A folding wing morphing aircraft should complete the folding and unfolding process of its wings while in flight. Calculating thehinge moments during the morphing process is a critical aspect of a folding wing design. Most previous studies on this problemhave adopted steady-state or quasi-steady-state methods, which do not simulate the free-flying morphing process. In this study,we construct an aeroelastic flight simulation platform based on the secondary development of ADAMS software to simulate theflight-folding process of a folding wing aircraft. A flexible multibody dynamic model of the folding wing structure is establishedin ADAMS using modal neutral files, and the doublet lattice method is developed to generate aerodynamic influence coefficientmatrices that are suitable for the flight-folding process. The user subroutine is utilized, aerodynamic loading is realized inADAMS, and an aeroelastic flight simulation platform of a folding wing aircraft is built. On the basis of this platform, the flight-folding process of the aircraft is simulated, the hinge moments of the folding wings are calculated, and the influences of thefolding rate and the aircraft’s center of gravity (c.g.) position on the results are investigated. Results show that the steady-statemethod is applicable to the slow folding process. For the fast folding process, the steady-state simulation errors of the hingemoments are substantially large, and a transient method is required to simulate the flight-folding process. In addition, the c.g.position considerably affects the hinge moments during the folding process. Given that the c.g. position moves aft, themaximum hinge moments of the inner and outer wings constantly increase.
1. Introduction
As a typical morphing aircraft, the folding wing aircraft canchange its configuration autonomously by folding its wingsduring flight to fit variable flying environments and satisfymultimission demands. Its operational performance is muchhigher than that of a conventional fixed configuration aircraft[1, 2] and thus has elicited widespread attention.
Many studies on a folding wing aircraft have been con-ducted in recent years, and the focus of research includestwo aspects. One is the aeroelastic characteristic in fixed con-figuration. Related works have undergone a phase from lin-ear to nonlinear and from simulation to experiment [3–5].This aspect has been the subject of considerable research.The other is the dynamic response during the folding pro-cess. Related studies are based on simulation and require fur-
ther development. Zhao and Hu [6] established the flexiblemultibody dynamic model of a folding wing aircraft by com-bining the Craig–Bampton synthesis technique with the flex-ible multibody dynamic approach, and the accuracy of thismethod was verified by simulating the folding process. Jungand Kim [7] investigated the aerodynamic characteristics ofa wing during the folding process using the unsteady vortexlattice method and discussed the influences of the foldingangle and angular velocity in detail. Hu et al. [8] developedaerodynamic influence coefficient (AIC) matrices in the timedomain based on the doublet lattice method and coupled thisapproach with the structural model to simulate the aeroelas-tic response of a body-fixed folding wing aircraft during thefolding process. Lee and Weisshaar [9] regarded the foldingprocess of the folding wing aircraft as a steady-state processand used the steady-state approach of static trimming to
HindawiInternational Journal of Aerospace EngineeringVolume 2019, Article ID 9362629, 11 pageshttps://doi.org/10.1155/2019/9362629
calculate the hinge moments at different folding angles on thebasis of the ZAERO software. Reich et al. and Bowman et al.[10, 11] developed an integrated aeroelastic multibodymorphing simulation tool by using flexible multibodydynamics (ADAMS) and vortex lattice model aerodynamicloading (an in-house code). On the basis of this tool, Scarlettet al. [12] conducted a series of wind tunnel simulations andstudied the change in load paths and hinge moments underspecific motions.
Given the aforementioned research status, the calcula-tion of the hinge moment, as a representative issue of thefolding wing aircraft, has attracted extensive attention.Considering the complexity of flight-folding simulation,the majority of current studies on this problem adoptsteady-state or quasi-steady-state simplified methods, whichare suitable for a slow folding process. However, the appli-cability of such methods to a fast folding process has yet tobe verified.
In the current work, we construct an aeroelastic flightsimulation platform to simulate the flight-folding process ofa folding wing aircraft. A flexible multibody dynamic modelof the folding wing structure is established using ADAMSsoftware, and the aerodynamic force is calculated via theAIC matrices and loaded in ADAMS through the user sub-routine. The dynamic response during the flight-folding pro-cess of the aircraft is simulated on the basis of this platform,
the hinge moments of the folding wings are calculated, andthe influences of the folding rate and the center of gravity(c.g.) position on the results are investigated.
2. Simulation Platform
2.1. Model of Folding Wing Aircraft. A typical half model of afolding wing aircraft is shown in Figure 1. This half modelcan be treated as a flexible multibody structure, as each flex-ible body is a substructure (i.e., fuselage (I), inner wing (II),and outer wing (III)). In addition, an aileron (IV) is arrangedat the trailing edge of the outer wing to maintain stabilityduring the folding process. The folding angle θ is defined asthe angle between the fuselage and the inner wing. To guar-antee the lift performance of the folded wing, the outer wingis designed to be parallel to the fuselage.
2.2. Framework of the Platform. The aeroelastic flight simula-tion platform constructed in this paper is shown in Figure 2.To effectively complete the dynamic simulation of the foldingwing aircraft during the flight-folding process, the simulationplatform has the following functions: flexible multibodydynamic modeling, unsteady aerodynamic load modeling,and flight control modeling.
The platform integrates the capabilities of ADAMS andNASTRAN in structural modeling, combining advanced
IIIIV
II
x
I
y
z
I
II
III IV
y
z
θ
Figure 1: Sketch of a folding wing.
ADAMS view(multi-body systemdynamics modeling)
ADAMS solver(solver)
NASTRAN(finite element
modeling)
User subroutine(aerodynamic
modeling)
AIC matricesin time domain
ADAMS control(PID control method)
Time
Time
Doublet-latticemethod
ADAMS postprocessor(result analysis)
Figure 2: Aeroelastic flight simulation platform.
2 International Journal of Aerospace Engineering
finite element models with multibody dynamic models tosimulate complex flexible multibody systems. ADAMS cur-rently allows incorporating flexible bodies from finite ele-ment analyses by the Craig–Bampton component modesynthesis method [13]. By using the NASTRAN software, amodal analysis of each aircraft component is conducted withthe classical Craig–Bampton modal synthesis method to gen-erate modal neutral files suitable for importing into theADAMS modeling environment. Each component must becreated separately in order for ADAMS to treat them as sep-arate parts. Subsequently, the components introduced intothe system are assembled by revolute joints in ADAMS,and rotational motion is applied to each revolute joint. Theentire model (assembled machine) for simulation is shownin Figure 3.
Thus far, we have completed the flexible multibodydynamic modeling of the folding wing aircraft, and it needsto integrate the unsteady aerodynamic force to achievedynamic simulation. However, ADAMS itself does not havean aerodynamic calculation module and cannot directly cal-culate complex unsteady aerodynamic loads. Fortunately,ADAMS provides a friendly second-development interface,and we can use the user subroutine to realize aerodynamicupdating and loading during the folding process. Thus, weset GFORCE-generalized forces at the structural nodes andwrite the SUBROUTINE GFOSUB user subroutine accord-ingly. The subroutine flow is shown in Figure 4. The sub-routine uses SYSARY and SYSFNC macros to read thedisplacement, velocity, and acceleration information of thestructural nodes from ADAMS solver and passes them tothe aerodynamic calculation program. By using the AICmethod, the aerodynamic load is calculated and then fedback to ADAMS solver.
In addition, mass distribution and aerodynamic loaddistribution change considerably during the folding process.
To maintain flight stability during folding, we establish alongitudinal stabilization control system in the ADAMSbuilt-in control module. The stability control system ofthe aircraft uses the angle of attack, pitch rate, altitude,and plunging velocity as the feedback input signals andthen generates a control law by using the PID controlmethod.
2.3. Aerodynamic Model. The modeling of the unsteadyaerodynamic load is a critical work in the dynamicsimulation of the folding process. Based on the ideaof rational approximation, we develop the doublet lat-tice method and propose a calculation method for theAIC matrices in the time domain suitable for the calcu-lation of unsteady aerodynamic loads during the flight-folding process.
The conventional unsteady AIC matrix Q based on thedoublet lattice method can be obtained in simple har-monic conditions. The matrix is a function of the reducedfrequency ω and related to the Mach number Ma [14].The aerodynamic model can be expressed in the followingform:
fk = q∞Qk Ma, ω uk, 1
where q∞ is the dynamic pressure, Qk is the AIC matrixfor the aerodynamic nodes, Ma is the Mach number, ωis the reduced frequency, and fk and uk are the aerody-namic force vector and the displacement vector, respec-tively, for aerodynamic nodes (for the folding wingaircraft, containing components in both directions of y-and z-axes).
2.3.1. Interpolation Technique. In aeroelastic analysis, toadapt to the specific theory of structural dynamics and
Revolute joint Rotational motion
+
Figure 3: ADAMS model of the folding wing aircraft.
3International Journal of Aerospace Engineering
aerodynamics, the division of the aerodynamic mesh andthe structural mesh are conducted independently. To com-plete the analysis of aeroelasticity, two transformations areneeded: one transforms the displacement of the structuralnodes into the displacement of the aerodynamic nodes,while the other transforms the aerodynamic force of theaerodynamic nodes into an equivalent force that acts onthe structural nodes. This transformation can be achievedby applying spline matrix Gkg [15, 16], which can beexpressed as
uk =Gkgug,
fg =GTkgfk,
2
where fg and ug are the aerodynamic force vector and thedisplacement vector, respectively, for the structural nodes.By substituting equation (2) into equation (1), we canobtain the relationship between aerodynamic force anddisplacement for the structural nodes.
fg = q∞Qg Ma, ω ug, 3
where Qg =GTkgQkGkg is the AIC matrix for the struc-
tural nodes.
2.3.2. Rational Approximation. The conventional AICmatrix Q based on the doublet lattice method can beobtained in simple harmonic conditions. However, theoutput can only be used to calculate frequency domain,and it cannot be directly applied to the aerodynamic cal-culations in the time domain. To calculate the unsteady
aerodynamic force in arbitrary motion, the Roger method[17] is used in this study to derive a rational approxima-tion of the unsteady AIC matrices in the Laplace domain.The fitting formula is
Qg s =A0 +A1bVs +A2
bVs
2+ 〠
n
i=1Eis s + V
bri
−1,
4
where s is the Laplace variable, b is the reference length, Vis the air speed, A and E are the undetermined matrices,and ri are the undetermined coefficients. The n values inthe reduced frequency range of interest are taken as ri.For the general fitting accuracy requirement, n is 4.
By transforming equation (4) for application in the timedomain environment, we express the aerodynamic force fgfor the structural nodes as
fg = q∞ A0ug +A1ugbV
+A2ugb2
V2 + 〠4
i=1Eiyi 5
For y in equation (5), the following relationship exists:
yi +Vbriyi = ug 6
By iterating the above equation into time steps, the equa-tion can be transformed into
yi t + Δt = yi t 1 − VbriΔt + ug t + Δt − ug t 7
StartSUBROUTINE
StartADAMS solver
ITER++
Call displacement,velocity, andacceleration
Aerodynamiccalculation
CFOSUB return
T > tend
Processmanagement
SYSARY/SYSFNC
Dynamicsolver
Aerodynamicupdating
ITER++
T > tend
End
fg fg
ug, ug, üg ug, ug, üg
Yes Yes
No No
End
‧ ‧
Figure 4: Flowchart of the aerodynamic subroutine.
4 International Journal of Aerospace Engineering
Thus far, we have obtained the AIC matrices in the timedomain for a specific configuration. On the basis of the small-perturbation assumption of the doublet lattice method, theobtained AIC matrices in the time domain have becomesuitable for the calculation of aerodynamic force in the smallperturbation range near the given configuration. Here, thesmall perturbation range includes the elastic deformation ofthe wings and a small angle of folding and pitching motion.In addition, considering that the aerodynamic force is calcu-lated relative to the reference plane, the reference plane canbe dynamically selected as the plane at the aircraft’s altitude.Thus, the AIC matrices can also be used to calculate aerody-namic force in heavy plunging motion.
2.3.3. Interpolation of the Folding Angle. The folding processof the folding wing aircraft is a large-scale morphing process.The AIC matrices in the time domain in single configurationcannot fully describe the aerodynamic load during the entirefolding process. Thus, we calculate a set of AIC matricesevery 15° of the folding angle. The AIC matrices Aθ and Eθ
of each folding angle θ are obtained by linearly interpolatingtwo sets of AIC matrices (i.e., before and after θ). The inter-polation algorithm is defined as follows:
Aθ =Aθi + Aθi+1 −Aθi
θi+1 − θiθ − θi ,
Eθ = Eθi + Eθi+1 − Eθi
θi+1 − θiθ − θi θi < θ < θi+1
8
Figure 5 shows the flowchart of building the aerodynamicmodel of the folding wing aircraft. By using the method, wecan obtain the AIC matrices in the time domain that contin-uously changes with the folding angle. The aerodynamicforce at any folding angle θ can be expressed as
fg = q∞ Aθ0ug +Aθ
1ugbV
+Aθ2ug
b2
V2 + 〠4
i=1Eθi yi , 9
where y can be obtained in time steps with equation (7).
3. Numerical Examples
3.1. Geometrical Analysis. The geometries and dimensions ofthe folding wing in the present study are illustrated inFigure 6. The fuselage and wings are composed of a skinand inner skeleton and assume an NACA0006 airfoil shapethat defines the upper and lower surfaces. The aileron is setas the control surface with 20% of the outer-wing chordand is located between 11% and 83% of the outer-wing span[9]. These substructures are composed of aluminum andconnected by rotating hinges, as indicated by the circles inFigure 6(c). The drivers are placed at the hinges of the foldingwings. Counterweights are added to the fuselage and wings torepresent the weight of the engine, actuators, and transmis-sions, as indicated by the triangles in the figure. The intersec-tions of the spars and ribs are selected as the structural nodeswhere aerodynamic forces are applied and are identified inFigure 6(c). The aerodynamic panel model is presented in
Figure 5: Flowchart of building the aerodynamic model.
5International Journal of Aerospace Engineering
Figure 6(b). The aircraft mass is 823 kg, and other mainparameters are listed in Table 1.
3.2. Dynamic Response during the Flight-Folding Process.Based on the aeroelastic flight simulation platform built inthis study, a simulation of dynamic response of the foldingwing aircraft during the flight-folding process is conducted.We set the flight altitude to 2 km, maintain the airspeedto 100m/s, and add a driver to the aileron for the controlsystem. The aircraft is designed to maintain an unfoldedwing configuration at the initial 10 s for the preliminarytrimming calculations, and then, the wings begin to foldat 10 s. When the wings are folded in place, the aircraftmaintains a leveled flight in the folded wing configuration.
In the simulation, we first investigate the general vari-ation of the dynamic parameters of the aircraft during the
folding process, and then, we study the influence of thefolding rate and the c.g. position on the results. Thus,we first simulate the folding process of the aircraft witha folding time of 30 s. To reduce the impact at the startand end of the folding process, a cosine driving law wasapplied to the folding. The main dynamic parameters ofthe aircraft are depicted in Figure 7.
Figures 7(a)–7(d) show that the aircraft is in equilibriumat approximately 6 s. The initial trimmed angle of attack is1.87°, and the deflection angle of the ailerons is −5.15°. At10 s, the wings begin to fold. As the folding angle increases,the effective wing area decreases and the aircraft drops(Figure 7(c)). To recover the altitude, the ailerons aredesigned to deflect upward movement (Figure 7(d)), whichthen generates a head-up torque that can increase the angleof attack (Figure 7(b)) and maintain balance of lift. The wingsare folded in place at 40 s, followed by a short stabilizationprocess. After approximately 2 s, the aircraft stabilizes toreach a new equilibrium state; that is, the trimmed conditionof the folded wing configuration is achieved. Relative to thetrimmed condition of the initial unfolded wing configura-tion, the angle of attack increases to 6.83°, and the aileronsdeflect upward to −8.78°.
Aerodynamic panel model
45.0°
0.8
0.5
4.6
0.9
1.4
2.1
Unit: m
10 kg × 2
20 kg × 4
100 kg × 2
Spars and ribsStructural node
(a)
(b) (c)
Rotating hingeCounterweight
Figure 6: Geometries and dimensions of the folding wing aircraft. (a) Internal and external structure. (b) Aerodynamic panel model. (c)Structural model.
Table 1: Main parameters of the aircraft.
c.g. coordinate, xc, yc, zc(m, m, m)
Inertia tensor,Iyy (kgm
2)
Unfolded 2.12, 0.00, 0.00 4502
Folded 2.12, 0.00, 0.24 4675
6 International Journal of Aerospace Engineering
In addition, we focus on the change in hinge momentsduring the folding process (see Figures 7(e) and 7(f)). In theinitial unfolded state, the driver of the inner wing must simul-taneously maneuver the inner and outer wings. The hingemoment in this state is relatively large at 1537Nm, whereasthe hinge moment of the outer wing is relatively small at351Nm. With the folding of the wings, two importantchanges occur in the aerodynamic force that acts on the wings.First, the arm of the aerodynamic force that acts on the outerwing to the inner wing hinges is reduced, thereby also reduc-ing the hinge moment of the inner wing. Second, the aerody-namic force that acts on the outer wing increases with theangle of attack, thus also increasing the hinge moment of theinner wing. Under the common variation of the two changesat 30.75 s at an approximately 94° folding angle, the hingemoment of the inner wing reaches a maximum of 2637Nm.The hinge moment of the outer wing increases with the aero-dynamic load in the outer wing and reaches a maximum valueof 1762Nm when the wings are folded in place.
Considering the complexity of the dynamic behaviorduring the flight-folding process, we verify the above conclu-sions only for the equilibrium state of the unfolded and
folded wing configurations. The statically trimmed resultscalculated by the aeroelastic flight simulation platform(ADAMS platform) and by the NASTRAN platform arelisted in Table 2.
The comparison result shows that the statically trimmedcalculation based on the NASTRAN platform supports theconclusion of the ADAMS platform, which also demon-strates the reliability of the dynamic simulation resultsderived from the simulation platform proposed in this paper.
3.3. Effect of the Folding Rate. To complete the folding pro-cess at a faster speed and within a shorter time, achieving
0 10 20 30 40 500
50
100
150
Time (s)
Ang
le (d
eg)
Preliminarytrimming
Folding process Levelflight
(a) Folding angle
0 10 20 30 40 500
2
4
6
8
Time (s)
Ang
le (d
eg)
(b) Angle of attack
0 10 20 30 40 501990
1992
1994
1996
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2000
Time (s)
Leng
th (m
)
(c) Altitude
0 10 20 30 40 50−10
−8
−6
−4
−2
0
Time (s)A
ngle
(deg
)
(d) Deflection angle of aileron
0 10 20 30 40 50
5001000
1500
2000
2500
3000
Time (s)
Mom
ent (
N m
)
(e) Hinge moment of the inner wing
0 10 20 30 40 50
500
1000
1500
2000
Time (s)
Mom
ent (
N m
)
(f) Hinge moment of the outer wing
Figure 7: Dynamic response during the flight-folding process.
Table 2: Comparison of statically trimmed results.
Foldingangle (deg)
PlatformAngle of
attack (deg)Deflection angle ofthe ailerons (deg)
0ADAMS 1.87 -5.15
NASTRAN 1.87 -5.17
120ADAMS 6.83 -8.78
NASTRAN 6.76 -8.54
7International Journal of Aerospace Engineering
rapid attack-defense conversion is undoubtedly one of theobjectives of a folding wing design. However, most previ-ous studies have regarded the folding process as slowand have adopted steady-state or quasi-steady-state simpli-fied methods. When folding speed is increased, the appli-cability of such methods has yet to be verified. Here, wesimulated the folding process with four different foldingtimes (5, 10, 20, and 30 s) to study the influence of thefolding rate on the simulation results and the applicabilityof the steady-state method. The dynamic response of theflight-folding process is shown in Figure 8. The responseindicates that as the folding speed increases, the timerequired for an aircraft to complete the folding process isshortened, while the transient effect of the folding processbecomes increasingly evident and a longer stabilizationprocess is required after folding in place, which also makesthe steady-state method face a test.
Figure 9 shows the specific time required to complete thefolding and reach an equilibrium state. The blue bar in thefigure indicates the time required for folding, and the redone indicates the time spent in the stabilization process.The figure shows that the time required for the stabilizationprocess increases as the folding speed increases, whereas thetotal time decreases. The total time required to completethe folding and reach an equilibrium state is 32 s when thefolding time is 30 s. The total time required is decreased to14.8 s when the folding time is 5 s.
The hinge moments of the intermediate configurationsat different folding speeds are compared to illustrate thetransient effect during the folding process and the applica-bility of the steady-state method. The result is presentedin Figure 10, where the hinge moment at a folding angleof 120° is the maximum hinge moment after folding inplace. When the folding process is relatively slow (folding
0 10 20 30 40 500
20406080
100120
Time (s)
Ang
le (d
eg)
5 s 10 s
20 s 30 s
(a) Folding angle
5 s 10 s
20 s 30 s
0 10 20 30 40 500
2
4
6
8
Time (s)
Ang
le (d
eg)
(b) Angle of attack
5 s 10 s
20 s 30 s
0 10 20 30 40 50
1992
1994
1996
1998
2000
Time (s)
Leng
th (m
)
(c) Altitude
5 s 10 s
20 s 30 s
0 10 20 30 40 50−12
−10
−8
−6
−4
−2
0
Time (s)
Ang
le (d
eg)
(d) Deflection angle of aileron
5 s 10 s
20 s 30 s
0 10 20 30 40 50
500
1000
1500
2000
2500
3000
Time (s)
Mom
ent (
N m
)
(e) Hinge moment of the inner wing
5 s 10 s
20 s 30 s
0 10 20 30Time (s)
40 50
500
1000
1500
2000
Mom
ent (
N m
)
(f) Hinge moment of the outer wing
Figure 8: Response at different folding speeds.
8 International Journal of Aerospace Engineering
time: 30 s), the transient effect during the process is weak,and the steady-state and transient simulation results ofthe hinge moments are similar. The steady-state simulationerror (the difference between the hinge moments calculatedby the steady-state and transient method) in this case is lessthan 1%. When the folding process is fast (folding time:5 s), the transient effect during the process is evident, andthe steady-state method results in considerable simulationerrors. The hinge moments of the inner and outer wingsare 16.4% and 12.1% larger than the transient simulationresults, respectively, at a folding angle of 60°, and 12.4%and 12.2% smaller than the transient simulation results,respectively, at a folding angle of 120°.
The preceding analysis shows that increasing folding speedcan achieve fast attack-defense conversion. However, theflight-folding process is complex and dynamic and involvesthe coupling of aerodynamics, inertia, and control. Thesteady-state method only considers the balance of the constantaerodynamic loads and gravity. When the folding speed isincreased to a certain level, the transient effect during thefolding process cannot be ignored. At this point, the steady-state method will generate a large simulation error, and a tran-sient method is required to simulate the flight-folding process.
3.4. Effect of the c.g. Position.We know that the c.g. position isan important factor that affects the distribution of aerody-
namic forces on the wings. The distributed aerodynamicforces and corresponding moments also have an effect onthe hinge moments during the folding process. Therefore,studying the effect of the c.g. position on the abovementionedsimulation results is important. Considering the maneuver-ability of the folded wing configuration, we examine a widerange of the c.g. position. We adjust the fore and aft locationsof the fuselage counterweights so that the aircraft c.g. posi-tion varies from 2.12m to 2.80m (the aerodynamic centerof the extended and fully folded wing configurations is 2.47and 2.65m, respectively). Figure 11 depicts the time historyof the main dynamic parameters of the aircraft during thefolding process at three c.g. positions.
Figure 11 demonstrates that the c.g. position exhibitsgreat influence on the simulation. Given that the c.g. positionmoves aft, the angle of attack decreases, and the deflectionangle of aileron increases. This result is mainly due to the fol-lowing reasons. First, when the c.g. position moves aft, itrequires substantial aileron deflection to balance the momentin the pitch direction. Second, the downward aileron deflec-tion increases the lift coefficient of the outer wing, and a smallangle of attack can satisfy the balance of the normal load.Finally, the reduced angle of attack reduces the aerodynamicload of the fuselage and inner wing, at which point a consid-erable aerodynamic force will act on the outer wing, therebyresulting in an increase in the hinge moments of the inner
Time
Dynam
ic p
aram
eter
Folding time Stabilization time
Start folding
0
Stop folding
Equilibrium state
(a) Description of the time spent
5 s 10 s 20 s 30 s0
10
20
30
40
Folding speed
Tim
e (s)
Folding timeStabilization time
(b) Time spent
Figure 9: Time spent at different folding speeds.
0 20 40 60 80 100 1200
500
1000
1500
2000
2500
3000
Folding angle (deg)
Mom
ent (
N m
)
5 s 10 s
30 s Steady
(a) Hinge moment of the inner wing
0 20 40 60 80 100 1200
500
1000
1500
2000
Folding angle (deg)
Mom
ent (
N m
)
5 s 10 s
30 s Steady
(b) Hinge moment of the outer wing
Figure 10: Comparison of hinge moments at different folding speeds.
9International Journal of Aerospace Engineering
and outer wings. Interestingly, we found that for the hingemoment of the inner wing, unlike the three other sets of mea-surements, the changing trend with the folding angle changesdramatically when the c.g. position moves aft and fore. Theresults show an increasing trend for a fore location of thec.g. but a decreasing trend for an aft c.g. location. Here, weprovide an explanation. As mentioned previously, with thefolding of the wings, two important changes occur in theaerodynamic force that acts on the wings. When the c.g. posi-tion is located fore, the aerodynamic load on the aircraftmainly acts on the fuselage and the inner wing at the initialunfolded state. As the wings are folded, the aerodynamic loadof the outer wing increases sharply. At this time, the secondchange dominates, thus eventually increasing the hingemoment of the inner wing. When the c.g. position islocated aft, the aerodynamic load of the aircraft mainlyacts on the outer wing at the initial unfolding state. Theaerodynamic load on the outer wing increases limitedlywith the folding of the wings, but the arm to the hingesof the inner wing decreases. At this point, the first changedominates, thereby causing the hinge moment of the innerwing to continue to drop.
Figure 12 exhibits the effect of the c.g. position on themaximum hinge moments. At the c.g. position = 2 12m,the maximum hinge moments of the inner and outer wingsare the smallest at 2637 and 1762Nm, respectively. Whenthe c.g. position moves aft, the maximum hinge momentsof the inner and outer wings increase continuously. When
the c.g. position = 2 80m, the maximum hinge moments ofthe inner and outer wings are the largest at 3706 and2563Nm, correspondingly.
4. Conclusion
To simulate the flight-folding process and calculate the hingemoments of a folding wing aircraft, this study develops a cal-culation method of the AIC matrices in the time domain andconstructs an aeroelastic flight simulation platform based onADAMS. The effectiveness of the platform is verified bynumerical examples, and the influences of the folding rateand the c.g. position on the results are investigated. Results
10 15 20 25 30 35 40 45 500
2
4
6
8
10
Time (s)
Ang
le (d
eg)
c.g. = 2.12 m c.g. = 2.45 m c.g. = 2.80 m
(a) Angle of attack
c.g. = 2.12 m c.g. = 2.45 m c.g. = 2.80 m
10 15 20 25 30 35 40 45 50−10
−5
0
5
10
Time (s)
Ang
le (d
eg)
(b) Deflection angle of aileron
c.g. = 2.12 m c.g. = 2.45 m c.g. = 2.80 m
10 15 20 25 30 35 40 45 500
1000
2000
3000
4000
Time (s)
Mom
ent (
N m
)
(c) Hinge moment of the inner wing
c.g. = 2.12 m c.g. = 2.45 m c.g. = 2.80 m
10 15 20 25 30 35 40 45 500
1000
2000
3000
Time (s)
Mom
ent (
N m
)
(d) Hinge moment of the outer wing
Figure 11: Influence of the c.g. position on the response.
2.2 2.3 2.4 2.5 2.6 2.7 2.80
1000
2000
3000
4000
5000
c.g. position (m)
Max
imum
hin
ge m
omen
t (N
m)
Inner wingOuter wing
Figure 12: Influence of the c.g. position on the hinge moments.
10 International Journal of Aerospace Engineering
show that, for the fast folding process, the steady-state simu-lation errors of the hinge moments are substantially large,and a transient method is required to simulate the flight-folding process. In addition, the c.g. position considerablyaffects the hinge moments during the folding process. Giventhat the c.g. position moves aft, the maximum hingemoments of the inner and outer wings constantly increase.
Data Availability
The data used to support the findings of this study are avail-able from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grant No. 11472133).
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