multibody
Multibody Dynamic Modeling and Simulation of a Tailless
Folding Wing Morphing Aircraft
Ting Yue * and Lixin Wang ?
School of Aeronautic Science and Engineering Beijing University of Aeronautics and Astronautics
Beijing, China, 100191
Junqiang Ai ?
The First Aircraft Institute of AVIC-I, Xi’an, China, 710089
The purpose of this paper is to model and simulate a tailless folding wing morphing aircraft in wing folding process. The wing area and other parameters of morphing aircraft largely change when morphing, which will affect the aerodynamic force acting on the aircraft and lead to a variation of movements in the morphing process. The dynamic model of tailless folding wing morphing aircraft in wing folding process was founded and the six-DOF nonlinear equation was deduced. Furthermore, the decoupled longitudinal dynamic equation of morphing process was presented by simplifying. And the result of numeric simulation of aerodynamic force in wing folding process shows that the aerodynamic force is approximate to the steady state. The longitudinal response of morphing process in different wing fold angular velocities was numerically simulated by quasi-steady aerodynamic assumption when taking no account of the unsteady aerodynamic effect in small wing fold angular velocities. And the key factors which affect the dynamical characteristics of morphing process of the tailless folding wing morphing aircraft were investigated by quantity. The research results can be as great reference for the flight control system design in morphing and evaluating the morphing flight safety at low altitude of morphing aircraft.
Nomenclature
0D C = drag coefficient at zero
L C DV C =
?/()D C V ??D C α = /D C α?? 0L C = lift coefficient at zero α
LV C =
?/()L C V ??Lq C =
?/L C q ??L C α = /L C α??
*
Ph.D. Student, AIAA Student Member, School of Aeronautic Science and Engineering, email: yueting_buaa@https://www.sodocs.net/doc/c69402192.html, ?
Professor, School of Aeronautic Science and Engineering, email: bhu_wlx@https://www.sodocs.net/doc/c69402192.html, ?
Researcher
1
AIAA Atmospheric Flight Mechanics Conference 10 - 13 August 2009, Chicago, Illinois
AIAA 2009-6155
L C α =
?/L C α?? L C δ = /L C δ?? 0m C = pitching-moment coefficient at zero α
mV C =
?/()m C V ??mq C
= damp in pitch, ?/m C q ??m C α = /m C α??
m C α =
?/m C α?? m C δ = /m C δ??
c
= length of mean aerodynamic chord D = drag F = force acting on aircraft w G = wing gravity
g
= acceleration due to gravity Η = moment of momentum vector I = inertia moment of an aircraft about the origin i I = inertia moment of the wings about their own gravity center L = lift M = moment acting on aircraft m = aircraft mass, kg b b b b o x y z = body coordinates
g g g g o x y z = ground
coordinates p = momentum
vector q
= dynamic pressure or pitch angular velocity S = wing area S = static moment of an aircraft about the origin T = thrust V = velocity of aircraft
α = angle of attack θ = pitch angle
ω = rotational angular velocity body coordinates relative to ground coordinates i ω = rotational angular velocity of the about their own rotating shaft
fold ω
= wing fold angular velocity M Δ = additional pitching moment caused by wing’s center of gravity shift w z Δ = wing center of gravity shift along axis
b oz δ = control surface deflection angle T ?
= angle between thrust and
b b o x
Superscripts
(.) =
/d dt (..) = 2
2
/d dt
Subscripts
2
A =
aerodynamic
i=0,1,2,3,4 = fuselage, right inner and outer wing, left inner and outer wing
thrust
T =
x y z= components of respective variable in body coordinates
,,
I.Introduction
M
orphing is that the aircraft can adapt the different flight environments and combat missions by changing the aerodynamic configuration, using advanced materials and actuators.1,2For example, the morphing aircraft can have big wing span and area in cruise flight or take off, and small wing span and area in high speed. So compared with conventional configuration aircraft, morphing aircraft has multiobjected mission adaptability and higher combat efficiency.
Morphing aircraft mainly change the aerodynamic configuration by wing morphing. And when the wings are morphing, some of the aircraft’s parameters such as wing area, moments of inertia and center of gravity will change. Moreover, the aerodynamic force acting on the aircraft also change, which will affect the movement of the morphing aircraft. So wing morphing is a very important process for morphing aircraft. If some kinetic parameters of the morphing aircraft change unexpectedly, the flying quality may be affected and will even threaten the flight safety of the morphing aircraft. So it is necessary to analysis the dynamic response of morphing aircraft in the morphing process.
The wing morphing process of morphing aircraft is similar with the process of changing the sweep angle of variable swept wing aircraft. Ref. 3 investigated the dynamic response of a variable swept wing aircraft in the course of changing the sweep angle. But for morphing aircraft, the range of parameters changing will be wider. In the wing morphing process, the morphing aircraft should be regarded as a multibody system which is composed by the fuselage and several moving parts of wing. Thomas M Seigle4 analyzed the modeling of large-scale morphing aircraft in theory. However, few studies have been done on the modeling and simulation of a specific morphing aircraft. This paper focuses on modeling and simulation of a tailless folding wing morphing aircraft during the wing fold process, which can be based as the flight control design for morphing aircraft.
II.Multibody Dynamic Modeling
A.Tailless Folding Wing Morphing Aircraft
For a tailless folding wing morphing aircraft investigated in this paper, the whole wings are composed of inner wings and outer wings, which can rotate by the axis that is parallel to the fuselage axes. Then the aircraft can be divided to three parts: the fuselage, inner wings and outer wings (see Fig. 1a). In the process of wing fold, the inner wings rotate and the outer wings keep level (see Fig. 1b). After the wing folds, the reference wing area
I; the center of gravity decreases; the inertia moment of aircraft changes but mainly presents the decrease of
x
shift upward and the aerodynamic center shifts back.
3
4
a) Components of folding wing.
inner wing outer wing
b) Wing fold process
Figure 1. Tailless folding wing morphing aircraft.
B. Muitibody Dynamic Modeling of Tailless Folding Wing Morphing Aircraft
In the wing morphing process of morphing aircraft, the movement of wings will cause a shift of the aircraft’s center of gravity. So in this paper the origin of body coordinate system is located at a certain fixed point on the aircraft’s fuselage, and the ground coordinate system is described as b b b b o x y z g g g g o x y z (see Fig. 2). When
the aircraft’s aerodynamic configuration is changing, the fuselage and several moving parts of wings whose
center of gravities are described by
().
i c 1,i n =
…
b x
Figure 2. Coordinate axis system.
In the body coordinates
, the momentum and moment of momentum of morphing aircraft are 3
b b b b o x y z 11[]
n
i i i i i m m t δδ=?=++×??
i =×+?+×+???
∑p V S
ωS S ΗS V I ωS I ω (1) y g
Where x
xy zx xy
y yz yz yz
z I I I I I I I I I ??
????
=?????????
?
I (2) The origin of body coordinates is not located at the center of morphing aircraft, by the momentum law and moment of momentum law related to any moving point, so that 5
=??=+×?
F p
M H V S (3) By equations (1) and (3), the dynamic equations for a morphing aircraft with wing morphing can be expressed in vector form as
22122()2()()(){1
()[()]}n i i i i i i i i i i i i i t t t t t t t t m t t δδδδδδδδδδδδδδδδδδδδ=?=+×+×+×+××+??
??
=?
+?+×?+×+××+?+???
?+×?+×+××???
∑ωS S F m V
ωV S ωωωS ωI ωI V M I ωωI ωS S ωV I ωS S ωI ωS ωS
(4) In Eq. (4), each stands for the right inner wing, right outer wing, left inner wing and left outer wing. In the wing morphing process of tailless folding wing aircraft, the left and right wings rotate symmetrically, so the fold angular velocity of which becomes
1,2,3,4i =11113111x y z x y z ωωωωωω=++???
=?+???ωi j k
ωi j k
(5) Since the left and right outer wings always keep level in the folding process, their rotate angular velocities are zero.
The inner and outer wings’ static moments are also symmetric, so
1111222231114
222x y z x y z x y z x y z S S S S S S S S S S S S =++??
=++??
=?+??=?+?S i j k S i j k
S i j k S i j k
(6) And the wing fold angular velocity is constant, therefore 1y ω=0,1z ω=0,1x S =0,2x S =0,1x S =0 and 2x
S =0. Take Eqs. (5) and (6) into Eq. (4) then Eq.(4) reduced in to the form of the component as (7) and (8):
()2()(()2()(
()()()-))x x y z z y y z z y y z y x y y x z z x x z
y y z x x z z x x z x z z y z z y x x y y x z z x y y x x y y x x z x x z y y z z y z F m V V V S S S S S S S F m V V V S S S S S S S F m V V V S S S S S S S ωωωωωωωωωωωωωωωωωωωωωωωωω
ωωωωωωω?=+?+?++???=+?+??+????=+?+?+???+ ???? (7) 5
111222 ()()()2/2/()()()x x x zx z x x zx z y zx x z z z y y
y z z y y x y y x z z x x z z x z z x z
y y y y y z x x zx z x zx x z z z x x z z y z z y M I I I I I I I S V S V S V V S V V S S m m S S M I I I I w I I S V S V S V V ωωωωωωωωωωωωωωωωωωωωωωωω=?+?+?+?+?+???++?=++???++?+? 111222111222()2/2/()()()2/2/x x y y x x z x z z zx x z z zx x z z x y y y x x zx z x y y x x z x x z y y z z y x x z x x z S V V S S m m S S M I I I I I I I S V S V S V V S V V S S m m S S ωωωωωωωωωωωωωωωωω??
??????????=?+?++????+?+??????
- - (8)
Equations (7) and (8) show that there is additional effect factor in the equations which is the shift of center of gravity (caused by wing fold), which is the difference of equations between wing folding process of morphing aircraft and conventional configuration aircraft. In Eq.(7) the terms including x S , and mean the additional forces produced by the shift of aircraft’s CG position in wing folding; and the terms
including and mean the additional forces produced by the velocity and acceleration of aircraft’s CG position shift in wing folding. In Eq. (8) the terms including y S z S z S z
S x S , and mean the additional moments produced by the shift of aircraft’s CG position in wing folding; and the terms including first and second order of
(or ) mean the additional moments produced by the velocity and acceleration of inner wings’ (or outer wings’) CG position shift in wing folding. If the wings are fixed, which means (y S z S 1z S 2z S k S ,,,1,1,2,2k x y z x z x z =)
and its derivative are zero, Eqs. (7) and (8) are the same with the six-DOF nonlinear equations of conventional configuration aircraft. There is no effect of wing fold angular velocity in the equations that is because the wings of tailless folding wing morphing aircraft fold symmetrically by the axis and the effects counteract.
b b o x III. Longitudinal Dynami
c Response to Morphing
A. Simulation of Longitudinal Dynamic Response in Wing Folding
If only consider the longitudinal response in wing folding process, then decouple and simplify the equations (7) and (8), becomes
111222()2 ()()2/2/x x y z y y z
z z y x y y z y y y y y z x z y z z
x z x F m V V Sz S F m V V S S z
M I I S V S V S S m m S S ωωωωωωω
ωω?=+++?
=??+??
=+++????? - (9) Where
cos sin cos sin sin cos sin cos (cos sin )x T z A T x z F D L T mg F D L T mg M M M S S g
T αα?θαα?θθ=?++???
=???+??=??+?
θ (10) The fold of wings will produce unsteady aerodynamic force for tailless folding wing morphing aircraft. To investigate the unsteady aerodynamic effect in wing folding process, the aircraft’s aerodynamic performances in different wing fold angular velocities are numerically simulated, which is based on the method of non-structure dynamic overlap grid(see Fig.3).
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a) grids of surface b) grid section at x=1.9m
Figure 3. Non-structure overlaps grid system.
Figure 4 shows the lift, drag and pitching moment variety of morphing aircraft when the wing fold angular velocities are 100deg/s, 50deg/s, 25deg/s, 12.5deg/s and 5deg/s. It can be seen from Fig.4 that the lift, drag and pitching moment in different wing fold angular velocities are approximate with that of the steady state and approximately linear vary along with the wing fold angle. When the wing fold angle increases, as decrease of wing area and increase zero-lift pitching moment, the lift, drag and the nose-down pitching moment will decrease. The wing fold angle is bigger, the lift and the nose-down pitching moment decrease more. And when the wing fold angular velocities are 5deg/s and 100deg/s, the lifts of whose are only about 7% difference and the pitching moments are only 4% difference.
D r a g , N
Wing fold angle, deg
L i f t , N
Wing fold angle, deg
a) Lift b) Drag
7
-27000
-26000-25000-24000-23000-22000-21000-20000-19000P i t c h m o m e n t , N ·m
Wing fold angle, deg
c) pitching moment （reference by the origin of body coordinates ）
Figure 4. Morphing aircraft’s force and moment varieties in different wing fold angular velocities. The aircraft dynamic response affected by unsteady aerodynamic force can refer to the unsteady aerodynamic force contribution to the dynamic response of variable swept wing aircraft in the process of changing sweep angle. Ref. 3 compared the quasi-steady and unsteady aerodynamic characteristics of variable swept wing aircraft in wing sweep angular velocities by 8, 10, 15, 20deg/s, and the aircraft dynamic response results were rather similar. Consequently, when the wing fold angular velocity is not very big, the dynamic response affected by unsteady aerodynamic force caused in folding can be ignored.
So in this paper the unsteady aerodynamic effect is not considered and the aerodynamic force and moment are gained by quasi-steady assumption: the aerodynamic force of the morphing aircraft under some certain configuration in wing folding process nearly equals to that of corresponding static configuration. Then the lift, drag and pitching moment of the morphing aircraft in wing folding process can be expressed as
000(/)(//(2//(2/)//(2/)/(2/)D D DV L L LV Lq L L A m m mV mq m m D qS C C C V V L qS C C C V V C q V c C V c C M qSc C C C V V C q V c C V c C αααααδααδαδααδ?=++Δ?
=++Δ??
+++??
=++Δ??+++?
(11) The aerodynamic derivatives of the morphing aircraft in wing folding process can be linearly interpreted by those of static configuration which are calculated by CDF.
The center of gravity of tailless folding wing morphing aircraft shifts along the axis in wing folding process, so we can choose the gravity center of unfolding configuration as the origin of body coordinates when modeling and simulation. Figure 5 shows the dynamic response to wing fold under different wing fold angular velocities which are 5, 8 and 10 deg/s(＝10000m, b b o z H 0.8M =). The response of tailless folding wing morphing aircraft in wing folding can be divided into two phases: the main fold phase and fold completed phase.
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a) b)
d)
c)
e) f)
Fig.5 Morphing aircraft dynamic responses in different wing fold angular velocities.
1) The main fold phase
When the wing begins to fold, the initial response mainly represents the variation of pitching moment of the aircraft, and the aerodynamic forces acting on the aircraft changed not too much. The dynamic response mainly represents the variation of angle of attack, pitch angular velocity and pitch angle. The velocity and altitude varies limitedly. In this phase pitch angular velocity is minus and the numerical value increase, the reason is that the aerodynamic center of aircraft continues to shift back in wing folding, which will produce a nose-down pitching moment. When the wing area continues to decrease, the lift and drag will decrease. The altitude descends and the velocity increases as the thrust is bigger than the drag.
2) The fold completed phase
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After the wing fold completed, the dynamic response represents the variation of all parameters. When the wing stops folding, the aerodynamic center also stops shifting back, so the pitch angular velocity gradually restores. And the drag is still smaller than the thrust, which will cause the velocity increase. Moreover, the lift is not big enough to balance with the gravity of aircraft, so the altitude decreases quickly. Hereafter, the parameters will oscillate in long duration. Every parameter will converge to a stable value through long period of time and the aircraft will achieve a new balance to flight. Since the wing area decreases by wing fold, the new balanced velocity and angle of attack will be bigger.
Figure 5 also compares the dynamic response under different wing fold angular velocities. We can see that in bigger wing fold angular velocity the oscillation of parameters is more violent but the dynamic response of aircraft will not deteriorate. When the wing fold angular velocity is bigger, the parameters will restore faster after the wing fold completed. The reason is that the lift will restore faster in bigger wing fold angular velocity. And in a long time period the parameter’s difference is small in different wing fold angular velocities, so the tailless folding wing morphing aircraft can use bigger wing fold angular velocity to morph.
In the wing folding process of tailless folding wing aircraft, the altitude decreases a lot and needs a long time to restore. So it is dangerous to fold wing at low altitude without control. To ensure a good flying quality in the process of wing folding, the flight control system is needed.
B.Influences on Longitudinal Dynamic Response in Wing Folding
This section investigates the factors’ influence on the longitudinal dynamic response of morphing aircraft in wing folding process. Different from the dynamic equations of wing fixed configuration aircraft, the tailless folding wing morphing aircraft is a multibody dynamic system in the process of wing folding. The factors that influence on longitudinal response are the center of gravity shift (caused by wing shift), change of aerodynamic characteristics of the aircraft, et al.
1) Influence of center of gravity shift
The influences on dynamic response by the aircraft’s center of gravity shift are shown in Fig.6. The altitude is 10000m, Mach number is 0.8, and the wing fold angular velocity is 10deg/s. The dynamic response affected by the center of gravity shift can be seen from Fig.6 obviously. The reason is that the wing fold leads to the wing’s center of gravity shifts upward, which will cause an additional pitching moment by gravity related to the origin that fixed on the fuselage of aircraft when the pitch angle is not zero. And this consequently affects the dynamic response (see Fig.7). In another word, when the aircraft’s center of gravity shifts, the thrust will produce an additional pitching moment related to the center of gravity. However, although the shift of center of gravity is small, the additional pitching moment is big enough to cause a certain effect on dynamic response.
a)
10
b)
c)
d) e) Figure 6. Effects of center of gravity shift to response.
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Figure 7. Principle of additional pitching moment caused by wing’s center of gravity shift. 2) Influence of aircraft aerodynamic change
The main derivatives that represent the aircraft’s aerodynamic characteristic are , 0m C m C α and L C α. The influences of these derivatives on dynamic response in wing folding process can be analyzed by changing the varying range of these derivatives.
b x
M
Δθ
w
z Δb
z w
G wing’s center of gravity
origin of body coordinates
a)
b)
c)
d) e)
Figure8. Effects of aerodynamic derivative variation to morphing aircraft motion parameters. Figure 8 shows the dynamic response of tailless folding aircraft in wing folding process when enlarging the varying range of the derivatives , 0m C m C α and L C α by 10%. The aircraft movement is most influenced by
variety. The varieties of 0m C m C α and L C α influence the movement is almost. That means the variety of pitching moment in wing folding will largely affect the dynamic response. And the variation of m C αshows the
static longitudinal stability changes of morphing aircraft when wing folds, which is the cost of flight performance benefits. So in morphing aircraft design the variation of pitching moment should keep small variety based on flight performance benefit.
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IV.Conclusions
This paper has presented the modeling and simulation of a tailless folding wing morphing aircraft in the process of wing folding. The dynamic equations in wing morphing process including the additional effect of center of gravity shift were induced. And the simulation results showed that the oscillation of kinetic parameters were more violent under bigger fold angular velocity but would not deteriorate and it was dangerous to fold wing without control at low altitude. The dynamic response to wing folding mainly depends on the aerodynamic characteristics of the aircraft. The flight control system should be designed in wing morphing process for tailless folding wing morphing aircraft.
References
1Wilson, J.R., “Morphing UAVs change the shape of warfare”, Aerospace America, Vol. 42, No. 2, 2004, pp. 28-29.
2Moorhouse, D., Sanders, B., von Spakovsky, M. and Butt, J., “Benefits and design challenges of adaptive structures for morphing aircraft”, Aeronautical Journal, Vol. 110, No. 1105, 2006, pp.157-162.
3An, J.G., Yan, M., Zhou, W.B., “Aircraft dynamic response to variable wing sweep geometry”, Journal of Aircraft, Vol. 25, No.3, 1988, pp.216-221.
4Seigler, T.M., Neal, D.A, Bae J. S. and Inman, D.J., “Modeling and flight control of large-scale morphing aircraft”, Journal of Aircraft, Vol. 44, No. 4, 2007, pp1077-1087.
5Liu, Y.Z, Hong J.Z. and Yang H.X., Multibody system dynamics, Advanced Education Press, Shanghai, 1989(in Chinese)
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