aerodynamic performance of a passive pitching model on bionic flapping...

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

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

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

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


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.


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.


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


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.


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.


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.


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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