Optimizing the Forces and Propulsive Efficiency in Bird-Scale Flapping Flight Aditya A. Paranjape∗ and Soon-Jo Chung† Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801.
Harry H. Hilton‡ Department of Aerospace Engineering and Private Sector Program Division, National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801.
This paper presents first-principle and numerical studies of flapping flight with the objective of optimizing the force production and propulsive efficiency. Strouhal number is identified as a critical parameter affecting all of these variables, and the optimum ranges of Strouhal number are calculated. The results of the calculations, in particular, explain why a value in the range of 0.2−0.4 is usually preferred by nature’s flyers. Next, an attempt is made to quantify the effects of wing flexibility on force generation and propulsive efficiency, and it is shown that there exists a critical value of wing elasticity at which propulsive efficiency is maximized, and moreover, this value is driven primarily by the dynamics of the bending motion of the wing rather than twisting.
Nomenclature b, c Cl , Cd L, D, F Sr V∞ , V α θ ξ ψ φ φA , θ A
wing span and wing chord coefficients of lift and drag per unit span lift, drag and thrust Strouhal number free-stream and local wind speed, respectively angle of attack feathering angle of the wing (due to rigid rotations and elastic twisting) vertical displacement at a spanwise position on the wing (due to rigid body motion and elastic bending) lead-lag (sweep) angle flapping angle amplitude of the periodic motions corresponding to flapping and feathering
I.
Introduction
The objective of this paper is to analyse the flapping motion of a wing with the objective of deriving insights into how forces and produced by the wing and the propulsive efficiency can be optimized for efficient flight. The flapping motion is assumed to be 2-dimensional, such that the wing is allowed to flap (i.e., ∗ Post-Doctoral
Research Associate; Email:
[email protected]; Member, AIAA Professor; Email:
[email protected]; Senior Member, AIAA. ‡ Professor Emeritus and Senior Academic Lead for Computational Structural/Solid Mechanics at NCSA; Email:
[email protected]. Fellow, AIAA. † Assistant
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beat up-and-down) and feather (i.e., change the angle of incidence at the root). This case is reasonably representative of forward flight in birds. The objective is to identify conditions under which the aerodynamic forces, propulsive efficiency, or a combination of these quantities is optimized. Optimization is also performed with respect to the elastic properties of the wing, particularly the Young’s modulus.
Tip Control: Flap
Root Control: Wing Twist (R ) and Wing Dihedral (R )
Figure 1. The results presented in this paper can aid the development of robotic flapping wing aircraft. This picture shows a robotic bat testbed developed by Chung and Dorothy.1
The development of flapping wing aircraft has attracted considerable interest from the aeronautical and the robotics community.1–5 A typical flapping wing has several degrees of freedom, e.g., see Fig. 1 which shows a robotic bat testbed developed by Chung and Dorothy.1 From the point of view of control, the flapping frequency and the phase differences between the different degrees of freedom are the most important. At the same time, if the flapping wings are flexible (and that is most likely to be the case if the wing is to be kept light), then flexibility needs to be considered during design, both from the point of view of stability, as well as control.2, 6 The elastic response of the wing can be fine-tuned using active control and actuators such as those depicted in Fig. 1. This work has two objectives: firstly, to contribute to the broader understanding of flapping flight, and secondly, to provide a set of tools which can be used in the design of flapping wing aircraft. It is a well-known fact from the literature7, 8 that nature’s flyers and swimmers prefer to maintain a non-dimensional quantity known as the Strouhal numbera in a range of 0.2 − 0.4 (see Fig. 2 from [7]). Experimental evidence shows that propulsive efficiency, defined as the ratio of the power produced by the wing to the power required to flap it, is maximized in the aforementioned Strouhal number range, while the value of thrust is considerable as well.7, 8 The explanation provided for this observation is mainly along the lines of fluid mechanics2, 8 - vortex shedding, energy dissipation in the wake, etc. - rather than the kinematics of the wing which play a primary role in determining the forces generated by it. In this paper, we derive analytical expressions for the average forces generated in a flapping cycle, as well as the propulsive efficiency. Exact closed-form expressions are computed for a special case where the lift is expressed as a sine of the angle of attack (α) with the stall angle of attack (αstall ) set to 45 deg. It is shown that the forces and propulsive efficiency are determined largely by two factors: the Strouhal number and the lift curve slope of the airfoil (captured via the stall angle of attack αstall ). The analysis presented here shows that the propulsive efficiency is indeed high at low Strouhal numbers, but the range depends heavily on the lift curve slope of the airfoil section. The thrust initially increases with Strouhal number before decreasing as the Strouhal number is increased. The key observation, though, is that the Strouhal number at which the thrust is optimized also depends on the lift curve slope of the airfoil. Moreover, we demonstrate that the lift decreases with increasing Strouhal number and becomes zero, thereby creating an important constraint on the flight Strouhal number. To the best of our knowledge, the analysis presented in this paper has no precedent in the literature, and although experimental results exist to quantify force generation and propulslve efficiency of flapping flight, the following analysis provides a neat theoretical explanation for it. Continuing along the lines of force and propulsive efficiency optimization, an effort is made to identify the optimum elastic properties of the wing, assuming spanwise flexibility and linear elasticity. Yet again, experimental results2, 9, 10 showed that there exists an optimum amount of flexibility for maximizing the efficiency as well as thrust production. Using an analytical model of a flapping flexible wing, we demonstrate that this is indeed the case, and identify factors which affect the optimum elasticity. a The
Strouhal number is given by f bφA /V∞ where f is the flapping frequency of the wing, b is the wing span, φA is the amplitude of the flapping motion, and V∞ is the forward speed.
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sufficient homoscedasticity for the procedure to be valid. Only 53 of the 50,000 randomized combinations of residuals (,0.11%) had a lower s.d. of St than the original sample. The actual s.d. of St (0.10) was therefore significantly smaller than expected by chance
sample size for insects limited its efficiency. Using a Wilcoxon rank sum test to compare directly birds and bats showed instead that St is significantly lower in birds than in bats (two-tailed, P ¼ 0.020). We then dropped all families of n ¼ 1 and used a
7 Figure Strouhal number swimming Figure 2 Strouhal number for 42 species of birds, bats and2. insects in unconfined, cruisingof birds peaks;and dashed line marks theanimals. modal peak at St ¼ 0.3. Unbroken lines indicate the range 3,4 flight. Published ranges of St in cruising fish and dolphins are included for comparison. of variation in St across other non-zero flight speeds, where such data exist. Dotted lines mark the range 0.2 , St , 0.4, in which propulsive efficiency usually
The paper is organized as follows. The equations of motion and the aerodynamic model are briefly αstall = 45 deg are presented in Section III. Numerical results for other values of αstall , where the expressions for the forces cannot be simplified as in the aforementioned special case, are presented in Section IV. Section V concludes this draft. Results for a flexible wing are planned for inclusion in the final version of the manuscript.
recapitulated Section II. Theoretical results for the special Group case of © 2003 Nature Publishing NATURE | VOL 425 | 16 OCTOBERin 2003 | www.nature.com/nature
II. A.
Preliminaries
Equations of Motion
The model used in this paper is a subset of a more advanced model designed by the authors.11 The wing is modelled as an Euler-Bernoulli beam. The bending and twisting elastic equations of motion for the right wing are given by " #" # " # " # (EIb ξ 00 )00 F m ˜ −mx ˜ ec ξ¨ + = (1) ˜ 0 )0 θ¨ −(GJθ M −mx ˜ ec Ip The force and moment are given by 1 ρ∞ V 2 cCz + unsteady aerodynamics 2 1 M = ρ∞ V 2 c2 (xe Cz + Cmac ) + unsteady aerodynamics (2) 2 where the exact expressions for the aerodynamic coefficients and the unsteady terms may be found in a prior paper by the authors.11 The term V is the local wind speed approximated by v !2 !2 u u ˙ ψy ξ˙ t V = V∞ cos α − + sin α + V∞ V∞ F
=
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709
where ψ˙ is the lead-lag angular velocity. The local angle of attack is given by ! ˙ V sin α + ξ ∞ αlocal = θ + tan−1 ˙ V∞ cos α − ψy
(3)
Note that V∞ ≈ 0 in the hovering mode, while ψ˙ ≈ 0 in forward flight. Note also that the terms ξ˙ and θ include contributions from the rigid body motion as well as elastic deformations. B.
Aerodynamics
The aerodynamic model presented here is a simplified version of the complete, but more cumbersome, model described in Ref. [11]. The expression for Cl is obtained from a sinusoidal approximation to lift. The inertial terms are obtained from Theodorsen’s model.12 The coefficient of lift is assumed to be of the form π α Cl = 4αstall sin 2 αstall This gives a lift curve slope of 2π at α = 0, and this can be scaled for other airfoils. The scaling factor is not nearly as important as αstall , or to be precise αstall measured with respect to the zero lift angle of attack.
III.
Analytical Model of a Rigid Wing
Lift and thrust on a flapping wing are calculated by first computing the local angle of attack from Eq. (3), then substituting it into a sinusoidal approximation for the lift - α relationship, and finally by resolving and summing the local lift across the wing to obtain the net lift and thrust. An alternative to this approach is to use a more rigorous model such as that put forth by Garrick.13 A.
Lift
Consider forward flight with quasi-steady speed V∞ . In order to compute the aerodynamic lift produced by the wing, it is essential to compute the force produced at a spanwise station located at a distance y from ˙ where φ˙ is the flapping angular velocity. the wing root. The plunge speed at this section is given by ξ˙ = y φ, Assuming that the “free stream” angle of attack is zero, the local angle of attack is given by ! ξ˙ −1 +θ (4) α = tan V∞ We assume that
Cl = 4αstall sin
π α 2 αstall
For simplifying the notation, define k = 4αstall and p = π/(2αstall ), so that Cl = k sin(pα) and kp = 2π. The sectional lift is then given by 1 (5) l = ρV 2 ck sin(pα) 2 We are interested in determining the conditions under which the lift is maximized at a given flight speed. Therefore, we define a normalized sectional lift, ln =
V 2 sin(pα) 2 V∞ p
and try to maximize it. The expression for ln has been derived in the appendix: !2 −(p−2)/2 !2 2 ˙ ˙ ˙ V sin(pα) ξ˙ ξ ξ ξ θ + ln = ≈ 1+ − (p − 1) θ − (p − 1) θ2 V∞ p V∞ V∞ V∞ V∞ The phase difference between bending and twist is set to 90 deg (twist in the lead), so that we get ξ˙ = yφA ω cos(ωt), θ = θ0 + θA cos(ωt) 4 of 12 American Institute of Aeronautics and Astronautics
(6)
(7)
This particular phase relationship maximizes the thrust as well as the lift. The effect on lift is evident from a preliminary examination of the lift equation. Equations for thrust will be introduced later in this paper. The above equation for ln is, in general, very difficult to simplify further. One case for which we can analytically optimize ln is p = 2 (i.e., αstall = 45 deg). We look for secular contributions over a unit time. Note that odd powers of cos(ωt) integrate to zero. For convenience, define Sr(y) = yφA ω/V∞ which gives the average sectional lift per unit time: ω Z 2π/ω ln dt < ln >1 = 2π 0 1 = θ0 1 − Sr(y)2 − Sr(y)θA (8) 2 Integrating both sides with respect to y gives the total lift generated by the wing over a unit time, denoted by < Ln >1 : Z b < ln >1 dy < Ln >1 = 0 1 2 Sr0 θA = bθ0 1 − Sr0 − (9) 6 2 ωφA b . This is interesting because we are now able to express lift exclusively in terms of the V∞ Strouhal number, V∞ and θ0 . For other values of p, a numerical analysis is required, although as Eq. (7) shows, the Strouhal number plays an over-arching role in determining the qualitative profile of the lift as a function of the flapping frequency. 2f φA b (the mid-span The Strouhal number in the flight mechanics literature is usually defined as Sr = 2V∞ amplitude is used, along with the flapping frequency f ), so that Sr = Sr0 /(2π). From Eq. (9), it is evident that the lift reduces uniformly as Sr is increased from zero. For larger values of p, it turns out that the lift reduces initially before recovering for larger values of Sr. It is interesting to note that the qualitative nature of the secular value of lift, as a function of Sr, depends only on αstall (via p), while θ0 and θA serve to “scale” it. √ Note that the lift goes to zero when Sr0 = 6 for θA = 0, i.e., Sr = 0.39. In general, it depends on θA : r 3 9 Sr0 (L = 0) = − θA + 6 + θA 2 2 4 where Sr0 =
Thus, increasing θA causes the lift to approach zero at lower values of Sr. For a given bird or aircraft, this translates to zero lift at lower values of flapping frequency and/or amplitude. B.
Thrust at Low Flapping Frequencies
The sectional value of thrust can be written as ˙ ˙ ∞ )) = l q ξ f ≈ l sin(tan−1 (ξ/V 2 + ξ˙2 V∞
(10)
2 2 The thrust model is clearly much more complicated. To simplify our analysis, assume that ξ˙2 + V∞ ≈ V∞ , and αstall = π/4, so that p = 2. !2 ξ˙ ξ˙ ξ˙ ξ˙ ξ˙ 2 fn ≈ ln ≈ θ+ − θ− θ (11) V∞ V∞ V∞ V∞ V∞
As we did for lift, we look for secular contributions over a flapping cycle, and remind ourselves that ξ˙ = yφA ω cos(ωt), θ = θ0 + θA cos(ωt),
ξ˙ = Sr(y) cos(ωt) V∞
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Thus, fn = Sr(y) cos(ωt) θ + Sr(y) cos(ωt) − Sr(y)2 cos2 (ωt)θ − Sr(y) cos(ωt)θ2
Therefore, we get the sectional average-thrust-per-unit time as < fn >1
=
2 Sr(y)2 (1 − θ02 − 3θA /8) 3 Sr(y)θA Sr(y)2 3 Sr(y)θA + − Sr(y)3 θA ≈ + − Sr(y)3 θA (12) 2 2 8 2 2 8
This gives the resultant thrust per unit time as Z
b
< fn >1 dy = b
< Fn >1 = 0
Sr0 θA Sr2 3 + 0 − Sr30 θA 4 6 32
The optimum Strouhal number (for maximum thrust) SrT0 is found by differentiating both sides with respect to Sr0 and setting the derivative to zero, so that SrT0 satisfies SrT 9 32 θA + 0 − θA (SrT0 )2 = 0 =⇒ SrT0 ≈ 4 3 32 27 θA T Sr 0.2 0 =⇒ SrT = ≈ 2πθA θA
(13)
We conclude that the optimum value of Strouhal number for thrust depends strongly on θA . Since θA is typically on the order of 0.1 − 0.4 rad, it follows in fact that the Strouhal number for optimum thrust is in the range 0.5 − 2, as shown in Fig. 3. Incidentally, this Strouhal number is beyond the range of the low Strouhal number approximation where we assumed that Sr0 1 = 0
2 4 = (θA + Sr(y)) − θA Sr(y)2 π 3π Z b b 4θA Sr20 =⇒ < F >1 = < f >1 dy = 2θA + Sr0 − π 9 0
(14)
Thus, the thrust is optimized at a Strouhal number of SrT =
9 0.18 ≈ ∈ (0.6, 1.8) typically 16πθA θA
Clearly, thrust is optimized at Strouhal numbers higher than that at which lift becomes zero. Therefore, the constraint of positive lift is the single most important factor governing the choice of the Strouhal number. Incidentally, for a fixed Strouhal number, thrust increases linearly with θA , once again underscoring the importance of θA as a thrust-control input. D.
Power and Efficiency
The efficiency of flapping flight can be measured by η=
Power Available < T > V∞ = Power Required < D > V∞ + Pbeat
where < · > indicates that the terms are calculated by cycle averaging. It is obvious that flapping wing aircraft generally cannot maximize lift and thrust simultaneously. Instead, the optimum value depends on drag. However, drag is not very easy to describe by a closed form expression because of the unsteady effects involved in flapping. A numerical analysis is therefore warranted. In order to get a “feel” for the efficiency, we ignore the effects of drag, and calculate η=
Pavailable < T > V∞ = Rb ˙ Pbeat > < 0 |ξl(y)|dy
We first calculate the beating power: b
Z Pbeat
Z
1
= 0
˙ |ξl(t, y)|dt dy
0
The integrand, incidentally, is the absolute value of the sectional thrust computed assuming low flapping frequencies in Eq. (12). Thus, Pbeat = V∞ b
Sr0 θA Sr2 3Sr30 θA 2Sr30 θ0 + 0+ + 2Sr0 θ0 + 4 6 32 3
(15)
Since the numerator and the denominator at low Strouhal numbers represent the same quantity, we conjecture that the efficiency is very high and almost constant at low Strouhal numbers.
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At moderate to high Strouhal numbers, from Eq. (14) b 4θA Sr20 V∞ < T >= V∞ 2θA + Sr0 − π 9 Thus, we get
4θA Sr20 2θA + Sr0 − 1 9 η = π Sr0 θA Sr20 3 3 2Sr30 θ0 + + Sr0 θA + 2Sr0 θ0 + 4 6 32 3
(16)
0.4 0.6
0.35
0.5
0.3
0.4
0.25 0.2
0.3
0.15 0.2 0.1 0.1
0.05
0
0 0.2
0.4
0.6
0.8 Sr
1
1.2
1.4
0.2
(a) θ0 = 0.2 rad
0.4
0.6
0.8 Sr
1
1.2
1.4
(b) θ0 = 0.4 rad
Figure 4. Propulsive efficiency plotted as a function of Sr for θA ∈ (0, 0.4) rad, for two different values of θ0 . A greyscale is used for θA with increasing shade denoting an increasing value of the variable.
Fig. 4 plots η as functions of Sr for various values of θA , for p = 2 and for two values of θ0 . The efficiency reduces monotonically with increasing Str. The variation with θA is smaller at small values of Sr, and becomes significant rapidly thereafter. E.
Interim Summary
The propulsive efficiency becomes zero at Sr = 0.8, as expected from Eq. (14). The range from 0.2 − 0.8 sees a sharp drop in propulsive efficiency. Note, however, that lift becomes zero at Sr ≈ 0.39. Hence, we conclude that the flight regime of Sr ≈ 0.2 − 0.4 is chosen by birds for its relatively high (and constant) propulsive efficiency, as well as for ensuring a positive lift. Around Sr ≈ 0.3, the propulsive efficiency is independent of θA , i.e., robust to changes in wing incidence amplitudes. Neither thrust nor lift are individually optimized in this range. An important point to note is the dependence of Sr, and its dependence, in turn, on αstall , in determining the operating regime. Interestingly, from the analysis carried out with αstall = 45 deg and Eq. (7), which gives the most general expression for lift as a function of p (i.e., αstall ), we deduce that the aerodynamic forces and propulsive efficiency are qualitatively governed by the Strouhal number directly, while the flight speed and wing incidence angles (θ0 and θA ) act as scaling factors. In the next section, we analyse how the aerodynamic forces and the propulsive efficiency chance with p (i.e., with αstall ).
IV.
Numerical Results for a Rigid Wing: Influence of p
In order to quantify the effect of p (i.e., αstall , since p = π/2αstall ), we compute the aerodynamic forces and the propulsive efficiency as functions of the Strouhal number using a high fidelity model, as compared to theoretical calculations above. 8 of 12 American Institute of Aeronautics and Astronautics
0.4
0.4 0.3
0.2
0.2 0.1 D, T,
D, T,
0 -0.2 -0.4
0 -0.1 -0.2
-0.6 -0.8
-0.3 0
-0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sr
0
(a) αstall = 0.4 rad (p = 4)
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sr
(b) αstall = 0.5 rad (p = 3.1)
0.5
0.8 0.7
0.4
0.6 0.5 D, T,
D, T,
0.3 0.2
0.4 0.3 0.2
0.1
0.1 0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sr
0 0
(c) αstall = 0.6 rad (p = 2.6)
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sr
(d) αstall = 0.7 rad (p = 2.2)
1.4 1.2
D, T,
1 0.8 0.6 0.4 0.2 0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sr
(e) αstall = 0.7 rad
Figure 5. Plots of thrust (blue), drag (black) and propulsive efficiency (red) for a rigid wing as functions of Sr for different values of αstall . The values of θ0 and θA were set to 0.3 and 0.1, respectively.
It is evident that thrust and drag both increase with increase in αstall . This should be interpreted, though, as the effect of an increase in the available CL . However, the maximum propulsive efficiency consistently
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remains around 0.2, and is independent of αstall . Moreover, as αstall increases, so does the Strouhal number at which thrust (and hence the propulsive efficiency) become zero. Note also that the thrust-maximizing Strouhal number increases at a uniform rate with αstall , starting with a value of nearly 0.1 for αstall = 0.4, and increasing to nearly 0.4 at αstall = 0.8.
V.
1.4
Effects of Wing Flexibility
E = 0.01 E = 0.05 E = 0.1 E = 0.5
1.2
1.6
GPa GPa GPa GPa
E = 0.01 E = 0.05 E = 0.1 E = 0.5
1.4
GPa GPa GPa GPa
1 1.2 Drag
Lift
0.8 0.6 0.4
1
0.8
0.2 0.6
0 -0.2 0.05
0.1
0.15
0.2
0.25 Sr
0.3
0.35
0.4
0.4 0.05
0.1
0.15
(a) Lift
0.2
0.25 Sr
0.3
0.35
0.4
(b) Drag
1
E = 0.01 E = 0.05 E = 0.1 E = 0.5
GPa GPa GPa GPa
0.8
Thrust
0.6
0.4
0.2
0 0.05
0.1
0.15
0.2
0.25 Sr
0.3
0.35
0.4
(c) Thrust
Figure 6. Plots showing the lift, drag and thrust as functions of Strouhal number (or equivalently, the flapping frequency) for various values of the Young’s modulus (E). The plots were obtained by simulation at V∞ = 6 m/s with αstall = 0.7 rad. Note that the black curve (E = 0.1 GPa) is broken between Sr = 0.2 and Sr = 0.35. This is because the wing is dynamically unstable in this regime.
The analysis presented above can be extended to a flexible wing with the objective of measuring, on the one hand, the effects of flexibility on the lift, drag, and the thrust generated by the wing, and, on the other hand, identifying a suitable elastic modulus (E). We perform simulations using a Galerkin approximation of the model in Eq. (1) which converts it into a set of ordinary differential equations. Figure 6 shows the values of the lift, the drag, and the thrust as functions of the Strouhal number for four candidate values of the Young’s modulus. The simulations were performed at V∞ = 6 m/s, so that they covered flapping frequencies between 2 Hz and 8 Hz. The phase difference between the commanded pitching and heaving 10 of 12 American Institute of Aeronautics and Astronautics
(flapping) motions was set to 90 deg. It is amply clear that the effects of flexibility are a highly nonlinear function of the Young’s modulus (E). At one extreme, for E = 0.5 GPa, a comparison with Fig. 5(e) shows that the the values of drag and thrust are already nearly equal to those for a rigid wing. A noticeable difference, though, is that whereas the drag curve flattens out for a rigid wing, the drag on a flexible wing increases quite rapidly as the Strouhal number increases. The value of thrust produced by the flexible wing is smaller by nearly 20 % as compared to the rigid wing. At the other extreme, for E = 0.01 GPa, the value of thrust is extremely small, making this value of Young’s modulus unsuitable at least for the prescribed phase difference of 90 deg between the two degrees of freedom. The intermediate range of elasticity is represented in Fig. 6 by E = 0.05 GPa (red curves) and E = 0.1 GPa (black). For E = 0.05 GPa, the respective values of thrust as well as drag are maximum around Sr = 0.2, after which they reduce steeply. The lift remains near 1, although it increases with increasing Strouhal number. For E = 0.1 GPa, the vibrations become unstable between Sr = 0.2 and Sr = 0.35. The plots, however, suggest the possibility of qualitative trends similar to the previous case, with the difference that the thrust and drag are maximized in the range where the wing is dynamically unstable. However, the lift in the unstable range also drops significantly as compared to the case of E = 0.05 GPa. The aforementioned analysis leads to the following hypotheses which require further testing: • The net lift of flexible wings increases with increasing flexibility, i.e., with reducing Young’s modulus, and becomes increasingly independent of the Strouhal number (and the wing flapping frequency). • Highly flexible wings (such as E = 0.01 GPa in Fig. 6), however, suffer from poor thrust generation while corresponding reductions in drag are incommensurate with the reduction in thrust. • Wings with moderate flexibility (E = 0.05, 0.1 GPa in Fig. 6) are suitable for flight at intermediate Strouhal numbers where they outperform rigid as well as highly flexible wings. Although the wing considered in the present analysis demonstrated unstable vibrations for E = 0.1 GPa, it must be noted that the onset of instabilities is highly case-dependent. Moreover, as illustrated in Paranjape, Guan, Chung and Krstic,6 it may be possible to use active feedback control at the boundary of the wing to suppress these instabilities should they occur in a regime of optimum flight performance. The conclusions presented here do find some support in experiments9, 10 but need further theoretical modelling.
VI.
Conclusions
The analysis presented in this paper presents a quantitative correlation between Strouhal number and the aerodynamic profile of the airfoil (through αstall ), and the aerodynamic forces and propulsive efficiency. It was shown that the lift reduces rapidly with increasing Strouhal number and becomes zero, setting a binding constraint on the operating Strouhal number. In particular, for αstall around 45 deg, the propulsive efficiency was shown to be nearly constant in the Strouhal number range of 0.2 − 0.4, the range preferred by flyers and swimmers in the animal kingdom, while the exact value of the thrust-maximizing Strouhal number was shown to depend strongly on αstall and the amplitude of the feathering motion. The effects of wing flexibility were illustrated numerically and suggest how an optimum value for a wing may be chosen.
Acknowledgement Support by the National Science Foundation (IIS-1253758) and the Private Sector Program Division of the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign is gratefully acknowledged.
References 1 Chung,
S.-J. and Dorothy, M., “Neurobiologically Inspired Control of Engineered Flapping Flight,” Journal of Guidance, Control and Dynamics, Vol. 33, No. 2, 2010, pp. 440–453. 2 Shyy, W., Aono, H., Chimakurthi, S. K., Trizila, P., Kang, C.-K., Cesnik, C. E. S., and Liu, H., “Recent Progress in Flapping Wing Aerodynamics and Aeroelasticity,” Progress in Aerospace Sciences, Vol. 46, 2010, pp. 284 – 327. 3 Taha, H., Hajj, M., and Nayfeh, A., “Flight Dynamics and Control of Flapping-Wing MAVs: A Review,” Nonlinear Dynamics, 2012, pp. 1–33.
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4 Orlowski, C. T. and Girard, A. R., “Dynamics, Stability, and Control Analysis of Flapping Wing Micro-Air Vehicles,” Progress in Aerospace Sciences, Vol. 51, 2012, pp. 18 – 30. 5 Paranjape, A. A., Dorothy, M., Chung, S.-J., and Lee, K. D., “A Flight Mechanics-Centric Review of Bird-Scale Flapping Flight,” International Journal of Aeronautical and Space Sciences, Vol. 13, No. 3, 2012, pp. 267 – 281. 6 Paranjape, A. A., Guan, J., Chung, S.-J., and Krstic, M., “PDE Boundary Control for Flexible Articulated Wings on a Robotic Aircraft,” IEEE Transactions on Robotics, Vol. 29, No. 3, 2013, pp. 625 – 640. 7 Taylor, G. K., Nudds, R. L., and Thomas, A. L. R., “Flying and Swimming Animals Cruise at a Strouhal Number Tuned for High Power Efficiency,” Nature, Vol. 425, 2003, pp. 707 – 711. 8 Schouveiler, L., Hover, F. S., and Triantafyllou, M. S., “Performance of Flapping Foil Propulsion,” Journal of Fluids and Structures, Vol. 20, 2005, pp. 949 – 959. 9 Heathcote, S., Martin, D., and Gursul, I., “Flexible Flapping Airfoil Propulsion at Zero Freestream Velocity,” AIAA Journal, Vol. 42, No. 11, 2004, pp. 2196 – 2204. 10 Heathcote, S., Wang, Z., and Gursul, I., “Effect of Spanwise Flexibility on Flapping Wing Propulsion,” Journal of Fluids and Structures, Vol. 24, 2008, pp. 183 – 199. 11 Paranjape, A. A., Chung, S.-J., Hilton, H. H., and Chakravarthy, A., “Dynamics and Performance of Tailless MAV with Flexible Articulated Wings,” AIAA Journal, Vol. 50, No. 5, 2012, pp. 1177 – 1188. 12 Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism of Flutter,” NACA report 496, 1935. 13 Garrick, I. E., “Propulsion of a Flapping and Oscillating Airfoil,” Tech. rep., NACA, 1936, NACA TR 567.
Appendix: Formulaic Expansion of (V /V∞ )2 (sin(pα)/p) We start by expanding sin(pα). First, note that p = higher order terms, we approximate
π . Using DeMoivre’s formula, and ignoring 2αstall
sin(pα) ≈ p cosp−1 (α) sin(α) q ˙ 2 + ξ˙2 . Furthermore, we assume that θ is small, so that Note that α = tan−1 Vξ∞ + θ, and V = V∞ ξ˙ V∞ V∞ ξ˙ + θ, cos α = − θ V V V V
sin α = Thus, we get
=⇒
p p−1 sin(pα) 1 ˙ = V∞ − ξθ ξ˙ + V∞ θ p V !p−1 ! p ˙ ξθ V∞ ξ˙ 1− = θ+ V V∞ V∞ !2 p ˙ ˙ ˙ V∞ ξ ξ ξ θ + ≈ − (p − 1) θ − (p − 1) θ2 V V∞ V∞ V∞ !2 2 p−2 ˙ ˙ ˙ sin(pα) V∞ ξ ξ ξ V θ + = − (p − 1) θ − (p − 1) θ2 V∞ p V V∞ V∞ V∞
Note that
V∞ V
p−2
=
V V∞
−(p−2)
= 1 +
ξ˙ V∞
!2 −(p−2)/2
Thus,
V V∞
2
sin(pα) ≈ 1+ p
ξ˙ V∞
!2 −(p−2)/2
˙ θ + ξ − (p − 1) V∞
ξ˙ V∞
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!2
˙ξ θ − (p − 1) θ2 V∞
(17)
(18)
