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Research ArticleAerodynamic Performance of a Passive Pitching
Model on BionicFlapping Wing Micro Air Vehicles
Jinjing Hao, Jianghao Wu , and Yanlai Zhang
School of Transportation Science and Engineering, Beihang
University, Beijing 100191, China
Correspondence should be addressed to Yanlai Zhang;
[email protected]
Received 30 June 2019; Revised 15 November 2019; Accepted 23
November 2019; Published 18 December 2019
Academic Editor: Raimondo Penta
Copyright © 2019 Jinjing Hao et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
Reducing weight and increasing lift have been an important goal
of using flapping wing micro air vehicles (FWMAVs). However,FWMAVs
with mechanisms to limit the angle of attack (α) artificially by
active force cannot meet specific requirements. This studyapplies a
bioinspired model that passively imitates insects’ pitching wings
to resolve this problem. In this bionic passive pitchingmodel, the
wing root is equivalent to a torsional spring. α obtained by
solving the coupled dynamic equation is similar to that ofinsects
and exhibits a unique characteristic with two oscillated peaks
during the middle of the upstroke/downstroke under theinteraction
of aerodynamic, torsional, and inertial moments. Excess rigidity or
flexibility deteriorates the aerodynamic force andefficiency of the
passive pitching wing. With appropriate torsional stiffness,
passive pitching can maintain a high efficiency whileenhancing the
average lift by 10% than active pitching. This observation
corresponds to a clear enhancement in instantaneousforce and a more
concentrated leading edge vortex. This phenomenon can be attributed
to a vorticity moment whosecomponent in the lift direction grows at
a rapid speed. A novel bionic control strategy of this model is
also proposed. Similar tothe rest angle in insects, the rest angle
of the model is adjusted to generate a yaw moment around the wing
root without losinglift, which can assist to change the attitude
and trajectory of a FWMAV during flight. These findings may guide
us to deal withvarious conditions and requirements of FWMAV designs
and applications.
1. Introduction
The requirements for the design of flapping wing micro
airvehicles (FWMAVs) include excellent aerodynamic perfor-mance,
high efficiency, and satisfactory maneuverability.However,
balancing all these standards is difficult for existingFWMAVs.
Fortunately, flying creatures have been consid-ered as a basis for
proposing new innovations related to fly-ing. For example, insects
can manipulate their wings tocomplete a series of complex
movements, such as hovering,climbing, braking, accelerating, and
turning. Inspired bythese phenomena, researchers have attempted to
adopt thephysiological characteristics of insects and apply a
bionicmodel to artificial FWMAVs. Researchers have also con-ducted
a series of studies on this topic. For example, Ennos[1] stated
that torsion is necessary to design insect wingsbecause insects
have to twist their wings between wingbeatsto optimize the
performance of an aerofoil. Nevertheless,
the kinematic mechanism of insect wings is difficult to
fullyunderstand because of the complex structure of organisms.On
the one hand, this scenario is a typical type of a fluid-structure
coupling problem, and the interaction betweenwings and the unsteady
flow field generated during theirmovement is highly complicated. On
the other hand, themechanism through which insects control their
wingsinvolves numerous muscle structures and neural activitiesbut
remains poorly understood. Beatus and Cohen [2, 3]summarized this
intractable behavior by applying areduced-order approach in which
the wing hinge of insectsand fluid-structure interactions are
represented by simpli-fied models. Then, a passive pitching model
based on thetorque exerted by insects on their wings was proposed.
Inthis model, the wing root of an insect is equivalent to a
tor-sional spring [4]. The pitching dynamics of wings areassumed to
be passively determined by combining aerody-namic, torsional, and
inertial moments. Bergou et al. [5]
HindawiApplied Bionics and BiomechanicsVolume 2019, Article ID
1504310, 12 pageshttps://doi.org/10.1155/2019/1504310
https://orcid.org/0000-0002-1642-0910https://orcid.org/0000-0002-2823-2837https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/1504310
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also confirmed that pitching is passive by showing
thataerodynamic and inertial forces are sufficient to pitch awing
without the aid of muscles.
Numerous theories and experiments have shown that apassive
pitching model is generally accepted. Ishihara et al.[6, 7] applied
a novel fluid-structure interaction similaritylaw to two- and
three-dimensional wings and analyzed themotion of a passive
pitching wing through computationaland experimental methods. They
mainly discussed the con-tributions of a wing’s elastic,
aerodynamic, and inertial forcesand tried to find the important
control parameters of passivepitching motion. Chen et al. [8]
successfully used this passivepitching model to estimate
aerodynamic forces with quasis-teady and numerical methods. They
found that wings withstiff hinges achieve a favorable pitching
kinematic that leadsto large mean lift forces. This model is
applicable not onlyto a hovering state but also to a maneuvering
state. Beatusand Cohen [3] explained wing pitch modulation in
maneu-vering fruit flies by an interplay between aerodynamics anda
torsional spring. Zeyghami et al. [9] studied the passivepitching
of a flapping wing in turning flight and concludedthat passive wing
kinematic modulations are fast and ener-getically efficient.
Similarly, our study equated the wing’sflexibility to a torsional
spring at the wing root located closeto the leading edge. This
study is mainly aimed at determin-ing whether aerodynamic force and
efficiency could beimproved if we used this passive pitching model
to designFWMAVs and identifying whether the maneuverability
ofFWMAVs would be compromised.
In this study, we investigate the aerodynamic perfor-mance of a
FWMAVwith a Reynolds number of 104. A seriesof analyses is
conducted on the basis of a bionic passive pitch-ing model through
a 3D numerical simulation and a system-atic comparison among them.
To develop a desirableoutcome of a FWMAV design, we discuss the
effect of severaldominant parameters, such as torsional stiffness
and restangle of torsional spring, on aerodynamic performance.
Wefind that a FWMAV with passive pitching wings more likelyreduces
weight, increases lift, and shows great potential forflight
control.
2. Modeling and Method
2.1. Wing Model and Kinematics. Insect wings have adynamic
geometry. They are made of different materialsand exhibit varying
structures to adapt to different flightenvironments. In practical
applications, artificial wings can-not achieve the same effect as
insect wings. Consequently,simplifications are frequently adopted.
In this study, we usea rectangle to approximate a planar shape and
regard a flap-ping wing as a thin plate with a uniform density
(Figure 1).The reason why the rectangular model wings are used is
asfollows. Luo and Sun [10] have investigated the effect of
wingplanform on the aerodynamic force production of modelinsect
wings in rotating at Reynolds numbers 200 and 3500at an angle of
attack of 40° in 2005 and revealed that the var-iation in wing
shape and aspect ratio (from 2.84 to 5.45) hasminor effects on the
lift and drag coefficients. Based on theirconclusions, we neglected
the effect of planar shape and
focused on other important parameters such as torsionalstiffness
in this paper. Besides, the rectangular model winghas been
extensively used in many numerical simulations[7, 11], which can be
regarded as a typical case to illustratea universal conclusion.
To clearly describe the 3D motion of a flapping wing
andaccurately analyze its force, we establish two coordinate
sys-tems with the same origin located on the wing root(Figure 2).
The inertial system O‐XYZ is located on theground, whereas the OXY
plane is parallel to the horizontalplane. The OX axis is oriented
toward the trailing edge, theOZ axis is opposite to the direction
of gravity, and the OYaxis is determined on the basis of the
right-hand rule. Thecoordinate systemO‐xyz is fixed on the wing.Ox
andOy axesare along the chordwise and spanwise directions,
respec-tively. The Oz axis is determined on the basis of the
right-hand rule.
Insects generally have three degrees of freedom whilehovering.
The motion perpendicular to the flapping planeis relatively small
and frequently overlooked during simpli-fication. Therefore, the
motion of a wing can be approxi-mately decomposed into flapping and
pitching, which aredescribed by the flapping angle φ and the angle
of attackα, respectively. Flapping refers to the rotation around
theOZ axis, whereas pitching corresponds to the rotationaround the
Oy axis.
x c
�훥R b
R
r
R2
Crot
Figure 1: Geometric parameters of a flapping wing. b is
theunilateral wingspan, c is the mean chord length, crot is the
distancebetween the leading edge and the rotation axis, R is the
radius ofthe wing tip, ΔR is the distance between the wing root and
theflapping axis, and R2 is the radius of the second moment of
thewing area.
�훼
�훼∗
�휑
X
Y
Z
x
y
z
O(o)Torsion
Figure 2: Bioinspired passive pitching model and
coordinatesystem.
2 Applied Bionics and Biomechanics
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The flapping motion can be described by a trigonometricfunction
as follows:
_φ = π360Φ sin 2πTð Þ, ð1Þ
where Φ and T are the flapping amplitude and nondimen-sional
time, respectively. Wing kinematic parameters
arenondimensionalized. The mean chord length and the
averagevelocity at the span location R2 are taken as the
referencelength c and the velocity U , respectively. U is defined
as 2Φf λc/180, where f and λ are the flapping frequency and thewing
aspect ratio, respectively. Reference time is defined asc/U , and
the nondimensional time T is t/ðc/UÞ. These refer-ence values are
used to nondimensionalize wing kinematicparameters, forces, and
moments in this study. Unless other-wise specified, the physical
quantities in the following sec-tions are in a dimensionless
form.
In previous studies, the wing is thought to pitch inaccordance
with a preset form (e.g., sinusoidal curve andtrapezoidal curve).
In general, α takes a constant valueexcept at the beginning or near
the end of a half-stroke[12]. _α is given by
_α = 0:5ωr 1 − cos2π t − trð Þ
Δτr
� �� �, tr ≤ t ≤ tr + Δτr , ð2Þ
where ωr is the mean angular velocity, tr is the time atwhich
the pitching motion starts, and Δτr is the nondi-mensional time
interval over which the rotation lasts.The constant α in the
upstroke and downstroke aredefined as αu and αd , respectively. In
the time interval ofΔτr , the wing α changes from αu to αd .
An active pitching model artificially decouples φ fromα, which
considerably simplifies the analysis and calcula-tion processes.
This model is also widely used in quasis-teady estimations.
However, this model also exhibitsunavoidable drawbacks in the
design and application ofFWMAVs. It creates additional burdens to
mechanismsand does not reflect actual pitching motion. Under
thiscircumstance, a passive pitching model based on bionicsbecomes
widely recognized. This model was first proposedbecause
deformations play an important role on the aero-dynamic performance
of flapping wings, but it is difficultto directly simulate the
deformation process as a resultof the interaction between flexible
wing with the surround-ing flow and the complex structure of the
insect wing. Inthis paper, we considered the effect of deformation
witha reduced-order approach [3]. For most dipteran insects,the
narrow root region of wings is flexible, thereby allow-ing them to
rotate around the axis in the leading edge [6].On the basis of this
structural feature, we compress thetorsional flexibility of a
flapping wing to the wing rootand simulate it with a torsional
spring [5]. The variationin α can be obtained as follows.
In a passive pitching model, α is determined in accor-dance with
the coupled dynamic equations of aerodynamicand elastic forces. A
flapping wing is considered as a rigid
plate, and the moment generated by the torsional spring ata
rotating axis can be expressed as
Mtorsion = −k α − α0ð Þ, ð3Þ
where k and α0 are the elastic coefficient and rest angle of
thetorsional spring, respectively.
The initial state of a flapping wing can be artificially
spec-ified. In our study, it is set perpendicular to the OXY
plane(α0 = 90°). When the wing begins to flap, the aerodynamicforce
is substantially perpendicular to the wing surface,thereby
generating a moment around the wing leading edgeand causing the
wing to rotate. At this time, the torsionalspring applies a moment
opposite to the aerodynamicmoment. Thus, the two moments interact
with the inertialmoment and reach equilibrium. In comparison with
theaerodynamic force, the weight of the wing is essentially
neg-ligible because it is typically less than 0.5% of the
entireweight [13]. The aerodynamic and torsional spring
momentsincrease as the average flapping speed increases, resulting
in alarge pitch angle.
The coordinate system fixed on the wing rotates at anangular
velocity _φ during motion. Thus, the transformationrelationship
between coordinates O‐XYZ and O‐xyz mustbe considered when the
equation of α is derived:
〠τ = dLwdt
� �OXYZ
= dLwdt
� �oxyz
+ ω × Lw, ð4Þ
where ∑τ is the external moment, Lw is the momentummoment of the
wing relative to the origin of the coordinatesystem, and ω is the
angular velocity of the wing.
In the coordinate O‐xyz, the projection of angular veloc-ity in
three directions can be expressed as
p
q
r
0BBB@
1CCCA =
ωx
ωy
ωz
0BBB@
1CCCA =
0
_α
0
0BBB@
1CCCA +
cos α 0 sin α
0 1 0
−sin α 0 cos α
0BBB@
1CCCA
0
0
_φ
0BBB@
1CCCA
=
_φ sin α
_α
_φ cos α
0BBB@
1CCCA:
ð5Þ
The component form of the dynamic equation can beexpressed as
follows:
Ixxdpdt
+ Iyy − Izz�
qr − Ixy pr +dqdt
� �= τx,
Iyydqdt
+ Izz − Ixxð Þpr + Ixy qr −dpdt
� �= τy,
Izzdrdt
+ Ixx − Iyy�
pq + Ixy p2 − q2�
= τz ,
8>>>>>>>><>>>>>>>>:
ð6Þ
3Applied Bionics and Biomechanics
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where τx, τy , and τz are the components of the externalmoment
in the directions ox, oy, and oz, respectively. Themoment of the
inertia of the wing to different axes and theinertial product can
be expressed as
Ixx =ðy2 + z2�
dm,
Iyy =ðx2 + z2�
dm,
Izz =ðx2 + y2�
dm,
Ixy =ðxydm,
Iyz =ðyzdm,
Ixz =ðxzdm:
ð7Þ
When the wing is regarded as a flat plate and placed onthe Oxy
plane, the wing is thin and can be disregarded. Thus,z = 0. The
preceding equation can be simplified as
Ixz = Iyz = 0,Izz = Ixx + Iyy:
ð8Þ
An elastic restoring torque, which acts on the rotatingaxis of
the wing, is generated when the torsional spring isdeformed by an
external force. Therefore, only the spanwisedirection should be
considered:
Maero − k α − α0ð Þ = Iyy €α + prð Þ − Ixy _p − qrð Þ: ð9Þ
Finally, the equation of α can be written as
€α = Maero − k α − α0ð ÞIyy
+IxyIyy
€φ sin α − _φ2 sin α cos α, ð10Þ
whereMaero is the aerodynamic moment acting on the wing.This
equation is solved using the improved Euler scheme,and α is
computed from the time integration.
2.2. Governing Equations and Solution Method. The govern-ing
equations of the flow are 3D incompressible unsteadyNavier-Stokes
equations, which are written in the coordinatesystem O‐XYZ in the
following dimensionless form [14]:
∇ ⋅ u = 0,∂u∂t
+ u ⋅ ∇ð Þu+∇p − 1Re∇2u = 0,
8<: ð11Þ
where u is the velocity vector and p is the static pressure. Re
isdefined as Uc/υ, where υ is the kinematic viscosity of thefluid.
The governing equations are solved using a pseudo-compressibility
method based on the upwind scheme [15,16]. We introduce a partial
derivative term of pressure versus
pseudotime in the continuous equation and transform theelliptic
continuous equation into a hyperbolic continuousequation. Thus, the
dimensionless flow control equation istransformed into a hyperbolic
equation, which considerablyimproves the efficiency of the
solution. We verified thenumerical solution method in our past
relevant research,and our previous conclusions are directly used in
the presentwork [12, 14, 17–19].
Once the Navier-Stokes equations are numericallysolved, the
fluid velocity components and pressure at discre-tized grid points
for each time step are available. The aerody-namic forces acting on
the wing are calculated from thepressure and the viscous stress on
the wing surface [14].The force and moment coefficients are
computed by
CF =F
1/2ρU2S ,
CM =M
1/2ρU2Sc,
ð12Þ
where ρ is the fluid density and S is the wing area. The
com-ponent of CF in theOZ direction is the lift coefficient CL.
Theaerodynamic power coefficient CP is given as Cp = CM ⋅ ω,where ω
is the angular velocity vector in the coordinate sys-tem O‐XYZ. The
average lift coefficient CL and the aerody-namic power coefficient
CP are computed by averaging CLand CP in a flapping period,
respectively. Aerodynamic effi-ciency η, which measures the wing
aerodynamic power con-sumption to produce a certain amount of lift,
is defined as
η = CL3/2
CP: ð13Þ
As a result of interaction between flapping wing and itsown
steady flow, the equation of α (equation (10)) and theNavier-Stokes
equations (equation (11)) are coupled in thesolution process. In
order to solve this coupled dynamicproblem, we refer to the Euler
predictor-corrector method.Supposing that α of the wing is known at
a certain time step,the boundary condition of the Navier-Stokes
equations canbe known and the flow equations can be solved to
providethe aerodynamic forces and moments at this time step.
Then,the value of α would be updated and the equations of
motionwould be marched to the next time step. This process
isrepeated in the following time steps. In theory, the
iterationneeds to be continued at a certain time step until the
aerody-namic moments and α of the wing no longer change. But Wuet
al. confirmed that the Euler predictor-corrector methodhas
sufficient accuracy in practical application [20].
2.3. Validation. The velocity and the pressure in the flow
fieldaround the wing are obtained using an O-H grid (Figure 3).
Atypical case is selected and tested in which the domainparameters
are as follows: Re = 16100, λ = 3, Φ = 120°, andT = 7:255.
The Reynolds number of most insects and flapping crea-tures
generally lies within the range of 102~103 because of
4 Applied Bionics and Biomechanics
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their small size and weight. For example, the Reynolds num-ber
of Drosophila is approximately 160, its total weight is lessthan
20mg, and its wing length is only approximately2.5mm. For a
bumblebee, these parameters are 1100,175mg, and 13mm, respectively.
In this study, we aim todesign FWMAVs with a good load capacity in
which theReynolds number is slightly larger and reaches 104.
However,a laminar flow transition problem may occur under this
sce-nario. Isogai et al. [21] compared the calculation results
oflaminar and turbulent flows to investigate issues related
toflapping thrust and propulsion efficiency. They determinedthat
the difference between the results is small when thereduced
frequency is large. Moreover, no evident flow sepa-ration is
observed, and the flow structure is similar to laminar
and turbulent flows with only slight differences in
severaldetails. On the basis of the results of Isogai et al., we
use lam-inar flow without introducing a turbulence model under
aReynolds number of 104 in our calculation because thereduced
frequency of our aircraft is within their conclusions.
In numerical solutions, results and efficiency are affectedby
grid quality. As such, an appropriate grid density, a
com-putational domain size, and a step value should be deter-mined
to ensure the accuracy and speed of calculation.Three sets of grids
are evaluated to select the appropriate griddensity: (a) 51 × 57 ×
63 (around the wing section, in the nor-mal direction of the wing
surface, and in the spanwise direc-tion of the wing), (b) 64 × 73 ×
79, and (c) 80 × 93 × 99.These sets differ in density but have the
same domain size
(a) (b)
Figure 3: (a) Complete grid and (b) surface mesh.
CL
CD
–1
0
1
2
3
4
T0 0.5 1
–4
–2
0
2
4
51×57×6364×73×79
80×93×99
(a)
CL
CD
T0 0.5 1
0.020.01
0.005
–1
0
1
2
3
4
–4
–2
0
2
4
(b)
Figure 4: Comparison of three grids with different (a) densities
and (b) time steps.
5Applied Bionics and Biomechanics
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of 40 times the chord length and a nondimensional time stepvalue
of 0.02. The time course of the aerodynamic force coef-ficients (CL
and CD) in one cycle is shown in Figure 4, indi-cating that the
relatively coarse grid exhibits a remarkabledeviation at the peak.
The other parts of the three grids pres-ent good agreement.
Similarly, grids with different time step values are veri-fied.
A grid with a density of 64 × 73 × 79, a domain size of40c, and a
step value of 0.01 is selected to balance the calcu-lation accuracy
and the time cost.
3. Results and Discussions
The cases under typical conditions are chosen first to
ensurecomparability of the active and passive pitching wings: Re=
16100, λ = 3, Φ = 120°, and T = 7:255. α is an importantparameter
that influences the wing aerodynamic perfor-mance, so it is set to
be changeable in this study. For theactive pitching wing, αu and αd
increase or decrease by 1.5times on the basis of 45°. For the
passive pitching wing, kincreases or decreases by 8 times on the
basis of 1.2, indirectlyleading to the change in α.
3.1. Instantaneous α of the Passive Pitching Flapping
Wing.Studies on the mechanism of insect motion have shown
thatpassive pitching is common during flight. A typical
charac-teristic of α is “double peak oscillation” [11]. In
particular,α∗ continues to increase during the first quarter of a
wingbeatcycle and then gradually reaches the maximum value,
wherethe first peak occurs. Subsequently, α∗ starts to decrease
andrebounds slightly near the end of upstroke/downstroke,where the
second peak occurs. Lastly, α∗ continues to declineand returns to
its initial value. In Figure 5, the solution for thecoupled dynamic
equation corresponding to the simplifiedpassive pitching model is
similar to experimental results[22, 23] and computational results
[9] listed in the previousliterature, which exhibits a tendency
quite different fromthe active pitching.
To investigate the reason why the curve of α has twopeaks, we
analyze the variations in aerodynamic, torsional,and inertial
moments within a wingbeat cycle to determinetheir interaction.
Given that α changes continuously duringflapping, a flapping wing
has a positive pitching angularvelocity, although it is in
equilibrium at the beginning ofupstroke (Figure 6). Initially, the
effect of the inertial momentis stronger than those of aerodynamic
and torsionalmoments. This condition causes the wing to move
fartherfrom the initial position, and α∗ increases continuously
untilit reaches the peak. Then, the effect of the inertial
momentdeclines, whereas the effect of the torsional moment
becomesconsiderable. As such, the flapping wing slowly returns to
itsinitial position, which causes α∗ to decline. However,
anexception occurs when the magnitude of the aerodynamicmoment is
the largest. The tendency of the wing to restoreequilibrium is
hindered, and α∗ increases slightly. Thus,another small peak can be
observed in the curve. Subse-quently, inertial moment prevails,
thereby causing α∗ todecrease rapidly to the initial value. The
situation in down-stroke is similar.
3.2. Effect of Torsional Stiffness on the AerodynamicPerformance
of the Passive Pitching Model. In the passivepitching model, k is
an important parameter that consider-ably affects aerodynamic force
and power consumption.Excess rigidity or flexibility deteriorates
the performance.From Table 1, we can see that the torsional spring
generatesconsiderable elastic recovery moments when k is
excessivelylarge; i.e., the flapping wing is too rigid. Torsional
momentoffsets the effect of the aerodynamic moment within a
shortperiod each time the flapping wing rotates. Thus, the wingcan
only oscillate near the initial α. Although this conditioncan
produce a certain amount of lift, it can also lead to a
T
�훼 (°
)
�휑 (°
)
0 0.5 10
45
90
135
180 (Experimental data [23])�훼fruitfly
�훼passive
�훼active
�훼passive(numerical data [9])
–90
–45
0
45
90
Figure 5: Curve of α from the active pitching model and the
passivepitching model.
T
Mom
ent c
oeffi
cent
0 0.5 1–1
–0.5
0
0.5
1
Aerodynamic momentTorsional momentInertial moment
(b)
(a)
Figure 6: (a) Angle of attack and (b) three dimensionless
momentsof the passive pitching wing in one cycle.
6 Applied Bionics and Biomechanics
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distinct increase in drag, thereby causing aerodynamic
powerconsumption to become extremely high. Consequently, theoverall
aerodynamic efficiency is low. If k is excessively small,i.e., the
flapping wing is too flexible, then the aerodynamicmoment is
clearly dominant. Once the wing starts to flap, αrapidly increases,
and the wing becomes parallel to the inflowdirection. The effect of
torsional moment is weak and unableto maintain a stable periodic
motion. Although drag andaerodynamic power are small, lift is
considerably lower thanthe required value.
Figure 7(a) shows the time history of α for cases with
dif-ferent k. These curves have similar trends with that
reportedpreviously by Kolomenskiy et al. [24]. They changed the
tor-sional stiffness to obtain the one that coincides best with
theexperiment measurement, proving that this kind of simpli-fied
passive pitching model successfully reproduces the maindynamical
features of some insects.
The preceding analysis shows that a suitable k should beselected
to design a FWMAV with good load capacity andhigh efficiency.
Different values are taken at approximatelyequal intervals within
the limitation of 0:15 ≤ k ≤ 6:4 to fur-ther explore the effect of
this parameter on aerodynamic per-formance. For comparison, the
related results of the activepitching wing are also plotted. The
points of CL, CP , and ηare fitted by the curves. The maximum CL of
the active pitch-ing model is chosen as the baseline. The dashed
line definesthe lift constraint, and the points of the red curve
above itrepresent the target lift that can be satisfied. Similarly,
thedash dot line defines the aerodynamic efficiency constraint,and
the points of the green curve above it indicate a higheraerodynamic
efficiency. In Figure 8, the ideal range of kmay be in the
intersection of the two regions with an approx-imate value of
1–2.
3.3. Comparison of the Passive and Active Pitching
WingAerodynamic Performance. Based on the previous analysis,a
conclusion can be drawn that the passive pitching wingcan maintain
a high aerodynamic efficiency while generatingmore lift, which is
beneficial to FWMAVs to enhance thepayload and implement the
maneuver flight. Although asmall loss of lift is observed at the
beginning and the end ofthe upstroke/downstroke, the instantaneous
lift at the middlestage significantly increases by nearly 30%
(Figure 9) and theaverage lift in one cycle improves by 10%, with
the coefficientchanges from 1.519 to 1.671. For instantaneous
power, thepassive pitching wing consumes much more power in the
ini-tial phase of the upstroke/downstroke but greatly savespower in
the phase of rotation. Overall, the average aerody-namic power
consumption slightly differs between the activeand passive pitching
wings in one cycle; their coefficients are2.287 and 2.291,
respectively.
Several differences can be observed in the flow fieldaround the
wings in the two models. The periodic motioncauses LEV to develop
and then decline. Subsequently, theLEV in the opposite direction
begins to expand. During theentire process, the LEV attached to the
wing surface ensuresthe distribution of aerodynamic forces. Figure
9 shows thatno evident vorticity is observed around the flapping
wingduring the initial stage of the upstroke, and the generated
liftis small. The LEV of the twomodels becomes increasingly
sig-nificant as time progresses. However, the intensity of
theactive pitching model rapidly increases, and the lift is
largerthan that of the passive pitching model during the
initialperiod. Subsequently, the LEV of the passive pitching
modeldevelops rapidly. A clear enhancement in lift is
observedbecause vorticity is concentrated, attached to the
surface,and continuous. This condition can also be explained by
pres-sure distribution. Figure 10 shows that the pressure
differencebetween the upper and lower surfaces of the passive
pitchingwing is more considerable than that of the active
pitchingwing. LEV gradually sheds at the end of upstroke, and the
liftdeclines. During this process, the vorticity of the
passivepitching model remains relatively concentrated, whereas
thevorticity of the active pitching model becomes dispersed.
We associate the aerodynamic force with vorticity in theflow
field and attempt to explain the aforementioned phe-nomenon from
another perspective. In an incompressibleviscous flow, the
relationship between aerodynamic forceand vorticity is defined as
[25]
γ∗f,b =ðV f +Vb
r∗ × ω∗dV , ð14Þ
where ω∗ is vorticity; r∗ is the position vector; V f and Vb
arethe volumes of fluid and solid, respectively; and γ∗f,b is the
firstmoment of vorticity.
The aerodynamic force vector F∗ can be written as
F∗ = − 12 ρdγ∗f,bdt∗
+ ρ ddt∗
ðVb
v∗dV , ð15Þ
where v∗ represents the speed of a certain point in Vb.
Itsdimensionless form is expressed as
F= − dγf,bdτ
+ 2ρc
ddτ
ðVb
vdV , ð16Þ
where F= 2F∗/ρU2S, γf,b = γ∗f,b/UcS, and v = v∗/U .If the wing
rotates at a constant speed, then the first term
at the right of equation (16) can be written as −4
_φ2ðVb/ScÞðrm/cÞ, where rm is the position of the wing centroid,
and thesecond term at the right of equation (16) can be writtenas
−2 _φ2ðVb/ScÞðrm/cÞ. Vb/Sc is small when the wing isthin. Thus, the
two terms are small. Equation (16) canbe approximated as
F= − dγdτ
, ð17Þ
Table 1: α, CL, CP , and η corresponding to different k.
k Max/min α CL CP η
6.4 105°/74° 1.196 7.394 0.177
1.2 132°/48° 2.109 4.476 0.684
0.15 165°/9° 0.481 1.033 0.323
7Applied Bionics and Biomechanics
-
T0 0.5 1
0
45
90
135
180
�훼 (°
)
k = 0.15k = 1.2
k = 6.4
(a)
–2
0
2
4
6
CL
T0 0.5 1
k = 0.15k = 1.2
k = 6.4
(b)
CP
–5
0
5
10
15
20
T0 0.5 1
k = 0.15k = 1.2
k = 6.4
(c)
Figure 7: Instantaneous (a) α, (b) CL, and (c) CP under
different k.
CL
0.5 1 1.5 2 2.50
0.5
1
ActivePassive
�휂
(a)
k⁎
CL
0 2 4 60
1
2
3
0
0.5
1
CL
�휂
�휂
(b)
Figure 8: (a) Comparison between the two models of η versus CL.
(b) CL and CP as a function of k.
8 Applied Bionics and Biomechanics
-
where γ is the sum of the first moments of vorticity in
thefluid. The lift and drag coefficients can be written as
CL =d −γy
�dτ
,
CD =dγxdτ
cos φ + dγzdτ
sin φ,
ð18Þ
where γx, γy, and γz are the components of γ in the x, y,and z
directions, respectively.
Equation (18) indicates that aerodynamic force is pro-portional
to the time rate of change in the first moment ofvorticity. Since
the γy curve of passive pitching has alarger slope in the middle of
the upstroke/downstroke(T ≈ 0:2–0.4/T ≈ 0:7–0.9) than that of
active pitching(Figure 11), the lift of the passive pitching wing
is greaterthan that of the active pitching wing during this period.
Incombination with the characteristic of α (Figure 5), weassume
that the rapid change in vorticity may be attrib-uted to the second
small peak, indicating the occurrenceof a sudden reverse pitch
motion.
3.4. Control Strategies in the Passive Pitching Model. Despiteof
a higher lift compared to active pitching wing, the passivewing
kinematic modulations are energetically efficient [9].Early studies
on fruit flies have drawn conclusions from var-ious observations
and experiments that fruit flies asymmetri-cally change the twist
angle of their left and right wings anddrive their body to complete
a lateral movement [22]. Giventhat the passive pitching model is
based on the characteristicof insects, we infer that a similar
effect can be achieved in thedesign of FWMAV [3].
In our calculation, the flapping wing is in an
equilibriumposition when α = 90°. At this time, the torsional
springexhibits no angular displacement and the recovery momentis 0.
In the previous analysis, α0 = 90° and the initial positionof the
wing is the equilibrium position. However, the initialposition of
the wing deviates from the equilibrium positionwhen α0 ≠ 90°. The
symmetry of α during the upstroke anddownstroke is broken, thereby
increasing horizontal and ver-tical forces and resulting in a
moment around the wing root.Almost no lift loss is observed when a
moment is produced.
Figure 12 shows that relative speed and drag increaseduring the
upstroke as α0 decreases, thereby causing a posi-tive variation in
horizontal force. During downstroke, rela-tive speed and drag
decrease, thereby causing a positivevariation in horizontal force.
Thus, a large yaw moment isgenerated around the wing root.
Simultaneously, the lift
c
Pres
sure
coeffi
cien
t
0 0.5 1
–6
–4
–2
0
Upper surfaces
Lower surfaces
2
4
ActivePassive
Figure 10: Comparison of the pressure distributions of the
wingcross section at R2position in the middle of the upstroke.
T
–�훾y
0 0.5 10
10
20
30
40
ActivePassive
d�훾y d�훾ydt( passive dt( )) active>
Figure 11: Comparison of the first moment of vorticity in one
cyclebetween the two models.
CL
Active
Passive
T0 0.5 1
Vorz504540353025201510
50
–5–10–15–20–25–30–35–40–45–50
4
2
0
CP
6
3
0
Figure 9: Comparison of CL, CP , and flow fields during
upstrokebetween two models.
9Applied Bionics and Biomechanics
-
increases during upstroke, thereby increasing the verticalforce.
Then, the lift decreases during downstroke, conse-quently
decreasing the vertical force. As such, the variationsin vertical
forces during the upstroke and downstroke canceleach other.
However, their distribution contributes to thepitch moment around
the wing root.
In Figure 13, the average aerodynamic power almostremains the
same when α0 changes from 70
° to 110°. The restangle of the torsional spring can be used as
a control variablein applying the passive pitching model. The
adjustment of α0on the left and right wings controls the attitude
and trajectory
of the aircraft during flight. This process requires
neithercomplex auxiliary mechanisms nor additional power input,and
this characteristic is an advantage that is not exhibitedby the
active pitching model.
4. Conclusions
We investigate the aerodynamic performance of the
passivepitching model on FWMAVs via 3D numerical simulationand
demonstrate that the angle of attack exhibits the charac-teristic
of “double peak oscillation” under the combination of
T
�훥C
Z
0 0.5 1–1
–0.5
0
0.5
1
(a)
�훥C
X
T0 0.5 1
–0.5
0
0.5
1
1.5
(b)
Figure 12: Differences in (a) horizontal force coefficient and
(b) vertical force coefficient between α0 = 70° and α0 = 90°.
�훼0 (°)60 90 120
–0.6
0.3
0
0.3
0.6
CZ
CX
CP
Diff
eren
ce fr
om eq
uilib
rium
(a)
�훼0 (°)60 90 120
–1
–0.5
0
0.5
1
RollYaw
Pitch
Diff
eren
ce fr
om eq
uilib
rium
(b)
Figure 13: (a) Horizontal force, vertical force, and power
coefficient at different α0 and (b) roll, yaw, and pitch moments at
different α0.
10 Applied Bionics and Biomechanics
-
aerodynamic, spring, and inertial moments in the
simplifiedpassive pitching model, which simulates the motion of
insectwings well. Torsional stiffness considerably affects
aerody-namic force and efficiency in the passive pitching
model.Excess rigidity or flexibility deteriorates the
performance.According to the comparison between active and
passivepitching wings, with appropriate torsional stiffness, the
aver-age lift can be enhanced by 10% at the same aerodynamic
effi-ciency when the wing pitches passively. Simultaneously, theyaw
moment around the wing root can be obtained to assistthe control
system without losing lift by setting different restangles for the
left and right wings. These results show that thepassive pitching
model positively contributes to the improve-ment of the hovering
and maneuverability of FWMAVs. Inthe future, we will conduct a
series of studies about the effecton the stability caused by
passive pitching wing to furtherinvestigate this bionic model.
Data Availability
The data used to support the findings of this study areincluded
within the article. The detailed calculation resultsare available
from the corresponding author upon request.The program and source
code have not been made availablebecause of privacy protection.
Disclosure
The results were originally presented at ICBE 2019.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this paper.
Acknowledgments
This research was primarily supported by the National Natu-ral
Science Foundation of China (grant numbers are11672028 and
11672022).
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