter analysis would be more accurate and allow for other important factors such as the trim impact as well as the deflection, to be accounted for. Over the last decade, few researches have attempted to develop the following analysis despite its increased accuracy and accountability. Without a doubt, the analysis of the response for MALE aircraft, including those that account for the six degrees, will only help to provide a clearer and more accurate study of MALE aircraft as a whole.
5. Conclusion
In this study, an aeroelastic analysis of an aircraft with high aspect ratio wings, developed for medium altitude and long endurance capabilities, was performed. In order to accurately and thoroughly analyze this aircraft and these inherent qualities, various structural models were implemented and analyzed. First, the crosssectional analysis was conducted by using VABS. This was complimented by the high aspect ratio wing model, which was modeled as a onedimensional beam. Additionally, the structural analysis for the main and tail wing were performed and the natural frequencies and the mode shapes were validated and compared with the threedimensional NASTRAN model. The results obtained through this comparison demonstrated a strong and accurate correlation between the two models, in terms of both the natural frequency and mode shapes. These figures correlated with striking accuracy; for example, the natural frequencies of the onedimensional beam demonstrated a maximum difference of 10% from those obtained through the threedimensional model. In addition to this, flutter results of the main wing were obtained through combining the linear beam and the unsteady aerodynamics model in ZAERO. The flutter result for the main wing was validated with threedimensional finite elementbased aeroelastic analysis. After analyzing the onedimensional beam, the results obtained were close to those predicted by the NASTRAN model. For the main wing, there was only a difference of 7.94% in terms of the flutter speed, and only a 2% difference was observed in the tail wing. What’s more, a new approach was suggested to provide a more accurate, succinct, and thorough representation of and comparison between the two models. This model, a nonlinear beam model, would account for several important factors that remain largely unaccounted for in these two models, which include the trim effect, the steady state, the deflection, and the ability to overcome limitations produced by linearity.
Acknowledgements
This work was sponsored by Brain Korea 21 Project, Mechanical Research Division, School of Mechanical and Aerospace Engineering, Seoul National University, South Korea. This work was also sponsored by the New and Renewable Energy Program of the Korean Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government, Ministry of Knowledge Economy (No.20104010100490).
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