Nonlinear Dyn (2014) 78:2399–2408DOI 10.1007/s11071-014-1596-6

ORIGINAL PAPER

Effect of the aerodynamic-induced parametricexcitation on the longitudinal stability of hoveringMAVs/insects

Haithem E. Taha · Ali H. Nayfeh ·Muhammad R. Hajj

Received: 23 October 2013 / Accepted: 9 July 2014 / Published online: 2 August 2014© Springer Science+Business Media Dordrecht 2014

Abstract Longitudinal flight dynamics of hoveringmicro-air-vehicles and insects is considered. The nat-ural oscillatory flapping motion of the wing leads totime-periodic stability derivatives (aerodynamic loadsdue to perturbation in the body-motion variables).Hence these terms play the role of parametric excitationon the flight dynamics. The main objective of this workis to assess the effects of these aerodynamic-inducedparametric excitation terms that are neglected by theaveraging analysis. The method of multiple scales isused to determine a second-order uniform expansionfor the response of the time-periodic system at hand.The proposed approach is applied to the hovering flightdynamics of five insects that cover a wide range of oper-ating frequency ratios to assess the applicability of theaveraging analysis.

Keywords Flapping flight · Hovering insects · Microair vehicles · Time-periodic systems · Perturbationtechniques · Method of multiple scales · Flightdynamics · Averaging theorem · Parametric excitation

H. E. Taha (B) · A. H. Nayfeh · M. R. HajjVirginia Tech, Blacksburg, VA, USAe-mail: hezzat@vt.edu

A. H. NayfehUniversity of Jordan, Amman, Jordane-mail: anayfeh@vt.edu

M. R. Hajje-mail: mhajj@vt.edu

1 Introduction

The dynamics of flapping flight has been a researchtopic of interest for more than a decade. In addition tothe complex dynamics associated with flapping flight,the periodicity aspect of the flapping motion gives theflapping flight dynamics a time-varying characteristic.Since hovering is usually performed at relatively highfrequencies with respect to the body’s natural frequen-cies, it is usually claimed that the body of a hoveringinsect feels only the cycle-averaged aerodynamic loads.In fact, this is a very common assumption in the analy-sis of flapping flight dynamics. Some research reportsutilize this assumption on the basis of physical intuition[1–7]. Others use its rigorous mathematical represen-tation [8–11], namely, the averaging theorem. Thereexist interesting discussions about the range of valid-ity of this assumption by Taylor and Thomas [1] andTaha et al. [12]. However, there have been no mathe-matical investigations that rigorously support or refutethe direct use of averaging in the stability analysis ofhovering insects and micro-air-vehicles (MAVs).

One major issue with the direct application of theaveraging theorem is the neglection of the periodic,parametric excitation terms with zero-mean, whichmay provide a stabilizing action. In fact, this is awell-known characteristic of high-frequency, high-amplitude periodic forcing, known as vibrational stabi-lization, see Bullo [13] and Sarychev [14]. One exam-ple is the linear, time-periodic Mathieu equation, seeNayfeh [15]. To assess the effects of these neglected

123

2400 H. E. Taha et al.

terms and whether they lead to a change in the sys-tem’s dynamic behavior, one has to take into accountthe effect of the flow along the nonautonomous vectorfield comprising these terms. In this work, we assessthe validity of the direct application of the averagingtheorem to the hovering flight dynamics of flappingMAVs/insects. Particularly, we determine the range offrequency ratios over which this theorem is valid. Weuse higher order perturbation techniques (namely themethod of multiple scales) to determine the effect ofthe aerodynamic-induced parametric excitation termson the longitudinal stability of hovering MAVs/insects.

2 Flight dynamic model of hovering MAVs/insects

In the present dynamic analysis, only rigid-bodydegrees of freedom are taken into account and the iner-tial effects due to the wing motion are neglected. Infact, these assumptions are ubiquitous in the literatureof dynamics and control of flapping flight [3,5–10,16].The conventional set of body axes xb, yb, and zb that arecommonly used in flight dynamics analysis [17–19] isalso used here. That is, the xb-axis points forward, theyb-axis points to the right wing, and the zb-axis com-pletes the triad. Due to flapping, a wing-fixed framexw, yw, and zw would be required. The wing-fixedframe is considered to coincide with the body-fixedframe for zero wing kinematic angles. Most hoveringinsects perform a stroke in an approximate horizontalplane without an out-of-plane motion [20,21]. As such,and because the aerodynamic loads are expressed inthe relative wind frame (normal and tangential to thehorizontal flapping velocity), only the back and forthflapping angle ϕ would be of interest to transfer theaerodynamic loads to the body frame. Figure 1 showsa schematic diagram for a flapping MAV whose wingsweeps in a horizontal plane.

Using the above representation, one finds that theequations of the longitudinal body motion are similarto those of a conventional aircraft [17]; that is,

⎛⎜⎜⎝

uw

qθ

⎞⎟⎟⎠ =

⎛⎜⎜⎝

−qw − g sin θ

qu + g cos θ

0q

⎞⎟⎟⎠ +

⎛⎜⎜⎜⎝

1m X1m Z1Iy

M

0

⎞⎟⎟⎟⎠ (1)

or in vector form as χ = f (χ) + ga(χ , t), where gis the gravitational acceleration; m and Iy represent

the body mass and moment of inertia about the yb-axis, respectively; χ = [u, w, q, θ ]T is the vector ofstate variables of the longitudinal motion with u andw being the velocity of the body center of mass inthe xb- and zb-directions, respectively; and θ and qbeing the pitching angle and angular velocity aboutthe 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.

Taha et al. [11,22] developed a compact flightdynamic model for hovering MAVs using a quasi-steady aerodynamic model that accounts for the lead-ing edge vortex and rotational contributions. The fullunsteady version of the adopted aerodynamic modelhas been developed in Refs. [23,24] and validated overthe range of Reynolds number of interest for hover-ing insects (75 - 4000). Based on this aerodynamicmodel, Taha et al. [22] derived analytical expressionsfor the first-order aerodynamic contributions (time-varying stability derivatives) in terms of the systemparameters and, as such, wrote the longitudinal flap-ping flight dynamics as

⎛⎜⎜⎝

u(t)w(t)q(t)θ(t)

⎞⎟⎟⎠ =

⎛⎜⎜⎝

−q(t)w(t) − g sin θ(t)q(t)u(t) + g cos θ(t)

0q(t)

⎞⎟⎟⎠ +

⎛⎜⎜⎜⎝

1m X0(t)1m Z0(t)1Iy

M0(t)

0

⎞⎟⎟⎟⎠

+

⎡⎢⎢⎣

Xu(t) Xw(t) Xq (t) 0Zu(t) Zw(t) Zq (t) 0Mu(t) Mw(t) Mq (t) 0

0 0 1 0

⎤⎥⎥⎦

⎛⎜⎜⎝

u(t)w(t)q(t)θ(t)

⎞⎟⎟⎠ (2)

Assuming piecewise-constant variation for the wingpitch angle (η = 0), one obtains

X0 = −2K21ϕ|ϕ| cos ϕ sin2 η, Z0 =−K21ϕ|ϕ| sin 2η

M0 = 2ϕ|ϕ| sin η[K22�x cos ϕ + K21xh cos η

+K31 sin ϕ cos η] ,

where Kmn = 12ρ AImn , Imn = 2

∫ R0 rmcn(r) dr , �x is

the normalized chordwise distance between the centerof pressure and the hinge location, and xh is the dis-tance between the vehicle center of mass and the rootof the wing hinge line along the xb-axis, as shown inFig. 1. The time-varying stability derivatives are writ-ten directly in terms of the system parameters as [22]

Xu = −4K11

m|ϕ| cos2 ϕ sin2 η,

Xw = − K11

m|ϕ| cos ϕ sin 2η,

123

Effect of the aerodynamic-induced parametric excitation 2401

CG

yw

yb

xw

xb

xw

Horizontal Stroke plane

xh

Top view for the body and wingsSide view for a wing section

Fig. 1 Schematic diagram for a hovering MAV flapping in a horizontal stroke plane

Xq = K21

m|ϕ| sin ϕ cos ϕ sin 2η − xh Xw

Zu = 2Xw, Zw = −2K11

m|ϕ| cos2 η,

Zq = 2K21

m|ϕ| sin ϕ cos2 η − Krot12

mϕ cos ϕ − xh Zw

Mu = 4K12�x

Iy|ϕ| cos2 ϕ sin η + m

Iy

(2Xq − xh Zu

)

Mw = 2K12�x

Iy|ϕ| cos ϕ cos η

+2K21

Iy|ϕ| sin ϕ cos2 η − mxh

IyZw

Mq = −2�x

Iy|ϕ| cos ϕ cos η (K12xh + K22 sin ϕ)

+ 1

Iyϕ cos ϕ

(Krot13�xcos ϕ cos η+Krot22 sin ϕ

)+

− 2

Iy|ϕ| cos2 η sin ϕ (K21xh + K31 sin ϕ)

− Kvμ1 f

Iycos2 ϕ − mxh

IyZq ,

where Krotmn = πρ( 12 − �x)Imn and Kv = π

16ρ I04.

3 Direct application of the averaging theorem

In this section, we apply the averaging theorem directlyon the nonlinear time-periodic (NLTP) system pre-

sented in Eq. (2). Then, the stability results will becompared with those obtained using higher order per-turbation techniques in the next sections. The averagingtheorem allows us to investigate the dynamic stabilityof this NLTP system by obtaining a nonlinear time-invariant (NLTI) system for which the periodic orbit ofthe NLTP system is reduced to a fixed point. As such,linearization of the obtained NLTI about this fixed pointallows stability analysis of the fixed point. Based on theaveraging theorem, if the averaged system has an expo-nentially stable fixed point, the NLTP system will havean exponentially stable periodic orbit.

A nonautonomous dynamical system is representedby

χ = εY(χ , t) (3)

According to Khalil [25], if Y is T -periodic in t , theaveraged dynamical system corresponding to Eq. (3) iswritten as

χ = εY(χ), (4)

where Y(χ) = 1T

∫ T0 Y(χ , τ ) dτ . According to the

averaging theorem, if ε is small enough, then expo-nential stability of the averaged system implies expo-nential stability of the original time-periodic system.

123

2402 H. E. Taha et al.

Considering the abstract form χ = f (χ) + ga(χ , t)and introducing a new time variable τ = ω

ωnt , where

ω is the flapping frequency and ωn is the natural fre-quency of the body motion, we express the dynamicsin this new time variable as [10,11,22]

dχ

dτ= ωn

ω(f (χ) + ga(χ , τ )) (5)

The dynamic equations in the new time variable, Eq.(5), have the form of Eq. (3), with ε ≡ ωn

ω, that is

amenable to the averaging theorem. Since the averag-ing theorem is valid for small enough ε, this approachof using the averaging theorem in analyzing the dynam-ics of flapping flight works well only for high ratios ofthe flapping frequency to the body natural frequency.However, the limit frequency (or the ε-parameter)below which the averaging theorem does not hold isnot known. Since hovering frequencies typically rangebetween 25 and 250 Hz, the averaging theorem maynot be applicable for analyzing relatively low flappingfrequency flyers, such as the hawkmoth (26 Hz). Thus,higher order techniques may be required to provide arobust stability analysis analytically, which is the mainmotivation for this work.

Applying the averaging theorem on the NLTP sys-tem given by Eq. (5) and transforming it back tothe original time variable, t , we obtain the followingautonomous system

χ = f (χ) + ga(χ), (6)

where χ is the averaged state vector and ga(χ) is theaverage of the vector field ga(χ, t) over the flappingperiod; that is, ga(χ) = 1

T

∫ T0 ga(χ , τ ) dτ . Thus, the

averaged dynamics of Eq. (2) is written as⎛⎜⎜⎝

u(t)w(t)q(t)

θ(t)

⎞⎟⎟⎠ =

⎛⎜⎜⎝

−q(t)w(t) − g sin θ(t)q(t)u(t) + g cos θ(t)

0q(t)

⎞⎟⎟⎠ +

⎛⎜⎜⎜⎝

1m X01m Z01Iy

M0

0

⎞⎟⎟⎟⎠

+

⎡⎢⎢⎣

Xu Xw Xq 0Zu Zw Zq 0Mu Mw Mq 00 0 0 0

⎤⎥⎥⎦

⎛⎜⎜⎝

u(t)w(t)q(t)θ(t)

⎞⎟⎟⎠ (7)

The advantage of the averaging theorem is that itconverts the NLTP system, Eq. (2), into an autonomoussystem, Eq. (7). As such, the process of finding theperiodic orbit reduces to finding a fixed point of the

averaged system. For example, the periodic orbit ofthe steady hovering configuration corresponds to theorigin of the autonomous system (7). Thus, to find thetrim conditions for steady hovering of the MAV/insect,we require the vector field describing the dynamics of(7) to vanish at the origin; that is,

X0 = 0, −Z0 = L0 = mg, M0 = 0

A typical approach to trim MAVs at hover isto use symmetric flapping (identical downstroke andupstroke) which automatically ensures trim in the for-ward direction (X0 = 0). In addition, to ensure selfpitch trim (M0 = 0), we need to align the hingeline with the vehicle’s center of mass (xh = 0).Then, we are left with only one trim equation tobe satisfied; the condition L0 = mg which ensuresthat the generated averaged lift balances the weight.This dictates a certain combination of flapping ampli-tude, frequency, and mean angle of attack (αm). Forexample, if a triangular waveform is used for ϕ(t),

αm = 12 sin−1

(mgT 2

8ρ AI21Φ2

). Using a harmonic wave-

form results in αm = 12 sin−1

(mgT 2

π2ρ AI21Φ2

), instead.

After ensuring trim at hover (the origin is a fixedpoint for the averaged system), one investigates thestability of this equilibrium position. By the statementof the averaging theorem, exponential stability of thisfixed point implies exponential stability of the hoveringperiodic orbit for the original time-varying system. Anecessary and sufficient condition for local exponentialstability of the origin of the system in Eq. (7) is that theJacobian of its vector field evaluated at the origin beHurwitz. This yields the following matrix representa-tion for the averaged, linearized system:

⎡⎢⎢⎣

Xu Xw Xq −gZu Zw Zq 0Mu Mw Mq 00 0 0 0

⎤⎥⎥⎦ (8)

4 Effects of aerodynamic-induced parametricexcitation

We assess the effects of the aerodynamic-induced para-metric excitation terms that have a zero-mean; that is,terms that are neglected when applying the averagingtheorem directly. We write the time-varying aerody-namic vector field ga in Eq. (2) as

ga(χ , t) = g0(t) + [G(t)]χ,

123

Effect of the aerodynamic-induced parametric excitation 2403

where g0 represents the aerodynamic loads due to theflapping motion of the wing and the matrix G repre-sents the time-varying stability derivatives; that is, theaerodynamic loads due to body motion. We split g0(t)and G(t) into cycle-averaged components g0 and G andzero-mean components g1(t) and G1(t), respectively,and write

g0(t) = g0 + g1(t) and G(t) = G + G1(t).

These latter quantities are the ones that are neglectedwhen performing the direct averaging analysis. Thus,the system in Eq. (2) is re-written in terms of thesecomponents as

χ(t) = f (χ(t)) + [G + G1(t)]χ(t) + g0 + g1(t). (9)

We note that the time-periodic stability derivatives,represented by the term [G1(t)]χ , induce parametricexcitation on the body flight dynamics. Ignoring thezero-mean periodic forcing g1(t), we consider the fol-lowing system:

χ(t) = f (χ(t)) + [G + G1(t)]χ(t) + g0. (10)

Adopting the same trim/balance approach for the MAVat hover as the one discussed above using direct aver-aging (g0 + f (0) = 0) ensures that the origin is a fixedpoint for the system in Eq. (10) at all times. Linearizingthe drift vector field f around the ensured fixed point,we write

χ(t) = [A]χ(t) + [G1(t)]χ(t), (11)

where A = Df (χ = 0) + G is the system matrixobtained by direct application of the averaging theoremand is presented in Eq. (8).

To determine the effects of the aerodynamic-inducedparametric excitation term [G1(t)]χ , we consider thetransformation χ = Pζ , where P is the invertiblematrix constructed from the eigenvectors of the matrixA as its columns. As such, we write the dynamics ofthe system in Eq. (11) in the new coordinates as

ζ (t) = [P−1AP]ζ (t) + [P−1G1(t)P]ζ (t).

The diagonal (Jordan diagonal) matrix J = P−1AP canbe partitioned into two parts. The first block representsthe stable part (having eigenvalues with negative realpart) and the second block represents the unstable part.The above equation can be written as

(ζ s(t)ζ u(t)

)=

[Js 00 Ju

] (ζ s(t)ζ u(t)

)

+[

V11(t) V12(t)V21(t) V22(t)

](ζ s(t)ζ u(t)

)(12)

Next, we use the method of multiple scales [15,26]to obtain a second-order uniform expansion for Eq.(12). We rewrite Eq. (12) as

ζ s(t) = [Js]ζ s(t)+ε([V11(t)]ζ s(t)+[V12(t)]ζ u(t)

)

ζ u(t) = ε([V21(t)]ζ s(t) + [V22(t)]ζ u(t)

)

+ε2[Ju]ζ u(t), (13)

where ε is a bookkeeping parameter that represents therelative weighting between different terms. We expandthe time variable as Tn = εnt . As such, the time differ-ential is written as

d

dt= D0 + εD1 + ε2 D2 + · · · (14)

where Dn = ∂∂Tn

. For a uniform expansion, we requirethat ζ s and ζ u to be independent of T1. Thus, we expandζ s and ζ u as

ζ s,u(T0, T2) = ζ s,u0(T0, T2) + εζ s,u1

(T0, T2)

+ε2ζ s,u2(T0, T2) + · · · (15)

Substituting Eqs. (15) and (14) into Eq. (13) and sepa-rating coefficients of like powers of ε, we obtain threesets of equations, which can be solved successively.

The first set is expressed as

D0ζ s0= [Js]ζ s0

D0ζ u0= 0

(16)

Since Js is a Hurwitz matrix, ζ s0will exponentially

decay to a zero steady-state value. On the other hand,ζ u0

= ζ u0(T2). The second set of equations is given by

D0ζ s1= [Js]ζ s1

+ [V11(T0)]ζ s0+ [V12(T0)]ζ u0

D0ζ u1= [V21(T0)]ζ s0

+ [V22(T0)]ζ u0

(17)

Since [V12(T0)] is periodic in T0, it can be representedin a Fourier series as

[V12(T0)] = �(�n[V12]neiωnT0

),

where [V12]n is a matrix of complex entries and �refers to the real part of this matrix. Thus, the particular(steady-state) solution of Eq. (17) is written as

123

2404 H. E. Taha et al.

ζ s1(T0, T2) = �

(�nZneiωnT0

)ζ u0

ζ u1(T0, T2) = ∫ T0

0 [V22(τ )] dτζ u0(T2),

(18)

where Zn = [iωnI −Js]−1[V12]n and I represents theidentity matrix. Finally, the third set is given by

D0ζ s2+ D2ζ s0

= [Js]ζ s2+ [V11]ζ s1

+ [V12]ζ u1

D0ζ u2+ D2ζ u0

= [V21]ζ s1+ [V22]ζ u1

+ [Ju]ζ u0,

(19)

The first system (ζ s2-dynamics) is a stable autonomous

system that is subjected to periodic forcing. Thus, ζ s2

will not diverge. On the other hand, the second sys-tem (ζ u2

-dynamics) has trivial dynamics and is alsosubjected to periodic forcing. Thus, to have a boundedsolution, the sum of all of the T0-independent termsmust vanish. As such, we obtain the following solv-ability condition:

ζ ′u0(T2) = 1

T

∫ T

0

{[V21(t)]ζ s1(t, T2)

+[V22(t)]ζ u1(t, T2)

}dt + [Ju]ζ u0

(T2),

(20)

where ζ ′u0 = dζ u0

dT2. Equation (20) represents a linear

time-invariant system for ζ u2. Hence, the matrix

[BMMS] =[

1

T

∫ T

0

{[V21(t)]�

(�n[iωnI − Js ]−1[V12]neiωnt

)

+[V22(t)]∫ t

0[V22(τ )] dτ

}dt + Ju

](21)

must be Hurwitz for stability.

5 Stability of hovering insects

We apply the above methodology using the method ofmultiple scales on the longitudinal stability of five hov-ering insects; namely, the hawkmoth, cranefly, bumble-bee, dragonfly, and hoverfly. The morphological para-meters and the wing planforms for these insects aregiven in Appendix 1. Of particular interest is the hawk-moth, which has the lowest flapping frequency forwhich the direct averaging is expected to be insuffi-cient.

For simplicity and not to obfuscate the main target ofthe paper, we use the following simplified kinematics.A triangular wave form for the back and forth flappingangle is used along with a piecewise- constant pitch-ing angle to maintain a constant angle of attack (αm)

throughout the entire stroke. It is noteworthy to men-tion that this combination results in hovering with min-imum aerodynamic power [27]. It should also be notedthat adopting this simplified kinematics does not leadto a considerable change in the stability characteris-tics of hovering MAVs/insects in comparison to a morerealistic set of kinematics, as shown in [22]. Given theflapping frequency f and amplitude Φ, we calculatethe angle of attack αm required to achieve balance athover.

5.1 Stability of the hawkmoth at hover

Using the above setup and considering the hawkmothcase, we find that the eigenvalues of the matrix repre-senting the directly averaged, linearized system givenin Eq. (8) are

λavg = −11.89, −3.30, 0.19 ± 5.74i

That is, direct averaging results in an unstable system,which is consistent with the conclusions of Taylor andThomas [1], Sun and Xiong [3], Sun et al. [4], Chengand Deng [28], and others that hovering insects areunstable.

Using the method of multiple scales (MMS), as dis-cussed in Sect. 4, we obtain the following eigenvalues:

λMMS = −11.89, −3.30, −0.53 ± 4.34i

The first two stable eigenvalues are the same as theones obtained by direct averaging. The latter two(−0.53 ± 4.34i), which are the eigenvalues of thematrix in Eq. (21), are also stable. This result pointsto a shortcoming of the direct use of the averaging the-orem in analyzing stability of hovering insects flap-ping at relatively low frequencies. We note that thefrequency of oscillation that resulted from direct aver-aging is 5.74 rad/s, which is much less than the flap-ping frequency ω = 165.25 rad/s. This large ratio(around 30) of the excitation frequency to the naturalfrequency of the system has been used to support thedirect use of the averaging theorem in analyzing thedynamic stability of such insects. However, our analy-sis shows that, in spite of this large ratio, direct aver-aging may give false conclusions about stability ofhovering MAVs/insects. To further support the stabil-ity results obtained by accounting for the parametricexcitation using the method of multiple scales, we per-form numerical simulation for the flight dynamics of

123

Effect of the aerodynamic-induced parametric excitation 2405

Fig. 2 Numericalsimulation of the linearizedand nonlinear flightdynamic models (11) and(9)

0 50 100 150−0.05

0

0.05

0.1

0.15

u (m

/sec

)

Dynamics With Fixed Point

0 50 100 150−0.05

0

0.05

0.1

0.15

w (

m/s

ec)

0 50 100 150−1

−0.5

0

0.5

q (r

ad/s

ec)

t/T0 50 100 150

−1

−0.5

0

0.5

1

θ (d

eg)

t/T

Linearized

Nonlinear

the hovering hawkmoth. Figure 2 shows the numeri-cal simulation for the systems represented in Eqs. (10,11). We note that both systems have the origin as a fixedpoint.

The above analysis and simulation are further sup-ported by applying Floquet theorem on the linear,time-periodic system (11). The system in Eq. (11) issimulated using four orthonormal vectors of initialconditions. Then, the solutions obtained using theseinitial conditions are evaluated at the period T . Thefour obtained vectors are collected in a matrix calledthe Monodromy matrix, see Nayfeh and Balachandran[29] for example. The eigenvalues of the Monodromymatrix are called the Floquet multipliers. If all of theFloquet multipliers lie inside the unit circle, then theorigin is an exponentially stable fixed point for thesystem (11), which implies exponential stability of theorigin of the system in Eq. (10) by the Lyapunov indi-rect method [25]. The following Floquet multipliers areobtained for the hawkmoth case:

0.96 ± 0.11i, 0.6881, 0.8889

All of these multipliers lie inside the unit circle and,hence, indicate stability of the system in Eq. (10). Fig-ure 3 shows the resulting eigenvalues that determinethe stability of the system in Eq. (10) using direct

−12 −10 −8 −6 −4 −2 0 2−6

−4

−2

0

2

4

6

Real Axis

Imag

inar

y A

xis

Eigenvalues determining the System Stability

FloquetAveragingMMS

Fig. 3 Eigenvalues determining the stability of the time-periodicsystem (10) using direct averaging, second-order MMS, and theFloquet theorem approaches

averaging, second-order MMS, and the Floquet the-orem approaches. To have the same eigenvalue rep-resentation, we transform the eigenvalues obtained byusing the Floquet theorem approach from the Z-planeto the S-plane [30] via the common transformationz = eT s .

123

2406 H. E. Taha et al.

Table 1 The eigenvalues revealing the stability of the system in Eq. (10) using direct averaging and the method of multiple scales forthe five insects along with the ratio of the flapping frequency to the natural frequency ωn of the flight dynamics

Insect 2π fωn

λavg λMMS

Hawkmoth 28.78 [−11.89, −3.30, 0.19 ± 5.74i] [−11.89, −3.30, −0.53 ± 4.34i]Cranefly 50.62 [−47.71, −17.31, −1.13 ± 5.53i] [−47.71, −17.31, −12.42, 7.41]Bumblebee 144.46 [−11.63, −4.39, 1.58 ± 6.55i] [−11.63, −4.39, 1.39 ± 6.24i]Dragonfly 145.50 [−13.11, −7.03, 1.34 ± 6.65i] [−13.11, −7.03, 1.06 ± 6.09i]Hoverfly 113.98 [−14.01, −7.27, 2.13 ± 8.56i] [−14.01, −7.27, 1.81 ± 8.03i]

0 100 200 300−0.5

0

0.5

u (m

/sec

)

Simulation of The Bubmblebee Nonlinear Flight Dynamics With Fixed Point

0 100 200 3000.1

0.2

0.3

0.4

0.5

w (

m/s

ec)

0 100 200 300−10

−5

0

5

10

q (r

ad/s

ec)

t/T

0 100 200 300−20

−10

0

10

θ°

t/T

0 500 1000−0.1

−0.05

0

0.05

0.1

u (m

/sec

)

Simulation of The Bubmblebee Nonlinear Flight Dynamics With Fixed Point

0 500 1000−0.05

0

0.05

0.1

0.15

w (

m/s

ec)

0 500 1000−1

−0.5

0

0.5

q (r

ad/s

ec)

t/T0 500 1000

−2

−1

0

1

2

θ°

t/T

(a) (b)

Fig. 4 Simulation of the bumblebeebalanced nonlinear flight dynamics, Eq. 10, using different flapping amplitudes. a Natural flappingamplitude (Φ = 58◦). b Large flapping amplitude (Φ = 89◦)

5.2 Hovering stability of other insects

Next, we test the range of applicability of the averag-ing theorem by considering other four insects, namely,the cranefly, bumblebee, dragonfly, and hoverfly. Table1 shows the ratio of the flapping frequency to thebody natural frequency for the five insects, and theeigenvalues revealing the stability of the system in Eq.(10) using direct averaging and the method of multiplescales. It is noteworthy to mention that the ratio 2π f

ωnis not monotonically increasing with f as the increasein the flapping frequency f may be associated with alarger increase in the natural frequency ωn , as shownin the hoverfly case in comparison to the dragonfly. Wenote that the body mass of the hoverfly is considerablysmaller than that of the dragonfly, as shown in Appen-dix Table 2.

Table 1 shows that for large values of 2π fωn

(typicallyabove 100), the effects of the parametric excitation onthe eigenvalues and, consequently, on the stability char-acteristics of hovering insects are less important. Threetypes of behaviors are identified from Table 1: caseswhere the parametric excitation induces destabilizingeffects (cranefly), cases where the parametric excita-tion provides enough stabilizing action (hawkmoth),and cases where the parametric excitation induces astabilizing action but not large enough to stabilize theunstable hovering dynamics (bumblebee, dragonfly,and hoverfly). It should be noted that in the craneflycase where direct averaging shows stability, the matrixJu is taken to represent the least stable eigenvalue(s);that is, the nearest to the imaginary axis.

It can be shown that, even for the latter case, the weakinduced stabilizing action can be strengthened to pro-

123

Effect of the aerodynamic-induced parametric excitation 2407

vide stability if the flapping amplitude Φ is increased;that is, if the trim is achieved at higher amplitudesand, consequently, smaller angles of attack. Recall thathigher Φ leads to more stable averaged dynamics [22].In addition, a higher Φ naturally leads to parametricexcitation terms of higher amplitudes. Thus, using suf-ficiently large flapping amplitudes, the unstable hover-ing dynamics in Eq. (10) can be stabilized. The obtainedstabilizing thresholds of Φ for the bumblebee, dragon-fly, and hoverfly are 88.3◦, 73.1◦, and 62.5◦, respec-tively. The required flapping amplitude for stabiliza-tion decreases as the flapping frequency increases. Fig-ure 4a, b provides simulations for the nonlinear flightdynamics with fixed point, Eq. (10), for the bumblebeecase using its natural flapping amplitude (Φ = 58◦) anda larger flapping amplitude (Φ = 89◦), respectively. Asconcluded from the analysis above, at the natural Φ, thestabilizing action is not strong enough to ensure stabil-ity for the hovering fixed point (the origin), as shownin Fig. 4a. However, increasing Φ strengthens the para-metric excitation stabilization and leads to a stable fixedpoint, as shown in Fig. 4b. Because of the very weaklydamped oscillations shown in the stabilized case, Fig.4b, one may conclude that the aerodynamic-inducedparametric excitation provides a stabilizing stiffnessaction and no damping.

6 Conclusion

The longitudinal flight dynamics of hovering micro-air-vehicles (MAVs) and insects is considered. Themethod of multiple scales is used to determine asecond-order uniform expansion for the response of thetime-periodic system at hand. This approach allows usto determine the effects of the aerodynamic-inducedparametric excitation terms. It is shown that theseparametric excitation terms may provide a stabilizingaction for the hawkmoth flight dynamics near hover.

Hence, direct application of the averaging theorem isnot sufficient to capture the true stability characteris-tics of hovering MAVs/insects flapping at relativelylow frequencies. By considering other fast-flappinginsects, we show that direct averaging is valid over aregion of quite large ratios of the flapping frequencyto the natural frequency (above 100) and fails for arelatively lower ratio (up to 50). Additionally, it isshown that even for the cases where the stabiliza-tion effect due to parametric excitation is not largeenough, increasing the flapping amplitude enhancesthis effect and leads to stability of the time-periodicsystem.

Appendix 1: Morphological parameters

See Table 2.Table 2 shows the morphological parameters of the

five studied insects as given in [4] and [31].The moments of the wing chord distribution r1 and

r2 are defined as

Ik1 = 2∫ R

0rkc(r) dr = 2S Rkr k

k

As for the wing planform, the method of moments usedby Ellington [31] is adopted here to obtain a chorddistribution for the insect that matches the documentedfirst two moments r1 and r2; that is,

c(r) = c

β

( r

R

)α−1 (1 − r

R

)γ−1,

where

α = r1

[r1(1 − r1)

r22 − r2

1

− 1

], γ = (1 − r1)

[r1(1 − r1)

r22 − r2

1

− 1

]

and β =∫ 1

0rα−1(1 − r)γ−1 dr

Table 2 The morphological parameters for the five studied insects

Insect f (Hz) Φ◦ S (mm2) R (mm) c (mm) r1 r2 m (mg) Iy (mg cm2)

Hawkmoth 26.3 60.5 947.8 51.9 18.3 0.440 0.525 1648 2080

Cranefly 45.5 61.5 30.2 12.7 2.38 0.554 0.601 11.4 0.95

Bumblebee 155 58.0 54.9 13.2 4.02 0.490 0.550 175 21.3

Dragonfly 157 54.5 36.9 11.4 3.19 0.481 0.543 68.4 7.0

Hoverfly 160 45.0 20.5 9.3 2.20 0.516 0.570 27.3 1.84

123

2408 H. E. Taha et al.

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