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Integrated Aerodynamic and Trajectory Studies
of a Long-Range Morphing Missile
Siyu Hua,∗Xugang Wang,†Zhongyuan Wang,‡and Qi Chen§
Nanjing University of Science and Technology, 210094 Nanjing, People’s Republic of China
https://doi.org/10.2514/1.A35345
A morphing missile aims to get optimal flight performances at different flight conditions by a variable
aerodynamic shape according to the requirements of multiple missions. For a long-range missile wi th morphing
wings, an integrated method with a morphing geometry, an aerodynamic analysis, and a trajectory simulation
program was studied for a wider range. The morphing configuration was designed to improve aerodynamic
properties by variable-s weep wings. Those aerodyna mic features were then div ided into three parts, in cluding both
flight Mach numbers and lift-to-drag ratios through detailed aerodynamic analyses. Based on such classifications,
we first proposed the combination rules to morph by combining the maximal lift-to-drag ratio with the minimal
drag or the maximum pitching moment gradient coefficient. The trajectory simulation program used within the
gliding and morphing strategies was then launched. Our results demonstrated both strategies were beneficial for
extending the attack ranges of long-range missiles subject to the constraints of stabilities and variable flight
conditions. The gliding strategy provided 119.5–45.6% increases in range over the baseline geometry; the
morphing scheme also succeeded in increasing the range over the deployment of the gliding strategy by an
additional 20.7–46.8%.
Nomenclature
CD = drag coefficient
CD∕CL = drag-to-lift ratio
CL = lift coefficient
CM= pitching moment coefficient
Cma = pitching moment gradient coefficient
CN= total normal force
C0
D= drag coefficient derivations with zero angle of attack
Cα
L= lift coefficient derivations with angle of attack
Cδ
L= lift coefficient derivations with elevator defection
Cα
M= pitching moment coefficient derivations with angle
of attack
Cδ
M= pitching moment coefficient derivations with eleva-
tor deflection
C0
M= pitching moment coefficient derivations with zero
angle of attack
D= drag
g= gravity acceleration
k= induce drag coefficient
L= lift
Lref = reference length of a missile
M= pitching moments
Ma = Mach number
m= mass of a missile
mc= mass consuming ratio
m0= initial mass of a missile
ny= normal overload
P= engine thrust
Sref = reference area of a missile
t= time
Vig = initial gliding velocity
v= flight velocity
vf= final velocity
v0= initial flight velocity
XCF = center of pressure
XCG = center of gravity
XCP = stable margin
Xmax = attack range
x= range position
x0= initial range position
y= attitude position
y0= initial altitude position
α= angle of attack
δ= elevator defection
θ= inclination angle
θ0= initial inclination angle
ρ= air density
χ= sweep angle
Subscripts
bal = balance
CF = neutral point
CG = center of gravity
CP = center of pressure
c= consume
f= final
ig = initial gliding
ref = reference
I. Introduction
AS THE future battlefield and combat mode change, a new
generation of aircraft should have the autonomous capability
of performing a variety of tasks from takeoff, landing, cruising,
maneuvering, hover, and attack under a highly variable flight envi-
ronment [1]. An aircraft can be well suited for a cruise or a dash
mission leg, yet it would suffer a performance penalty when operat-
ing in a loiter mission leg [2]. Similarly, a missile equipped with
subsonic wings will suffer under supersonic flight conditions. Con-
fronting those challenges, a fixed-wing missile does not easily meet
the mission requirements in the same way a morphing missile does.
This research actively pursues all aspects of morphing technologies,
Received 8 January 2022; revision received 19 May 2022; accepted for
publication 16 July 2022; published online 15 August 2022. Copyright ©
2022 by the American Institute of Aeronautics and Astronautics, Inc. All
rights reserved. All requests for copying and permission to reprint should
be submitted to CCC at www.copyright.com; employ the eISSN 1533-
6794 to initiate your request. See also AIAA Rights and Permissions
www.aiaa.org/randp.
*Ph.D. Student, School of Energy and Power Engineering, 200 Xiaoling-
wei, Jiangsu Province.
†Professor, School of Energy and Power Engineering, 200 Xiaolingwei,
Jiangsu Province (Corresponding Author).
‡Professor, Ballistic Research Laboratory, 200 Xiaolingwei, Jiangsu
Province.
§Associate Professor, School of Energy and Power Engineering, 200
Xiaolingwei, Jiangsu Province.
Article in Advance / 1
JOURNAL OF SPACECRAFT AND ROCKETS
Downloaded by NANJING UNIVERSITY OF SCIENCE on August 22, 2022 | http://arc.aiaa.org | DOI: 10.2514/1.A35345
including morphing configuration, aerodynamic analysis, and trajec-
tory aspects [3].
Many aircraft designers aim to achieve developing morphing
configurations. Available ideas were provided for them by a sum-
mary of how changing a wing parameter affects the flight perfor-
mance of an aircraft [4]. Taking variable-sweep wings as an example,
changing the sweep angle can increase critical Mach numbers,
decrease high-speed drag, or increase the maximal lift to obtain better
lift and drag properties, which have a great effect on the final
trajectory performances. Thus, morphing strategies have received
much attention and intensive study by loads of scholars to make
efforts to work out beneficial solutions on how a morphing configu-
ration should start morphing.
On the one hand, the aerodynamic analysis may be performed to
figure out some beneficial morphing rules.
Through the computational fluid dynamics calculation of aerody-
namic characteristics for a morphing missile, the lift and drag proper-
ties of variable-sweep-wing configurations were summarized under
subsonic and supersonic flight conditions, respectively [5,6]. Some
conclusions were drawn: in subsonic flight, a smaller sweep angle
increased the lift-to-drag ratio; and as Mach numbers rose, the larger
sweep angle increased the lift-to-drag ratio when 1<Ma ≤3. Those
conclusions may provide one morphing strategy in which varying
sweep angles need to be matched with flight Mach numbers. Apart
from the lift and drag properties, the pitching moment gradient
coefficient of a morphing wave rider with variable-sweep wings
was analyzed in both low and high velocities [7]. Through computing
the aerodynamic coefficients of a morphing unmanned aerial vehicle
with variable-span and -sweep-angle wings at various altitudes, the
most efficient morphing configurations were found during all flight
phases [8]. This research indicates another morphing strategy in
which varying sweep angles or spans may be matched with flight
altitudes.
On the other hand, a trajectory simulation program is performed to
testify to the effectiveness of morphing strategies: that is, to evaluate
whether morphing rules improve the final trajectory performances
according to their requirements for flight missions, which are coupled
with aerodynamic analyses and flight conditions. There were some
effective morphing strategies after trajectory simulations that suc-
ceeded in reducing computing time and increasing the final velocity
and the attack range [9–12].
For a morphing configuration, trajectory simulations were per-
formed by adopting the morphing strategy of matching sweep angles
with flight velocities and using the multifidelity kriging model to
maximize the lift-to-drag ratio, respectively. Their results indicated
that morphing rules were more efficient than trajectory optimization
by reducing 86% of the computing time [9]. Considering the lift-to-
drag ratio, a morphing strategy combined with offline trajectory
optimization in the diving phase increased the final velocity of a
morphing missile with variable-sweep wings [10]. In the gliding
phase, morphing combined with trajectory optimization also
increased the final velocity [11]. For a hypersonic missile, a combi-
nation of morphing and gliding schemes achieved a 9.5% increase in
range over a fixed-wing configuration [12].
Previous studies demonstrated morphing technologies had scien-
tific significance and research value for a missile or an aircraft
through the design of a morphing configuration, aerodynamic analy-
sis, and trajectory simulation.
To the best of our knowledge, for morphing missiles with vari-
able-swee p wings, most of the curr ent research was on aer odynamic
characteristics and morphing strategies under subsonic or super-
sonic conditions. The research focused on lift and drag properties in
a flight trajectory. There was little evidence that showed morphing
rules were related to the pitching moment gradient coefficient under
transonic flight conditions. To be specific, when a morphing missile
flies under different trajectory phases and different Mach numbers,
morphing strategies were studied in consideration of the lift-to-drag
ratio or by using trajectory optimization with the maximal lift-to-
drag ratio and other constraints to achieve morphing flight.
Drawing on the achievements of previous studies, this work
takes a long-range morphing missile as a subject and proposes a
novel integrated method of aerodynamic and trajectory studies to
increase the attack range. The combination rules to morph are first
proposed through studying morphing configuration, gliding, and
morphing strategies in the gliding phase; that is, according to the
flight Mach numbers of a designed morphing missile ranging from
subsonic to supersonic speeds through aerodynamic analysis, it
proposes the maximal lift-to-drag ratio, combinations of the maxi-
mal lift-to-drag ratio, and the pitching moment gradient coefficient
(or the minimal drag coefficient under those three conditions by
dividing the Mach number and lift-to-drag ratio into three parts).
This provides a solution to the coupling problem of aerodynamic
analysis, gliding, and morphing strategies. This idea is available for
the morphing design of long-range morphing missiles in the glid-
ing phase.
The remainder of our work is organized as follows:
Section II introduces our subject; expresses morphing problems
under assumed mission requirements; and provides a state-of-the-art
morphing process, including geometry design, aerodynamic analy-
sis, and trajectory simulation schemes. Section III proposes combi-
nation rules to morph through aerodynamic analyses for all the
morphing configurations at different flight conditions. Section IV
presents a trajectory model. Section V shows the numerical results of
all of the trajectories and analyses. Concluding remarks are provided
in Sec. VI.
II. Morphing Problem
A. Morphing Missile Subject
The design of configuration changes for a missile has a direct
effect on the aerodynamic and trajectory performances. Between
stability and control ability, we should make a compromise accord-
ing to different flight stages. The recent results of flight trajectory
tests indicated that tailfin control flight vehicles performed better in
extending the attack range as compared to canard-control configu-
rations [13]. Using this experience for reference, our baseline
geometry can be the tailfin control type including pitching and
yaw motions. And, proposed morphing wings under asymmetric
sweep changes can achieve control of the roll motion. The maxi-
mum morphing configuration is as follows (Fig. 1).
Alleviating shock drag is benefici al to improving the lift and drag
properties for a long-range missile. So, we employed a nose shape
of the von Kármán type [14] and a critical airfoil of the NACA 0008
type [14,15]. Besides, we were partially motivated by Xiaoyu et al.
[9], who focused on the best aerodynamic performance through
changing axial positions and sweep angles of morphing wings; so,
we took the integrated wing position and stable properties for our
proposed morphing missile into consideration. The top idea of our
morphing strategies is limiting the morphing wing positions behind
the body gravity position. With the sweep angles increasing, the
center-of-pressure points of the morphing wings move afterward,
which lead to the distances increasing from the center-of-pressure
point to that of the gravity point. Under these conditions, the stable
Fig. 1 The maximum morphing configuration.
2Article in Advance / HUA ET AL.
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margin also increases; and the aircraft can keep stable while
morphing.
B. Problem Description
Both gliding control and a morphing strategy can increase the
attack range [2,16]. We mainly study extending the attack range to
meet the higher mission requirements of a morphing missile and
assume a trajectory determined by a certain flight mission including
preapogee and postapogee legs (Fig. 2). Under a preapogee phase, the
missile takes off from the ground with a higher initial speed by the
rocket pushing in a specific low-drag and zero-lift baseline configu-
ration, and then it climbs up to the apogee. Once the apogee is
reached, it flies into a postapogee phase where gliding and morphing
strategies are taken over by tailfins and morphing wings, respectively.
1. Mission Requirements
During the whole flight stage, subsonic, transonic and supersonic
aerodynamic performances need to be considered. More importantly,
how do we design morph wings? When does a missile start morph-
ing? What types of morphing configurations are reasonable? Those
are profound questions for morphing research if properly addressed.
For such a trajectory profile, the mission requirements are assumed as
follows:
1) The attack range Xmax must be over 190 km.
2) The initial gliding velocity Vig and the final velocity Vfmust be
over Mach numbers Ma of 1.5 and 0.4, respectively.
2. Aerodynamic Performance
This section aims to develop evaluation performances for morphing
missiles involving the lift-to-drag ratio, stability, and maneuverability.
a. Lift-to-Drag Ratio.The lift-to-drag ratio (CL∕CD) is required to
be maximal under stable conditions, and this can obtain the optimal
energy consumption to get a wider range.
b. Stability.Our gliding and morphing research requires the
morphing configurations to fly trajectories at various angles of attack
(AOAs) and sweep angles. Remaining stable is a must. The two
following performances are available to evaluate stabilities:
First, the stable margin XCP is defined as [14]
XCP XCF −XCG
Lref
(1)
Second, the pitch moment gradient coefficient Cma can be defined
as [17]
Cma ΔCM
ΔαCMjαΔα−CMjα−Δα
2Δα(2)
The sideslip angles are assumed to be zero, and roll effects are
neglected. The pitch moment coefficient can be expressed as
CMCNXCF −XCG.
c. Maneuverability.For a statically stable missile, the slope of the
pitching moment coefficient versus the AOA is negative, with an
aerodynamic center aft of the center of gravity; an increase in the
AOA (nose up) causesa negative incremental pitching moment (nose
down), which then tends to decrease the AOA; and control surface
deflections allow the missile to trim (CM0) at the desired AOA
[17]. Stability and control characteristics can be expressed as [18]
α
δbal
−Cδ
M
Cα
M
(3)
Equation (3) indicates a dependent relationship between stability
and maneuverability. The higher the maneuverability-to-stability
ratio of a missile, the better maneuverability it gets; and the lower
the maneuverability-to-stability ratio, the better stability it obtains.
So, a compromise between stability and control is needed.
C. Morphing Process
This section is intended to propose a state-of-the-art morphing
process. This is an integrated method of aerodynamic analysis and
trajectories. The whole diagram for that is depicted in Fig. 3.
There are four main steps of the proposed morphing procedure
involving the baseline geometry under the preapogee stage, specific
morphing configurations under postapogee phases, aerodynamics inter-
polation under whole trajectories, and numerical simulation schemes
and analyses. The last one is the most critical step introduced in Sec. V.
During this process, aerodynamic forces and moments need to be
calculated to provide inputs for solving trajectory equations itera-
tively. There are loads of work on aerodynamic calculations by using
computational fluid dynamics, which are unbearable, especially for a
morphing missile. However, the semiempirical method using the
Missile Datcom tool can fix this problem with considerable wind-
tunnel data [14], and many scholars succeed in the aerodynamic
modeling of morphing aircraft using this tool [18–22]. Compared
with a wind-tunnel method, Missile Datcom predicted a normal force
with errors of 7 and 3% for Mach 1.42 and 3.08, respectively [18]. In
our work, by considering the calculation time and efficiency, Missile
Datcom is necessary to get a quick and economical aerodynamic
estimation for morphing missiles in preliminary design.
Launch
Start gliding
Stretch wings
Climb
Postapogee
Preapogee
Strike a target
Morph
Fig. 2 Illustration of a trajectory profile.
Article in Advance / HUA ET AL. 3
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III. Aerodynamic Analyses and Morphing Rules
A. Aerodynamic Performances Analyses
The aerodynamic performances can help us to estimate the rea-
sonability of proposed configurations under δ−10 deg. A sweep
angle represents a type of configuration. There are seven morphing
wings with sweep angles from 0 to 65 deg and one baseline geometry
with an angle of 90 deg, which were all analyzed under different flight
conditions (Figs. 4–6).
There are three aerodynamic performances analyzed for estimat-
ing whether a morphing geometry is preferable or not under the
postapogee phase (sweep 0–65 deg) and preapogee phase
(sweep 90 deg). Because the mission requirement of a certain
missile varies under different flight trajectories, corresponding
aerodynamic requirements have some radical differences between
the preapogee and postapogee stages (Table 1).
Design baseline geometry and test
aerodynamic performance
Aerodynamic
requirements at preapogee
phase
NO Adjust body and
tail’s shape and size
YES
NO
YES
Select morphing forms and calculate
aerodynamic performance among
morphing configurations
Adjust the size of
morphing wings
Aerodynamic
requirements at postapogee
phase
Make interpolation tables of preapogee
trajectory with baseline shape and
postapogee phase with morphing shapes
Set gliding and morphing strategies with
postapogee configurations and
perform trajectory simulations to
show their effectiveness
Start
End
Mission requirements
YES
NO
Fig. 3 A diagram for a morphing missile.
a) Ma=0.6
b) Ma=0.9
c) Ma=1.5
Fig. 4 The stable margins of all configurations.
4Article in Advance / HUA ET AL.
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B. Aerodynamic Coefficients’Analyses
Aerodynamic coefficients commonly indicate flight properties
that enable us to set both gliding and morphing trajectories that are
reasonable. For a longitudinal motion, the drag, the lift, and the
pitching moment gradient coefficients are more important. Those
aerodynamic characteristics were analyzed in detail, including the
lift-to-drag ratio (CL/CD) (Fig. 7), the drag coefficient CD (Fig. 8),
the lift coefficient CL (Fig. 9), as well as the pitch moment gradient
a) Ma=0.6
b) Ma=0.9
c) Ma=1.5
Fig. 5 The lift-to-drag ratios of all configurations.
a) Ma=0.6
b) Ma=0.9
c) Ma=1.5
Fig. 6 The maneuverability-to-stability ratios of all configurations.
Article in Advance / HUA ET AL. 5
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coefficient Cma (Fig. 10). More important, this work can guide our
morphing rules along the gliding trajectories.
In Figs. 7–10, the Mach numbers’increase leads to CL∕CD loss;
when Mach numbers increase, other aerodynamic coefficients go up
first and then fall; and when the sweep angle goes up, both CL and
CD have a downward trend. Differently, CL∕CD has some fluctua-
tions and Cma changes irregularly. Those data are shown in Table 2.
The figure for the lift-to-drag ratio (CL/CD) can be over six (and
even up to nine) when Mach numbers are less than Mach 0.8 (Fig. 7).
The lift-to-drag ratio (CL/CD) is the most important morphing
reason. CL/CD ranges are different under different Mach numbers for
the proposed morphing subject. Under subsonic flight conditions,
there is the maximal CL/CD. But, under supersonic flight conditions,
there are minimal CL/CD; and under transonic flight conditions, the
values are irregular. Because of this, it is not reasonable for morphing
to only depend on the maximal CL/CD. Selecting integrated morph-
ing rules is a must, and this is our creative point and method.
C. Morphing Rules
Partially motivated by the results of Ryan and Lewis [2] and Ryan
[16], their trajectory optimization results indicated that classifying
Mach ranges into two sections is necessary for a morphing aircraft.
Differently, we divided Mach numbers into three pieces (subsonic,
transonic, and supersonic flight conditions) according to flight prop-
erties ranging from subsonic to supersonic speeds of a designed
morphing missile through aerodynamic analyses from different
aspects: not depending on optimization method. Also, we not only
did Mach number classification but lift-to-drag classification.
Such combination rules to morph are first proposed as follows.
They can be divided into three ranges.
Table 1 Aerodynamic performances
Trajectory phase XCP ,% CL∕CD α∕δbal
Preapogee >8—— ——
Postapogee >4>6≈0.5
Fig. 7 The lift-to-drag ratios.
Fig. 8 The drag coefficients.
Fig. 9 The lift coefficients.
Fig. 10 Pitch moment gradient coefficients.
Table 2 Effects of Mach numbers and sweep angles on longitudinal
aerodynamic characters for morphing configurations
Aerodynamic
property When Mach increases When sweep increases
CL/CD Decreases Decreases first and then
fluctuates lastly keeps
stable
CD Increases first and then decreases Decreases
CL Increases first and then decreases Decreases
Cma Increases first and then decreases ——
6Article in Advance / HUA ET AL.
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1. First Range in CL∕CD > 6and Ma ≤0.8
When the CL/CD is over six, a morphing missile can get better drag
and lift properties at subsonic conditions according to the maximal
CL/CD.
2. Second Range in 4<CL∕CD ≤6and 0.8 <Ma ≤1.2
At transonic conditions, when the CL/CD is from four to six, this
value is irregular. Under this flight phase, what a missile to morph is
according to needs to be reconsidered.
For such a problem, the drag coefficient CD, the lift coefficient CL,
or the pitch moment gradient coefficient Cma would be compared.
Both the drag and lift coefficients change dramatically; the pitch
moment gradient coefficient Cma indicates the stabilities while
morphing. So, when the CL/CD is from four to six, the pitch moment
gradient coefficient Cma is also considered at these transonic
conditions.
3. Third Range in CL∕CD ≤4;Ma > 1.2
At supersonic conditions, the CL/CD are all below four; and their
figures are nearly the same with sweep angle changes. The maximal
CL/CD may be less worthy. The drag may havea dramatic increase in
such high flight velocities according to Eq. (6). When the CL/CD is
below four, the drag coefficient CD is also considered at transonic
flight velocities.
The figure for the drag coefficient CD keeps stable when Mach
numbers are below Mach 0.8. But it goes up sharply when Mach
numbers are from 0.8 to 1.2. If the Mach numbers are over 1.2, it falls
rapidly. Besides, a smaller sweep angle causes a lower drag (Fig. 8).
The lift coefficient CL has such a similar trend. Morphing wings
with variable-sweep angles have a critical impact on lift character-
istics (Fig. 9). With sweep angles going up, a morphing missile has
some lift to lose.
The figure for the pitch moment gradient coefficient Cma remains
negative (Fig. 10). The morphing configuration with a sweep angle
of 45 deg has the higher pitching moment gradient coefficient
Cma, which is slightly ranging from 0.09 to 0.12. So, this morph-
ing configuration can be considered under transonic flight
conditions.
IV. Aerodynamic and Trajectory Modeling
A morphing aircraft can be regarded as a controllable particle [9].
Totestify to the proposed gliding and morphing strategies, the motion
of the mass center at the longitudinal plane is studied. The trajectory
equations at the vertical surface are written as
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
mdv
dt Pcos α−D−mg sin θ
mv dθ
dt Psin αL−mg cos θ
dx
dt vcos θ
dy
dt vsin θ
dm
dt −mc
(4)
The aerodynamic coefficients CL;C
D, and CMare mainly influ-
enced by the AOAs and Mach numbers. According to Eqs. (5) and
(6), the aerodynamic forces and moments can be provided as the
inputs of a whole trajectory:
0
B
@
CDC0
D1kα2
CLCα
LαCδ
Lδ
CMC0
MCα
MαCδ
Mδ
(5)
0
B
B
B
B
B
B
@
D1
2ρv2Sref CD
L1
2ρv2Sref CL
M1
2ρv2Sref Lref CM
(6)
V. Results and Discussion
A. Simulation Conditions
This section aims to testify to whether proposed gliding and
morphing strategies succeed in extending the attack range. To assure
the reasonability and effectiveness, some simulation limitations and
the initial conditions are provided in Eq. (7):
0
B
B
B
B
B
B
B
B
B
@
x00m;y
00m
v0800 m∕s;θ055°
m0100 kg;m
c0.1 kg∕s
nymax 4g
−8°≤α≤8°
−15°≤δ≤15°
(7)
B. Analysis and Simulation
There are three cases (case 1, case 2, and case 3) intended to verify
the effectiveness of gliding strategies and morphing rules; this is one
step of the novel morphing process in Fig. 3. Among that, case 1 is the
baseline trajectory without gliding and morphing; case 2 runs with
fixed-wing configurations with just gliding; and case 3 runs with a
morphing configuration with gliding and morphing.
1. Case 1: No Gliding and No Morphing
Case 1 assumes a trajectory that runs with a baseline geometry in a
specific “low-drag and zero-lift”configuration, shown in Fig. 11.
Case 1 simply launches the low-drag and zero-lift configuration.
And, it flies in such a baseline geometry without gliding control
during the whole flight trajectory. The results of the baseline trajec-
tories are presented in Figs. 12 and 13.
Along the baseline trajectory, the attack range is about 78 km and
the maximal altitude is about 25.74 km (Fig. 12). The flight Mach
number at the apogee is about Mach 1.81, where the start of gliding is
so beneficial in extending the attack range in case 2 and case 3.
Postapogee
Preapogee
Range
Altitude
Fig. 11 Example of case 1 trajectory.
Article in Advance / HUA ET AL. 7
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2. Case 2: Gliding but No Morphing
Case 2 assumes a trajectory that runs with a baseline missile in a
specific low-drag and zero-lift configuration before the apogee. Once
it reaches this point, fixed wings stretch, and gliding strategies take
over the trajectory control by tailfins until it is on the ground (Fig. 14).
Those six fixed configurations were tested; and their sweep angles
were 0, 10, 20, 30, 45, and 60 deg.
Case 2 aims to testify to the effectiveness of the proposed gliding
strategies by evaluating increases in the attack ranges. Besides, some
trajectory properties of different postapogee configurations can guide
us to set more effective morphing trajectories in case 3. The trajectory
results of case 2 are depicted in Figs. 15 and 16.
Those six fixed-wing configurations were tested as postapogee
ones. Their attack ranges were all between 169 and 188 km; all of
them that were under gliding control had dramatic increases in attack
ranges from about 92 to 111 km (Figs. 15 and 16).
Among those configurations, the one with a sweep angle of 45 deg
had a wider range of 188 km (Fig. 15). Besides, the postapogee
configurations with sweep angles of 60 and 45 deg suffered from
more gliding ranges under transonic flight conditions (Fig. 16).
3. Case 3: Gliding and Morphing
Case 3 assumes a trajectory that begins with a baseline missile in a
specific low-drag and zero-lift configuration before the apogee. Once
it reaches this point, both gliding and morphing strategies take over
the trajectory control by tailfins working and morphing wings
stretching through providing the commands of elevator defection
and sweep angle until it is on the ground. The morphing configura-
tions are tested with the variable-sweep angles from 0 to
65 deg (Fig. 17).
Both gliding and morphing can extend the attack range [2,16]. In
general, a morphing missile starts gliding depending on the tailfins.
Once it reaches the apogee, the tailfins work and bring an elevator
deflection. Then, this elevator defection can change aerodynamic
coefficients, which can bring another trimmed AOA.
Proposed morphing configurations start morphing by relying on
their morphing wings with sweep angle, wing area, average chord,
and span changes. When morphing wings stretch out, both the wing
area and span increase; and the sweep angle goes down. Those
generate the additional lift of morphing configurations. At the same
time, other aerodynamics properties under different flight Mach
numbers can be improved effectively. A study on morphing rules is
intended to figure out how to change wings according to what flight
conditions can extend the attack range more. By selecting the optimal
morphing configurations with the maximal lift-to-drag ratio, the
minimal drag, or the maximal pitch moment gradient coefficient
under different flight conditions, a morphing missile could get the
optimal aerodynamic properties and provide the widest attack range.
The proposed morphing rules have been figured out in Sec. III.
The main idea is that a morphing missile can fly in the optimal
morphing configuration under subsonic, transonic, and supersonic
flight conditions (Table 3). The simulation results are demonstrated in
Fig. 12 Case 1 baseline trajectory (altitude vs range).
Fig. 13 Case 1 baseline trajectory (Mach number vs range).
Glide and stretch fixed wings
Postapogee
Preapogee
Range
Altitude
Fig. 14 Example of case 2 trajectory.
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Fig. 15 Case 2 trajectories in fixed-wing configurations (altitude vs range).
Fig. 16 Case 2 trajectories in fixed wings configurations (Mach number vs range).
Glide and stretch morphing wings
Postapogee
Preapogee
Range
Altitude
Fig. 17 Example of case 3 trajectory.
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Figs. 18 and 19. The proposed morphing results are indicated clearly
and marked with circles (○). There were some other morphing strat-
egies [2,9,16] for comparing tests with our simulation examples
(Tables 4 and 5). Their trajectory results are marked with dots (⋅)
and tick marks (j), respectively.
In Table 4, their optimal results showed that morphing only
depended on the maximal CL∕CD and the divided ranges of Mach
numbers into two parts. In Table 5, their morphing rules also relied on
the maximal CL∕CD and divided ranges of Mach numbers into
four parts.
Differently, our proposed morphing rules have been figured out:
not based on optimal methods but based on an integrated aerody-
namic analysis and trajectory tests. The creative point is selecting the
“optimal”configurations by dividing both Mach numbers and the
CL∕CD into three parts. The specific meaning of optimal in this
paper is not just a single rule (the maximal CL∕CD). We first
proposed integrated morphing rules (combinations of the maximal
CL∕CD with the minimal CD or the maximal Cma).
Table 3 Proposed morphing rules with three Mach
numbers and three CL∕CD ranges
Mach CL/CD Morphing rules Optimum shape
0.5–0.8 6–10 CL∕CDmax Sweep 10 deg
0.8–1.2 4–6CL∕CDmax and jCmajmax Sweep 45 deg
1.2–22–4CL∕CDmax and CDmin Sweep 30 deg
Proposed
Method in Ref. [2, 16]
Method in Ref. [9]
Fig. 19 Case 3 trajectories in morphing configuration (Mach number vs range).
Proposed
Method in Ref. [2, 16]
Method in Ref. [9]
Fig. 18 Case 3 trajectories in morphing configurations (altitude vs range).
Table 4 Morphing rules with two Mach ranges
[2,16]
Mach CL/CD Morphing rules Optimum shape
0.5–1.1 —— CL∕CDmax Sweep 0 deg
1.1–2.5 —— CL∕CDmax Sweep 20 deg
Table 5 Morphing rules with multiple Mach
ranges [9]
Mach CL/CD Morphing rules Optimum shape
0.5–1.1 —— CL∕CDmax Sweep 0 deg
1.1–1.2 —— CL∕CDmax Sweep 10 deg
1.2–2—— CL∕CDmax Sweep 20 deg
2–2.5 —— CL∕CDmax Sweep 45 deg
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In Figs. 18 and 19, our trajectory results with the proposed morph-
ing strategy are analyzed. At the preapogee leg, case 3 runs in the
same baseline geometry (blue). At the postapogee phase, the initial
gliding velocity is under supersonic flight conditions (magenta); at
the same time, it stretches morphing wings with a sweep angle of
30 deg at first. Then, with flight time goes by and with energy
consumes, it loses some velocity under transonic flight conditions
(black); and morphing wings with a sweep angle of 45 deg are
deployed to keep gliding. When it runs under the subsonic conditions
(cyan), morphing wings stretch the wings with a sweep angle of
10 deg until it is on the ground.
The trajectory performances of case 1, case 2, and case 3 are given
in Table 6.
Our proposed morphing method enables the widest attack range of
205 km, and it meets the mission requirements.
Comparing case 2 with case 1, we figured out that the proposed
gliding strategies succeed in extending attack ranges from 92
to 112 km.
By comparing case 2 with case 3, the morphing rules in Refs. [2,9]
can extend the additional attack ranges from 8 to 27 km and 0 to
18 km, respectively. In contrast, our proposed morphing strategy has
extended more attack ranges from 16 to 36 km.
VI. Conclusions
The primary objective of this research is to develop a novel
morphing process along gliding trajectories for long-range morphing
missiles in a preliminary design including the design of morphing
configuration, study of aerodynamic analysis, and trajectory
research. To achieve this, the current work makes combinations of
aerodynamic and trajectory properties. To work out a clear and
specific solution on how to morph and when to start morphing, many
complex tradeoffs through the iteration process are needed.
A. Design of Morphing Configuration
The top idea of the design of the morphing configurations is taking
both the wing position and stability into consideration by setting
morphing wings behind the body’s gravity position. This improved
the iterative efficiency. To reach better aerodynamic performance, a
NACA 0008 type was employed as a critical airfoil. This airfoil not
only alleviated the shocking drag led by the morphing wings but also
increased the lift-to-drag ratio dramatically under subsonic flight
conditions.
B. Study of Aerodynamic Analysis
The aerodynamic properties of different morphing configurations
with variable-sweep wings were analyzed in detail at various flight
Mach numbers. It was found that the lift-to-drag ratio was quite
distinct at different flight Mach numbers; under transonic and super-
sonic flight conditions, it was not well matched with the morphing
rules. A missile to morph needs the integrated rules. Combinations of
the maximal lift-to-drag ratio and the minimum drag coefficient
under supersonic flight conditions or integrating the maximal lift-
to-drag ratio with the maximal pitching moment gradient coefficient
under transonic flight conditions were first proposed in this work.
C. Trajectory Research
To testify to the proposed morphing strategy along gliding trajec-
tories, three simulation cases were carried out. The results showed
they were beneficial for extending the attack range. Both gliding
and morphing strategies were effective for extending the attack
ranges of a morphing missile with 119.5–145.6% increases in the
attack range and an additional 20.7–46.8% increase in the attack
range, respectively.
Acknowledgments
This work has mainly received funding from the Fundamental
Research Funds for the Central Universities (number 30919011401),
and this was partially supported by the Natural Science Foundation of
the Jiangsu Province (number BK20200498). The authors would like
to thank the reviewers for their detailed technical comments and for
helping make this paper clearer.
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