JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 37, No. 3, May–June 2014
Engineering Notes Longitudinal Flight Dynamics of Hovering MAVs/Insects
considerably affect pitch damping. Also, they considered only one trim configuration at hover that yielded four nonzero longitudinal stability derivatives out of the full nine. In this work, a formal derivation of the full longitudinal stability derivatives for hovering MAVs/insects is presented. A quasi-steady aerodynamic model that accounts for the dominant leading edge vortex contribution and the rotational effects is used. As such, the final expressions for the stability derivatives are provided directly in terms of the system parameters. For validation purposes, a numerical simulation for the aerodynamic model is performed over one cycle and a complex-step finite differencing is used to determine the stability derivatives of the hawkmoth using the kinematics and trim data considered by Sun et al. [4] in their direct numerical simulation of Navier–Stokes equations. The effects of trim configuration on the cycle-averaged stability derivatives are then determined. The averaging theorem is used to analytically assess the stability of this time-periodic system. Finally, a parametric study for the cycleaveraged stability derivatives and the eigenvalues of the averaged, linearized system using the symmetric flapping trim configuration (SFTC) is performed.
Haithem E. Taha,∗ Muhammad R. Hajj,† and Ali H. Nayfeh‡ Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
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DOI: 10.2514/1.62323
I.
Introduction
The dynamics of flapping flight has been a research topic of interest with the objectives of better characterization of insect flight or realization of successful flight of microair vehicles (MAVs). In addition to the complex dynamics associated with flapping flight, the periodicity aspect of the flapping motion gives flapping flight dynamics a time-varying characteristic. Because hovering is usually performed at relatively high frequencies, with respect to the body’s natural frequencies, it is usually claimed that the body of a hovering insect feels only the cycle-averaged aerodynamic loads. In fact, this is a very common assumption in the analysis of flapping flight dynamics. Some research reports use this assumption on the basis of physical intuition [1–7]. Others use its rigorous mathematical representation [8–10]; namely, the averaging theorem. There still exists some debate about the range of validity of this assumption; see Taha et al. [11] for a discussion about the topic. If the averaging assumption is acceptable, or the dynamics of the insect under study are amenable to the averaging theorem, then one can convert the time-varying system into an autonomous one (averaged system). The stability of the averaged system is indicative of the stability of the original nonlinear time-periodic (NLTP) system. The inherent instability of any flying vehicle is mainly dictated by the aerodynamic loads due to the vehicle’s motion (stability derivatives). Several experimental [2,12] and computational [3–5,13] investigations have been performed to determine the cycle-averaged stability derivatives for hovering insects and MAVs. However, there have been few trials that aimed at deriving analytical expressions for the stability derivatives in terms of the system parameters. Cheng and Deng [14] derived the cycle-averaged stability derivatives for hovering insects in terms of their morphological parameters. However, because they did not adopt a specific aerodynamic model, they provided general expressions in terms of the cycle-averaged aerodynamic lift and drag coefficients that, in turn, need to be expressed in terms of the system parameters. Moreover, they did not account for the aerodynamic rotational effects [15], which may
II.
Flight Dynamics Modeling
A. Dynamic Model
In the present dynamic analysis, only rigid-body degrees of freedom are taken into account and the inertial effects due to the wing motion are neglected. In fact, these assumptions are ubiquitous in the literature of dynamics and control of flapping flight [3,5–10,16]. The conventional set of body axes xb , yb , and zb that are commonly used in flight dynamics analysis [17–19] are also used here. That is, the xb axis points forward, the yb axis points to the right wing, and the zb axis completes the triad. Because the interest here is in the longitudinal motion, only one body rotation is considered, namely, the pitching angle θ. In addition, due to flapping, a wing-fixed frame xw , yw , and zw is required. The wing-fixed frame is considered to coincide with the body-fixed frame for zero wing kinematic angles. Most hovering insects perform a stroke in an approximate horizontal plane without an out-of-plane motion [20,21]. As such, and because the aerodynamic loads are expressed in the relative wind frame (normal and tangential to the horizontal flapping velocity), only the back and forth flapping angle φ would be of interest to transfer the aerodynamic loads to the body frame. Figure 1 shows a schematic diagram for a flapping MAV whose wing sweeps in a horizontal plane. Using the preceding representation, the equations of the longitudinal body motion are written similarly to those of a conventional aircraft [17]; that is 1 0 1 0 1 1 u_ −qw − g sin θ mX B w_ C B qu g cos θ C B 1 Z C B CB C B 1m C @ q_ A @ A @ MA 0 Iy θ_ q 0 0
Presented as Paper 2013-1707 at the 54th Structures, Structural Dynamics, and Materials and Co-Located Conferences, Boston, MA, 8–11 April 2013; received 7 March 2013; revision received 12 September 2013; accepted for publication 28 August 2013; published online 9 April 2014. Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-3884/14 and $10.00 in correspondence with the CCC. *Ph.D. Candidate, Engineering Science and Mechanics, Norris Hall;
[email protected]. Student Member AIAA. † Professor, Engineering Science and Mechanics, Norris Hall; mhajj@vt. edu. Member AIAA. ‡ University Distinguished Professor, Engineering Science and Mechanics, Norris Hall;
[email protected]. Fellow AIAA.
(1)
or in vector form as χ_ fχ ga χ ; t, where g is the gravitational acceleration; m and I y represent the body mass and moment of inertia about the yb axis, respectively; and χ u; w; q; θT is the vector of state variables of the longitudinal motion with u and w being the velocity of the body center of mass in the xb and zb directions, respectively, and θ and q being the pitching angle and angular velocity about the yb axis, respectively. The generalized forces X, Z, and M are the aerodynamic forces in the xb and zb directions and the aerodynamic moment about the yb axis, respectively. 970
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B. Aerodynamic Model
must be taken into consideration. To do so, the effects of the body motion variables on the velocity of the airfoil section and its angle of attack are included; that is
The aerodynamics of flapping flight are characterized by a very complex flowfield; that is, an unsteady nonlinear flowfield with nonconventional contributors to the aerodynamic forces. Dickinson et al. [15] pointed out that the main contributors are the translatory effect (leading edge vortex effect), the rotational effect, and the effect of wake capture. The latter is very difficult to be modeled analytically. As for the translatory effect, Wang et al. [22] showed that the static lift coefficient for a translating wing, taking into account the leading edge vortex effect, can be fit by CL A sin 2α, where A is a constant coefficient and α is the angle of attack. Taha and Hajj [23] showed that taking A as half of the lift curve slope; i.e.
U
αη t
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η_ → η_ q cos φ
η; U>0 π − η; U < 0
(3)
and αi tan−1
w − qr sin φ xh U
is the angle of attack induced by the body motion, where xh is the distance between the vehicle center of mass and the root of the wing hinge line along the xb axis, as shown in Fig. 1. In their derivation for the cycle-averaged stability derivatives, Cheng and Deng [14] did not account for the rotational lift and, as such, their q contribution appears only as an induced angle of attack qr sin φ xh U This assumption may overestimate the pitch damping (known as flapping countertorque), because the q contribution in the rotational lift component results in a negative contribution to the pitch damping for some locations of the hinge line. Recalling the basic definitions of the lift and drag forces per unit span, Eq. (2), expanding their trigonometric functions of α around αη , keeping up to first-order terms in αi , and assuming that αi ≃ weff ∕jUj, where weff w − qr sin φ xh , the lift and drag forces per unit span are written as
where c is the airfoil chord length, η_ is the pitching angular velocity of the wing, and x^ 0 is the chord-normalized position of the pitch axis from the leading edge. Using the preceding representation for the aerodynamic loads, the lift and drag forces per unit span on an airfoil section that is a distance r from the wing root are written as 1 lr; t ρAcrU2 r; t sin 2αr; t 2 3 πρ − x^ 0 Ur; tc2 r_ηt 4 dr; t ρAcrU r; tsin αr; t
and
where αη is the angle of attack induced by the wing pitching angle and defined as
matches well the experimental results of Berman and Wang [24], where AR and a0 are the wing aspect ratio and lift curve slope of the two-dimensional (2-D) airfoil, respectively. As for the drag coefficient, one could assume a similar representation to the aerodynamic drag on delta wings where the effect of leading edge vortex is pronounced [23,25], i.e., CD CL tan α 2A sin2 α. The viscous drag is neglected here according to the experiments of Dickinson et al. [15] and Dickinson and Gotz [26]. As for the rotational lift, according to the experimental results of Dickinson et al. [15], Sane and Dickinson [27], and Andersen et al. [28], it is reasonable to use the potential flow result for the 2-D rotational circulation 3 Γrot πc2 η_ − x^ 0 4
2
q rφ_ u cos φ2 w2 ;
α αη αi ;
πAR q A 2 21 πAR a0 1
2
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ENGINEERING NOTES
lχ ; t; r
1 w χ ; t; r ρAcrU2 χ ; t; r sin 2αη t 2 cos αη t eff 2 jUχ ; t; rj 3 πρ − x^ 0 c2 rUχ ; t; r_ηt q cos φt 4
(2)
where ρ is the air density and U is the velocity of the wing section relative to the air.
dχ ; t; r
w χ ; t; r ρAcrU2 χ ; t; r sin2 αη t sin 2αη t eff jUχ ; t; rj
C. Aerodynamic–Dynamic Coupling
Because the objective here is to analyze the stability of the body motion, the contribution of the body motion to the aerodynamic loads yw
yb
xw
xw
xh
Horizontal stroke plane xb
Side view for a wing section
CG
Top view for the body and wings
Fig. 1 Schematic diagram for a hovering MAV flapping in a horizontal stroke plane (CG refers to the center of gravity).
(4)
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It is noted that the additional angle of attack αi induced by the body motion would tilt the lift and drag forces so that the X and Z forces per unit span would be given by X 0 −sgnUH 0 cos φ and
Z 0 −V 0
(5)
where U is assumed positive in the forward (downstroke) direction, and H 0 and V 0 are the horizontal (opposing the instantaneous velocity) and vertical (upward) aerodynamic forces per unit span, respectively, which are written as
ENGINEERING NOTES
0
_ ut
1
0
B C B B wt C B B _ C B qtut g cos B CB B qt C B 0 @ _ A @ _θt qt 2 Xu t Xw t Xq t 6 6 Zu t Zw t Zq t 6 6 6 Mu t Mw t Mq t 4 0
H 0 d cos αi − l sin αi ≃ d − lαi
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1 X 0 − cos φ ρAcjUj2U sin2 η weff sin 2η 2 3 − πρ − x^0 c2 weff η_ 4 1 Z 0 − ρAcjUjU sin 2η 2 cos2 ηtweff 2 3 2 ^ πρ − x0 c U_η q cos φ 4
_ φj _ cos φ sin2 η; X0 −2K 21 φj
1 m X 0 t C B B 1 Z0 t C C Bm C B1 B M0 t C A @ Iy
30
1
0
ut C 7B B 7 0 7B wt C C C 7B B 7 0 5@ qt C A 0
(8)
θt
_ φj _ sin 2η Z0 −K21 φj
_ φj _ sin ηK 22 Δx^ cos φ K21 xh cos η M0 2φj K31 sin φ cos η where K mn 1∕2ρAI mn and I mn 2∫ R0 rm cn r dr. The timevarying stability derivatives are written directly in terms of the system parameters as Xu −4 (6)
K11 2 φ sin2 η; _ jφjcos m
Xw −
K11 _ cos φ sin 2η; jφj m
K 21 _ sin φ cos φ sin 2η − xh Xw ; jφj Zu 2Xw ; m K 2 _ η; Zw −2 11 jφjcos m K rot12 K _ sin φ cos2 η − Zq 2 21 jφj φ_ cos φ − xh Zw m m K Δx 2 φ sin η m 2X − x Z _ Mu 4 12 jφjcos q h u Iy Iy Xq
The pitching moment per unit span about the yb axis is written as M 0 Mh0 cos φ − Z 0 xh MAF where the hinge moment Mh0 is given by Mh0 χ ; t; r V 0 χ ; t; r cos ηt
Mw 2 ^ ηtΔxcr τv χ ; t; r
where Δx^ is the normalized chordwise distance between the center of pressure and the hinge location and τv is the viscous rotational damping torque per unit span. The center of pressure is assumed to be located at the quarter-chord point. As for τv, the formula provided by Berman and Wang [24] is used; that is, τv −π∕16ρc4 μ1 f μ2 j_ηj_η, where μ1 and μ2 depend on the viscosity of the fluid. Berman and Wang [24] recommended a value of 0.2 for both of them. It is noted that although τv may be too small to be accounted for in this medium level of fidelity analysis, its inclusion is highly recommended for the stability analysis near hover. This is because the pitch damping plays a very important role in determining the stability and because there exists two counteracting mechanisms for pitch damping and this latter one may help one outweigh the other. Finally, the pitching moment due to asymmetric flapping is written as 0 MAF −Z 0 r sin φ. Thus, the total pitching moment per unit span about the yb axis is written as ^ M 0 χ ; t; r −Z 0 χ ; t; rΔxcr cos φt cos ηt ^ xh r sin φt −X 0 χ ; t; r sin ηtΔxcr τv χ ; t; r cos φ
0
C θt C C C C A 0
1
0
where, assuming piecewise constant variation for the wing pitch angle (_η 0), one obtains
Thus, the first-order terms in χ for X 0 and Z 0 are written as
0
1
and
V 0 l cos αi d sin αi ≃ l dαi
sgnUH 0 χ ; t; r sin
−qtwt − g sin θt
(7)
Because only first-order terms in χ are considered, assume that U ≃ rφ_ u cos φ. Then, Eqs. (6) and (7) are integrated over the wing span to obtain the aerodynamic loads X, Z, and M. Substituting these loads into Eq. (1), one obtains a tight coupling between the aerodynamic and dynamic models. Thus, the flight dynamic model is written as
K12 Δx K mx _ cos φ cos η 2 21 jφj _ sin φ cos2 η − h Zw jφj Iy Iy Iy
Mq −
2Δx _ cos φ cos ηK12 xh K22 sin φ jφj Iy 1 φ_ cos φK rot13 Δx cos φ cos η Krot22 sin φ Iy
2 K v μ1 f 2 η sin φK x K _ jφjcos cos2 φ 21 h 31 sin φ − Iy Iy mx − h Zq Iy
−
π ^ mn and Kv 16 where Krotmn πρ12 − ΔxI ρI 04
III.
Stability Analysis
Equation (8) represents a NLTP system. Two different approaches have been proposed in the literature to investigate the dynamic stability of this NLTP system. The first approach, adopted in the early studies [1–5,29] is to use the averaging theorem to obtain a nonlinear time invariant (NLTI) system for which the periodic orbit of the NLTP system is reduced to a fixed point. As such, linearization of the obtained NLTI system about this fixed point allows stability analysis of the fixed point. Based on the averaging theorem, if the averaged system has an exponentially stable fixed point, the NLTP system will have an exponentially stable periodic orbit. The second approach, first adopted independently by Dietl and Garcia [16] and Bierling and Patil [30] and then by Su and Cesnik [31], is to use numerical techniques to obtain the periodic solution of the NLTP system. Then, linearization about the obtained periodic orbit yields a linear timeperiodic (LTP) system whose stability analysis can be performed using Floquet theory. The Floquet theory dictates solving the LTP
J. GUIDANCE, VOL. 37, NO. 3:
system to obtain the state transition matrix (fundamental matrix solution) evaluated at the minimal period, called the monodromy matrix. Then, stability analysis of the periodic orbit is performed by checking the eigenvalues of this monodromy matrix. Dietl and Garcia [16] stated that the first approach, which encompasses averaging of the dynamics, is not suitable for a large ornithopter whose beat frequency is close to the natural frequencies of its body motion. In contrast, Floquet theory is suitable for analyzing the stability of LTP systems, independent of their characteristic time scales. However, the difficulty in determining the periodic orbit and the corresponding monodromy matrix analytically makes the first approach more suitable when analytical results are required. Because the goal here is to perform an analytical study for the dynamic stability of hovering MAVs/insects, the first approach is adopted. A. Averaged Dynamics Downloaded by VIRGINIA TECHNICAL UNIVERSITY on April 30, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.62323
A nonautonomous dynamical system is represented by χ_ ϵfχ ; t
(9)
According to Khalil [32], if f is T-periodic in t, the averaged dynamical system corresponding to Eq. (9) is written as χ χ_ ϵf
dχ 1 fχ ga χ ; τ dτ ω
(11)
The dynamic equations in the new time variable, Eq. (11), have the form of Eq. (9), with ϵ ≡ 1∕ω, which is amenable to the averaging theorem. Because the averaging theorem is valid for small enough ϵ, this approach of using the averaging theorem in analyzing the dynamics of flapping flight works well only for high flapping frequencies. Applying the averaging theorem on the NLTP system given by Eq. (11), and transforming it back to the original time variable t, the following autonomous system is obtained χ_ fχ g a χ
the process of finding the periodic orbit reduces to finding a fixed point of the averaged system. For example, the periodic orbit of the steady hovering configuration corresponds to the origin of the autonomous system in Eq. (13). Thus, to find the trim conditions for steady hovering of the MAV/insect, the vector field describing the dynamics of Eq. (13) must vanish at the origin, i.e.
(12)
where χ is the averaged state vector and g a χ is the average of the vector field ga χ; t over the flapping period; i.e., g a χ 1∕T∫ T0 ga χ ; τ dτ. Thus, the averaged dynamics of Eq. (8) are written as 0 1 1 0 1 0 1 _ − g sin θt wt −qt ut m X0 B C C B C B B wt C B qt C B 1 Z 0 C g cos θt B _ C C B ut C Bm B C C B CB 1 B qt B C B M 0C _ C 0 @ A A @ A @ Iy 0 qt θt 1 30 2 ut X u X w X q 0 C 7B 6 C 6 Z u Z w Z q 0 7B wt 7B C 6 6 (13) C 7B 7B C 6M A 4 u Mw Mq 0 5@ qt θt 0 0 0 0
B. Trim/Balance of the Averaged System
The advantage of the averaging theorem is that it converts the NLTP system, Eq. (8), into an autonomous system, Eq. (13). As such,
−Z 0 L 0 mg;
X 0 0;
0 0 M
Two approaches can be used to trim MAVs at hover. If symmetric flapping (identical downstroke and upstroke) is used, X 0 will vanish automatically. This is intuitively expected because symmetric flapping yields zero cycle-averaged forward thrust at hover. However, there might be a nonzero net pitching moment (the cycleaverage of the second term of M0 proportional to xh ). Thus, to make 0 0, the hinge line needs to be aligned with the vehicle’s center of M mass (xh 0). Finally, the condition L 0 mg ensures that the generated averaged lift balances the weight. This dictates a certain combination of flapping amplitude Φ, frequency f 1∕T, and mean angle of attack (αm ). For example, if a triangular waveform is used for φt 1 mgT 2 αm sin−1 2 8ρAI 21 Φ2
(10)
1∕T∫ T χ ; τ dτ. According to the averaging where fχ 0 theorem, if ϵ is small enough, then exponential stability of the averaged system concludes exponential stability of the original timeperiodic system. Considering the abstract form of Eq. (1), χ fχ ga χ ; t and introducing a new time variable τ ωt, where ω is the flapping frequency, the dynamics in this new time variable are given by [10,33]
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ENGINEERING NOTES
Using a harmonic waveform results in 1 −1 mgT 2 αm sin 2 π 2 ρAI 21 Φ2 This trim approach has been adopted by Doman et al. [6] and Oppenheimer et al. [7]. As a second approach, if it is not possible to align the hinge line with the center of mass, one can use asymmetric flapping (with an offset in φt, referred to as φ0 , and possibly different angles of attack during downstroke and upstroke αd ≠ αu ) and solve the three trim conditions to find three of the five controlling parameters αd , αu , Φ, φ0 , and f. This trim approach has been adopted by Sun and Xiong [3] and Sun et al. [4]. C. Stability of the Averaged System
After ensuring trim at hover (the origin is a fixed point for the averaged system), the stability of this equilibrium position is investigated. By the statement of the averaging theorem, exponential stability of this fixed point yields exponential stability of the hovering periodic orbit for the original time-varying system. A necessary and sufficient condition for local exponential stability of the origin of Eq. (13) is that the Jacobian of its vector field evaluated at the origin be Hurwitz. The Jacobian of the nondimensional form of the vector field of Eq. (13), evaluated at the origin and expressed in the terminology used by [3–5,14], is written 2 Xu
m 6 Zu 6 m 6 4 Mu Iy
0
X w m Z w m M w I y
X q m Zq m M q I y
−g
0
1
0
3
7 0 7 7 0 5
(14)
I
gc y where m ρSmc I y ρSc3 g U2 , and 2
X; Z u;w M u;w
mX; Zq mX; Zu;w ; X; Z ; q ρU2 S ρU2 Sc I y Mu;w I y Mq ; M q ρU2 Sc ρU2 Sc2
where S is the area of one wing, c is the mean chord length, and U2 is the maximum wing speed at the section of the second moment of the wing chord distribution. The use of the preceding formulation is validated by comparing the hawkmoth stability results with those of Sun et al. [4] obtained by
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ENGINEERING NOTES Eigenvalues of the averaged, linearized system
solving Navier–Stokes equations. The morphological parameters of the hawkmoth are [4,34]
R 51.9 mm; r^2 0.525
c 18.3 mm;
m 1.648 g;
R
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0
I y 2.08 g · cm2 ;
(15)
c r P−1 r Q−1 1− β R R
where r^1 1 − r^1 r^1 1 − r^1 −1 ; Q 1 − r^1 −1 and P r^1 r^22 − r^21 r^22 − r^21 Z1 ^ Q−1 dr^ r^P−1 1 − r β
0
Sun et al. [4] obtained the following trim parameters αd 25.5 deg
and
αu 28 deg
In the present analysis, the exact aerodynamic model, discussed in Sec. II.B, is used without any approximations. Then, the complexstep finite difference is used to obtain accurate approximations for the cycle-averaged stability derivatives using the morphological parameters of the hawkmoth and the kinematics used by Sun et al. [4]. For example, X u is calculated as 1 ∂X ; ρU2 S ∂u
−0.1
−0.3
where Δtr is the duration of each rotational phase, and tr is the time at which this phase starts. Knowing αd and αu is enough to determine Δη, because the wing rotates from αd to π − αu or vice versa. Setting Δtr equal to 0.25T, and considering symmetric rotation, tr can be determined. As for the wing planform, the method of moments used by Ellington [34] is adopted here to obtain a chord-distribution for the insect that matches the documented first two moments r^1 and r^2 , i.e.
X u
0
−0.2
Δη Δt 2πt − tr ηt t − tr − r sin Δtr Δtr 2π
ϕ0 9 deg;
0.1
rk cr dr 2SRk r^k1
Because the insects, in general, do not have zero xh , Sun et al. [4] adopted the second trim configuration (asymmetric flapping). They considered a horizontal stroke plane flapping with φt φ0 − Φ cos2π ft. As for the pitching kinematics, they considered a constant value in each half stroke, referred to as αd and αu , except near the beginning and end of each half stroke. During the rotation phase, the η variation is described by
cr
Cheng and Deng [14]
0.2
r^1 0.44;
where f is the flapping frequency, Φ is the flapping amplitude, and r^1 and r^2 are defined as Z
Current Numerical
0.3
S 947.8 mm2 ;
xh 0.22R
I k1 2
Sun et al. [4]
Imaginary Axis
Φ 60.5 deg;
f 26.3 Hz;
0.4
ih ∂X imagXu ∂u h
Figure 2 shows a comparison between the eigenvalues of the nondimensional form of the averaged, linearized system matrix, given in Eq. (14), using the stability derivatives obtained by complexstep finite difference and those of Sun et al. [4] and Cheng and Deng [14]. Although the adopted aerodynamic model lacks the unsteadiness, it gives a good estimate for the cycle-averaged stability derivatives, as the eigenvalues of the averaged system are much closer to the benchmark results of Sun et al. [4] than those of Cheng and Deng [14]. The deviation of the results obtained by Cheng and Deng
−0.4 −0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Real Axis
Fig. 2 Eigenvalues of the averaged, linearized dynamics of the five insects.
may be attributed to their neglection of the rotational lift contribution and some of the stability derivatives.
IV.
Stability Characterization
A. Effect of Trim Configuration
In the previous section, the asymmetric flapping trim configuration (AFTC) has been considered. The trim configuration has a considerable effect on the cycle-averaged stability derivatives. If the other trim configuration (symmetric flapping along with zero xh ) is considered, only four derivatives remain nonzero; X u , Zw , M u , and M . Considering this latter trim configuration, a triangular waveform q for the back and forth flapping angle φ, and a piecewise constant wing pitching angle to maintain a constant angle of attack αm , the nondimensional, cycle-averaged stability derivatives are written in terms of the system parameters as K 11 2Φ sin 2Φsin2 αm ρU2 ST K 11 Φ cos2 αm Z w −8 ρU2 ST K12 Δx 2Φ sin 2Φ sin αm M u 4 ρU2 ScT 1 2Φ sin 2Φ M q ρU2 ST K μ × −2K 31 cos2 αm Krot13 Δx cos αm − v 1 (16) 4Φ X u −4
Table 1 presents a comparison between the cycle-averaged stability derivatives for the hawkmoth case study during hover using three sets of kinematics and trim configurations. The first set is the one considered in the previous subsection (AFTC and kinematics used by Sun et al. [4]). The second set includes the AFTC using a triangular waveform and piecewise constant wing pitching angle. The third set includes the SFTC using triangular waveform and piecewise constant wing pitching angle. Thus, comparison between the first two sets will show the effects of kinematics on the cycle-averaged stability derivatives and comparison between the last two sets will show the effects of trim configuration. Table 1 shows that the waveform does not have a considerable effect on the stability derivatives. It also shows that the SFTC leads to a higher damping X u in the longitudinal direction. This is because the flapping offset associated with the AFTC decreases the component of any u perturbation perpendicular to the leading edge. An analytical investigation using a triangular waveform shows that
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Table 1
Cycle-averaged stability derivatives for the hawkmoth case study using different kinematics and trim configurations
Trim configuration AFTC using kinematics of Sun et al. [4] AFTC using triangular φ and constant α SFTC using triangular φ and constant α
X X Z Z Z M M M X u w q u w q u w q −1.31 0.028 −0.021 0.05 −2.14 −0.22 0.05 1.81 −1.90 −1.15 0.028 −0.022 0.06 −2.18 −0.14 0.05 1.82 −2.11 −1.69 0 0 0 −1.53 0 0.13 0 −1.39
motions. In addition, because Z w is negative, it yields one of the stable eigenvalues. This is similar to the results of Sun et al. [4] where the eigenvector analysis showed decoupling between the w degreeof-freedom and the other ones. The other three eigenvalues may be analyzed using the Routh–Hurwitz criterion, which requires checking the signs of the sequence
K 11 2Φ sin 2Φ ρU2 ST K 11 2Φ sin 2Φ cos 2φ0 −4 ρU2 ST
X u SFTC −4 sin2 αm X u AFTC
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×
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ENGINEERING NOTES
sin2 αd sin2 αu 2
One should note the cos 2φ0 factor in the expression for the asymmetric flapping case. Also, the speed stability derivative M u is higher for the SFTC than the asymmetric one for the same reason. On the other hand, the AFTC leads to higher damping in both of the vertical direction Z w and pitching direction M q . It should be noted that the nonzero xh associated with the AFTC results in a considerable pitch damping. This is because any q perturbation induces a change in the angle of attack on the wing sections and, consequently the produced lift that, in turn, produces a restoring pitching moment that is proportional to the given perturbation. B. Parametric Study
In this section, the SFTC is used with a triangular waveform for the flapping angle φt and piecewise constant wing pitching angle η. Using such trim configuration and kinematics, the analytic expressions for the cycle-averaged stability derivatives are provided in Eq. (16) in terms of the system parameters. Exploiting the analytic nature of the current formulation, the effect of the system parameters on these derivatives is assessed. Figure 3 shows the variation of the cycle-averaged stability derivatives for the hawkmoth at different mean angles of attack αm , flapping frequencies f, and hinge locations x^ 0 . It includes four subfigures, each for a different value of x^ 0 . Three plots corresponding to different values of f are presented in each subfigure. Recalling the nondimensional Jacobian of the averaged dynamics evaluated at the fixed point, Eq. (14), and eliminating the vanishing stability derivatives due to the SFTC, the characteristic polynomial of this Jacobian matrix is given by 3 2 λ − Z w λ − X u M q λ X u M q λ g X u
(17)
The negative signs in the analytic expressions of the derivatives X u and Z w indicate positive damping in the xb and zb directions. As for the rotational damping about the yb axis (dictated by the sign of M q ), there are three sources of pitch damping. Recall the analytic expression for M q presented in Eq. (16). The first term yields positive pitch damping (flapping countertorque); the one discussed by Cheng and Deng [14]. The other two terms were not considered in their work. The second term, which provides a negative contribution to the pitch damping, is a result of the induced pitch rate at the wing sections due to q perturbation. This creates a rotational lift component that may result in a destabilizing pitching moment. Therefore, one should be cautious to include models for almost all the considerable aspects of pitch damping. Hence, a third component is added. This is a viscous component adopted from the model of Berman and Wang [24] and definitely augments the pitch damping. It is noted that the second destabilizing component is proportional to Krot13 Δx that, in turn is proportional to 1∕2 − ΔxΔx. Thus, negative Δx (fore hinge locations) or Δx > 1∕2 (backward hinge locations far enough from the quarter-chord; center of pressure) results in a stabilizing contribution instead. It is clear from the characteristic polynomial in Eq. (17) that the stability of the motion in the zb direction is decoupled from the other
−X u M q ;
X u Mq
g M u M q
X u
and g M u . They all have to be positive for a stable system. In most cases, the first and the latter are positive but the middle is not, which indicates two unstable poles. Although this trim configuration is characterized by zero pitch stiffness (Mα or Mw ), stability is still achievable via proper damping in all directions and speed stability. Figure 3a shows that fore hinge locations lead to speed instability (negative M u ) that, according to the preceding stability analysis, concludes instability for the averaged system irrespective of the damping values in all directions. This can be explained by looking at the analytic expression for M u presented in Eq. (16). It is noted that M u is proportional to the chord-normalized distance between the hinge line and the center of pressure along the xb axis, Δx. Therefore, unlike conventional aircraft, using the SFTC for hovering MAVs/ insects leads to the fact that if the hinge line (coincident with the center of gravity) is ahead of the wing’s center of pressure, negative M u results and, consequently the system is unstable. On the other hand, for all hinge locations behind the center of pressure (located at the quarter-chord), Fig. 3 shows positive M u that increases as αm and f increase, and increases considerably as the hinge is located further backward. Figure 3 also shows that the damping in the longitudinal direction −X u increases as αm and f increase. This is because any increase in αm or f leads to a decrease in the flapping amplitude Φ to maintain balance. Thus, the flapping wing sweeps smaller angles that, in turn, increases both of the components of the u perturbation perpendicular to the wing leading edge and the resulting wing drag along the xb axis. Also, the drag curve slope increases with the operating angle of attack in the range of interest. As for the damping in the vertical direction −Z w , it is independent of the hinge location and frequency. The hinge location affects only the moment derivatives. Z w is proportional to the flapping speed 4Φ∕T, which is fixed upon specifying αm to achieve balance. Thus, any change in the frequency at a fixed αm is associated with a change in Φ so as to maintain the same Φ∕T and consequently, the same Z w . In addition, the damping in the vertical direction decreases as the mean angle of attack increases. This is because any increase in αm leads to a considerable decrease in the ratio Φ∕T to maintain balance. Also, the lift curve slope decreases as the operating angle of attack increases. Finally, Fig. 3 shows a positive pitch damping for all cases that decreases as αm and f increase for the same reasons. In addition, there is a sweet range of hinge locations over which a considerable pitch damping is obtained. This range can be specified based on the analytical investigation of M q contributors, as discussed previously. This range is for fore hinge locations 1 Δx < 0 x^ 0 < x^ cp 4 or for backward hinge locations far enough from the quarter-chord 1 3 x^ 0 > Δx > 2 4
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ENGINEERING NOTES
Cycle-averaged stability derivatives vs the mean AOA for different flapping frequencies
Cycle-averaged stability derivatives vs the mean AOA for different flapping frequencies
−5
20
40
−20
60
0
w
+
20
40
Z+ −3
60
20
40
u
−30
60
M+
f = 26 Hz f = 50 Hz f = 100 Hz
−20
0
−15 0
20
0
20
° αm
40
0
60
f = 26 Hz f = 50 Hz f = 100 Hz
0
20
40
−30
60
0
20
° α m
40
60
° m
α
Cycle-averaged stability derivatives vs the mean AOA for different flapping frequencies 0
0 −5
−20
60
1.5
Z+ −2
0
20
40
−3
60
10
q
−10 f = 26 Hz f = 50 Hz f = 100 Hz
−20 20
40
60
α°m
−30
0
20
40
0
20
40
60
3
0
2
−10
1
0
20
40
60
0
f = 26 Hz f = 50 Hz f = 100 Hz
−20
0
α°m
c) x 0 = 60%
−20
60
u
M+
u
M+
0.5
−15
M+
0
1
−10
q
40
w
+
−10 −15
20
−5
−1
M+
−2
0
Xu
Z+
u
X+
w
−1
0
40
b) x 0 = 30%
0
0
20
−20
0.05
60
0
−10
0.1
Cycle-averaged stability derivatives vs the mean AOA for different flapping frequencies
0
−20
60
0
° αm
a) x 0 = 10%
−3
40
0.15
+
−0.4
−10
0.2
Mq
+
Mu
0
−10
−0.6
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−2
0
−0.2
−0.8
−10 −15
0
−5
−1
q
−2
0
Xu
Z+
Xu+
w
−1
−3
0
0
M+
0
20
40
60
−30
0
α° m
20
40
60
α°
m
d) x 0 = 90% Fig. 3
Variation of the cycle-averaged stability derivatives with the flapping parameters.
where the rotational contribution augments the other translatory and viscous contributions for pitch damping. If the averaging assumption is acceptable, or the system is amenable to the application of the averaging theorem, the current study (due to its analytic formulation) would be useful in determining the effect of the system parameters on the hovering stability. Figure 4 includes root locus plots of the characteristic polynomial of the dimensional form of the averaged, linearized system for the hawkmoth for varying mean angles of attack αm . It includes four subfigures, each for a different value of hinge location x^ 0 . Three root locus plots corresponding to different values of flapping frequency f are presented in each subfigure. For each combination of αm and f, a balancing amplitude is calculated to ensure trim. Then, stability is studied around this newly established hovering equilibrium position. The dimensional form of the system is used here to produce more clear plots. Some conclusions can be drawn from these root locus plots. First, increasing the frequency and/or the mean angle of attack has a destabilizing effect because the preceding discussion about the variation of the stability derivatives with the system parameters shows a decrease of the pitch damping (crucial in maintaining stability) with αm and f. It should be noted that according to Taha et al. [35], higher f leads to lower aerodynamic power consumption.
However, the preceding dynamic analysis shows that higher f values lead to a more unstable system. This is a common contradiction between performance and stability found in flight vehicles. Second, lack of pitch stability (negative M u ) for fore hinge locations explains the instability shown in Fig. 4a for all values of αm and f. On the other hand, the relatively weak pitch damping shown in Fig. 3c may explain the instability shown in Fig. 4c for all values of αm and f. Although backward hinge locations far enough from the quarter-chord leads to favorable speed stability and pitch damping as discussed previously, the term X u Mq
g M u X u Mq
which must be positive for stability, requires some proper combination of the involved stability derivatives. A positive, small enough M u and a negative, large enough (X u M q ) must exist for stability to be concluded. Thus, the considerable large values of M u observed at x^0 90%, as shown in Fig. 3d, may also explain the instability shown in Fig. 4d for all values of αm and f. Thus, small positive values of x^0 and f, though they lead to relatively weak pitch
J. GUIDANCE, VOL. 37, NO. 3:
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ENGINEERING NOTES
Effect of η on the eigenvalues, hinge line at 30% c
Effect of ηm on the eigenvalues, hinge line at 10% c
m
15
8
f = 26 Hz f = 50 Hz 10 f = 100 Hz
6
f = 26 Hz f = 50 Hz f = 100 Hz
4
Imaginary Axis
Imaginary Axis
5
0
−5
2 0 −2 −4
−10
−15 −35
−30
−25
−20
−15
−10
−5
0
5
10
−8 −35
15
−30
−25
−20
a) x 0 = 10%
−10
−5
0
5
b) x 0 = 30% Effect of η on the eigenvalues, hinge line at 90% c
Effect of ηm on the eigenvalues, hinge line at 60% c
m
20
20
f = 26 Hz f = 50 Hz
15
10
10
5
5
Imaginary Axis
15 f = 100 Hz
0 −5
−5 −10
−15
−15
−30
−25
−20
−15
−10
−5
0
5
10
f = 26 Hz f = 50 Hz f = 100 Hz
0
−10
−20 −35
−15
Real Axis
Real Axis
Imaginary Axis
Downloaded by VIRGINIA TECHNICAL UNIVERSITY on April 30, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.62323
−6
−20 −40
Real Axis
c) x 0 = 60%
−30
−20
−10
0
10
Real Axis
d) x 0 = 90% Fig. 4
Root locus plots of the eigenvalues of the averaged linearized system for varying αm , f, and x^ 0 .
damping, may provide the required combination. Hence, Fig. 4b shows a narrow region of stability for the hawkmoth when x^ 0 30% at low values of ηm (less than 38 deg) and consequently larger Φ (higher than 62.4 deg) at the natural f 26 Hz. This region becomes narrower at higher frequencies as expected; ηm < 10 deg (Φ > 55.3 deg) at f 50 Hz. This observation shows the potential for passive stabilization of hovering MAVs.
V.
Conclusions
The longitudinal flight dynamics of hovering MAV/insects have been investigated. A quasi-steady aerodynamic model that captures the dominant leading edge vortex and the rotational effects is used to obtain an analytical representation of the aerodynamic loads. Stability derivatives are derived and presented in terms of the system parameters. The aerodynamic–dynamic interaction is then validated in terms of the stability results against those obtained via direct numerical simulation of Navier–Stokes equations on the hawkmoth wings. Despite the lack of unsteadiness in the model, a good agreement for the eigenvalues of the averaged, linearized system is obtained. The analytical formulation of the problem has led to some interesting results. First, two approaches are found to trim MAVs at hover. The first one is to flap symmetrically with aligning the hinge line with the vehicle center of gravity. If this alignment is not possible,
the MAV has to flap asymmetrically with a prescribed flapping offset angle to satisfy balance at hover. The SFTC leads to the vanishing of five cycle-averaged stability derivatives out of nine. Also, it leads to a higher damping in the longitudinal direction, higher speed stability derivative, and lower damping in both of the vertical and pitching directions. Second, increasing the mean angle of attack and/or flapping frequency has similar effects on the stability derivatives; i.e., it leads to a higher damping in the longitudinal direction, higher speed stability derivative, and lower damping in both of the vertical and pitching directions. Third, backward hinge locations are favorable for speed stability. Realizing, via Routh–Hurwitz stability analysis, that speed stability is a must to conclude stability for the whole system, fore hinge locations (specifically ahead of the wing’s center of pressure) lead to instability of the system, irrespective of the values of the other parameters and derivatives. Thus, unlike conventional aircraft, stability of the SFTC necessitates backward center of gravity position (coincident with the hinge location). Fourth, rotational lift contributions lead to lack of pitch damping, except over some range of the hinge locations (fore hinge locations or backward hinge locations far enough from the center of pressure). Finally, passive stability of the hawkmoth is achievable at lower mean angles of attack and flapping frequencies (higher flapping amplitudes) and hinge locations that are backward and too close to the wing’s center of pressure.
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Acknowledgments The authors would like to acknowledge the thorough comments and valuable suggestion of the reviewers.
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ENGINEERING NOTES
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