Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping Wing Micro Air Vehicles

Jinjing Hao; Jianghao Wu; Yanlai Zhang

doi:10.1155/2019/1504310

Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping Wing Micro Air Vehicles

Hao, Jinjing;Wu, Jianghao;Zhang, Yanlai 2019-12-18 00:00:00 Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 1504310, 12 pages https://doi.org/10.1155/2019/1504310 Research Article Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping 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; zhangyanlai@buaa.edu.cn 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 study applies a bioinspired model that passively imitates insects’ pitching wings to resolve this problem. In this bionic passive pitching model, the wing root is equivalent to a torsional spring. α obtained by solving the coupled dynamic equation is similar to that of insects and exhibits a unique characteristic with two oscillated peaks during the middle of the upstroke/downstroke under the interaction of aerodynamic, torsional, and inertial moments. Excess rigidity or ﬂexibility deteriorates the aerodynamic force and eﬃciency of the passive pitching wing. With appropriate torsional stiﬀness, passive pitching can maintain a high eﬃciency while enhancing the average lift by 10% than active pitching. This observation corresponds to a clear enhancement in instantaneous force and a more concentrated leading edge vortex. This phenomenon can be attributed to a vorticity moment whose component in the lift direction grows at a rapid speed. A novel bionic control strategy of this model is also proposed. Similar to the rest angle in insects, the rest angle of the model is adjusted to generate a yaw moment around the wing root without losing lift, which can assist to change the attitude and trajectory of a FWMAV during ﬂight. These ﬁndings may guide us to deal with various conditions and requirements of FWMAV designs and applications. 1. Introduction the kinematic mechanism of insect wings is diﬃcult to fully understand because of the complex structure of organisms. The requirements for the design of ﬂapping wing micro air On the one hand, this scenario is a typical type of a ﬂuid- vehicles (FWMAVs) include excellent aerodynamic perfor- structure coupling problem, and the interaction between wings and the unsteady ﬂow ﬁeld generated during their mance, high eﬃciency, and satisfactory maneuverability. However, balancing all these standards is diﬃcult for existing movement is highly complicated. On the other hand, the FWMAVs. Fortunately, ﬂying creatures have been consid- mechanism through which insects control their wings ered as a basis for proposing new innovations related to ﬂy- involves numerous muscle structures and neural activities ing. For example, insects can manipulate their wings to but remains poorly understood. Beatus and Cohen [2, 3] complete a series of complex movements, such as hovering, summarized this intractable behavior by applying a climbing, braking, accelerating, and turning. Inspired by reduced-order approach in which the wing hinge of insects these phenomena, researchers have attempted to adopt the and ﬂuid-structure interactions are represented by simpli- physiological characteristics of insects and apply a bionic ﬁed models. Then, a passive pitching model based on the model to artiﬁcial FWMAVs. Researchers have also con- torque exerted by insects on their wings was proposed. In ducted a series of studies on this topic. For example, Ennos this model, the wing root of an insect is equivalent to a tor- [1] stated that torsion is necessary to design insect wings sional spring [4]. The pitching dynamics of wings are because insects have to twist their wings between wingbeats assumed to be passively determined by combining aerody- to optimize the performance of an aerofoil. Nevertheless, namic, torsional, and inertial moments. Bergou et al. [5] 2 Applied Bionics and Biomechanics also conﬁrmed that pitching is passive by showing that aerodynamic and inertial forces are suﬃcient to pitch a rot wing without the aid of muscles. x R Numerous theories and experiments have shown that a 2 c passive pitching model is generally accepted. Ishihara et al. [6, 7] applied a novel ﬂuid-structure interaction similarity law to two- and three-dimensional wings and analyzed the 훥R b motion of a passive pitching wing through computational and experimental methods. They mainly discussed the con- tributions of a wing’s elastic, aerodynamic, and inertial forces Figure 1: Geometric parameters of a ﬂapping wing. b is the and tried to ﬁnd the important control parameters of passive unilateral wingspan, c is the mean chord length, c is the distance rot pitching motion. Chen et al. [8] successfully used this passive between the leading edge and the rotation axis, R is the radius of pitching model to estimate aerodynamic forces with quasis- the wing tip, ΔR is the distance between the wing root and the teady and numerical methods. They found that wings with ﬂapping axis, and R is the radius of the second moment of the stiﬀ hinges achieve a favorable pitching kinematic that leads wing area. to large mean lift forces. This model is applicable not only to a hovering state but also to a maneuvering state. Beatus and Cohen [3] explained wing pitch modulation in maneu- vering fruit ﬂies by an interplay between aerodynamics and a torsional spring. Zeyghami et al. [9] studied the passive pitching of a ﬂapping wing in turning ﬂight and concluded Torsion Y O(o) that passive wing kinematic modulations are fast and ener- getically eﬃcient. Similarly, our study equated the wing’s ﬂexibility to a torsional spring at the wing root located close to the leading edge. This study is mainly aimed at determin- ing whether aerodynamic force and eﬃciency could be improved if we used this passive pitching model to design FWMAVs and identifying whether the maneuverability of FWMAVs would be compromised. In this study, we investigate the aerodynamic perfor- Figure 2: Bioinspired passive pitching model and coordinate mance of a FWMAV with a Reynolds number of 10 . A series system. of analyses is conducted on the basis of a bionic passive pitch- ing model through a 3D numerical simulation and a system- focused on other important parameters such as torsional atic comparison among them. To develop a desirable stiﬀness in this paper. Besides, the rectangular model wing outcome of a FWMAV design, we discuss the eﬀect of several has been extensively used in many numerical simulations dominant parameters, such as torsional stiﬀness and rest [7, 11], which can be regarded as a typical case to illustrate angle of torsional spring, on aerodynamic performance. We a universal conclusion. ﬁnd that a FWMAV with passive pitching wings more likely To clearly describe the 3D motion of a ﬂapping wing and reduces weight, increases lift, and shows great potential for accurately analyze its force, we establish two coordinate sys- ﬂight control. tems with the same origin located on the wing root (Figure 2). The inertial system O‐XYZ is located on the 2. Modeling and Method ground, whereas the OXY plane is parallel to the horizontal plane. The OX axis is oriented toward the trailing edge, the 2.1. Wing Model and Kinematics. Insect wings have a OZ axis is opposite to the direction of gravity, and the OY dynamic geometry. They are made of diﬀerent materials axis is determined on the basis of the right-hand rule. The and exhibit varying structures to adapt to diﬀerent ﬂight coordinate system O‐xyz is ﬁxed on the wing. Ox and Oy axes environments. In practical applications, artiﬁcial wings can- not achieve the same eﬀect as insect wings. Consequently, are along the chordwise and spanwise directions, respec- tively. The Oz axis is determined on the basis of the right- simpliﬁcations are frequently adopted. In this study, we use hand rule. a rectangle to approximate a planar shape and regard a ﬂap- ping wing as a thin plate with a uniform density (Figure 1). Insects generally have three degrees of freedom while hovering. The motion perpendicular to the ﬂapping plane The reason why the rectangular model wings are used is as follows. Luo and Sun [10] have investigated the eﬀect of wing is relatively small and frequently overlooked during simpli- ﬁcation. Therefore, the motion of a wing can be approxi- planform on the aerodynamic force production of model mately decomposed into ﬂapping and pitching, which are insect wings in rotating at Reynolds numbers 200 and 3500 at an angle of attack of 40 described by the ﬂapping angle φ and the angle of attack in 2005 and revealed that the var- α, respectively. Flapping refers to the rotation around the iation in wing shape and aspect ratio (from 2.84 to 5.45) has minor eﬀects on the lift and drag coeﬃcients. Based on their OZ axis, whereas pitching corresponds to the rotation around the Oy axis. conclusions, we neglected the eﬀect of planar shape and Applied Bionics and Biomechanics 3 plate, and the moment generated by the torsional spring at The ﬂapping motion can be described by a trigonometric function as follows: a rotating axis can be expressed as M = −kðÞ α − α , ð3Þ torsion 0 φ _ = Φ sin 2πT , ð1Þ ðÞ where k and α are the elastic coeﬃcient and rest angle of the torsional spring, respectively. where Φ and T are the ﬂapping amplitude and nondimen- The initial state of a ﬂapping wing can be artiﬁcially spec- sional time, respectively. Wing kinematic parameters are iﬁed. In our study, it is set perpendicular to the OXY plane nondimensionalized. The mean chord length and the average (α =90 ). When the wing begins to ﬂap, the aerodynamic velocity at the span location R are taken as the reference 0 force is substantially perpendicular to the wing surface, length c and the velocity U, respectively. U is deﬁned as 2Φ thereby generating a moment around the wing leading edge f λc/180, where f and λ are the ﬂapping frequency and the and causing the wing to rotate. At this time, the torsional wing aspect ratio, respectively. Reference time is deﬁned as spring applies a moment opposite to the aerodynamic c/U, and the nondimensional time T is t/ðc/UÞ. These refer- moment. Thus, the two moments interact with the inertial ence values are used to nondimensionalize wing kinematic moment and reach equilibrium. In comparison with the parameters, forces, and moments in this study. Unless other- aerodynamic force, the weight of the wing is essentially neg- wise speciﬁed, the physical quantities in the following sec- ligible because it is typically less than 0.5% of the entire tions are in a dimensionless form. weight [13]. The aerodynamic and torsional spring moments In previous studies, the wing is thought to pitch in increase as the average ﬂapping speed increases, resulting in a accordance with a preset form (e.g., sinusoidal curve and large pitch angle. trapezoidal curve). In general, α takes a constant value The coordinate system ﬁxed on the wing rotates at an except at the beginning or near the end of a half-stroke angular velocity φ _ during motion. Thus, the transformation [12]. α _ is given by relationship between coordinates O‐XYZ and O‐xyz must be considered when the equation of α is derived: 2π t − t ðÞ α _ =0:5ω 1 − cos , t ≤ t ≤ t + Δτ , ð2Þ r r r r Δτ dL dL w w 〠τ = = + ω × L , ð4Þ dt dt OXYZ oxyz where ω is the mean angular velocity, t is the time at r r which the pitching motion starts, and Δτ is the nondi- mensional time interval over which the rotation lasts. where ∑τ is the external moment, L is the momentum The constant α in the upstroke and downstroke are moment of the wing relative to the origin of the coordinate deﬁned as α and α , respectively. In the time interval of system, and ω is the angular velocity of the wing. u d In the coordinate O‐xyz, the projection of angular veloc- Δτ , the wing α changes from α to α . r u d An active pitching model artiﬁcially decouples φ from ity in three directions can be expressed as α, which considerably simpliﬁes the analysis and calcula- 0 1 0 1 0 1 0 10 1 tion processes. This model is also widely used in quasis- p ω 0 cos α 0 sin α 0 teady estimations. However, this model also exhibits B C B C B C B CB C B C B C B C B CB C q = ω = α _ + 01 0 0 unavoidable drawbacks in the design and application of B C B C B C B CB C @ A @ A @ A @ A@ A FWMAVs. It creates additional burdens to mechanisms r ω 0 −sin α 0 cos α φ _ and does not reﬂect actual pitching motion. Under this 0 1 circumstance, a passive pitching model based on bionics φ _ sin α becomes widely recognized. This model was ﬁrst proposed B C B C = α _ : because deformations play an important role on the aero- B C @ A dynamic performance of ﬂapping wings, but it is diﬃcult φ _ cos α to directly simulate the deformation process as a result of the interaction between ﬂexible wing with the surround- ð5Þ ing ﬂow and the complex structure of the insect wing. In this paper, we considered the eﬀect of deformation with The component form of the dynamic equation can be a reduced-order approach [3]. For most dipteran insects, expressed as follows: the narrow root region of wings is ﬂexible, thereby allow- ing them to rotate around the axis in the leading edge [6]. dp dq On the basis of this structural feature, we compress the I + I − I qr − I pr + = τ , xx yy zz xy x > dt dt torsional ﬂexibility of a ﬂapping wing to the wing root and simulate it with a torsional spring [5]. The variation dq dp ð6Þ I +ðÞ I − I pr + I qr − = τ , yy zz xx xy y in α can be obtained as follows. dt dt In a passive pitching model, α is determined in accor- dr dance with the coupled dynamic equations of aerodynamic > 2 2 I + I − I pq + I p − q = τ , zz xx yy xy z and elastic forces. A ﬂapping wing is considered as a rigid dt 4 Applied Bionics and Biomechanics pseudotime in the continuous equation and transform the where τ , τ , and τ are the components of the external x y z elliptic continuous equation into a hyperbolic continuous moment in the directions ox, oy, and oz, respectively. The equation. Thus, the dimensionless ﬂow control equation is moment of the inertia of the wing to diﬀerent axes and the transformed into a hyperbolic equation, which considerably inertial product can be expressed as improves the eﬃciency of the solution. We veriﬁed the numerical solution method in our past relevant research, 2 2 I = y + z dm, xx and our previous conclusions are directly used in the present ð work [12, 14, 17–19]. 2 2 Once the Navier-Stokes equations are numerically I = x + z dm, yy solved, the ﬂuid velocity components and pressure at discre- tized grid points for each time step are available. The aerody- 2 2 I = x + y dm, zz namic forces acting on the wing are calculated from the ð7Þ pressure and the viscous stress on the wing surface [14]. The force and moment coeﬃcients are computed by I = xydm, xy C = , I = yzdm, F yz 1/2ρU S ð12Þ C = , I = xzdm: xz 1/2ρU Sc When the wing is regarded as a ﬂat plate and placed on where ρ is the ﬂuid density and S is the wing area. The com- the Oxy plane, the wing is thin and can be disregarded. Thus, ponent of C in the OZ direction is the lift coeﬃcient C . The F L z =0. The preceding equation can be simpliﬁed as aerodynamic power coeﬃcient C is given as C = C ⋅ ω, P p M where ω is the angular velocity vector in the coordinate sys- I = I =0, xz yz tem O‐XYZ. The average lift coeﬃcient C and the aerody- ð8Þ I = I + I : namic power coeﬃcient C are computed by averaging C zz xx yy P L and C in a ﬂapping period, respectively. Aerodynamic eﬃ- ciency η, which measures the wing aerodynamic power con- An elastic restoring torque, which acts on the rotating sumption to produce a certain amount of lift, is deﬁned as axis of the wing, is generated when the torsional spring is deformed by an external force. Therefore, only the spanwise 3/2 direction should be considered: η = : ð13Þ M − k α − α = I α € + pr − I p_ − qr : ð9Þ ðÞ ðÞ ðÞ aero 0 yy xy As a result of interaction between ﬂapping wing and its Finally, the equation of α can be written as own steady ﬂow, the equation of α (equation (10)) and the Navier-Stokes equations (equation (11)) are coupled in the M − kðÞ α − α aero 0 xy α € = + φ € sin α − φ _ sin α cos α, ð10Þ solution process. In order to solve this coupled dynamic I I yy yy problem, we refer to the Euler predictor-corrector method. Supposing that α of the wing is known at a certain time step, where M is the aerodynamic moment acting on the wing. aero the boundary condition of the Navier-Stokes equations can This equation is solved using the improved Euler scheme, be known and the ﬂow equations can be solved to provide and α is computed from the time integration. the aerodynamic forces and moments at this time step. Then, the value of α would be updated and the equations of motion 2.2. Governing Equations and Solution Method. The govern- would be marched to the next time step. This process is ing equations of the ﬂow are 3D incompressible unsteady repeated in the following time steps. In theory, the iteration Navier-Stokes equations, which are written in the coordinate needs to be continued at a certain time step until the aerody- system O‐XYZ in the following dimensionless form [14]: namic moments and α of the wing no longer change. But Wu et al. conﬁrmed that the Euler predictor-corrector method ∇⋅ u =0, has suﬃcient accuracy in practical application [20]. ð11Þ ∂u 1 +ðÞ u ⋅∇ u+∇p − ∇ u =0, 2.3. Validation. The velocity and the pressure in the ﬂow ﬁeld ∂t Re around the wing are obtained using an O-H grid (Figure 3). A where u is the velocity vector and p is the static pressure. Re is typical case is selected and tested in which the domain deﬁned as Uc/υ, where υ is the kinematic viscosity of the parameters are as follows: Re = 16100, λ =3, Φ = 120 , and T =7:255. ﬂuid. The governing equations are solved using a pseudo- compressibility method based on the upwind scheme [15, The Reynolds number of most insects and ﬂapping crea- 2 3 16]. We introduce a partial derivative term of pressure versus tures generally lies within the range of 10 ~10 because of Applied Bionics and Biomechanics 5 (a) (b) Figure 3: (a) Complete grid and (b) surface mesh. 4 4 3 3 1 1 –1 –1 2 2 0 0 –2 –2 –4 –4 0 0.5 1 0 0.5 1 T T 51×57×63 80×93×99 0.02 0.005 64×73×79 0.01 (a) (b) Figure 4: Comparison of three grids with diﬀerent (a) densities and (b) time steps. their small size and weight. For example, the Reynolds num- and turbulent ﬂows with only slight diﬀerences in several details. On the basis of the results of Isogai et al., we use lam- ber of Drosophila is approximately 160, its total weight is less than 20 mg, and its wing length is only approximately inar ﬂow without introducing a turbulence model under a 2.5 mm. For a bumblebee, these parameters are 1100, Reynolds number of 10 in our calculation because the 175 mg, and 13 mm, respectively. In this study, we aim to reduced frequency of our aircraft is within their conclusions. design FWMAVs with a good load capacity in which the In numerical solutions, results and eﬃciency are aﬀected Reynolds number is slightly larger and reaches 10 . However, by grid quality. As such, an appropriate grid density, a com- a laminar ﬂow transition problem may occur under this sce- putational domain size, and a step value should be deter- nario. Isogai et al. [21] compared the calculation results of mined to ensure the accuracy and speed of calculation. laminar and turbulent ﬂows to investigate issues related to Three sets of grids are evaluated to select the appropriate grid ﬂapping thrust and propulsion eﬃciency. They determined density: (a) 51 × 57 × 63 (around the wing section, in the nor- that the diﬀerence between the results is small when the mal direction of the wing surface, and in the spanwise direc- reduced frequency is large. Moreover, no evident ﬂow sepa- tion of the wing), (b) 64 × 73 × 79, and (c) 80 × 93 × 99. ration is observed, and the ﬂow structure is similar to laminar These sets diﬀer in density but have the same domain size C C D L C C D L 6 Applied Bionics and Biomechanics (Experimental data [23]) passive fruitfly (a) 훼 1 active 0.5 훼 휑 –45 –0.5 passive (numerical data [9]) –90 –1 0 0.5 1 0 0.5 1 Aerodynamic moment Figure 5: Curve of α from the active pitching model and the passive Torsional moment pitching model. Inertial moment (b) of 40 times the chord length and a nondimensional time step value of 0.02. The time course of the aerodynamic force coef- Figure 6: (a) Angle of attack and (b) three dimensionless moments ﬁcients (C and C ) in one cycle is shown in Figure 4, indi- L D of the passive pitching wing in one cycle. cating that the relatively coarse grid exhibits a remarkable deviation at the peak. The other parts of the three grids pres- To investigate the reason why the curve of α has two ent good agreement. Similarly, grids with diﬀerent time step values are veri- peaks, we analyze the variations in aerodynamic, torsional, ﬁed. A grid with a density of 64 × 73 × 79, a domain size of and inertial moments within a wingbeat cycle to determine 40c, and a step value of 0.01 is selected to balance the calcu- their interaction. Given that α changes continuously during lation accuracy and the time cost. ﬂapping, a ﬂapping wing has a positive pitching angular velocity, although it is in equilibrium at the beginning of upstroke (Figure 6). Initially, the eﬀect of the inertial moment 3. Results and Discussions is stronger than those of aerodynamic and torsional moments. This condition causes the wing to move farther The cases under typical conditions are chosen ﬁrst to ensure from the initial position, and α increases continuously until comparability of the active and passive pitching wings: Re it reaches the peak. Then, the eﬀect of the inertial moment = 16100, λ =3, Φ = 120 , and T =7:255. α is an important declines, whereas the eﬀect of the torsional moment becomes parameter that inﬂuences the wing aerodynamic perfor- considerable. As such, the ﬂapping wing slowly returns to its mance, so it is set to be changeable in this study. For the initial position, which causes α to decline. However, an active pitching wing, α and α increase or decrease by 1.5 u d exception occurs when the magnitude of the aerodynamic times on the basis of 45 . For the passive pitching wing, k moment is the largest. The tendency of the wing to restore increases or decreases by 8 times on the basis of 1.2, indirectly equilibrium is hindered, and α increases slightly. Thus, leading to the change in α. another small peak can be observed in the curve. Subse- quently, inertial moment prevails, thereby causing α to 3.1. Instantaneous α of the Passive Pitching Flapping Wing. decrease rapidly to the initial value. The situation in down- Studies on the mechanism of insect motion have shown that stroke is similar. passive pitching is common during ﬂight. A typical charac- teristic of α is “double peak oscillation” [11]. In particular, 3.2. Eﬀect of Torsional Stiﬀness on the Aerodynamic α continues to increase during the ﬁrst quarter of a wingbeat Performance of the Passive Pitching Model. In the passive cycle and then gradually reaches the maximum value, where pitching model, k is an important parameter that consider- ably aﬀects aerodynamic force and power consumption. the ﬁrst peak occurs. Subsequently, α starts to decrease and rebounds slightly near the end of upstroke/downstroke, Excess rigidity or ﬂexibility deteriorates the performance. where the second peak occurs. Lastly, α continues to decline From Table 1, we can see that the torsional spring generates and returns to its initial value. In Figure 5, the solution for the considerable elastic recovery moments when k is excessively coupled dynamic equation corresponding to the simpliﬁed large; i.e., the ﬂapping wing is too rigid. Torsional moment oﬀsets the eﬀect of the aerodynamic moment within a short passive pitching model is similar to experimental results [22, 23] and computational results [9] listed in the previous period each time the ﬂapping wing rotates. Thus, the wing literature, which exhibits a tendency quite diﬀerent from can only oscillate near the initial α. Although this condition the active pitching. can produce a certain amount of lift, it can also lead to a (°) (°) Moment coefficent Applied Bionics and Biomechanics 7 Several diﬀerences can be observed in the ﬂow ﬁeld Table 1: α, C , C , and η corresponding to diﬀerent k. L P around the wings in the two models. The periodic motion k Max/min α η C C causes LEV to develop and then decline. Subsequently, the L P ° ° LEV in the opposite direction begins to expand. During the 6.4 105 /74 1.196 7.394 0.177 ° ° entire process, the LEV attached to the wing surface ensures 1.2 132 /48 2.109 4.476 0.684 ° ° the distribution of aerodynamic forces. Figure 9 shows that 0.15 165 /9 0.481 1.033 0.323 no evident vorticity is observed around the ﬂapping wing during the initial stage of the upstroke, and the generated lift distinct increase in drag, thereby causing aerodynamic power is small. The LEV of the two models becomes increasingly sig- consumption to become extremely high. Consequently, the niﬁcant as time progresses. However, the intensity of the active pitching model rapidly increases, and the lift is larger overall aerodynamic eﬃciency is low. If k is excessively small, i.e., the ﬂapping wing is too ﬂexible, then the aerodynamic than that of the passive pitching model during the initial moment is clearly dominant. Once the wing starts to ﬂap, α period. Subsequently, the LEV of the passive pitching model rapidly increases, and the wing becomes parallel to the inﬂow develops rapidly. A clear enhancement in lift is observed direction. The eﬀect of torsional moment is weak and unable because vorticity is concentrated, attached to the surface, to maintain a stable periodic motion. Although drag and and continuous. This condition can also be explained by pres- aerodynamic power are small, lift is considerably lower than sure distribution. Figure 10 shows that the pressure diﬀerence the required value. between the upper and lower surfaces of the passive pitching wing is more considerable than that of the active pitching Figure 7(a) shows the time history of α for cases with dif- ferent k. These curves have similar trends with that reported wing. LEV gradually sheds at the end of upstroke, and the lift previously by Kolomenskiy et al. [24]. They changed the tor- declines. During this process, the vorticity of the passive sional stiﬀness to obtain the one that coincides best with the pitching model remains relatively concentrated, whereas the experiment measurement, proving that this kind of simpli- vorticity of the active pitching model becomes dispersed. We associate the aerodynamic force with vorticity in the ﬁed passive pitching model successfully reproduces the main dynamical features of some insects. ﬂow ﬁeld and attempt to explain the aforementioned phe- The preceding analysis shows that a suitable k should be nomenon from another perspective. In an incompressible selected to design a FWMAV with good load capacity and viscous ﬂow, the relationship between aerodynamic force high eﬃciency. Diﬀerent values are taken at approximately and vorticity is deﬁned as [25] equal intervals within the limitation of 0:15 ≤ k ≤ 6:4 to fur- ther explore the eﬀect of this parameter on aerodynamic per- ∗ ∗ ∗ = r × ω dV, ð14Þ f,b formance. For comparison, the related results of the active V +V f b pitching wing are also plotted. The points of C , C , and η L P ∗ ∗ are ﬁtted by the curves. The maximum C of the active pitch- L where ω is vorticity; r is the position vector; V and V are f b ing model is chosen as the baseline. The dashed line deﬁnes the volumes of ﬂuid and solid, respectively; and γ is the ﬁrst f,b the lift constraint, and the points of the red curve above it moment of vorticity. represent the target lift that can be satisﬁed. Similarly, the The aerodynamic force vector F can be written as dash dot line deﬁnes the aerodynamic eﬃciency constraint, and the points of the green curve above it indicate a higher 1 dγ d ∗ f,b ∗ aerodynamic eﬃciency. In Figure 8, the ideal range of k F = − ρ + ρ v dV, ð15Þ ∗ ∗ dt dt may be in the intersection of the two regions with an approx- imate value of 1–2. where v represents the speed of a certain point in V . Its 3.3. Comparison of the Passive and Active Pitching Wing dimensionless form is expressed as Aerodynamic Performance. Based on the previous analysis, a conclusion can be drawn that the passive pitching wing dγ 2 d f,b F = − + vdV, ð16Þ can maintain a high aerodynamic eﬃciency while generating dτ ρc dτ more lift, which is beneﬁcial to FWMAVs to enhance the payload and implement the maneuver ﬂight. Although a ∗ 2 ∗ ∗ where F=2F /ρU S, γ = γ /UcS, and v = v /U. f,b f,b small loss of lift is observed at the beginning and the end of If the wing rotates at a constant speed, then the ﬁrst term the upstroke/downstroke, the instantaneous lift at the middle at the right of equation (16) can be written as −4φ _ ðV /ScÞ stage signiﬁcantly increases by nearly 30% (Figure 9) and the b ðr /cÞ, where r is the position of the wing centroid, and the m m average lift in one cycle improves by 10%, with the coeﬃcient second term at the right of equation (16) can be written changes from 1.519 to 1.671. For instantaneous power, the as −2φ _ ðV /ScÞðr /cÞ. V /Sc is small when the wing is passive pitching wing consumes much more power in the ini- b m b thin. Thus, the two terms are small. Equation (16) can tial phase of the upstroke/downstroke but greatly saves be approximated as power in the phase of rotation. Overall, the average aerody- namic power consumption slightly diﬀers between the active dγ and passive pitching wings in one cycle; their coeﬃcients are F = − , ð17Þ 2.287 and 2.291, respectively. dτ 8 Applied Bionics and Biomechanics 180 6 –2 0 0.5 1 0 0.5 1 k = 0.15 k = 6.4 k = 0.15 k = 6.4 k = 1.2 k = 1.2 (a) (b) –5 0 0.5 1 k = 0.15 k = 6.4 k = 1.2 (c) Figure 7: Instantaneous (a) α, (b) C , and (c) C under diﬀerent k. L P 1 1 0.5 0.5 휂 휂 0 0 0 0.5 1 1.5 2 2.5 0246 (a) (b) Figure 8: (a) Comparison between the two models of η versus C . (b) C and C as a function of k. L L P Active Passive (°) L Applied Bionics and Biomechanics 9 Vorz 40 d훾 d훾 y y ( ) ( ) 30 active dt passive dt 훾 20 –5 –10 –15 0 0.5 1 –20 –25 Active –30 Passive –35 –40 Figure 11: Comparison of the ﬁrst moment of vorticity in one cycle –45 between the two models. –50 0 0.5 1 Equation (18) indicates that aerodynamic force is pro- Figure 9: Comparison of C , C , and ﬂow ﬁelds during upstroke L P portional to the time rate of change in the ﬁrst moment of between two models. vorticity. Since the γ curve of passive pitching has a larger slope in the middle of the upstroke/downstroke –6 (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 greater than that of the active pitching wing during this period. In –4 combination with the characteristic of α (Figure 5), we Upper surfaces assume that the rapid change in vorticity may be attrib- –2 uted to the second small peak, indicating the occurrence of a sudden reverse pitch motion. 3.4. Control Strategies in the Passive Pitching Model. Despite of a higher lift compared to active pitching wing, the passive wing kinematic modulations are energetically eﬃcient [9]. Early studies on fruit ﬂies have drawn conclusions from var- Lower surfaces ious observations and experiments that fruit ﬂies asymmetri- cally change the twist angle of their left and right wings and 0 0.5 1 drive their body to complete a lateral movement [22]. Given that the passive pitching model is based on the characteristic Active of insects, we infer that a similar eﬀect can be achieved in the Passive design of FWMAV [3]. In our calculation, the ﬂapping wing is in an equilibrium Figure 10: Comparison of the pressure distributions of the wing cross section at R position in the middle of the upstroke. position when α =90 . At this time, the torsional spring exhibits no angular displacement and the recovery moment where γ is the sum of the ﬁrst moments of vorticity in the is 0. In the previous analysis, α =90 and the initial position ﬂuid. The lift and drag coeﬃcients can be written as of the wing is the equilibrium position. However, the initial position of the wing deviates from the equilibrium position when α ≠ 90 . The symmetry of α during the upstroke and downstroke is broken, thereby increasing horizontal and ver- d −γ tical forces and resulting in a moment around the wing root. C = , dτ ð18Þ Almost no lift loss is observed when a moment is produced. dγ dγ Figure 12 shows that relative speed and drag increase x z C = cos φ + sin φ, during the upstroke as α decreases, thereby causing a posi- dτ dτ tive variation in horizontal force. During downstroke, rela- tive speed and drag decrease, thereby causing a positive where γ , γ , and γ are the components of γ in the x, y, variation in horizontal force. Thus, a large yaw moment is x y z and z directions, respectively. generated around the wing root. Simultaneously, the lift Passive Active Pressure coefficient L 10 Applied Bionics and Biomechanics 0.5 –0.5 –1 0 0.5 1 (a) 1.5 0.5 –0.5 0 0.5 1 (b) ° ° Figure 12: Diﬀerences in (a) horizontal force coeﬃcient and (b) vertical force coeﬃcient between α =70 and α =90 . 0 0 0.6 1 0.3 0.5 0 0 0.3 –0.5 –0.6 –1 60 90 120 60 90 120 훼 (°) 훼 (°) 0 0 C C Z P Roll Pitch Yaw (a) (b) Figure 13: (a) Horizontal force, vertical force, and power coeﬃcient at diﬀerent α and (b) roll, yaw, and pitch moments at diﬀerent α . 0 0 increases during upstroke, thereby increasing the vertical of the aircraft during ﬂight. This process requires neither force. Then, the lift decreases during downstroke, conse- complex auxiliary mechanisms nor additional power input, quently decreasing the vertical force. As such, the variations and this characteristic is an advantage that is not exhibited in vertical forces during the upstroke and downstroke cancel by the active pitching model. each other. However, their distribution contributes to the pitch moment around the wing root. 4. Conclusions In Figure 13, the average aerodynamic power almost ° ° remains the same when α changes from 70 to 110 . The rest We investigate the aerodynamic performance of the passive angle of the torsional spring can be used as a control variable pitching model on FWMAVs via 3D numerical simulation in applying the passive pitching model. The adjustment of α and demonstrate that the angle of attack exhibits the charac- on the left and right wings controls the attitude and trajectory teristic of “double peak oscillation” under the combination of Difference from equilibrium 훥C 훥C X Z Difference from equilibrium Applied Bionics and Biomechanics 11 [6] D. Ishihara, T. Horie, and M. Denda, “A two-dimensional aerodynamic, spring, and inertial moments in the simpliﬁed computational study on the ﬂuid–structure interaction cause passive pitching model, which simulates the motion of insect of wing pitch changes in dipteran ﬂapping ﬂight,” Journal of wings well. Torsional stiﬀness considerably aﬀects aerody- Experimental Biology, vol. 212, no. 1, pp. 1–10, 2009. namic force and eﬃciency in the passive pitching model. [7] D. Ishihara, T. Horie, and T. Niho, “An experimental and Excess rigidity or ﬂexibility deteriorates the performance. three-dimensional computational study on the aerodynamic According to the comparison between active and passive contribution to the passive pitching motion of ﬂapping wings pitching wings, with appropriate torsional stiﬀness, the aver- in hovering ﬂies,” Bioinspiration & Biomimetics, vol. 9, no. 4, age lift can be enhanced by 10% at the same aerodynamic eﬃ- article 046009, 2014. ciency when the wing pitches passively. Simultaneously, the [8] Y. Chen, N. Gravish, A. L. Desbiens, R. Malka, and R. J. Wood, yaw moment around the wing root can be obtained to assist “Experimental and computational studies of the aerodynamic the control system without losing lift by setting diﬀerent rest performance of a ﬂapping and passively rotating insect wing,” angles for the left and right wings. These results show that the Journal of Fluid Mechanics, vol. 791, no. 1, pp. 1–33, 2016. passive pitching model positively contributes to the improve- [9] S. Zeyghami, Q. Zhong, G. Liu, and H. Dong, “Passive pitching ment of the hovering and maneuverability of FWMAVs. In of a ﬂapping wing in turning ﬂight,” AIAA Journal, vol. 57, the future, we will conduct a series of studies about the eﬀect no. 9, pp. 3744–3752, 2019. on the stability caused by passive pitching wing to further [10] G. Luo and M. Sun, “The eﬀects of corrugation and wing plan- investigate this bionic model. form on the aerodynamic force production of sweeping model insect wings,” Acta Mechanica Sinica, vol. 21, no. 6, pp. 531– 541, 2005. Data Availability [11] H. Dai, H. Luo, and J. F. Doyle, “Dynamic pitching of an elastic The data used to support the ﬁndings of this study are rectangular wing in hovering motion,” Journal of Fluid included within the article. The detailed calculation results Mechanics, vol. 693, pp. 473–499, 2012. are available from the corresponding author upon request. [12] M. Sun and Y. Xiong, “Dynamic ﬂight stability of a hovering The program and source code have not been made available bumblebee,” Journal of Experimental Biology, vol. 208, no. 3, because of privacy protection. pp. 447–459, 2005. [13] D. Lentink and M. H. Dickinson, “Rotational accelerations sta- bilize leading edge vortices on revolving ﬂy wings,” Journal of Disclosure Experimental Biology, vol. 212, no. 16, pp. 2705–2719, 2009. The results were originally presented at ICBE 2019. [14] W. Jianghao, Z. Chao, and Z. Yanlai, “Aerodynamic power eﬃciency comparison of various micro-air-vehicle layouts in hovering ﬂight,” AIAA Journal, vol. 55, no. 4, pp. 1265–1278, Conflicts of Interest [15] S. E. Rogers, D. Kwak, and C. Kiris, “Steady and unsteady solu- The authors declare that there are no conﬂicts of interest tions of the incompressible Navier-stokes equations,” AIAA regarding the publication of this paper. Journal, vol. 29, no. 4, pp. 603–610, 1991. [16] S. E. Rogers and D. Kwak, “An upwind diﬀerencing scheme for Acknowledgments the incompressible navier–strokes equations,” Applied Numer- ical Mathematics, vol. 8, no. 1, pp. 43–64, 1991. This research was primarily supported by the National Natu- [17] M. Sun and X. Yu, “Aerodynamic force generation in hovering ral Science Foundation of China (grant numbers are ﬂight in a tiny insect,” AIAA Journal, vol. 44, no. 7, pp. 1532– 11672028 and 11672022). 1540, 2006. [18] J. H. Wu, “Unsteady aerodynamic forces of a ﬂapping wing,” References Journal of Experimental Biology, vol. 207, no. 7, pp. 1137– 1150, 2004. [1] A. R. Ennos, “The importance of torsion in the design of insect [19] J. Wu, D. Wang, and Y. Zhang, “Aerodynamic analysis of a wings,” Journal of Experimental Biology, vol. 140, no. 1, ﬂapping rotary wing at a low Reynolds number,” AIAA Jour- pp. 137–160, 1988. nal, vol. 53, no. 10, pp. 2951–2966, 2015. [2] A. J. Bergou, L. Ristroph, J. Guckenheimer, I. Cohen, and Z. J. [20] J. H. Wu, Y. L. Zhang, and M. Sun, “Hovering of model insects: Wang, “Fruit ﬂies modulate passive wing pitching to generate simulation by coupling equations of motion with Navier– in-ﬂight turns,” Physical Review Letters, vol. 104, no. 14, article stokes equations,” Journal of Experimental Biology, vol. 212, 148101, 2010. no. 20, pp. 3313–3329, 2009. [3] T. Beatus and I. Cohen, “Wing-pitch modulation in maneu- [21] K. Isogai, Y. Shinmoto, and Y. Watanabe, “Eﬀects of dynamic vering fruit ﬂies is explained by an interplay between aerody- namics and a torsional spring,” Physical Review E, vol. 92, stall on propulsive eﬃciency and thrust of ﬂapping airfoil,” AIAA Journal, vol. 37, no. 10, pp. 1145–1151, 1999. no. 2, article 022712, 2015. [22] D. Ishihara and T. Horie, “Passive mechanism of pitch recoil in [4] M. N. J. Moore, “Torsional spring is the optimal ﬂexibility arrangement for thrust production of a ﬂapping wing,” Physics ﬂapping insect wings,” Bioinspiration & Biomimetics, vol. 12, no. 1, article 016008, 2016. of Fluids, vol. 27, no. 9, article 091701, 2015. [5] A. J. Bergou, S. Xu, and Z. J. Wang, “Passive wing pitch reversal [23] S. N. Fry, R. Sayaman, and M. H. Dickinson, “The aerodynam- in insect ﬂight,” Journal of Fluid Mechanics, vol. 591, pp. 321– ics of hovering ﬂight in Drosophila,” Journal of Experimental 337, 2007. Biology, vol. 208, no. 12, pp. 2303–2318, 2005. 12 Applied Bionics and Biomechanics [24] D. Kolomenskiy, S. Ravi, R. Xu et al., “The dynamics of passive feathering rotation in hovering ﬂight of bumblebees,” Journal of Fluids and Structures, vol. 91, article 102628, 2019. [25] J. C. Wu and W. Zhenyuan, Elements of Vorticity Aerodynam- ics, Shanghai Jiaotong University Press, 2014. 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References (26)

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A two-dimensional computational study on the fluid–structure interaction cause of wing pitch changes in dipteran flapping flight
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M. Sun, Xin Yu (2006)
Aerodynamic Force Generation in Hovering Flight in a Tiny Insect
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The aerodynamics of hovering flight in Drosophila
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Hu Dai, Haoxiang Luo, J. Doyle (2012)
Dynamic pitching of an elastic rectangular wing in hovering motion
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Attila Bergou, Leif Ristroph, J. Guckenheimer, I. Cohen, Z. Wang (2009)
Fruit flies modulate passive wing pitching to generate in-flight turns.
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Experimental and computational studies of the aerodynamic performance of a flapping and passively rotating insect wing
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Hovering of model insects: simulation by coupling equations of motion with Navier–Stokes equations
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Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing
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An experimental and three-dimensional computational study on the aerodynamic contribution to the passive pitching motion of flapping wings in hovering flies
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D. Kolomenskiy, S. Ravi, Ru Xu, K. Ueyama, T. Jakobi, T. Engels, T. Nakata, J. Sesterhenn, K. Schneider, R. Onishi, Hao Liu (2019)
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Rotational accelerations stabilize leading edge vortices on revolving fly wings
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Aerodynamic Analysis of a Flapping Rotary Wing at a Low Reynolds Number
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Publisher: Hindawi Publishing Corporation
Copyright: 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.
ISSN: 1176-2322
eISSN: 1754-2103
DOI: 10.1155/2019/1504310
Publisher site: See Article on Publisher Site

Abstract

Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 1504310, 12 pages https://doi.org/10.1155/2019/1504310 Research Article Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping 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; zhangyanlai@buaa.edu.cn 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 study applies a bioinspired model that passively imitates insects’ pitching wings to resolve this problem. In this bionic passive pitching model, the wing root is equivalent to a torsional spring. α obtained by solving the coupled dynamic equation is similar to that of insects and exhibits a unique characteristic with two oscillated peaks during the middle of the upstroke/downstroke under the interaction of aerodynamic, torsional, and inertial moments. Excess rigidity or ﬂexibility deteriorates the aerodynamic force and eﬃciency of the passive pitching wing. With appropriate torsional stiﬀness, passive pitching can maintain a high eﬃciency while enhancing the average lift by 10% than active pitching. This observation corresponds to a clear enhancement in instantaneous force and a more concentrated leading edge vortex. This phenomenon can be attributed to a vorticity moment whose component in the lift direction grows at a rapid speed. A novel bionic control strategy of this model is also proposed. Similar to the rest angle in insects, the rest angle of the model is adjusted to generate a yaw moment around the wing root without losing lift, which can assist to change the attitude and trajectory of a FWMAV during ﬂight. These ﬁndings may guide us to deal with various conditions and requirements of FWMAV designs and applications. 1. Introduction the kinematic mechanism of insect wings is diﬃcult to fully understand because of the complex structure of organisms. The requirements for the design of ﬂapping wing micro air On the one hand, this scenario is a typical type of a ﬂuid- vehicles (FWMAVs) include excellent aerodynamic perfor- structure coupling problem, and the interaction between wings and the unsteady ﬂow ﬁeld generated during their mance, high eﬃciency, and satisfactory maneuverability. However, balancing all these standards is diﬃcult for existing movement is highly complicated. On the other hand, the FWMAVs. Fortunately, ﬂying creatures have been consid- mechanism through which insects control their wings ered as a basis for proposing new innovations related to ﬂy- involves numerous muscle structures and neural activities ing. For example, insects can manipulate their wings to but remains poorly understood. Beatus and Cohen [2, 3] complete a series of complex movements, such as hovering, summarized this intractable behavior by applying a climbing, braking, accelerating, and turning. Inspired by reduced-order approach in which the wing hinge of insects these phenomena, researchers have attempted to adopt the and ﬂuid-structure interactions are represented by simpli- physiological characteristics of insects and apply a bionic ﬁed models. Then, a passive pitching model based on the model to artiﬁcial FWMAVs. Researchers have also con- torque exerted by insects on their wings was proposed. In ducted a series of studies on this topic. For example, Ennos this model, the wing root of an insect is equivalent to a tor- [1] stated that torsion is necessary to design insect wings sional spring [4]. The pitching dynamics of wings are because insects have to twist their wings between wingbeats assumed to be passively determined by combining aerody- to optimize the performance of an aerofoil. Nevertheless, namic, torsional, and inertial moments. Bergou et al. [5] 2 Applied Bionics and Biomechanics also conﬁrmed that pitching is passive by showing that aerodynamic and inertial forces are suﬃcient to pitch a rot wing without the aid of muscles. x R Numerous theories and experiments have shown that a 2 c passive pitching model is generally accepted. Ishihara et al. [6, 7] applied a novel ﬂuid-structure interaction similarity law to two- and three-dimensional wings and analyzed the 훥R b motion of a passive pitching wing through computational and experimental methods. They mainly discussed the con- tributions of a wing’s elastic, aerodynamic, and inertial forces Figure 1: Geometric parameters of a ﬂapping wing. b is the and tried to ﬁnd the important control parameters of passive unilateral wingspan, c is the mean chord length, c is the distance rot pitching motion. Chen et al. [8] successfully used this passive between the leading edge and the rotation axis, R is the radius of pitching model to estimate aerodynamic forces with quasis- the wing tip, ΔR is the distance between the wing root and the teady and numerical methods. They found that wings with ﬂapping axis, and R is the radius of the second moment of the stiﬀ hinges achieve a favorable pitching kinematic that leads wing area. to large mean lift forces. This model is applicable not only to a hovering state but also to a maneuvering state. Beatus and Cohen [3] explained wing pitch modulation in maneu- vering fruit ﬂies by an interplay between aerodynamics and a torsional spring. Zeyghami et al. [9] studied the passive pitching of a ﬂapping wing in turning ﬂight and concluded Torsion Y O(o) that passive wing kinematic modulations are fast and ener- getically eﬃcient. Similarly, our study equated the wing’s ﬂexibility to a torsional spring at the wing root located close to the leading edge. This study is mainly aimed at determin- ing whether aerodynamic force and eﬃciency could be improved if we used this passive pitching model to design FWMAVs and identifying whether the maneuverability of FWMAVs would be compromised. In this study, we investigate the aerodynamic perfor- Figure 2: Bioinspired passive pitching model and coordinate mance of a FWMAV with a Reynolds number of 10 . A series system. of analyses is conducted on the basis of a bionic passive pitch- ing model through a 3D numerical simulation and a system- focused on other important parameters such as torsional atic comparison among them. To develop a desirable stiﬀness in this paper. Besides, the rectangular model wing outcome of a FWMAV design, we discuss the eﬀect of several has been extensively used in many numerical simulations dominant parameters, such as torsional stiﬀness and rest [7, 11], which can be regarded as a typical case to illustrate angle of torsional spring, on aerodynamic performance. We a universal conclusion. ﬁnd that a FWMAV with passive pitching wings more likely To clearly describe the 3D motion of a ﬂapping wing and reduces weight, increases lift, and shows great potential for accurately analyze its force, we establish two coordinate sys- ﬂight control. tems with the same origin located on the wing root (Figure 2). The inertial system O‐XYZ is located on the 2. Modeling and Method ground, whereas the OXY plane is parallel to the horizontal plane. The OX axis is oriented toward the trailing edge, the 2.1. Wing Model and Kinematics. Insect wings have a OZ axis is opposite to the direction of gravity, and the OY dynamic geometry. They are made of diﬀerent materials axis is determined on the basis of the right-hand rule. The and exhibit varying structures to adapt to diﬀerent ﬂight coordinate system O‐xyz is ﬁxed on the wing. Ox and Oy axes environments. In practical applications, artiﬁcial wings can- not achieve the same eﬀect as insect wings. Consequently, are along the chordwise and spanwise directions, respec- tively. The Oz axis is determined on the basis of the right- simpliﬁcations are frequently adopted. In this study, we use hand rule. a rectangle to approximate a planar shape and regard a ﬂap- ping wing as a thin plate with a uniform density (Figure 1). Insects generally have three degrees of freedom while hovering. The motion perpendicular to the ﬂapping plane The reason why the rectangular model wings are used is as follows. Luo and Sun [10] have investigated the eﬀect of wing is relatively small and frequently overlooked during simpli- ﬁcation. Therefore, the motion of a wing can be approxi- planform on the aerodynamic force production of model mately decomposed into ﬂapping and pitching, which are insect wings in rotating at Reynolds numbers 200 and 3500 at an angle of attack of 40 described by the ﬂapping angle φ and the angle of attack in 2005 and revealed that the var- α, respectively. Flapping refers to the rotation around the iation in wing shape and aspect ratio (from 2.84 to 5.45) has minor eﬀects on the lift and drag coeﬃcients. Based on their OZ axis, whereas pitching corresponds to the rotation around the Oy axis. conclusions, we neglected the eﬀect of planar shape and Applied Bionics and Biomechanics 3 plate, and the moment generated by the torsional spring at The ﬂapping motion can be described by a trigonometric function as follows: a rotating axis can be expressed as M = −kðÞ α − α , ð3Þ torsion 0 φ _ = Φ sin 2πT , ð1Þ ðÞ where k and α are the elastic coeﬃcient and rest angle of the torsional spring, respectively. where Φ and T are the ﬂapping amplitude and nondimen- The initial state of a ﬂapping wing can be artiﬁcially spec- sional time, respectively. Wing kinematic parameters are iﬁed. In our study, it is set perpendicular to the OXY plane nondimensionalized. The mean chord length and the average (α =90 ). When the wing begins to ﬂap, the aerodynamic velocity at the span location R are taken as the reference 0 force is substantially perpendicular to the wing surface, length c and the velocity U, respectively. U is deﬁned as 2Φ thereby generating a moment around the wing leading edge f λc/180, where f and λ are the ﬂapping frequency and the and causing the wing to rotate. At this time, the torsional wing aspect ratio, respectively. Reference time is deﬁned as spring applies a moment opposite to the aerodynamic c/U, and the nondimensional time T is t/ðc/UÞ. These refer- moment. Thus, the two moments interact with the inertial ence values are used to nondimensionalize wing kinematic moment and reach equilibrium. In comparison with the parameters, forces, and moments in this study. Unless other- aerodynamic force, the weight of the wing is essentially neg- wise speciﬁed, the physical quantities in the following sec- ligible because it is typically less than 0.5% of the entire tions are in a dimensionless form. weight [13]. The aerodynamic and torsional spring moments In previous studies, the wing is thought to pitch in increase as the average ﬂapping speed increases, resulting in a accordance with a preset form (e.g., sinusoidal curve and large pitch angle. trapezoidal curve). In general, α takes a constant value The coordinate system ﬁxed on the wing rotates at an except at the beginning or near the end of a half-stroke angular velocity φ _ during motion. Thus, the transformation [12]. α _ is given by relationship between coordinates O‐XYZ and O‐xyz must be considered when the equation of α is derived: 2π t − t ðÞ α _ =0:5ω 1 − cos , t ≤ t ≤ t + Δτ , ð2Þ r r r r Δτ dL dL w w 〠τ = = + ω × L , ð4Þ dt dt OXYZ oxyz where ω is the mean angular velocity, t is the time at r r which the pitching motion starts, and Δτ is the nondi- mensional time interval over which the rotation lasts. where ∑τ is the external moment, L is the momentum The constant α in the upstroke and downstroke are moment of the wing relative to the origin of the coordinate deﬁned as α and α , respectively. In the time interval of system, and ω is the angular velocity of the wing. u d In the coordinate O‐xyz, the projection of angular veloc- Δτ , the wing α changes from α to α . r u d An active pitching model artiﬁcially decouples φ from ity in three directions can be expressed as α, which considerably simpliﬁes the analysis and calcula- 0 1 0 1 0 1 0 10 1 tion processes. This model is also widely used in quasis- p ω 0 cos α 0 sin α 0 teady estimations. However, this model also exhibits B C B C B C B CB C B C B C B C B CB C q = ω = α _ + 01 0 0 unavoidable drawbacks in the design and application of B C B C B C B CB C @ A @ A @ A @ A@ A FWMAVs. It creates additional burdens to mechanisms r ω 0 −sin α 0 cos α φ _ and does not reﬂect actual pitching motion. Under this 0 1 circumstance, a passive pitching model based on bionics φ _ sin α becomes widely recognized. This model was ﬁrst proposed B C B C = α _ : because deformations play an important role on the aero- B C @ A dynamic performance of ﬂapping wings, but it is diﬃcult φ _ cos α to directly simulate the deformation process as a result of the interaction between ﬂexible wing with the surround- ð5Þ ing ﬂow and the complex structure of the insect wing. In this paper, we considered the eﬀect of deformation with The component form of the dynamic equation can be a reduced-order approach [3]. For most dipteran insects, expressed as follows: the narrow root region of wings is ﬂexible, thereby allow- ing them to rotate around the axis in the leading edge [6]. dp dq On the basis of this structural feature, we compress the I + I − I qr − I pr + = τ , xx yy zz xy x > dt dt torsional ﬂexibility of a ﬂapping wing to the wing root and simulate it with a torsional spring [5]. The variation dq dp ð6Þ I +ðÞ I − I pr + I qr − = τ , yy zz xx xy y in α can be obtained as follows. dt dt In a passive pitching model, α is determined in accor- dr dance with the coupled dynamic equations of aerodynamic > 2 2 I + I − I pq + I p − q = τ , zz xx yy xy z and elastic forces. A ﬂapping wing is considered as a rigid dt 4 Applied Bionics and Biomechanics pseudotime in the continuous equation and transform the where τ , τ , and τ are the components of the external x y z elliptic continuous equation into a hyperbolic continuous moment in the directions ox, oy, and oz, respectively. The equation. Thus, the dimensionless ﬂow control equation is moment of the inertia of the wing to diﬀerent axes and the transformed into a hyperbolic equation, which considerably inertial product can be expressed as improves the eﬃciency of the solution. We veriﬁed the numerical solution method in our past relevant research, 2 2 I = y + z dm, xx and our previous conclusions are directly used in the present ð work [12, 14, 17–19]. 2 2 Once the Navier-Stokes equations are numerically I = x + z dm, yy solved, the ﬂuid velocity components and pressure at discre- tized grid points for each time step are available. The aerody- 2 2 I = x + y dm, zz namic forces acting on the wing are calculated from the ð7Þ pressure and the viscous stress on the wing surface [14]. The force and moment coeﬃcients are computed by I = xydm, xy C = , I = yzdm, F yz 1/2ρU S ð12Þ C = , I = xzdm: xz 1/2ρU Sc When the wing is regarded as a ﬂat plate and placed on where ρ is the ﬂuid density and S is the wing area. The com- the Oxy plane, the wing is thin and can be disregarded. Thus, ponent of C in the OZ direction is the lift coeﬃcient C . The F L z =0. The preceding equation can be simpliﬁed as aerodynamic power coeﬃcient C is given as C = C ⋅ ω, P p M where ω is the angular velocity vector in the coordinate sys- I = I =0, xz yz tem O‐XYZ. The average lift coeﬃcient C and the aerody- ð8Þ I = I + I : namic power coeﬃcient C are computed by averaging C zz xx yy P L and C in a ﬂapping period, respectively. Aerodynamic eﬃ- ciency η, which measures the wing aerodynamic power con- An elastic restoring torque, which acts on the rotating sumption to produce a certain amount of lift, is deﬁned as axis of the wing, is generated when the torsional spring is deformed by an external force. Therefore, only the spanwise 3/2 direction should be considered: η = : ð13Þ M − k α − α = I α € + pr − I p_ − qr : ð9Þ ðÞ ðÞ ðÞ aero 0 yy xy As a result of interaction between ﬂapping wing and its Finally, the equation of α can be written as own steady ﬂow, the equation of α (equation (10)) and the Navier-Stokes equations (equation (11)) are coupled in the M − kðÞ α − α aero 0 xy α € = + φ € sin α − φ _ sin α cos α, ð10Þ solution process. In order to solve this coupled dynamic I I yy yy problem, we refer to the Euler predictor-corrector method. Supposing that α of the wing is known at a certain time step, where M is the aerodynamic moment acting on the wing. aero the boundary condition of the Navier-Stokes equations can This equation is solved using the improved Euler scheme, be known and the ﬂow equations can be solved to provide and α is computed from the time integration. the aerodynamic forces and moments at this time step. Then, the value of α would be updated and the equations of motion 2.2. Governing Equations and Solution Method. The govern- would be marched to the next time step. This process is ing equations of the ﬂow are 3D incompressible unsteady repeated in the following time steps. In theory, the iteration Navier-Stokes equations, which are written in the coordinate needs to be continued at a certain time step until the aerody- system O‐XYZ in the following dimensionless form [14]: namic moments and α of the wing no longer change. But Wu et al. conﬁrmed that the Euler predictor-corrector method ∇⋅ u =0, has suﬃcient accuracy in practical application [20]. ð11Þ ∂u 1 +ðÞ u ⋅∇ u+∇p − ∇ u =0, 2.3. Validation. The velocity and the pressure in the ﬂow ﬁeld ∂t Re around the wing are obtained using an O-H grid (Figure 3). A where u is the velocity vector and p is the static pressure. Re is typical case is selected and tested in which the domain deﬁned as Uc/υ, where υ is the kinematic viscosity of the parameters are as follows: Re = 16100, λ =3, Φ = 120 , and T =7:255. ﬂuid. The governing equations are solved using a pseudo- compressibility method based on the upwind scheme [15, The Reynolds number of most insects and ﬂapping crea- 2 3 16]. We introduce a partial derivative term of pressure versus tures generally lies within the range of 10 ~10 because of Applied Bionics and Biomechanics 5 (a) (b) Figure 3: (a) Complete grid and (b) surface mesh. 4 4 3 3 1 1 –1 –1 2 2 0 0 –2 –2 –4 –4 0 0.5 1 0 0.5 1 T T 51×57×63 80×93×99 0.02 0.005 64×73×79 0.01 (a) (b) Figure 4: Comparison of three grids with diﬀerent (a) densities and (b) time steps. their small size and weight. For example, the Reynolds num- and turbulent ﬂows with only slight diﬀerences in several details. On the basis of the results of Isogai et al., we use lam- ber of Drosophila is approximately 160, its total weight is less than 20 mg, and its wing length is only approximately inar ﬂow without introducing a turbulence model under a 2.5 mm. For a bumblebee, these parameters are 1100, Reynolds number of 10 in our calculation because the 175 mg, and 13 mm, respectively. In this study, we aim to reduced frequency of our aircraft is within their conclusions. design FWMAVs with a good load capacity in which the In numerical solutions, results and eﬃciency are aﬀected Reynolds number is slightly larger and reaches 10 . However, by grid quality. As such, an appropriate grid density, a com- a laminar ﬂow transition problem may occur under this sce- putational domain size, and a step value should be deter- nario. Isogai et al. [21] compared the calculation results of mined to ensure the accuracy and speed of calculation. laminar and turbulent ﬂows to investigate issues related to Three sets of grids are evaluated to select the appropriate grid ﬂapping thrust and propulsion eﬃciency. They determined density: (a) 51 × 57 × 63 (around the wing section, in the nor- that the diﬀerence between the results is small when the mal direction of the wing surface, and in the spanwise direc- reduced frequency is large. Moreover, no evident ﬂow sepa- tion of the wing), (b) 64 × 73 × 79, and (c) 80 × 93 × 99. ration is observed, and the ﬂow structure is similar to laminar These sets diﬀer in density but have the same domain size C C D L C C D L 6 Applied Bionics and Biomechanics (Experimental data [23]) passive fruitfly (a) 훼 1 active 0.5 훼 휑 –45 –0.5 passive (numerical data [9]) –90 –1 0 0.5 1 0 0.5 1 Aerodynamic moment Figure 5: Curve of α from the active pitching model and the passive Torsional moment pitching model. Inertial moment (b) of 40 times the chord length and a nondimensional time step value of 0.02. The time course of the aerodynamic force coef- Figure 6: (a) Angle of attack and (b) three dimensionless moments ﬁcients (C and C ) in one cycle is shown in Figure 4, indi- L D of the passive pitching wing in one cycle. cating that the relatively coarse grid exhibits a remarkable deviation at the peak. The other parts of the three grids pres- To investigate the reason why the curve of α has two ent good agreement. Similarly, grids with diﬀerent time step values are veri- peaks, we analyze the variations in aerodynamic, torsional, ﬁed. A grid with a density of 64 × 73 × 79, a domain size of and inertial moments within a wingbeat cycle to determine 40c, and a step value of 0.01 is selected to balance the calcu- their interaction. Given that α changes continuously during lation accuracy and the time cost. ﬂapping, a ﬂapping wing has a positive pitching angular velocity, although it is in equilibrium at the beginning of upstroke (Figure 6). Initially, the eﬀect of the inertial moment 3. Results and Discussions is stronger than those of aerodynamic and torsional moments. This condition causes the wing to move farther The cases under typical conditions are chosen ﬁrst to ensure from the initial position, and α increases continuously until comparability of the active and passive pitching wings: Re it reaches the peak. Then, the eﬀect of the inertial moment = 16100, λ =3, Φ = 120 , and T =7:255. α is an important declines, whereas the eﬀect of the torsional moment becomes parameter that inﬂuences the wing aerodynamic perfor- considerable. As such, the ﬂapping wing slowly returns to its mance, so it is set to be changeable in this study. For the initial position, which causes α to decline. However, an active pitching wing, α and α increase or decrease by 1.5 u d exception occurs when the magnitude of the aerodynamic times on the basis of 45 . For the passive pitching wing, k moment is the largest. The tendency of the wing to restore increases or decreases by 8 times on the basis of 1.2, indirectly equilibrium is hindered, and α increases slightly. Thus, leading to the change in α. another small peak can be observed in the curve. Subse- quently, inertial moment prevails, thereby causing α to 3.1. Instantaneous α of the Passive Pitching Flapping Wing. decrease rapidly to the initial value. The situation in down- Studies on the mechanism of insect motion have shown that stroke is similar. passive pitching is common during ﬂight. A typical charac- teristic of α is “double peak oscillation” [11]. In particular, 3.2. Eﬀect of Torsional Stiﬀness on the Aerodynamic α continues to increase during the ﬁrst quarter of a wingbeat Performance of the Passive Pitching Model. In the passive cycle and then gradually reaches the maximum value, where pitching model, k is an important parameter that consider- ably aﬀects aerodynamic force and power consumption. the ﬁrst peak occurs. Subsequently, α starts to decrease and rebounds slightly near the end of upstroke/downstroke, Excess rigidity or ﬂexibility deteriorates the performance. where the second peak occurs. Lastly, α continues to decline From Table 1, we can see that the torsional spring generates and returns to its initial value. In Figure 5, the solution for the considerable elastic recovery moments when k is excessively coupled dynamic equation corresponding to the simpliﬁed large; i.e., the ﬂapping wing is too rigid. Torsional moment oﬀsets the eﬀect of the aerodynamic moment within a short passive pitching model is similar to experimental results [22, 23] and computational results [9] listed in the previous period each time the ﬂapping wing rotates. Thus, the wing literature, which exhibits a tendency quite diﬀerent from can only oscillate near the initial α. Although this condition the active pitching. can produce a certain amount of lift, it can also lead to a (°) (°) Moment coefficent Applied Bionics and Biomechanics 7 Several diﬀerences can be observed in the ﬂow ﬁeld Table 1: α, C , C , and η corresponding to diﬀerent k. L P around the wings in the two models. The periodic motion k Max/min α η C C causes LEV to develop and then decline. Subsequently, the L P ° ° LEV in the opposite direction begins to expand. During the 6.4 105 /74 1.196 7.394 0.177 ° ° entire process, the LEV attached to the wing surface ensures 1.2 132 /48 2.109 4.476 0.684 ° ° the distribution of aerodynamic forces. Figure 9 shows that 0.15 165 /9 0.481 1.033 0.323 no evident vorticity is observed around the ﬂapping wing during the initial stage of the upstroke, and the generated lift distinct increase in drag, thereby causing aerodynamic power is small. The LEV of the two models becomes increasingly sig- consumption to become extremely high. Consequently, the niﬁcant as time progresses. However, the intensity of the active pitching model rapidly increases, and the lift is larger overall aerodynamic eﬃciency is low. If k is excessively small, i.e., the ﬂapping wing is too ﬂexible, then the aerodynamic than that of the passive pitching model during the initial moment is clearly dominant. Once the wing starts to ﬂap, α period. Subsequently, the LEV of the passive pitching model rapidly increases, and the wing becomes parallel to the inﬂow develops rapidly. A clear enhancement in lift is observed direction. The eﬀect of torsional moment is weak and unable because vorticity is concentrated, attached to the surface, to maintain a stable periodic motion. Although drag and and continuous. This condition can also be explained by pres- aerodynamic power are small, lift is considerably lower than sure distribution. Figure 10 shows that the pressure diﬀerence the required value. between the upper and lower surfaces of the passive pitching wing is more considerable than that of the active pitching Figure 7(a) shows the time history of α for cases with dif- ferent k. These curves have similar trends with that reported wing. LEV gradually sheds at the end of upstroke, and the lift previously by Kolomenskiy et al. [24]. They changed the tor- declines. During this process, the vorticity of the passive sional stiﬀness to obtain the one that coincides best with the pitching model remains relatively concentrated, whereas the experiment measurement, proving that this kind of simpli- vorticity of the active pitching model becomes dispersed. We associate the aerodynamic force with vorticity in the ﬁed passive pitching model successfully reproduces the main dynamical features of some insects. ﬂow ﬁeld and attempt to explain the aforementioned phe- The preceding analysis shows that a suitable k should be nomenon from another perspective. In an incompressible selected to design a FWMAV with good load capacity and viscous ﬂow, the relationship between aerodynamic force high eﬃciency. Diﬀerent values are taken at approximately and vorticity is deﬁned as [25] equal intervals within the limitation of 0:15 ≤ k ≤ 6:4 to fur- ther explore the eﬀect of this parameter on aerodynamic per- ∗ ∗ ∗ = r × ω dV, ð14Þ f,b formance. For comparison, the related results of the active V +V f b pitching wing are also plotted. The points of C , C , and η L P ∗ ∗ are ﬁtted by the curves. The maximum C of the active pitch- L where ω is vorticity; r is the position vector; V and V are f b ing model is chosen as the baseline. The dashed line deﬁnes the volumes of ﬂuid and solid, respectively; and γ is the ﬁrst f,b the lift constraint, and the points of the red curve above it moment of vorticity. represent the target lift that can be satisﬁed. Similarly, the The aerodynamic force vector F can be written as dash dot line deﬁnes the aerodynamic eﬃciency constraint, and the points of the green curve above it indicate a higher 1 dγ d ∗ f,b ∗ aerodynamic eﬃciency. In Figure 8, the ideal range of k F = − ρ + ρ v dV, ð15Þ ∗ ∗ dt dt may be in the intersection of the two regions with an approx- imate value of 1–2. where v represents the speed of a certain point in V . Its 3.3. Comparison of the Passive and Active Pitching Wing dimensionless form is expressed as Aerodynamic Performance. Based on the previous analysis, a conclusion can be drawn that the passive pitching wing dγ 2 d f,b F = − + vdV, ð16Þ can maintain a high aerodynamic eﬃciency while generating dτ ρc dτ more lift, which is beneﬁcial to FWMAVs to enhance the payload and implement the maneuver ﬂight. Although a ∗ 2 ∗ ∗ where F=2F /ρU S, γ = γ /UcS, and v = v /U. f,b f,b small loss of lift is observed at the beginning and the end of If the wing rotates at a constant speed, then the ﬁrst term the upstroke/downstroke, the instantaneous lift at the middle at the right of equation (16) can be written as −4φ _ ðV /ScÞ stage signiﬁcantly increases by nearly 30% (Figure 9) and the b ðr /cÞ, where r is the position of the wing centroid, and the m m average lift in one cycle improves by 10%, with the coeﬃcient second term at the right of equation (16) can be written changes from 1.519 to 1.671. For instantaneous power, the as −2φ _ ðV /ScÞðr /cÞ. V /Sc is small when the wing is passive pitching wing consumes much more power in the ini- b m b thin. Thus, the two terms are small. Equation (16) can tial phase of the upstroke/downstroke but greatly saves be approximated as power in the phase of rotation. Overall, the average aerody- namic power consumption slightly diﬀers between the active dγ and passive pitching wings in one cycle; their coeﬃcients are F = − , ð17Þ 2.287 and 2.291, respectively. dτ 8 Applied Bionics and Biomechanics 180 6 –2 0 0.5 1 0 0.5 1 k = 0.15 k = 6.4 k = 0.15 k = 6.4 k = 1.2 k = 1.2 (a) (b) –5 0 0.5 1 k = 0.15 k = 6.4 k = 1.2 (c) Figure 7: Instantaneous (a) α, (b) C , and (c) C under diﬀerent k. L P 1 1 0.5 0.5 휂 휂 0 0 0 0.5 1 1.5 2 2.5 0246 (a) (b) Figure 8: (a) Comparison between the two models of η versus C . (b) C and C as a function of k. L L P Active Passive (°) L Applied Bionics and Biomechanics 9 Vorz 40 d훾 d훾 y y ( ) ( ) 30 active dt passive dt 훾 20 –5 –10 –15 0 0.5 1 –20 –25 Active –30 Passive –35 –40 Figure 11: Comparison of the ﬁrst moment of vorticity in one cycle –45 between the two models. –50 0 0.5 1 Equation (18) indicates that aerodynamic force is pro- Figure 9: Comparison of C , C , and ﬂow ﬁelds during upstroke L P portional to the time rate of change in the ﬁrst moment of between two models. vorticity. Since the γ curve of passive pitching has a larger slope in the middle of the upstroke/downstroke –6 (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 greater than that of the active pitching wing during this period. In –4 combination with the characteristic of α (Figure 5), we Upper surfaces assume that the rapid change in vorticity may be attrib- –2 uted to the second small peak, indicating the occurrence of a sudden reverse pitch motion. 3.4. Control Strategies in the Passive Pitching Model. Despite of a higher lift compared to active pitching wing, the passive wing kinematic modulations are energetically eﬃcient [9]. Early studies on fruit ﬂies have drawn conclusions from var- Lower surfaces ious observations and experiments that fruit ﬂies asymmetri- cally change the twist angle of their left and right wings and 0 0.5 1 drive their body to complete a lateral movement [22]. Given that the passive pitching model is based on the characteristic Active of insects, we infer that a similar eﬀect can be achieved in the Passive design of FWMAV [3]. In our calculation, the ﬂapping wing is in an equilibrium Figure 10: Comparison of the pressure distributions of the wing cross section at R position in the middle of the upstroke. position when α =90 . At this time, the torsional spring exhibits no angular displacement and the recovery moment where γ is the sum of the ﬁrst moments of vorticity in the is 0. In the previous analysis, α =90 and the initial position ﬂuid. The lift and drag coeﬃcients can be written as of the wing is the equilibrium position. However, the initial position of the wing deviates from the equilibrium position when α ≠ 90 . The symmetry of α during the upstroke and downstroke is broken, thereby increasing horizontal and ver- d −γ tical forces and resulting in a moment around the wing root. C = , dτ ð18Þ Almost no lift loss is observed when a moment is produced. dγ dγ Figure 12 shows that relative speed and drag increase x z C = cos φ + sin φ, during the upstroke as α decreases, thereby causing a posi- dτ dτ tive variation in horizontal force. During downstroke, rela- tive speed and drag decrease, thereby causing a positive where γ , γ , and γ are the components of γ in the x, y, variation in horizontal force. Thus, a large yaw moment is x y z and z directions, respectively. generated around the wing root. Simultaneously, the lift Passive Active Pressure coefficient L 10 Applied Bionics and Biomechanics 0.5 –0.5 –1 0 0.5 1 (a) 1.5 0.5 –0.5 0 0.5 1 (b) ° ° Figure 12: Diﬀerences in (a) horizontal force coeﬃcient and (b) vertical force coeﬃcient between α =70 and α =90 . 0 0 0.6 1 0.3 0.5 0 0 0.3 –0.5 –0.6 –1 60 90 120 60 90 120 훼 (°) 훼 (°) 0 0 C C Z P Roll Pitch Yaw (a) (b) Figure 13: (a) Horizontal force, vertical force, and power coeﬃcient at diﬀerent α and (b) roll, yaw, and pitch moments at diﬀerent α . 0 0 increases during upstroke, thereby increasing the vertical of the aircraft during ﬂight. This process requires neither force. Then, the lift decreases during downstroke, conse- complex auxiliary mechanisms nor additional power input, quently decreasing the vertical force. As such, the variations and this characteristic is an advantage that is not exhibited in vertical forces during the upstroke and downstroke cancel by the active pitching model. each other. However, their distribution contributes to the pitch moment around the wing root. 4. Conclusions In Figure 13, the average aerodynamic power almost ° ° remains the same when α changes from 70 to 110 . The rest We investigate the aerodynamic performance of the passive angle of the torsional spring can be used as a control variable pitching model on FWMAVs via 3D numerical simulation in applying the passive pitching model. The adjustment of α and demonstrate that the angle of attack exhibits the charac- on the left and right wings controls the attitude and trajectory teristic of “double peak oscillation” under the combination of Difference from equilibrium 훥C 훥C X Z Difference from equilibrium Applied Bionics and Biomechanics 11 [6] D. Ishihara, T. Horie, and M. Denda, “A two-dimensional aerodynamic, spring, and inertial moments in the simpliﬁed computational study on the ﬂuid–structure interaction cause passive pitching model, which simulates the motion of insect of wing pitch changes in dipteran ﬂapping ﬂight,” Journal of wings well. Torsional stiﬀness considerably aﬀects aerody- Experimental Biology, vol. 212, no. 1, pp. 1–10, 2009. namic force and eﬃciency in the passive pitching model. [7] D. Ishihara, T. Horie, and T. Niho, “An experimental and Excess rigidity or ﬂexibility deteriorates the performance. three-dimensional computational study on the aerodynamic According to the comparison between active and passive contribution to the passive pitching motion of ﬂapping wings pitching wings, with appropriate torsional stiﬀness, the aver- in hovering ﬂies,” Bioinspiration & Biomimetics, vol. 9, no. 4, age lift can be enhanced by 10% at the same aerodynamic eﬃ- article 046009, 2014. ciency when the wing pitches passively. Simultaneously, the [8] Y. Chen, N. Gravish, A. L. Desbiens, R. Malka, and R. J. Wood, yaw moment around the wing root can be obtained to assist “Experimental and computational studies of the aerodynamic the control system without losing lift by setting diﬀerent rest performance of a ﬂapping and passively rotating insect wing,” angles for the left and right wings. These results show that the Journal of Fluid Mechanics, vol. 791, no. 1, pp. 1–33, 2016. passive pitching model positively contributes to the improve- [9] S. Zeyghami, Q. Zhong, G. Liu, and H. Dong, “Passive pitching ment of the hovering and maneuverability of FWMAVs. In of a ﬂapping wing in turning ﬂight,” AIAA Journal, vol. 57, the future, we will conduct a series of studies about the eﬀect no. 9, pp. 3744–3752, 2019. on the stability caused by passive pitching wing to further [10] G. Luo and M. Sun, “The eﬀects of corrugation and wing plan- investigate this bionic model. form on the aerodynamic force production of sweeping model insect wings,” Acta Mechanica Sinica, vol. 21, no. 6, pp. 531– 541, 2005. Data Availability [11] H. Dai, H. Luo, and J. F. Doyle, “Dynamic pitching of an elastic The data used to support the ﬁndings of this study are rectangular wing in hovering motion,” Journal of Fluid included within the article. The detailed calculation results Mechanics, vol. 693, pp. 473–499, 2012. are available from the corresponding author upon request. [12] M. Sun and Y. Xiong, “Dynamic ﬂight stability of a hovering The program and source code have not been made available bumblebee,” Journal of Experimental Biology, vol. 208, no. 3, because of privacy protection. pp. 447–459, 2005. [13] D. Lentink and M. H. Dickinson, “Rotational accelerations sta- bilize leading edge vortices on revolving ﬂy wings,” Journal of Disclosure Experimental Biology, vol. 212, no. 16, pp. 2705–2719, 2009. The results were originally presented at ICBE 2019. [14] W. Jianghao, Z. Chao, and Z. Yanlai, “Aerodynamic power eﬃciency comparison of various micro-air-vehicle layouts in hovering ﬂight,” AIAA Journal, vol. 55, no. 4, pp. 1265–1278, Conflicts of Interest [15] S. E. Rogers, D. Kwak, and C. Kiris, “Steady and unsteady solu- The authors declare that there are no conﬂicts of interest tions of the incompressible Navier-stokes equations,” AIAA regarding the publication of this paper. Journal, vol. 29, no. 4, pp. 603–610, 1991. [16] S. E. Rogers and D. Kwak, “An upwind diﬀerencing scheme for Acknowledgments the incompressible navier–strokes equations,” Applied Numer- ical Mathematics, vol. 8, no. 1, pp. 43–64, 1991. This research was primarily supported by the National Natu- [17] M. Sun and X. 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Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping Wing Micro Air Vehicles

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Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping Wing Micro Air Vehicles

Aerodynamic Performance of a Passive Pitching Model on Bionic Flapping Wing Micro Air Vehicles

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