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Aerodynamic modelling of flapping flight using lifting line theory Joydeep Bhowmik, Debopam Das and Saurav Kumar Ghosh Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India Abstract Purpose – The purpose of the work is to design a flapping wing that generates net positive propulsive force and vertical force over a flapping cycle operating at a given freestream velocity. In addition, an optimal wing is designed based on the comparison of the force estimated from the quasi-steady theory, with the wind-tunnel experiments. Based on the designed wing configuration, a flapping wing ornithopter is fabricated. Design/methodology/approach – This paper presents a theoretical aerodynamic model of the design of an ornithopter with specific twist distribution that results generation of substantial net positive vertical force and thrust over a cycle at non-zero advance ratio. The wing has a specific but different twist distribution during the downstroke and the upstroke that maintains the designed angle of attack during the strokes. The wing is divided into spanwise strips and Prandtl’s lifting line theory is applied to estimate aerodynamic forces with the assumptions of quasi-steady flow and the wings are without any dihedral or anhedral. Spanwise circulation distribution is obtained and hence lift is calculated. The lift is resolved along the freestream velocity and perpendicular to the freestream velocity to obtain vertical force and propulsive thrust force. Experiments are performed in a wind tunnel to find the forces generated in a flapping cycle which compares well with the theoretical estimation at low flying speeds. Findings – The estimated aerodynamic force indicates whether the wing geometry and operating conditions are sufficient to carry the weight of the vehicle for a sustainable flight. The variation of the aerodynamic forces with varying flapping frequencies and freestream velocities has been illustrated and compared with experimental data that shows a reasonable match with the theoretical estimations. Based on the calculations a prototype has been fabricated and successfully flown. Research limitations/implications – The theory does not take into account the unsteady effects and estimates the aerodynamic forces at wing level condition. It doesn’t predict stall and ignores structural deformations due to aerodynamic loads. The airfoil section is only specified by the chord, zero lift angle of attack, lift slope, profile drag coefficient and angle of attack as given inputs. To fabricate a light weight wing that maintains a very accurate geometric twist and camber distribution as per the theoretical requirement is challenging. Practical implications – Useful for designing ornithopter wing (preferably bigger) involving an unswept rigid spar with flapping and twisting. Originality/value – The novelty of the present wing design is the appropriate spanwise geometric twisting about the leading edge spar. Keywords Ornithopter, Lifting line, Flapping, MAV, Wing twist, Upstroke, Downstroke, Aerodynamics, Design Paper type Research paper
International Journal of Intelligent Unmanned Systems Vol. 1 No. 1, 2013 pp. 36-61 r Emerald Group Publishing Limited 2049-6427 DOI 10.1108/20496421311298134
Introduction Flapping wing flying machines have been envisaged as bio-mimicking mini/micro aerial vehicles (MAVs) and several models have been developed to date. The additional The authors would like to acknowledge the National Programme of Micro Ariel Vehicle (NPMICAV) of AR&DB and DST, India for partial support of this work.
requirement of an MAV is its invisibility to radars. This means it should be silent and small as far as possible. A bird or an insect type MAV would be an excellent choice in this respect. The MAVs are classified based on the size. The mini aerial vehicle size typically varies from 300 to 2,500 mm and that of micro aerial vehicle will be less than 300 mm. In pursuit of understanding how birds and insects fly, many theoretical modellings have been developed so far. But during the 19th century when fixed wing aircrafts were successfully flown, Prandtl’s lifting line theory served as a good method to predict the aerodynamic forces of the wing. This method has also been adopted by researchers like (Betteridge and Archer, 1974; Archer et al., 1979), in which they modeled a torsionally flexible rigid wing flapping about the spar axis. Unfortunately flow over a flapping wing is strictly not steady so attempts have been made in unsteady lifting line theory, for example (Phillips et al., 1981). On the other hand computational methods also predicted more generalized wing geometry with reasonable accuracy, for example, panel methods by Ashley and Landahl (1985), Ashby et al. (1988) and Katz and Plotkin (1991). An unsteady panel method has also been implemented to model the aerodynamics of flapping wing for moths by Smith et al. (1996). Another method is a two-dimensional method using thin airfoil theory developed by DeLaurier (1993) and DeLaurier and Harris (1993). Their method is, however, well applicable in stability of helicopter rotor blades having small deflections. Twisting of wings is an important aspect that greatly affects the efficiency and performance of flight. Twisting can be provided actively or passively. Aero-elastic models developed by Theodorsen (1935) modeled aerodynamic instability and flutter in aircraft wings. Jones (1980) mathematically predicted an ideal distribution of the additional lift during slow flapping. On the other hand several experimental works have also been carried out. Hertel (1963/1966) showed a correlation of the weight of different birds and their flapping frequency. Bigger birds like the pelican flap at low 1 Hz whereas small birds like humming birds can beat their wings up to 60 Hz. Dickinson et al. (1999) also presented a detailed work on insect flight performance due to delayed stall, rotational circulation and wake capture. Ho et al. (2003) presented a detailed study of force measurements and showed the variation of lift and thrust coefficients with varying flight conditions. Numerous experiments and flow visualizations have been carried out on tethered and free flying insects by Adrian et al. (2004) and humming birds by Tobalske et al. (2007). Slow motion cameras have also been used to capture the kinematics and geometric twist of the wings. Force and PIV measurements are conducted on a flapping model goose with a wingspan of 1.13 m by Hubel and Tropea (2009). Experiments on flapping wing models of different shapes and size are carried out by Banerjee et al. (2011) robots and insect flight by Ellington (1984) and Ellington et al. (1996). The present method extends the application of lifting line theory both during upstroke and downstroke with a specified distributed twist for estimation of forces and its verifications with experiments. Modelling of flapping flight using lifting line theory In the present work, the wing of an ornithopter has been designed with specific twist distribution that produces sufficient lift and thrust over a cycle at non-zero advance ratio. The twist distribution maintains the designed angle of attack during the strokes. The wing is divided into spanwise strips to estimate aerodynamic forces using Prandtl’s lifting line theory. Quasi-steady assumption is made to estimate the forces
Modelling of flapping flight
37
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when wings are without any dihedral or anhedral. The forces estimated using this theory are expected to reduce with dihedral or anhedral and hence a factor of safety is used in the design. The novelty of the present wing design is the appropriate spanwise geometric twisting about the leading edge spar axis. This simple theory turns out to be a tool for quick estimation of the performance of a flapping flight that has lead us to develop a flyable prototype MAV (Bhowmik and Das, 2011). Assumptions The wing planform is elliptical and unswept. The wing maintains a specific geometrical twist for the upstroke and also a specific but different geometrical twist during the downstroke. The geometries are not affected due to changing wing loads. The estimated vertical force and thrust force is assumed to be the average of the same generated at the upstroke and the downstroke, respectively. Wing stall is also not considered. The model considered is one of the simplest in nature where flow is considered quasi-steady, i.e. unsteady effects are not included. The flow is also potential but an overall profile drag coefficient is considered in addition to the induced drag of the wing. Nomenclature a aCL¼0 ai f, Ø O Vr od VN Vtip ft, fv FT, FV
local angle of attack (or AOA) angle of attack for zero lift induced angle of attack frequency and amplitude of flapping angular velocity of the wing spar resultant velocity at wing section downwash velocity freestream velocity wing tip velocity due to flapping only sectional thrust and vertical force total thrust and total vertical force
F V; F T as yt ~ G, Cd0 P, T r, a0 b c, C0, C S yi Le, De, Di, D0
Average vertical force and thrust on a flapping cycle angle between hinge axis and VN angle between VN and Vr transform variable circulation, profile drag coefficient power, torque on wing density of air, lift curve slope. tip to tip wing span at wings level Chord, root chord and geometric mean chord total wing area geometric twist of wing section with respect to the hinge axis effective lift, effective drag, induced drag and profile drag
Wing details and flapping parameters Figure 1 shows the schematic elliptical wing geometry, flapping parameters and the coordinate axes used as primary inputs for calculation in the present model. The wing has an elliptical planform and flaps through angle Ø. The angle Ø1 and Ø2 are defined
(a)
Modelling of flapping flight
b
39
C0
(b) Top dead center
Ø Level position
Ø1 Ø2
Bottom dead center
(c) Hinge axis
s V∞
Note: The hinge axis center line makes an angle (s) with the free stream velocity
with respect to the wing level position. A dihedral is obtained throughout the flapping cycle when Ø14Ø2 which enhances the role stability and used in the prototype. The wings are hinged to a common axis with a pin and this axis line makes an angle as with the freestream velocity. Further details of the wing fabrication are given in Bhowmik (2012).
Figure 1. (a) The wing planform (elliptical), (b) front view of the flapping plane showing the wing spars at the top and bottom dead centers, (c) side view of the wing at the top dead center
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The parameters used in this study are: .
Advance ratio: this is the ratio of the freestream velocity (VN) to the tip velocity of the wing.
.
Time period of flapping: this is the time required to complete one cycle.
.
Average angular velocity of flapping (O): this is defined as the ratio of flapping amplitude to the time period of flapping.
.
Effective lift and drag: by convention lift and drag is defined as the force normal and parallel to VN, respectively. But in case of flapping the local airfoil section sees a velocity Vr to which the lift is perpendicular. So to avoid confusion we call this lift as effective lift Le and the drag as effective drag De.
40
Advantage of flexible wing over rigid wing Figure 2 illustrates the differences in motion between the rigid wing as considered by Archer et al. (1979) and flexible wings of the present study. Figure 2(a) and (b) depict the scenario if the current model is subjected to a constant twist (yi) throughout the span similar to that of model of Archer et al. (1979). Figure 2(c) shows the variable spanwise twist (yi) during the downstroke as used in the present work. Similar spanwise twist of different twist distribution is also used in upstroke. Thus the present model closely mimics the observed kinematics of a bird’s wing during its flight. (a)
i
(b) i
(c)
Figure 2. A rigid wing
Figure 3 shows the spanwise variation of a for a rigid wing, operating at different advance ratios, keeping f ¼ 8 Hz, Ø ¼ 601 and at yi ¼ 7201 at as ¼ 01. The half span of the wing is 0.25 m. The corresponding advance ratio is shown in the bracket in the legend of the figure. It is observed that at zero advance ratios, the angle of attack is constant throughout the span. As the advance ratio increases, the root to tip variation of a (i.e. Da) initially increases and then decreases. Figure 3 follows from Equation (1) and can be derived easily from the vector diagram of Figure 5: a ¼ tan1
O:y yi VN
Modelling of flapping flight
41
ð1Þ
If a rigid wing operates at an advance ratio 1.433 (At 8 Hz and 6 m/s, see Figure 3) at which the variation of root to tip a is 15 to þ 151 (or Da is 301, see Figure 4) some part of the wing will be on the verge of stall or completely stalled under steady states. The wing can be given a specific geometric twist so that a desired a is maintained throughout. The main objective of this analysis is to estimate the power consumption for producing a required vertical force and thrust on a wing with a given distributed spanwise twist. A flexible wing thus will be capable of maintaining a desired spanwise angle of attack and therefore could be more efficient than a rigid wing. 80
V∞ 0 m/s 1 m/s 2 m/s 3 m/s 5 m/s 6 m/s Infinite
70 60 Increasing advance ratio
40 30 20
Figure 3. Spanwise a variation of a rigid wing if allowed to twist 7201 about the spar axis during the upstroke and downstroke at different speeds with 8 Hz flapping frequency
Δ
(degrees)
50
10 0
–10 –20
f=8 Hz
–30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 y/ wingspan
1
70 Δ (degrees)
60 50 40 30 20 10 0 0
1
2
3
4 J
5
6
7
Figure 4. Variation of Da with advance ratio
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Theory The method presented here is based on the Prandtl’s lifting line equation on a flapping wing at the level position similar to (Betteridge and Archer, 1974). A certain geometric twist distribution (spanwise aCl ¼ 0 and angle of attack is provided during the downstroke and upstroke for calculation of net vertical and thrust forces. The average of these is taken to estimate the net force in a flapping cycle. The forces are estimated using the generalized Prandtl’s lifting line equation as given below: Z b=2 2Gðy0 Þ 1 dG=dy aðy0 Þ ¼ þ aL¼0 þ dy ð2Þ a0 V1 Cðy0 Þ 4pV1 b=2 y0 y GðyÞ ¼ 2bV1
X
ð3Þ
An sinðnyÞ
Derivation is available in any standard textbook of aerodynamics. Equation (2) is an integra-differential equation and can be used for most unswept and level wings. During downstroke and upstroke, this equation is used to model the vortex wake and the spanwise circulation distribution is determined at the wing level condition, assuming elliptical planform of unswept wings with inviscid, steady flow. The downstroke equations. This is the movement of the wing from the top dead center to the bottom dead center and provides most of the vertical and thrust forces. The vector diagram of forces and velocities are shown in Figure 5. It may be noted that the heaving velocity component O.y is normal to the hinge axis. From the geometry of a wing section in Figure 5 the following equations are derived: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vr ¼ ðO:y: cosðas ÞÞ2 þ ðV1 þ O:y: sinððas ÞÞ2 ð4Þ O ¼ 2Øf
ð5Þ fT
De
fV
Le Hinge axis s
Figure 5. Wing section with velocity and force vectors during downstroke
d
V∞ i
t
Vr
Ω.y
The twist in the wing is depended on the angle (yt) between the (VN) and (Vr) and section wise angle of attack a. The wing twist yt, changes with advance ratio and written as: O:y: cosðas Þ ð6Þ yt ¼ tan1 V1 þ O:y: sinðas Þ
Modelling of flapping flight
43 In Figure 6, y varies from 01 (at tip) to 901 (at root). This angular sector is divided into “n” equal angular divisions, whose projections on the wing are at a distance “y” from the root where: b y¼ cosðyÞ ð7Þ 2 The wing is divided into equal angular divisions and projected on the wing planform as shown in Figure 7(a) and “y” is calculated from Equation (7). This produces a set of points having larger strip thickness near the root and gradually thinning up near the tip instead of having strips of equal lengths. The key difference between a fixed and a flapping wing is that in a flapping wing the wing section sees the relative velocity Vr instead of the freestream velocity VN. The G distribution given in Equation (3) can be written as: X GðyÞ ¼ 2:b:Vr : An : sinðnyÞ ð8Þ
Trailing edge Root
Tip
dy
Leading edge
y b/2
Figure 6. Back view of the wing at level condition showing the elementary strips and relation between y and y
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Figure 7. Dividing the wing into elementary strips
Notes: (a) Equal angular divisions and then projecting and (b) strips of equal length which is shown for comparison
Substituting Equation (8) in (2) and after simplification, a system of linear equations for different values of y is obtained as: X mðyÞ:aa ðyÞ: sinðyÞ ¼ A2nþ1 : sinfð2n þ 1Þyg:fð2n þ 1Þmy þ sinðyÞg ð9Þ where n varies from 0 to infinite for a continuous distribution and: mðyÞ ¼
a0 :cðyÞ 4b
aa ðyÞ ¼ aðyÞ aCL ¼0 :ðyÞ
ð10Þ
ð11Þ
Solve for the Fourier coefficients A1, A2, A3, etc. and substitute in Equation (8) to get G (y). Effective Lift (lift normal to Vr instead of VN) per unit span, ai and Di can be found as: Le ðyÞ ¼ GðyÞ:r:Vr :dy X
ð12Þ
sin ny sin y
ð13Þ
Di ¼ Le ðyÞ: sin ai
ð14Þ
1 2 De ¼ Di þ Cd0 :r:V1 :Astrip 2
ð15Þ
ai ¼
An :
The total drag is estimated as the sum of the induced drag Equation (14) and profile drag where, Cd0 is the profile drag coefficient.
Resolving the components of effective lift and drag along and normal to the freestream velocity, thrust ( fT) and vertical force ( fV) are obtained (see Figure 5): fV ¼ Le ðyÞ: cos yt þ De ðyÞ: sin yt
ð16Þ
fT ¼ Le ðyÞ: sin yt þ De ðyÞ: cos yt
ð17Þ
Equations (16) and (17) above can be plotted to see the spanwise vertical and thrust force distribution. Therefore, the total vertical and thrust force is obtained by integrating throughout the wing span. In this case the integration is done numerically by summing up fVdy and fTdy from root to the wing tip: Z þb=2 FV ¼ fV dy ð18Þ b=2
FT ¼
Z
þb=2
ð19Þ
fT dy b=2
Similarly the wing torque is obtained by integrating the elementary moment due to the component of the force on the flapping stroke plane: T¼
Z
þ2b b2
ffV : cos as ft : sin as g:y:dy
ð20Þ
Once the torque is obtained, the power can be calculated as a product of the torque and the angular speed of the wing: P ¼ T:O
ð21Þ
The upstroke equations. The upstroke is the movement of the wing from the bottom dead center to the top dead center. Figure 8 shows that any section of the wing during upstroke can either produce positive fV and negative fT (for positive cambered part of the wing) or negative FV and positive fT (for negative cambered part of the wing). Figure 9 shows the vector diagram of balance of forces and velocities on the elementary section of the wing during upstroke. During the upstroke, if there is net positive vertical force, the net torque will cause the wing to flap up on its own; the power for such instant will be negative. Formulae for different quantities during upstroke can be derived from Figure 9 similar to that during downstroke as shown in Equations (22)-(25). The rest of the quantities are the same as Equations (12)-(15). The upstroke vertical force, thrust and power are found similar to that explained in downstroke equations, Equations (18)-(21):
yt ¼ tan1
O:y: cosðas Þ V1 O:y: sinðas Þ
ð22Þ
Modelling of flapping flight
45
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De
– fT Le
Vr
+ fV
Ω.y
V∞
46
Vr
Ω.y Le
– fV V∞
Figure 8. A wing section during upstroke
+ fT
De
Notes: (a) Positive ( f V) and negative ( f T ), (b) negative ( f V) and positive ( f T)
De
fT
Le
fV
Hinge axis
s
α
V∞
Figure 9. Force and velocity vectors of a section with positive vertical force and negative thrust during upstroke
Vr
Ω.y t
i d
Vr ¼
V∞
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðO:y: cosðas ÞÞ2 þ ðV1 O:y: sinðas ÞÞ2
ð23Þ
fV ¼ Le ðyÞ: cos yt De ðyÞ: sin yt
ð24Þ
fT ¼ Le ðyÞ: sin yt þ De ðyÞ: cos yt
ð25Þ
Methodology. The set of equations are obtained for downstroke and upstroke separately and solved for the Fourier coefficients (A1, A3, A5 , etc.). This gives the spanwise circulation distribution during the downstroke and upstroke. Effective lift and thrust can then be calculated. Integrating these elementary quantities the total effective lift, thrust and power are obtained. The net vertical force F V and thrust force F T on a flapping cycle is the average of FV and FT during upstroke and downstroke, respectively.
The process may be repeated by adjusting various conditions till sufficient lift and thrust are produced to sustain flight. Once the flying conditions are decided, the geometric twist, yi can be found as shown in Equations (26) and (27) by setting as ¼ 01 in Figure 10. Once this design configuration is determined, its performance at different advance ratios and as can also be determined: yi ðupstrokeÞ ¼ yt þ a
ð26Þ
yi ðdownstrokeÞ ¼ yt a
ð27Þ
Modelling of flapping flight
47
Effect of advance ratio. As advance ratio increases, yt decreases and vice versa. The fabricated wing has a fixed geometric twist, i.e. yi is fixed. The wing then behaves like a fixed pitch propeller which operates best at the designed advance ratio. Due to this restriction, the angle of attack increases at low advance ratio and reduces at high advance ratio. For example in Figure 11(b) the freestream velocity is reduced to almost half as that in Figure 11(a) but O.y is same (less advance ratio) thereby increasing yt. Thus the angle of attack a increases as yt increases and vice versa. During upstroke, Figure 11(d)
α
V∞
V∞
i
V∞
i Vr
t
Ω.y
t
Ω.y
Vr
α
V∞
Notes: Sign convention: i measured anticlockwise is positive (a) upstroke |i| = |t| + ||, (b) downstroke |i| = |t| – ||
Hinge axis
Hinge axis
s=0 i
V∞ t
Ω.y
α
s=0 i
Vr
Ω.y
Vr
α s=0 Hinge axis
i t
α
Vr
–α
Hinge axis
i Vr t
t V∞
s=0
V∞
Ω.y
Figure 10. Geometric twist during upstroke and downstroke
V∞
Ω.y
Figure 11. (a) Velocity vectors of a wing section at downstroke for reference, (b) the same section operating at less advance ratio causing an increase in a, (c) velocity vectors of a wing section at upstroke for reference, (d) the same section operating at less advance ratio (same O.y and reduced VN) causing negative a
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the a changes to negative at low advance ratios. This airfoil would generate low lift but in the reverse camber section the vertical force would be downward producing positive thrust. The angles of attack at different advance ratios are given in Equations (28) and (29):
48
a ¼ jyt j jyi j ðDuring downstrokeÞ
ð28Þ
a ¼ jyi j jyt j ðDuring upstrokeÞ
ð29Þ
Effect of as. During downstroke, as the wing flaps down, a section of the wing sees an angle of attack due to the combination of VN and O.y with the hinge axis parallel to the freestream velocity as shown in Figure 12(a). Figure 12(b) illustrates the scenario when the hinge axis is set at an angle of as40 thereby increasing a by as. (the induced angle of attack is removed for clarity). This parameter in flight is controlled by the all movable tail. If the tail produces a pitch up moment, as would increase and vice versa. However, while designing the wing it should be ensured that the local a does not become so high that the wing stalls. Similarly during upstroke, as shown in Figure 13 when the wing tilts anticlockwise by an angle as the airfoil section in the lift producing part sees an increase in a as shown in Figure 13(b) compared to Figure 13(a). But in the reverse cambered part of the wing, a is likely to be negative as seen in Figure 13(c). The impact of as thus increases a by þ as as shown in Figure 13(d) compared to Figure 13(c). Due to combined variation of advance ratio and as the angle of attack becomes: a ¼ ðjyt j jyi j þ jas jÞ ðDuring downstrokeÞ
ð30Þ
a ¼ ðjyi j jyt j þ jas jÞ ðDuring upstrokeÞ
ð31Þ
Results This section contains the theoretical results based on the above derivations. Calculations are carried out for the upstroke and downstroke and used in predicting the aerodynamic forces on the wings of the different models. For analysis, wing size is taken as 50 cm tip to tip in order to give a resemblance of a small bird with an estimated weight of 32 gF. The wing planform is taken to be
Hinge axis
Hinge axis s=0
s>0 V∞
i
V∞ t
Figure 12. Velocity vectors at a wing section in downstroke
t
Ω.y Vr
V∞
α
Notes: (a) For s=0° and (b) for s>0
Ω.y Vr
α
i
Hinge axis
s=0
V∞ α
i
Vr
α
s>0
49
V∞
s=0
Hinge axis
V∞ s>0
i Vr α
Ω.y
t
V∞ Hinge axis
Modelling of flapping flight
Vr
i
Ω.y
t
Hinge axis
t V∞
Ω.y
i α
Vr
Ω.y
t V∞
Notes: (a) Positively cambered section at s=0°, (b) the same section at s>0°, (c) condition in a reverse cambered section of the wing at s=0°, (d) condition in the reverse cambered section of the wing at s>0°
elliptical with an expectation to maximize efficiency. The fuselage and tail are designed in proportion to this wing size. Using the theory, the aerodynamic forces and power consumption are calculated at different flapping frequencies and VN. Specific geometrical twist is assumed during downstroke and upstroke. The upstroke being more critical, three types of wing geometries are tried out for the upstroke keeping the downstroke same. The inputs during the downstroke are: r ¼ 1.2 kg/m3, b/2 ¼ 0.25 m, C0 ¼ 0.12 m, a0 ¼ 2~, No. of wing partitions ¼ 100, Cd0 ¼ 0.0135, acl ¼ 0 ¼ 41, a ¼ 61 throughout the wing from root to tip and flapping amplitude is 551. Substituting the Fourier coefficients in Equations (11), (12), (17), (18), the ( fV) and ( fT) are obtained as a function of (y), which is transformed back to normalized wing span to get the spanwise variation during the upstroke and downstroke. Figure 14(a) shows the computed sectional variation of vertical forces and thrust forces during the downstroke on one wing at a flapping frequency of 8 Hz and freestream velocity of 6 m/s. The downstroke thus produces most of the vertical force and propulsive thrust. Upstroke: the wing is twisted such a way that a part of the wing near the root has positive camber and positive a and remaining portion up to the tip has reverse camber and negative angle of attack. In Bhowmik and Das (2011) the calculations are carried out for a specific upstroke wing geometry. This paper explores more cases with different spanwise distributions of angle of attack and aCl ¼ 0 during the upstroke. Three such case studies have been presented here. The downstroke is same for all these cases as shown in Figure 14. Case 1 In Case 1 during the upstroke, the hand-wing maintains a constant positive camber and a whereas, the arm wing has a constant reverse camber and a. The spanwise distribution of a and aCl ¼ 0 is shown in Figure 15. The resulted
Figure 13. Velocity vectors at a wing section
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force distribution is shown in Figure 16. The upstroke reveals where there is positive vertical force, there is negative thrust and for negative vertical force, there is positive thrust. The negative cambered zone toward tip produces negative vertical force and positive thrust. Case 2 This case is similar to Case 1 but instead of a constant camber and a, a variable camber and a is used during the upstroke. Figure 17 shows the upstroke wing twist parameters. Variation of a and aCl ¼ 0 is given by: a ¼ m1 x2 þ m2 x þ m3 and aCl¼0 ¼ n1 x2 þ n2 x þ n3 where the coefficients are found based on the boundary conditions: at x ¼ 0, a and aCl ¼ 0 are known quantities at the root; at x ¼ x1, a and aCl ¼ 0 are zero; and at x ¼ 1, a and aCl ¼ 0 are known quantities at the tip.
2.5
10 Wing twist-downstroke
2
5
fT
0 1.5
i (degrees)
Figure 14. (a) Spanwise variation of ( fV) and ( fT) from root (0) to tip (1) during down stroke (b) wing twist during downstroke
Force distribution (N/m)
fV
1 0.5
-5 –10 –15 –20
0 –0.5
–25 0
0.2
0.4
0.6
0.8
1
–30
0
0.2
0.4
0.6
0.8
4
3
CL=0
Angle (degrees)
2 1 0 –1 –2
Figure 15. Section wise geometric and air foil lift properties during upstroke (Case 1)
–3 –4
0
0.2
0.4 0.6 Normalized wingspan
0.8
1
1
Once the coefficients are obtained, a distribution of a and aCl ¼ 0 can be obtained. Figure 18 shows the result where at x1 ¼ 0.6, a and aCL ¼0 are 0. This produced a more realistic output for the spanwise force distribution during the upstroke as in Figure 18. Case 3 In this case, the effect of the location of the reverse camber is studied. Let the reverse cambered zone of the arm wing be increased to 70 percent of the span from the tip as
51
35
1 fT
0.6 0.4 0.2
25 20 15
0
10
–0.2
5
–0.4 0
0.2 0.4 0.6 0.8 Normalized wingspan
Wing twist-upstroke
30
fV
0.8
i (degrees)
Force distribution (N/m)
Modelling of flapping flight
0
1
0
0.2 0.4 0.6 0.8 Normalized wingspan
1
Figure 16. For Case 1 (a) Spanwise variation of ( fV) and ( fT) from root (0) to tip (1) during upstroke, (b) wing twist during upstroke
4
Angle (degrees)
3
CL=0
2 1 0 –1 –2
Figure 17. Section wise geometric and aerofoil lift properties during upstroke (Case 2)
–3 –4
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Normalized wingspan
0.8
0.9
1
35 Wing twist-upstroke
fV
0.6
30
fT
25 i (degrees)
Force distribution (N/m)
0.8
0.4 0.2
20 15 10
0 5 –0.2 0
0.2
0.4 0.6 0.8 Normalized wingspan
1
0
0
0.2
0.4 0.6 0.8 Normalized wingspan
1
Figure 18. (a) Spanwise variation of ( fV) and ( fT) from root (0) to tip (1) during upstroke with the modified distribution of distribution of a and aCl ¼ 0 (Case 2), (b) wing twist during upstroke
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shown in Figure 19. The curve follows same equations as in y (I.2) but with different coefficients. It is observed that the spanwise loading also increases or decreases with the twist distribution. In this case, by increasing the reverse camber part, the thrust at tip increased but vertical force decreased. The net vertical force becomes negative but there is a net positive thrust. The positive thrust produced at the expense of huge negative vertical force near the tip as seen in Figure 20. The average vertical and thrust forces per wing at VN ¼ 6 m/s and flapping frequency of 8 Hz are tabulated in Table I which shows the upstroke Case 2 is the best choice. But any of the cases would certainly provide enough vertical force to sustain the estimated weight of 32 gF.
4 α
3
αCL=0
Angle (degrees)
2
Figure 19. Section wise geometric and aerofoil lift properties during upstroke with 70 percent reverse camber
1 0 –1 –2 –3 –4
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Normalized wingspan
0.8
0.9
1
35
0.4
Wing twist-upstroke
Table I. Average vertical forces and thrust forces at VN ¼ 6 m/s and f ¼ 8 Hz with as ¼ 01
0.2 25 i (degrees)
Figure 20. (a) Spanwise variation of ( fV) and ( fT) from root (0) to tip (1) during upstroke, (b) Wing twist during upstroke
Force distribution (N/m)
30
0 –0.2 fV
15 10
fT
–0.4 –0.6
20
5 0
0.2
F V (gF) per wing F T (gF) per wing
0.4 0.6 0.8 Normalized wingspan
1
0
0
0.2
0.4 0.6 0.8 Normalized wingspan
1
Case 1
Case 2
Case 3
25.1 3.16
25.18 4.733
20.16 6.15
Estimation of in-flight performance The geometric twist of the wing during downstroke is decided as shown in Figure 14(b) and for upstroke as in Figure 18(b) (i.e. Case 2). With this fixed geometry, any change in as or advance ratio will cause a change in the local angle of attack which is assumed to be constant for the designed operating condition. As a result F V and F T are re-calculated and plotted in Figure 21 at different flapping frequency, VN and as using the concept presented in y(4) and y (5). Figure 21(a) shows that the average vertical force
8.00 Hz
160
7.03 Hz
ƒ
5.65 Hz
120 100
7.03 Hz 5.65 Hz
20
4.36 Hz
s =7°
80
8.00 Hz
ƒ
30
FT (gF)
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4.36 Hz
s=7°
10 0
60 40
–10
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V∞ (m/s) 160 V∞
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40
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s=7°
100 80 60 40
s =7°
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4
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5 6 7 Flapping frequency (Hz)
8
50 V∞
V∞
45 40
ƒ = 8 Hz
1m/s 3m/s 5m/s
2m/s 4m/s 6m/s
35 FT (gF)
80 FV (gF)
Modelling of flapping flight
60 40
30 25 20 15 10
20
5 0
ƒ = 8 Hz
0 0
1
2
3 4 5 s (degrees)
6
7
8
0
1
2
3 4 5 s (degrees)
6
7
8
Figure 21. Theoretical variation of F V and F T with VN and flapping frequency
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gradually increases with VN and increases sharply at higher VN unlike the thrust force (Figure 21(b)) that decreases with increasing VN. Thus climb rate could be increased by increasing VN. Figure 21(c) shows that the average vertical force is increased by only about 1.2 gF per unit increase in flapping frequency, at VN ¼ 6 m/s and this slope reduces as VN reduces. However, from Figure 21(d), there is a drastic increase of average thrust with frequency increasing. With increasing speed, VN this increase in average thrust reduces. Thus to accelerate forward, flapping frequency should be increased. Figure 21(e) shows that the vertical force increases with increase in as but the slope reduces as VN decreases. But from Figure 21(f) the average thrust force reduces by only a few gF with increase in as and the slope becomes steeper at higher VN. Thus by increasing as the ornithopter can be made to gain altitude without changing VN. Fabrication of the protoype The wing The existing wing design consists of a carbon fiber spar and three stiffeners pasted on a polythene membrane. When the stiffener is pasted on a membrane, the membrane can have a twist about the stiffener. During the upstroke, there is about 50-60 percent of the wing (hand wing) near the root producing lift and the remaining (arm wing) producing thrust. Figure 22 shows the schematic of the present design where the cambered stiffeners “1” and “2” comprise the hand wing part and produce positive vertical force during upstroke and downstroke. The straight stiffener “3” is placed almost parallel to the wing spar provides positive camber during upstroke as well as reverse camber during the upstroke. A membrane coinciding through these stiffeners thus correspond to the nearest twist profile. During downstroke the high pressure below the wing gives a positive camber from root to tip whereas during upstroke, the high pressure over the wing causes the reverse camber near the tip but due to the stiffener arrangement in the hand wing “A,” a positive camber is maintained at all times. This is illustrated in Figure 23 showing the model ornithopter developed by the author during a wind tunnel experiment. The twisting and deformation takes place passively due to the wing loading so it also implies that once the twist of the wing is fixed this way, it will be providing the desired performance only at this designed speed and flapping frequency.
A
B
3
Figure 22. Schematic of the existing design showing the positive cambered “A” and negative cambered “B” during the upstroke
Hinge axis 2 Carbon fiber spar 1
Polythene membrane Balsa stiffeners
Upstroke
Modelling of flapping flight
Downstroke
55 Reverse camber
Positive camber
Notes: During the (a) upstroke and (b) downstroke. Filmed using a high-speed camera during wind tunnel testing
Figure 23. Camber distributions
The fuselage and mechanism This is a small size ornithopter which weighs 32 g with a wing span of 50 cm tip to tip. The structure is made with an integrated gearbox of 26:1 gear ratio. The Flapping motion is accomplished with a 4 bar mechanism. Both the wings flap symmetrically and are hinged on the same shaft. The fuselage is made from light weight balsa wood. The crankshafts and linkages are made from 1.2 mm steel wire. The wings are hinged to a common pin named the hinge axis and is used as a reference. The connecting rod is connected end to end by a ball joint. Figure 24 shows a schematic of the design layout and mechanism. Kinematic analysis At the full throttle, the crankshaft rotates at 504 rpm ( f ¼ 8.4 Hz). Motion analysis simulations using SolidWorkst-Motion has been performed and the rpm is set to 5.04 instead of 504 rpm crankshaft speed for better accuracy of the solver. The wing position, Ø with time data are obtained. The angular velocity and angular acceleration were determined by taking the first and second derivative of wing position, respectively. For a given flapping frequency f, the time axis is scaled depending on different flapping frequencies to get the angular velocities and acceleration as a function of time (T ): t¼
T f 60 100 504
ð32Þ
Wing spar
Hinge axis Connecting rod
Crank shaft
Figure 24. A SolidWorkst model of the gear box and mechanism integrated with the fuselage
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where f is the flapping frequency (Hz); T the time values at 5.04 rpm of the crank shaft; t the time values at a given rpm of the crankshaft. The plots in Figure 25 show the angular displacement, angular velocity and angular acceleration of the wing at a flapping frequency of 7.3 Hz. The wing starts from the topmost position, i.e. from the start of the downstroke. The arrow mark in the left most position indicates the start of the cycle with the wing at the topmost position, 01 indicates the wing level position. The wing attains a maximum of 351 in the topmost position and a minimum of 201 at the bottom most position. This provides an average dihedral angle during the flapping cycle and increases roll stability. In the angular velocity plot dØ/dt is 0 when the wings are at the topmost and bottom most position. At this point where dØ/dt is 0 the acceleration d2Ø/dt2 is maximum. The maximum d2Ø/dt2 occurs when the wing is the topmost position whereas minimum d2Ø/dt2 occurs when the wing is the lowest position. It may also be noted that the cycle
Wing position (degrees)
D
U
40
20
0
–20
0
0.05
0.1
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0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.35
Angular velocity (rad/sec)
30 20 10 0 –10 –20 –30
0.15 0.2 Time (s)
0.25
0.3
0.35
Figure 25. (a) A plot of angular displacement vs time, (b) angular velocity vs time and (c) angular acceleration vs time
Angular acceleration (rad/sec2)
2000
1000
0
–1000
0
0.05
0.1
Note: D, downstroke; U, upstroke
0.15 0.2 Time (s)
0.25
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average of d2Ø/dt2 is zero which obviously means there no net contribution of the reciprocating inertia forces of the wing in producing either lift or thrust. The effect of the inertia forces of the wing shall have some contribution separately during the upstroke and downstroke so the wing is made as light as possible. There is intake of energy when accelerating the wing and while decelerating the same energy is released only due to its inertia. Only in a conservative field the work done by the inertia forces will be zero. Horst Ra¨biger’s (n.d.) ornithopter used springs that stored energy during the upstroke and released during the downstroke in his radio-controlled ornithopter, EV7.
Modelling of flapping flight
57
Experimental validation Setup and procedure Experiments are carried out in a wind tunnel having contraction ratio 6, cross-section 914 mm 609 mm, tunnel operating speed of 1-35 m/s with turbulent intensity of less than 0.25 percent in the speed range of 1-6 m/s. At higher speed the turbulent intensity of the tunnel is o0.4 percent. The model is fixed on an ATI Nano-43 load cell and placed in the middle of the test section which is of 1,500 mm length. The resolution of force transducer is 1/640 N and frequency of measurement is 1 kHz. The data have been acquired with a simultaneous sample PCI DAQ card NI PCI6220 of National Instruments. A digital inclinometer (model-Clinotronic Plus IR Digital Inclinometer) is used to measure and set the model at a given as. A schematic view of the setup is shown in Figures 26 and 27. The camera is used to record the position of the wing as the load cell records the forces generated. For this paper, the average forces over a cycle are noted over a flapping cycle. A DC power supply was used instead of the existing lithium battery and the radio transmitter is used to control the flapping frequency only.
Tunnel wall
Front view of the model High speed camera
Load cell
RC transmitter DAQ card +
DC power source
Workstation 1
Workstation 2
Figure 26. Schematic view of the experimental setup
IJIUS 1,1
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Inclinometer
Pivot for changing s
58
Figure 27. Schematic side view of the model inside the wind tunnel
Start with adjustable length for changing s
Load cell
Tunnel wall
Experiments are carried out to determine the performance at design and off-design flying conditions. Flapping frequency, as and freestream velocity is varied in the experiments. To determine the force due to wings only, forces are also measured with the model frame removing the wings for the same parameters and subtracted from the forces obtained from the model tests. The details of the experiments can be obtained from (Ghosh et al., 2012). Comparison with theory The experimental results have been used to validate the predictions of the theory from yI.B. Figure 28 shows the variation of average vertical and thrust forces with VN at different flapping frequencies and as. The forces due to the wings are calculated by subtracting the contribution due to the fuselage tail and fixture. From Figure 28(a and c) it is observed that the average vertical force matches well with the theory at low freestream velocities. It is interesting to note that the theoretical average thrust force. The trend is also observed for different as varying from 01 to 71 as seen from Figure 28(b) and (d). The actual electrical power consumed to cruise is around 3 W. This includes the power required by the servos and electronics. There is also power loss due to inertia forces of the flapping wings. The drive train also would claim its friction losses. Besides, unsteady forces also would add up to the theoretical power estimation. The motor used could produce 10 Watts of power at its full potential. Weighing at only 3 g this motor could easily overcome the losses. Conclusion and future scope A simple quasi-steady model is used for designing the ornithopter with a novel aerodynamics twist distribution to achieve a sustained flight. The circulation distribution is obtained during the upstroke and downstroke which is used to calculate the spanwise distribution of vertical force and thrust force during the upstroke and downstroke, respectively. Most of the vertical force and thrust force is produced during the downstroke whereas due to the twist and camber distribution in upstroke, the part of the wing near the root produces positive vertical force but negative thrust (drag) whereas the part near the tip produces negative vertical force but positive
140 120 100
4.36 Hz
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7.03 Hz (expt)
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Modelling of flapping flight
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thrust. It is the portion of the wing near the tip also known as the hand-wing which mainly produces the thrust forces and the central part also known as arm wing produces most of the vertical force to sustain the weight. The theory predicted trends of performance is verified with the wind tunnel experiments which shows that such simple theory can provide a quick guideline for fabrication of bird size flapping model. Based on the calculations a prototype has been fabricated and successfully flown. The inflight predictions in y I.B reveal that climb rate could be increased by increasing VN. To accelerate forward, flapping frequency should be increased. Thus by increasing as the ornithopter can be made to gain altitude at the same VN. The theory presented here served to be a quick design tool for a preliminary design. The theory can be modified to take unsteady forces into account and estimate the forces over a complete flapping cycle. The wing twisting and camber distribution takes place passively in these wings and the performance of the wings is highly influenced by the position of the stiffeners. A more advanced theory is needed which can predict the optimum design for giving directional stiffness to the wing membrane. Flow visualization and PIV can be done to find out the spanwise circulation distribution and further explore the depths of aerodynamics of flapping flight at different flying conditions. Validation can be done in more detail by motion tracking systems to get the correct wing deformation and verified with the assumed geometry.
Figure 28. Comparison of theoretical and experimental results (effect of freestream velocity)
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The wing deforms at higher wing loads so the effect of aero-elasticity can also be included in the future study. The prototypes developed in this work fly by flapping and twisting of the wings. The wings flap up keeping the spar straight, unlike real birds that fold their wing during upstroke to reduce the upstroke effort and thus fly more efficiently and steadily. The aerodynamics of this effect can also be studied and implemented to make a better and efficient bio-mimicking flapping wing vehicle. There is a major area of study in the flight stability and control of flapping wing flight. The effect of change of stroke plane will also be useful as this can enable short take-off and landing possibility. A UAV capable of doing this will be of great demand. References Adrian, L.R.T., Taylor, G.K., Srygley, R.B., Nuddsand, R.L. and Bomphrey, R.J. (2004), “Dragonfly flight: free-flight and tethered flow visualizations reveal a diverse array of unsteady liftgenerating mechanisms, controlled primarily via angle of attack”, The Journal of Experimental Biology, Vol. 207 No. 24, pp. 4299-323. Archer, R.D., Sappupo, J. and Betteridge, D.J. (1979), “Propulsive characteristics of flapping wings”, Aeronaut. J, Vol. 83 No. 825, pp. 355-71. Ashley, H. and Landahl, M. (1985), Aerodynamics of Wings and Bodies, Dover Publications Inc, New York, NY. Betteridge, D.S. and Archer, R.D. (1974), “A study of the mechanics of flapping flight”, Aeronaut. Q, Vol. 25 No. 1, pp. 129-42. Bhowmik, J. (2012), “Design development and aerodynamic analysis of an efficient flapping wing flying machine”, Mtech thesis, IITK, Kanpur, June, available at: http://172.28.64.70:8080/ jspui/handle/123456789/13047 (accessed August 7, 2012). Bhowmik, J. and Das, D. (2011), “Development of a flapping wing MAV based on an efficient design of a wing with twist”, ICIUS, Chiba, October. DeLaurier, J.D. (1993), “An aerodynamic model for flapping-wing flight”, Aeronaut. J, Vol. 97 No. 964, pp. 125-30. DeLaurier, J.D. and Harris, J.M. (1993), “A study of mechanical flapping-wing flight”, Aeronaut. J, Vol. 97 No. 968, pp. 277-86. Dickinson, M.H., Lehmann, F.O. and Sane, S.P. (1999), “Wing rotation and the aerodynamic basis of insect flight”, Science, Vol. 284 No. 5422, pp. 1954-60. Ellington, C.P. (1984), “The aerodynamics of insect flight”, I-VI. Philos. Trans. R. Soc. Lond. B Biol. Sci, Vol. 305 No. 1122, pp. 1-181. Ellington, C.P., Van den Berg, C., Willmott, A.P. and Thomas, A.L.R. (1996), “Leading-edge vortices in insect flight”, Nature, Vol. 384 No. 6610, pp. 626-30. Ghosh, S.K., Bhowmik, J. and Das, D. (2012), “Aerodynamics of a flapping wing mav: an experimental study”, ICIUS, Singapore, October. Hertel, H. (1963/1966), Structure, Form, Movement, VanNostrand Reinhold, New York, NY (in German). Ho, S., Nassef, H., Pornsinsirirak, N., Tai, Y.C. and Ho, C.M. (2003), “Unsteady aerodynamics and flow control for flapping wing flyers”, Progress in Aerospace Sciences, Vol. 39 No. 8, pp. 635-81. Hubel, T.Y. and Tropea, C. (2009), “Experimental investigation of a flapping wing model”, Exp Fluids, Vol. 46 No. 5, pp. 945-61. Jones, R.T. (1980), “Wing flapping with minimum energy”, NASA Technical Memorandum No. 81 174, NASA, January.
Katz, J. and Plotkin, A. (1991), Low-Speed Aerodynamics: From Wing Theory to Panel Methods, McGraw-Hill Inc, New York, NY. Phillips, P.J., East, R.A. and Pratt, N.H. (1981), “An unsteady lifting line theory of flapping wings with application to the forward flight of birds”, J. Fluid Mech, Vol. 112 No. 1, pp. 97-112. Ra¨biger, H. (n.d.), “How ornithopters fly”, available at: www.ornithopter.de/english/index_en.htm (accessed August 7, 2012). Theodorsen, T. (1935), “General theory of aerodynamic instability and the mechanism of flutter”, NACA Report No. 496, National Advisory Committee for Aeronautics, Hampton, VA. Tobalske, B.W., Warrick, D.R., Clark, C.J., Powers, D.R., Hedrick, T.L., Hyder, G.A. and Biewener, A.A. (2007), “Three-dimensional kinematics of hummingbird flight”, The Journal of Experimental Biology, Vol. 210 No. 18, pp. 2368-82. Further reading Banerjee, A., Ghosh, S.K. and Das, D. (2011), “Aerodynamics of flapping wing at low Reynolds numbers: force measurement and flow visualization”, ISRN Mechanical Engineering, Vol. 2011, doi:10.5402/2011/162687. About the authors Joydeep Bhowmik is a graduate student in Aerospace Engineering, IIT Kanpur. Debopam Das is Associate Professor of the Department of Aerospace Engineering, IIT Kanpur. Debopam Das is the corresponding author and can be contacted at:
[email protected] Saurav Kumar Ghosh is a graduate student in Aerospace Engineering IIT Kanpur.
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Modelling of flapping flight
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