Optimal design of insect wing shape for hovering ...

AIAA 2017-1071 AIAA SciTech Forum 9 - 13 January 2017, Grapevine, Texas 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

Optimal design of insect wing shape for hovering nano air vehicles

Downloaded by UNIVERSITY OF COLORADO on January 18, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-1071

G. Throneberry 1, M. Hassanalian2, and A. Abdelkefi3

Insects are widely regarded as some of the best natural flyers when it comes to hovering flight. Three insect wing shapes are considered to analyze to determine the optimum wing shape for hovering capabilities. Wing shape and geometry of wings affect the aerodynamic properties of the flapping wing vehicle and can greatly alter the performance of the wing. The wings being considered are assumed to be of equal wingspan allowing more focus to be placed on the effect of wing shape rather than area. The aerodynamic modeling is carried out using a quasi-steady approximation and a steepest descent gradient method is utilized for the optimization. The optimum Euler angles and minimum aerodynamic power for each wing are determined. The optimum wing shape is selected based on the minimum required aerodynamic power to perform the mission. This study shows that the twisted parasite wing shape yields a lower required aerodynamic power than the other wings considered in this work. This methodology could be a useful technique to determine the desired wing shape for flapping wing nano air vehicles with the ability to hover.

I. Introduction In recent years, smaller classes of drones, classified as Nano and Pico Air Vehicles (NAVs and PAVs) have been studied by several researchers. These drones have gained attention because of their increased performance, mainly coming from hovering abilities, quick acceleration, and great maneuverability1. These attributes have prompted many researchers to study, design, and fabricate bio-inspired drones2. These micro drones are designed and fabricated with inspiration from the thousands of birds and bats and millions of insects that employ flapping motion for movement3. This has also brought on an increase in the focus on micro-robotic drones and flight. Dickinson et al.4 worked on building a drone based on the fruitfly. This drone included sensors at the base of a wing. These sensors are able to collect data and measure the time course of aerodynamic forces allowing the researchers to investigate insect flight, more specifically the aerodynamic basis of their flight. Other researchers followed suit and developed their own respective bio-inspired drones. These projects have led to much investigation of insect flight and flapping wing aerodynamic analysis. Researchers have proposed different methods of analysis for the aerodynamics of flapping wing flight. Wood et al. 5 adopted a design process for flapping wing, insect-sized drones with hovering ability. In this process, many assumptions were made including the use of both linear and lumped representations for modeling the flapping wing dynamics. The blade-element method was employed to model the aerodynamic forces of the system. Upon modeling the aerodynamic model, they determined the mass fractions of the battery and actuation mechanism for the design of their flapping wing. This approach allowed them to generate realistic designs for their drone. The sizing process is crucial in the overall design process of any bio-inspired drone. The sizing process includes defining wing shape, aspect ratio, and geometry of the wing. It also includes the calculation of various system parameters, such as frequency, weight, and span.The power requirements of the flapping wing can also 1

Undergraduate research student, Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. 2

PhD student, Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. 3 Assistant Professor, Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. 1 American Institute of Aeronautics and Astronautics

Copyright © 2017 by G. Throneberry, M. Hassanalian, A. Abdelkefi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


be estimated using studies of power requirements of birds and insects 3. A major focus of flapping wing vehicles is the optimization of the system. There are several critical parameters for NAVs and PAVs. These include planform, flapping kinematics, and the system’s actuation mechanism6,7. This study is focused on the optimization of insects’ wing shapes and their associated flapping kinematics to minimize the system’s required aerodynamic power. One of the beginning steps of most sizing processes and a crucial step is determining the system’s flight mode. Some of the different flight modes are gliding, soaring, flapping, hovering, etc. This work is focused on flapping wing NAVs in hovering flight. There are several analytical methods can be employed to investigate the aerodynamics of flapping wings during hovering flight. Commonly, a quasi-steady approximation is utilized for the analysis and optimization7,8. A quasi-steady approximation simplifies the system by allowing the system’s kinematics to be easily determined7. A drawback from using a quasi-steady approximation is that it does not account for unsteady wake effects. More complex and sophisticated methods can be used to model the system but they require more demanding computation methods to determine the aerodynamic loads. There are also various methods that can be utilizd for the optimization of the aerodynamic parameters of the drone. One popular method of optimization is calculus of variation; this is the method that was utilized by Taha et al. 7. Other researchers have employed different methods for optimization. For example, Berman and Wang 8 combined a genetic algorithm and a gradient optimization to find the minimum aerodynamic power for hovering flight. In this work, three insect’s wings with hovering capabilities are selected for investigation. The selected wings are from the cranefly, twisted parasite, and cicada. A quasi-steady approximation for the modeling of the aerodynamic loads for hovering flight is employed. In this study, the wings are considered with an equal wingspan. A gradient method is utilized for optimization of Euler angles and required aerodynamic power. Two polynomial functions for the leading and trailing edges are generated, allowing geometric and aerodynamic properties to be determined. In section II, the wing shapes and determination of aspect ratios are determined and explained. Section III includes the explanation of the modeling and analysis of the kinematics of the flapping wing nano air vehicles. The optimization for equal wingspan is presented in section IV. Finally, a conclusion, including summary of results, is presented in section V.

II. Selected insect wing shapes with hovering abilities Wing shape and placement, along with many other geometric properties, are two crucial factors in the performance of any flapping wing system9-11. Wing placement affects the stability of the system around the roll axle. Wing shape is important in the production of lift and thrust forces. This study investigates the difference in wing shape in the lift and thrust forces of the flapping wing vehicle. As previously mentioned, wing shapes of three insects capable of hovering are being considered in this study. The wings being investigated are the cranefly, twisted parasite, and cicada. In Figure 1, the three insects under consideration are shown. Their associated polynomial representations compared to their actual wing shapes are presented in Figure 2. Clearly, there is a very strong correlation between the generated polynomial functions in comparison with their corresponding actual wing shapes.

Figure 1. Insects under consideration; (a) cicada, (b) twisted parasite, and (c) cranefly.

2 American Institute of Aeronautics and Astronautics


Figure 2. Wing shape polynomial representation for: (a) cicada, (b) twisted parasite, and (c) cranefly.

III. Modeling and optimization of flapping wing nano air vehicles When wing tip frequency is low in comparison with the flight speed, quasi-steady flow is assumed 3. Quasisteady flow is considered for this work. Quasi-steady flow does not account for the unsteady flow regime which is experienced when wing tips move at a higher speed than the flight speed 3. There are four degrees of freedom seen in flapping wing flight. The four degrees of freedom commonly seen in flight are flapping, lagging, feathering, and spanning12. The up and down motion of the wing that is commonly seen in flight is referred to as flapping. Lagging is the back and forward motion of the wing perpendicular to the mentioned flapping. Feathering refers to the twisting of the wing during flight. Lastly, spanning is the expansion and contraction of the wing. Insect wings are considered rigid, therefore exhibit no spanning during flight. This simplifies the flapping system to three degrees of freedom through flight. For this study, three Euler angles are considered. They are η, φ, and θ representing the pitching angle, the back-and-forth flapping angle, and the plunging angle respectively. Insect’s in hovering flight exhibit a fully back-and-forth flapping motion, meaning θ=0. Figure 3 gives a visual representation of the Euler angles for an insect in hovering flight.

Figure 3. Insect’s Euler angles in hovering flight.

During hovering flight, the insect’s angle of attack (α) is equal to the pitching angle. The pitching angle is the measure to the wing from the horizontal stroke plane, about the pitch axis. The back-and-forth flapping angle comes from the rotation of the wing about the yaw axis. These described Euler angles can easily be seen from Figure 3. The upstroke and downstroke (can be viewed as the back and forward stroke) for the hovering flight under consideration is assumed to be symmetric for simplicity in modeling and optimization.To begin the modeling of the hovering motion, geometric parameters must first be determined for use in the aerodynamic formulation. The surface of the wing (S) and the second and third moments of inertia (I2 and I3) are some of the initial parameters that must be determined and can be calculated with the following equations: b (1)

S  2 2 c(r )dr 0


(2)

b

I2  2 2 r 2c(r)dr 0

b

I 3  2  2 r 3c(r )dr

(3)

0

In the equations above, r denotes the wingspan and c(r) represents the chord length along the insect’s wingspan. Since the flow is considered as quasi-steady, we can use averages over the cycle to express the lift (L) and required aerodynamic power (P). (4) 1 2 T /2 I 2 T /2 2 2


L  V SC L  L   Ldt   C L dt 2 T 0 T 0  I T /2 2 T /2 1 P  DV .   V 3SC D dt  31   3C D dt T 0 2 T 0

(5)

where ρ, V, CL, T, t, and CD represent the density, velocity, coefficient of lift, period time, time, and coefficient of drag, respectively. In this study, the velocity at every wing section radius r from the root of the wing is considered as . When considering the hovering constraint (weight is equal to lift) and cost functions, the Lagrangian function (Lf) for optimization can be modeled as: (6)  2W 

Lf t , (t ),(t )   3C D  (t )    2C L (t )   I 2  

where λ is a constant and W represents the weight of the flapping wing vehicle. For optimization in this study, the gradient method is utilized to find the ideal values of η and φ. Specifically, the steepest descent method is the gradient method used for optimization. The steepest descent method says that for = − ( ) where n≥0; ( ) ≥ ( ) ≥ ( ) ≥ ⋯ ≥ ( ). In this method, γn is a small step size that is allowed to change in each iteration of the optimization13. Using this method, we search for an optimization so that xn converges to a minimum. Employing the steepest descent method to Eq. (6) with = − ( ), we arrive at: (7) 3 2 (t )C [ (t )]  2 (t )C [ (t )]  0 D

L

 C D [ (t )]   C L [ (t )]  0 3

2

The results from Eq. 7 are identical to the equations found by Taha et al.7 using the calculus of variation approach for optimization. The boundary conditions needed in the calculus of variation are not needed when using the gradient method. Applying the results from Eq. 7, the required aerodynamic power is given by: (8) 3

P

I3 I 23/2

2 W



with Ψ, the performance index, as:



CD2 ( * ) CL3 ( * )

(9)

Taha et al.7 offer more details about the derivation and relations of Eqs. (8) and (9). It can be seen that Ψ is a function of η*, the optimized angle of attack. Ψ is unique to each wing and is a constant throughout each half stroke. Looking at Eq. (8), Ψ should be minimized in order to minimize the required aerodynamic power. The η* will be determined to effectively minimize Ψ. Once η* is determined, an optimum rate of flapping, ( )*, can be expressed from Eq. (7) and given as: (10) 2C L  ( m )

(t )  

3C D  (m )

When t is between 0 and T/2, the downstroke optimum flapping angle is given by the triangular wave as follows 7:


*(t ) 

2W  I 2C L  (m )

(11)

 T  t    4

IV. Optimal design and performance of insects wings with same wingspan The three selected insects, cicada, twisted parasite, and cranefly, are traced and polynomial functions are generated for both the leading edge and trailing edge of each wing. The length of the chord of each wing is a function of the distance r of the wingspan and is given by c(r). Examples of polynomials generated for the leading and trailing edges of the cicada are given by: 6 5 4 3 2 (12)

c(r )L  6.733r  17.95r  18.1r  8.363r  1.766r  0.1911r  0.1071


c(r )T  1.548r 4  3.348r 3  2.879r 2  1.149r  0.05219

(13)

Since the wing is modeled a two halves, the total chord, c(r), is calculated as: (14)

c (r )  c (r )L  c (r )T

Using c(r), the geometric parameters previously discussed, S, I2, I3, as well as aspect ratio (AR) can be determined. The values for aspect ratio are shown in Table 1 for the three insects under investigation. Table 1: Geometrical parameters of the inspired insects’ wings Insect’s wing

AR

Cicada

5.90

Twisted Parasite

3.34

Cranefly

9.82

Two-dimensional and three-dimensional lift coefficients can be related in different ways 13,14. In this study, the coefficient lift is considered as CL(α)=Asin2α, A is dependent on the flow and the wing geometry. Under the assumption of small angles of attack, the three-dimensional coefficient of lift can be considered as CL(α)=0.5CLαsin2α. The aspect ratio of the wing affects CLα, to determine this effect, the extending lifting line theory can be utilized. Using the extending lifting line theory results in 7: (15)  AR CL  sin 2  1  2 AR 2  1  1  4  Only the leading edge vortex effect is considered and therefore, Eq. (15) neglects the effect of the suction force. Drag force is primarily a result of the pressure force that is perpendicular to the wing surface. The resulting force is parallel to the wing in an opposing direction to the velocity. Similarly to the coefficient of lift, the coefficient of drag can be expressed as7: (16)  AR 2

C D  C D 0  C L tan   C D 0 

 1  AR 2  1  1   4 

sin 

with CD0 representing the viscous drag effect. The viscous drag coefficient is a function of the Reynolds number which is directly dependent on the geometry of the wing2. (17) V c

Re ref 

r m



C D 0  0.455  log10 Reref



2.58

(18)

Vr represents the relative velocity of the wing during hovering flight. The maximum relative velocity of hovering flight for a flapping wing can be taken as 1m/s15,16. The wingspans of each wing are taken to be equal to 20cm 5 American Institute of Aeronautics and Astronautics

for the calculation of the viscous drag coefficient. The wingspan alters the mean aerodynamic chord of each wing, cm, which directly effects CD0. In order to calculate the mean aerodynamic chord, XFLR5 software is utilized. The geometric properties of the wings with a wingspan consideration of 20cm are shown in Table 2.


Table 2. Geometrical parameters of the inspired insects’ wings when b=20cm. Insect’s wing I2(10-4 m4) I3(10-5 m5) S(10-2 m2)

cm(m)

Cicada

0.23039

0.16113

0.35417

0.039272

Twisted Parasite

0.62952

0.42744

1.16372

0.06626

Cranefly

0.12989

0.09145

0.20459

0.02196

As was previously mentioned, in order to minimize the required aerodynamic power, Ψ must be minimized. Plugging Eqs. (15) and (16) into Eq. (9), the performance index can be expressed as a function of angle of attack. Deriving the resulting equation in terms of the angle of attack and equating to zero results in an optimal value for the pitching angle, η. Figure 4 gives a visual representation of the relation between angle of attack and Ψ for the three considered wing shapes. It is clear that the wing shape has an impact on the relationship of the performance index and the angle of attack of the wing. The optimum value for the pitching angle is given in Table 3 along with the value for the viscous drag coefficient used for each wing.

Figure 4. Variations of CD2/CL3 as a function of the angle of attack for considered insects with equal wingspan. Table 3. CD0 and optimized η for each wing shape. Insect’s wing

CD0

η* (degree)

Cicada

0.0197

6.417

Twisted Parasite

0.0167

6.646

Cranefly

0.0241

6.646

It follows from Table 3 that the viscous drag coefficient has a negligible effect on the overall drag coefficient. The optimum value of the pitching angle is near 6.5 degrees for all wings under the hovering constraint. This differs greatly from the pitching angle resulting in maximum lift, 45 degrees. This difference can be attributed to the fact that the drag coefficient is a function of the pitching angle. From Eq. (11), the optimal flapping angle amplitude can be given as:


T

W 4 I 2CL sin 2*

(19)

It can be noted that wing shape plays a crucial role in determining the optimum amplitude for flapping angle as it is dependent of I2, CLα, and η*. Empirical formulas are proposed to determine the flapping frequency of flapping wing vehicles. Azuma17 proposes that flapping frequency of small insects is dependent on mass and given as: (20) f  28.7m 1/3


Using Eq. (20), the flapping frequency can simply be related to the weight by:

W  f  28.7   g 

1/3

(21)

This result allows the amplitude of flapping angle to be given as a function independent of time as shown below: (22) 5/3

  0.1

W g  I 2C L *  2/3

 

Both acceleration due to gravity, g, and air density, ρ, are constants (9.81m/s2 and 1.225kg/m3 respectively), therefore the optimum flapping angle can be written as a function of weight for each wing shape. The plotted curves in Figures 5 and 6 compare the optimum pitching and flapping angles as functions of time over a flapping cycle, respectively. Inspecting the plots in Figure 5, it is noted that the change of wing shape produces negligible effects on the optimum pitching angle. For all wing shapes under consideration, the optimal pitching angle is very comparable. In contrast, Figure 6 shows that there is a much greater effect of wing shape on the optimal flapping angle. This variation is an effect of flapping angle being dependent on both the second moment of inertia and the lift coefficient, both of which are dependent on the wing shape. Table 4 provides values for the optimum amplitudes of flapping angle and power for each of the considered wing shapes.

Figure 5: Optimum pitching angles as a function of the normalized time when considering the three considered wing shpes.



Figure 6. Optimum flapping angles as a function of the normalized time when considering the three considered wing shpes. Table 4. The optimum values of flapping amplitude (Φ) and required aerodynamic power (P) for considered wing shapes with equal wingspan. Insect’s wing P/√ Φ/√ / (rad) Cicada

12.432

3.9969

Twisted parasite

8.314

2.6872

Cranefly

15.246

5.1138

Table 4 shows that the wing shape plays an important role in both optimal flapping angle and required aerodynamic power. When considering all wing shapes with equal weight, it can be seen that the twisted parasite requires the least aerodynamic power for hovering. Figure 7 shows the relationship between aerodynamic power and weight for the three considered insect wings. Clearly, an increase in the weight of the the flapping wing micro air vehicle is accompanied by an increase in the required aerodynamic power. Moreover, it can be noted that when considering same wingspan and weight, the twisted parasite requires minimum aerodynamic power for hovering, followed by the cicada and cranefly.



Figure 7. Variations of the required aerodynamic power as a function of the weight for different inspired insects’ wings when having the same wingspan.

V. Conclusions The efficiency of flapping wing nano air vehicles with hovering capabilities have been analyzed in this study. The wing shapes of three insects with hovering capabilities have been considered with the constraint of equal wing span. A polynomial function was generated for the leading and trailing edges of each wing to determine the geometric properties of each wing to be used in the calculation of its aerodynamic properties. To optimize the aerodynamic properties analyzed in this work, a steepest descent gradient method was used. Using this optimization technique, the optimal pitching and flapping angles that are able to generate the minimum aerodynamic power needed to maintain hovering flight were calculated. It was found that the twisted parasite requires the least amount of aerodynamic power for hovering under these considerations. It was followed in performance by the cicada and cranefly, respectively. 1

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