Dynamic flight stability of a hovering model insect ... - Springer Link

Acta Mech Sin (2010) 26:175–190 DOI 10.1007/s10409-009-0303-1

RESEARCH PAPER

Dynamic flight stability of a hovering model insect: lateral motion Yanlai Zhang · Mao Sun

Received: 18 May 2009 / Revised: 5 August 2009 / Accepted: 5 August 2009 / Published online: 16 October 2009 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009

Abstract The lateral dynamic flight stability of a hovering model insect (dronefly) was studied using the method of computational fluid dynamics to compute the stability derivatives and the techniques of eigenvalue and eigenvector analysis for solving the equations of motion. The main results are as following. (i) Three natural modes of motion were identified: one unstable slow divergence mode (mode 1), one stable slow oscillatory mode (mode 2), and one stable fast subsidence mode (mode 3). Modes 1 and 2 mainly consist of a rotation about the horizontal longitudinal axis (x-axis) and a side translation; mode 3 mainly consists of a rotation about the x-axis and a rotation about the vertical axis. (ii) Approximate analytical expressions of the eigenvalues are derived, which give physical insight into the genesis of the natural modes of motion. (iii) For the unstable divergence mode, td , the time for initial disturbances to double, is about 9 times the wingbeat period (the longitudinal motion of the model insect was shown to be also unstable and td of the longitudinal unstable mode is about 14 times the wingbeat period). Thus, although the flight is not dynamically stable, the instability does not grow very fast and the insect has enough time to control its wing motion to suppress the disturbances. Keywords Insect · Dynamic flight stability · Hovering · Lateral motion · Natural modes of motion

The project was supported by the National Natural Science Foundation of China (10732030) and the 111 Project (B07009). Y. Zhang · M. Sun (B) Ministry-of-Education Key Laboratory of Fluid Mechanics, Institute of Fluid Mechanics, Beihang University, 100191 Beijing, China e-mail: [email protected]

1 Introduction In recent years, with the current understanding of the aerodynamic force mechanisms of insect flapping wings, researchers are beginning to devote more effort to understanding the area of stability and control of insect flight [1–4]. Taylor and Thomas studied dynamic flight stability in the desert locust Schistocerca gregaria at forward flight [2]. In their analysis, it was assumed that the wingbeat frequency was much higher than the natural oscillatory modes of the insect, thus when analyzing its flight dynamics, the insect could be treated as a rigid flying body with only 6◦ of freedom (the time variations of the wing forces and moments over the wingbeat cycle were assumed to average out; the effects of the flapping wings on the flight system were represented by the wingbeat-cycle-average aerodynamic forces and moments that could vary with time over the time scale of the flying rigid body). This assumption was termed rigid body assumption. It was further assumed that the animal’s motion consists of small disturbances from the equilibrium condition; as a result, the linear theory of aircraft flight dynamics was applicable to the analysis of insect flight dynamics. They showed that the longitudinal disturbed motion consisted of three natural modes of motion: one stable subsidence mode, one unstable divergence mode and one stable oscillatory mode. Sun and colleagues studied the dynamic flight stability of five model insects (bumblebee, hoverfly, dronefly, cranefly and hawkmoth) at hovering flight, also employing the rigid body approximation and the small disturbance theory [3,5]. They showed that the longitudinal disturbed motion of the hovering insects consisted of an unstable slow oscillatory mode, a stable fast subsidence mode and a stable slow subsidence mode.

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The disturbed flight has both longitudinal and lateral motions. As a result of the linearization, the longitudinal and lateral disturbance motions can be treated separately. As a first step, only the longitudinal motion was studied in the above stability analysis [2,3,5]. However, the stability of the flight can be determined only when both the longitudinal and lateral stability properties have been studied. Therefore, it is of great interest to study the dynamic flight stability properties of the lateral motion. In the present study, we conduct a quantitative analysis on the dynamic flight stability of the lateral motion of a hovering model insect. Morphological and certain kinematical data of droneflies are used for the model insect. We choose the data of droneflies for two reasons. One is that the longitudinal stability of a model dronefly has been studied previously by our group. The other is that for the analysis of lateral motion, moments of inertia about axes other than the pitching axis are needed and they can be computed using our recently measured data of droneflies. First, we use the method of computational fluid dynamics (CFD) to compute the flows and obtain the stability derivatives; then, we use the technique of eigenvalue and eigenvector analysis to study the dynamic stability.

2 Methods 2.1 Equations of motion As done by Taylor and Thomas and Sun and colleagues [2,3,5], we make the rigid body approximation: the insect is treated as a rigid body of 6◦ of freedom and the action of the flapping wings is represented by the wingbeat-cycleaverage forces and moments. Thus the equations of motion of the insect are the same of that of an airplane or helicopter. The equations of motion are linearized by approximating the body’s motion as a series of small disturbance from a steady, symmetric reference flight condition. As a result of the linearization, the longitudinal and lateral small disturbance equations are de-coupled and can be solved separately [6]. Let oxyz be a non-inertial coordinate system fixed to the body. The origin o is at the center of mass of the insect and axes are aligned so that at equilibrium, the x- and y-axis are horizontal, x-axis points forward, and y-axis points to the right of the insect (Fig. 1). The variables that define the lateral motion (Fig. 1) are the component of velocity along y-axis (denoted as v), the angular-velocity around the x-axis (denoted as p), the angular-velocity around the z-axis (denoted as r ), and the angle between the y-axis and the horizontal (denoted as γ ). The linearized equations of lateral motion are [6] mδ v˙ = δY + gδγ ,

(1)

I x δ p˙ − I x z δr˙ = δL ,

(2)

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Fig. 1 Definition of the state variables v, p, r and γ and sketches of the reference frames. The model dronefly is shown during a perturbation (v, p, r and γ are zero at equilibrium)

Iz δr˙ − I x z δ p˙ = δ N ,

(3)

δ γ˙ = δp,

(4)

where m is the mass of the insect (body and wings); g is the gravitational acceleration; I x and Iz are the moments of inertia about x- and z-axes, respectively, and I x z is a product of inertia; “·” represents differentiation with respect to time (t); the symbol δ denotes a small disturbance quantity; Y is the y-component of the mean (wingbeat-cycle-average) aerodynamic force, and L and N are mean aerodynamic moments around x- and z-axes, respectively (the mean inertial forces and moments, including the mean gyroscopic forces and moments, of the wings, are negligible [5]). The next step of the linearization process is to express the aerodynamic forces and moments as analytical functions of the disturbance motion variables. They are represented as [6] δY = Yv δv + Y p δp + Yr δr,

(5)

δL = L v δv + L p δp + L r δr,

(6)

δ N = Nv δv + N p δp + Nr δr,

(7)

where Yv , Y p , Yr , L v , L p , L r , Nv , N p and Nr are the lateral stability derivatives. In so doing, the effects of the whole body motion on the aerodynamic forces and moments are assumed to be quasi-steady (i.e. terms that include δ v, ˙ δ p, ˙ etc. are neglected). The whole body motion is assumed to be slow enough for its unsteady effects to be negligible. With δY , δL and δ N expressed as in Eqs. 5–7, Eqs. 1–4 can be written in the following matrix form ⎡

⎤ ⎡ ⎤ δ v˙ δv ⎢ δ p˙ ⎥ ⎢ δp ⎥ ⎢ ⎥ = A ⎢ ⎥, ⎣ δr˙ ⎦ ⎣ δr ⎦ δ γ˙ δγ where A is the system matrix

(8)

Dynamic flight stability of a hovering model insect: lateral motion

⎡

Yv ⎢ m ⎢ ⎢ ⎢ I z L v + I x z Nv ⎢ ⎢ A = ⎢ I x Iz − I x2z ⎢ ⎢ I x z L v + I x Nv ⎢ ⎢ I I − I2 x z ⎣ xz 0

Yp m

Yr m

Iz L p + I x z N p

Iz L r + I x z Nr

I x Iz − I x2z Ix z L p + Ix N p

I x Iz − I x2z I x z L r + I x Nr

I x Iz − I x2z

I x Iz − I x2z

1

0

177

⎤ g⎥ ⎥ ⎥ ⎥ 0⎥ ⎥ ⎥. ⎥ ⎥ 0⎥ ⎥ ⎦ 0

(9) In order to specify A , stability derivatives (Yv , Y p , etc.) and morphological data, m, I x , Iz and I x z , etc., are needed. 2.2 Determination of the stability derivatives The stability derivatives are determined using the same CFD method as that used by Sun and Xiong [3] and Sun et al. [5] in the study of longitudinal motions of several model insects. 2.2.1 The wing, the flapping motion and the flow solution method To obtain the aerodynamic forces and moments, in principle we need to compute the flows around the wings and the body. But near hovering, the aerodynamic forces and moments of the body are negligibly small compared to those of the wings because the velocity of the body is very small. Therefore, we only need to compute the flows around the wings. Under hover flight conditions, Sun and Yu [7] and Yu and Sun [8] showed that the left and right wings had negligible interaction except during “clap and fling” motion, and Aono et al. [9] and Yu and Sun [8] showed that interaction between wing and body was negligibly small: the aerodynamic force in the case with the body/wing interaction is less than 2% different from that without body/wing interaction. In the present study of small disturbance motion, the body motion is not far from that of hovering, and it is reasonable to assume that the above results on wing/wing and wing/body interactions are applicable. Thus we can assume that the contralateral wings do not interact aerodynamically and neither do the body and the wings. As a result, in the present flow computation, the body is neglected and the flows around the left and right wings are computed separately. The wing platform is the same as that of a dronefly (Fig. 2 [10]). The wing section is assumed to be a flat plate with rounded leading and trailing edges, the thickness of which is 3% of the mean chord length of the wing. On the basis of experimental observation [10,11], the flapping motion of the wing of a dronefly is modeled as follows. The motion mainly consists of two parts: the translation (azimuthal rotation) and the rotation (see Fig. 3). The time variation of the positional angle, representing the translation, is approximated by the simple harmonic function φ = φ¯ + 0.5 cos(2π nt),

(10)

where n is the wingbeat frequency, φ¯ the mean stroke angle and the stroke amplitude. The angle of attack of the wing (α) takes a constant value during the downstroke or upstroke translation (the constant value is denoted by αd for the downstroke translation and αu for the upstroke translation); around stroke reversal, the wing rotates and α changes with time, also according to the simple harmonic function. The function representing the time variation of α during the supination at the mth cycle is tr sin[2π(t − t1 )/tr ] , α = αd + a (t − t1 ) − 2π t1 ≤ t ≤ t1 + tr , (11a) where tr is the time duration of wing rotation during the stroke reversal and a is a constant a = (180◦ − αu − αd )/tr ,

(11b)

t1 is the time when the wing-rotation starts t1 = mT − 0.5T − tr /2.

(11c)

The expression of the pronation can be written in the same way. From Eqs. 10 and 11, we see that to prescribe the flapping motion, parameters , n, tr , αd and αu , and φ¯ must be given. The computational method used to solve the Navier– Stokes equations are the same as those described by Sun and Tang [12]. The algorithm was based on the method of artificial compressibility. The computational grid has dimensions 93 × 109 × 78 in the normal direction, around the wing section and in the spanwise direction, respectively, the normal grid spacing at the wall was 0.0015 (portions of the grids are shown in Fig. 2). The outer boundary was set at 20 chord lengths from the wing. The non-dimensional time step was 0.02 (non-dimensionalized by c/U where c is the mean chord of the wing of the insect and U is the mean flapping speed of the wing, see below for the definition of U ). A detailed study of the numerical variables such as grid size, domain size, time step, etc., was conducted and it was shown that the above values for the numerical variables were appropriate for the calculations (see Wu and Sun [13]). Boundary conditions are as follows. For far-field boundary condition, at inflow boundary, the velocity components are specified as free-streams conditions (determined by flight speed), whereas pressure is extrapolated from the interior; at the outflow boundary, pressure is set equal to the free-stream static pressure and the velocity is extrapolated from the interior. On the wing surfaces, impermeable wall and non-slip conditions are applied and the pressure is obtained through the normal component of the momentum equation written in the moving grid system. Once the flow equations are numerically solved, the fluid velocity components and pressure at discretized grid points for each time step are available and the aerodynamic

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Fig. 2 Portions of the grid for the dronefly wing [10]. a In the wing planform plane; b A sectional plane

Fig. 3 Definitions of the angles of the flapping wing that determine the wing orientation. (X Y Z ), coordinates in a system with its origin at the wing root and with Y -axis points to the side of the insect and X –Y plane coincides with the stroke plane

force acting on the wing can be calculated. Projecting resultant aerodynamic force of the wings into the y-axis, we obtain the lateral force (Y ). The moments about x- and z-axis due to the aerodynamic force on the wings (L and N , respectively) can be computed using the forces on the wings. 2.2.2 Stability derivatives Conditions in the equilibrium flight are taken as the reference conditions in the calculation of the stability derivatives. In order to estimate the stability derivatives (Yv , L v , etc.), similar to the case of longitudinal motion [3], we make three consecutive flow computations for the wings: a v-series in which v is varied whilst p, r and γ are fixed at the reference values (i.e. p, r and γ are zero); and similarly, a p-series and a r -series. Using the computed data, curves representing the variation of the aerodynamic forces and moments with each of the v, p and r variables are fitted. The partial derivatives are then estimated by taking the local tangent (at equilibrium) of the fitted curves. 2.3 Flight data Morphological and wing motion parameters for the model dronefly are taken from or estimated based on the data mea-

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sured in our lab [10]. Some of the data are: body mass (m bd ) is 87.76 mg; the mass of one wing (m wg ) is 0.56 mg; wing length (R) is 11.2 mm; mean chord length of wing (c) is 2.98 mm; radius of second moment of wing area (r2 ) is 0.55R; the radial distance between the wing-root to the center of mass of wing (r1,m ) is 0.35R; area of one wing (S) is 33.34 mm2 ; body length (lb ) is 14.11 mm; distance between two wing roots is 0.33lb ; distance from wing base axis to center of mass (l1 ) is 0.13lb ; free body angle (χ0 ) is 52◦ (note that l1 and χ0 determine the relative position of the center of mass and the wing roots [14], which is needed in the calculation of the aerodynamic moment about the center of mass); = 107.1◦ ; n = 164 Hz; tr is 30% of wingbeat cycle; the stroke plane angle is assumed to be horizontal; body angle (χ ) is 42◦ ; the Reynolds number of a wing (Re), which is needed in the non-dimensionalized Navier–Stokes equations, is 781.6 (Re is defined as Re = cU/ν, where ν is the kinematic viscosity of the air and U the mean flapping velocity, defined as U = 2nr2 ). The moments and products of inertia are computed as follows. First, we calculate the contribution of the body, using the method given by Ellington [14]. We use pictures showing the lateral and dorsolventral views of the wingless body [10]. A longitudinal body axis is drawn on the lateral view approximately cutting the body into dorsal and ventral halves (Fig. 4). The cross section of the body is taken as ellipse and a uniform density is assumed for the body. With these assumptions, the center of mass can be determined. Let (ox y z ) be a coordinate system fixed to the body; o is at the center of mass, the x -axis is along the longitudinal body axis; y -axis is to the right side of the body (Fig. 4b). The moments and products of inertia with respect to the (x y z ) axes can be calculated as ⎡

I

body

⎤ Ib,x 0 −Ib,x z ⎦ = ⎣0 Ib,y 0 Ib,z −Ib,x z 0 ⎡

⎤ 146.0 0 −27.99 ⎦ mg · mm2 , = ⎣0 1296 0 −27.99 0 1257

(12)


179

Fig. 4 a Pictures showing the lateral and dorsolventral views of the body and the corresponding outlines of the body; b The body coordinate systems used

where I body is the matrix of moments and products of inertia of body with respect to (x y z ) axes and

Ib,x = (y 2 + z 2 )dm, (13a) body

Ib,y =

(x 2 + z 2 )dm,

(13b)

(x 2 + y 2 )dm,

(13c)

body

I

b,z

= body

I

b,x z

=

x z dm.

(13d)

body

Note that a rotation about y -axis by angle χ carries the axes of the (o, x , y , z ) frame to that of the (o, x, y, z) frame (see Fig. 4b). Thus the matrix of moments and products of inertia with respect to the (o, x, y, z) frame, denoted as I body can be obtained as ⎡

I

body

⎤ 0 −Ib,x z Ib,x ⎦ Ib,y 0 = ⎣0 −Ib,x z 0 Ib,z ⎤ ⎤ ⎡ ⎡ cos χ 0 sin χ cos χ 0 − sin χ ⎦ I body ⎣ 0 ⎦ 1 0 1 0 = ⎣0 − sin χ 0 cos χ sin χ 0 cos χ ⎡ ⎤ 615.7 0 549.7 ⎦ mg · mm2 . 1296 0 = ⎣0 (14) 549.7 0 787.5

Next, we consider the contribution of the wings. In previous studies on longitudinal motion [2,3], only the moment of inertia about the pitch axis (I y ) appeared in the equation of motion; the contribution of the wings to I y was neglected because it was shown that it is much smaller than that of the body. In the present study on the lateral motion, moments of inertia about the x- and z-axes appear in the equations of motion. Although the wing mass is much smaller than that of the body, the distance of the mass elements of the wing to the x- and z-axes is generally much larger than that of mass elements of the body to these axes and the wings’ contribution might need to be considered. The wings’ contribution to the moments and products of inertia is complex and we

model it as follows. Each wing is represented as a line mass with a linear mass distribution ρ(r ) = b + kr ,

(15)

where r denotes the radial position along the line mass; ρ(r ), the mass density; b and k are two constants, so chosen that the total mass of the line mass is equal to that of the wing (m wg = 0.56 mg) and the radial position of the center of mass of the line mass equal to that of the wing (r1,m = 0.35R). With this representation, the moments and product of the wings with respect to the (ox yz) frame can be easily computed, since the line mass is in the stroke plane which is parallel to the x–y plane. Because the wing (or the line mass) moves in the stroke plane, its moments and products of inertia vary with time. We take their wingbeat-cycle-average values to represent the wings’ contribution. The computation gives ⎡ ⎤ Iw,x 0 −Iw,x z ⎦ Iw,y 0 I wing = ⎣ 0 Iw,z −Iw,x z 0 ⎡ ⎤ 42.0 0 0.3 = ⎣0 12.5 0 ⎦ mg · mm2 , (16) 0.3 0 47.1 where I wing denotes the matrix of moments and products of inertia of the wings with respect to the (oxyz) frame. The corresponding result with respect to the (x y z ) frame are ⎡ ⎤ Iw,x 0 −Iw,x z ⎦ I wing = ⎣ 0 Iw,y 0 Iw,z −Iw,x z 0 ⎤ ⎡ 43.9 0 −2.5 ⎦ mg · mm2 . = ⎣0 12.5 0 (17) −2.5 0 45.1 Comparing Iw,y in Eq. 16 with Ib,y in Eq. 14 shows that wings’ contribution to the pitch moment of inertia is negligibly small. But it should be noted that Iw,x (43.9 mg· mm2 , Eq. 17) is about 30% of Ib,x (146 mg· mm2 , Eq. 12). That is, the wings have a noticeable contribution to the moment of inertia about the long body-axis of the insect. From Eqs. 14 and 16, we obtain the moments and products of inertia of the body and wings with respect to (oxyz) frame, required in the lateral equations of motion (Eq. 8), as

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I x (= Ib,x + Iw,x ) = 657.7 mg · mm2 , I y (= Ib,y + Iw,y ) = 1308.5 mg · mm2 , Iz (= Ib,z + Iw,z ) = 834.6 mg · mm2 , I x z (= Ib,x z + Iw,x z ) = −550.0 mg · mm2 . For reference, the moments and products of inertia of the body and wings with respect to (x y z ) frame are also computed, from Eqs. 12 and 17, as I x = 189.9 mg · mm2 , Iz = 1302.1 mg · mm2 ,

3 Results and discussion

I y = 1308.5 mg · mm2 , I x z = −30.5 mg · mm2 .

3.1 The equilibrium flight and the stability derivatives

Now, all the needed morphological parameters are prescribed. From Eqs. 10 and 11 and the kinematic data listed above, we see that all the needed wing-kinematic parameters have been ¯ These three parameters are diffigiven, except αd , αu and φ. cult to be measured accurately and they will be determined using the force and moment balance conditions of equilibrium flight (see below). 2.4 Method of analysis After the stability derivatives are computed and the mass and moments and products of inertia are prescribed, A is determined, and Eq. 8 can be used to yield insights into the stability properties of the lateral motion of the hovering dronefly. Similar to the case of longitudinal motion [3,5], the technique of eigenvalue and eigenvector analysis is used to study the stability properties. A summary of the technique of eigenvalue and eigenvector analysis can be found in [3,6]. 2.5 Non-dimensional form of the equations of motion Let c, U and tw be the reference length, velocity and time, respectively (tw is the period of the wingbeat cycle, tw = 1/n). The non-dimensional form of Eq. 8 is ⎡ +⎤ ⎡ +⎤ δ v˙ δv ⎢ δ p˙ + ⎥ ⎢ δp + ⎥ ⎢ ⎥ ⎢ ⎥ (18) ⎣ δr˙ + ⎦ = A ⎣ δr + ⎦ , δ γ˙ + δγ + where ⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ A =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Y p+

Yv+ m+ + + Iz+ L + v + I x z Nv

I x+ Iz+

−

0

−

+ + Iz+ L + p + Ix z N p

I x+2 z

+ I x+z L + v + I x Nv

I x+ Iz+

Yr+ m+

m+ I x+ Iz+

−

+ + I x+z L + p + Ix N p

I x+2 z

I x+ Iz+ 1

−

Iz+ L r+ + I x+z Nr+

I x+2 z

I x+ Iz+ − I x+2 z I x+z L r+ + I x+ Nr+

I x+2 z

I x+ Iz+ − I x+2 z 0

⎤ g+ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎦ 0

(19) where m + = m/0.5ρU St tw (St is the area of two wings), I x+ = I x /0.5ρU 2 St ctw2 , Iz+ = Iz /0.5ρU 2 St ctw2 ,

123

I x+z = I x z /0.5ρU 2 St ctw2 and g + = gtw /U ; δv + = δv/U , δp + = δptw and δr + = δr tw ; Y + = Y/0.5ρU 2 St , L + = L/0.5ρU 2 St c and N + = N /0.5ρU 2 St c; t + = t/tw . Using the flight data given above, m + , I x+ , Iz+ , I x+z and g + are computed as m + = 92.7, I x+ = 5.055, Iz+ = 12.69, I x+z = −8.36 and g + = 0.0158(ρ is 1.25 kg/m3 and g is 9.81 m/s2 ).

In the calculation of the stability derivatives, equilibrium flight is taken as the reference condition. Therefore, we need to calculate the aerodynamic forces and moments of the equilibrium flight. Moreover, as mentioned above, values of αd , αu and φ¯ at equilibrium flight are not provided by measured data, but need to be determined by calculation using the force and moment balance requirements of the equilibrium flight. The calculation of the equilibrium flight proceeds as follows [3]. A set of values for αd , αu and φ¯ is guessed; the flow equations are solved and the corresponding mean vertical force (Z e ), mean horizontal force (X e ) and mean pitching moment (Me ) are calculated. If −Z e is not equal to the weight, or X e not equal to zero, or Me not equal to zero, αd , αu and φ¯ are adjusted; the calculation is repeated until the non-dimensional magnitudes of difference between −Z e and the weight, between X e and 0 and between Me and 0 are less than 0.01. The calculated results show that when αd = 31.8◦ , αu = 31.8◦ , and φ¯ = 2.8◦ , the equilibrium requirements are approximately met (note that experimental observations showed that α and φ¯ of a free-flying dronefly were about 35◦ and 0, respectively [10,11], and these values were used in the initial guess; by doing so, less iterations were needed; six iterations were used to obtain the above results). The calculated angles of attack at equilibrium compare favorably with those measured by Liu and Sun and Ellington based on the high-speed motion picture: Liu and Sun [10] measured αd and αu to be about 33◦ ; Ellington [11], about 35◦ . After the equilibrium flight conditions have been determined, aerodynamic forces and moments on the wings for each of v, p and r varying independently from the equilibrium value are computed. The corresponding Y + , L + and N + are obtained. In Fig. 5, the v-series, p-series and r -series data, respectively, are plotted. The variations of Y + , L + and N + with v + , p + and r + , respectively, have good linearity in a range of −0.15 ≤ v + , p + and r + ≤ 0.15, showing that for small disturbance motion, linearization of the aerodynamic forces and moments is justified. The aerodynamic derivatives, Yv+ , L + v , etc., estimated using the data in Fig. 5, are shown in Table 1. For the derivatives with respect to v + (side translation motion), Yv+ is negative and L + v positive, and their magnitudes are much larger


181 Table 2 Eigenvalues of the system matrix Mode 1 λ1

Mode 2 λ2,3

Mode 3 λ4

0.079

−0.089 ± 0.057i

−0.510

negative and its magnitude much larger than those of Y p+ and N p+ , showing that this rotation mainly produces a large damping moment. For the derivatives with respect to r + (rotation about z-axis), Nr+ is negative and its magnitude much larger than those of Yr+ and L r+ ; again, this rotation mainly produces a large damping moment. How these derivatives are produced will be discussed in a latter section. 3.2 The eigenvalues and eigenvectors With the aerodynamic derivatives computed, elements in the system matrix A (Eq. 19) are now known ⎡ ⎤ −0.0094 −0.0011 0 0.0158 ⎢ 0.1741 ⎥ −0.2496 0.2930 0 ⎥. A =⎢ (20) ⎣ −0.1117 0.1551 ⎦ −0.3500 0 0 1 0 0 Solving Eq. 18 using the techniques of eigenvalue and eigenvecter analysis yields insights into the lateral stability of the insect. The eigenvalues of A , the system matrix of Eq. 18, and the corresponding eigenvectors are computed. The results of the eigenvalues are given in Table 2. There are one positive real eigenvalue (λ1 ), one pair of complex eigenvalues with a negative real part (λ2,3 ) and one negative real eigenvalue (λ4 ). They represent an unstable divergence mode (mode 1), an stable oscillatory mode (mode 2) and a stable subsidence mode (mode 3), respectively. The period (T ) for the oscillatory mode and the time for disturbances to increase or decrease to half the initial values (tdouble or thalf ) for the three modes are computed using the following formulas [6] T = 2π/ω, ˆ Fig. 5 The a v-series; b p-series and c r -series force and moment data

Table 1 Non-dimensional stability derivatives Yv+

L+ v

Nv+

Y p+

L+ p

N p+

Yr+ L r+

−0.876 0.806 0.038 −0.104 −1.198 −0.119 0

Nr+

0.003 −1.991

than that of Nv+ , showing that the side translation mainly produces a side force opposing the motion and a moment acting to tilt the insect in the direction of the motion. For the derivatives with respect to p + (rotation about x-axis), L + p is

(21)

tdouble or thalf = 0.693/|λ| (for real λ),

(22a)

tdouble or thalf = 0.693/|n| ˆ

(22b)

(for complex λ),

where nˆ and ωˆ are the real and imaginary parts of a complex eigenvalue. The results are given in Table 3. Hereafter, we call modes 1, 2, and 3 unstable slow divergence mode, stable slow oscillatory mode and stable fast subsidence mode, respectively. The eigenvectors, expressed in polar form, are given in Table 4 (since the actual magnitude of an eigenvector is arbitrary and only its direction is unique, we have scaled them to make δγ = 1). For a natural mode, the eigenvector determines the magnitudes and phases of the disturbance

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Y. Zhang, M. Sun

Table 3 Non-dimensional time constants of the natural modes Mode 1 tdouble

Mode 2

8.8

Mode 3 thalf

T

thalf

110.7

7.8

1.36

from that of δγ by about 150◦ , showing that in most part of the body motion cycle, δv + and δγ + have opposite sign. Thus, in this mode, the insect conducts slow horizontal side motion with its body (and stroke plane) rolled to the opposite direction to that of the side motion. As discussed below (see the Discussion section), rolling left (or right) while moving to opposite direction has a stabilizing effect.

Table 4 Eigenvectors of the system matrix Mode 1

Mode 2

Mode 3

δv +

0.179 (0◦ )

0.163 (−144.7◦ )

0.033 (180◦ )

δp +

0.079

(0◦ )

δr +

0.018 (180◦ )

δγ

1

(0◦ )

0.106

(147.5◦ )

0.072 (74.6◦ ) 1

(0◦ )

0.510 (180◦ ) 0.472 (0◦ ) 1 (0◦ )

quantities relative to each other. These properties can be clearly displayed by the eigenvector in polar form. 3.2.1 The unstable slow divergence mode (mode 1) For the unstable slow divergence mode, tdouble is 8.8 (Table 3). Note that the reference time used in the non-dimensionalization of the equations of motion is the period of wingbeat cycle. Thus disturbances double the starting values in about 9 wingbeats (the wingbeat period is tw = 1/n = 6.1 ms); this mode is called unstable slow divergence mode because tdouble is relatively large. As seen in Table 4, the unstable slow divergence mode is a motion in which δv + , δp + and δγ are the main variables (δr + is relatively small). The phase angle of δv + is the same as that of δγ and δp + , showing that δv + and δγ + have same sign. Thus, in this mode the insect conducts slow horizontal side motion with its body (and stroke plane) rolled to the same direction as that of the horizontal motion. As discussed below (see the Discussion section), moving left (or right) while rolling to the same direction has a destabilizing effect. 3.2.2 The stable slow oscillatory mode (mode 2) For the stable slow oscillatory mode, T = 110.7 and thalf = 7.8 (Table 3). The period of the oscillation is about 111 times the wingbeat period and the amplitude of the oscillation decreases to half the starting value in about 8 wingbeats. This mode is called stable slow oscillatory mode because the period is very long. Since T is much larger than thalf , the disturbance motion would die out long before one complete oscillation cycle finishes. As a result, this mode behaves much like a monotonic subsidence mode. As seen in Table 4, similar to mode 1, the stable slow oscillatory mode is a motion in which δr + , δp + and δγ are the main variables, but the phase angle of δv + is different

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3.2.3 The stable fast subsidence mode (mode 3) For this mode, thalf is 1.36 (Table 3); it takes about one and a half wingbeats for the disturbance to decrease to half of its initial value. As seen in Table 4, in this mode, δp + , δγ and δr + are the main variables (δv + is smaller by one order of magnitude). The magnitudes of δp + and δr + are the approximately the same, whilst the phase angle of δr + is different from that of δp + by 180◦ , i.e. δp + and δr + have apposite sign. Since the angle between x-axis and the long axis of the body is around 45◦ (χ = 42◦ ), the resultant angular velocity of δp + and δr + is almost along the long axis of the body (see Fig. 4b), i.e., in this mode, the insect rotates about the long axis of its body. As discussed below (see the Discussion section), the large damping moments produced by δp + and δr + act to stabilize the motion.

3.3 Physical interpretation of the natural modes of motion The eigenvalue and eigenvector analysis of the lateral disturbance motion has identified an unstable slow divergence mode, a stable slow oscillatory mode and a stable fast subsidence mode. The lateral disturbance motion is a linear combination of these modes of motion. In order to have physical insight into the genesis of the natural modes of motion, it is desirable to have approximate analytical expressions for the eigenvalues. Here, we first obtain the expressions by simplifying the equations of motion on the basis of the known modal characteristics, and then discuss the genesis of the natural modes motion.

3.3.1 Approximate expressions of the eigen values For mode 1 (unstable slow divergence mode) and mode 2 (stable slow oscillatory mode), δr + is relatively small (Table 4); and we simplify the equations of motion (Eq. 18) by neglecting the yaw moment equation (the third equation in Eq. 18) and putting δr + to zero in the other three equations. This results in the following simplified equation


⎡

⎤ Y p+ Yv+ + g ⎥ ⎢ + δ v˙ + m+ ⎢m ⎥ ⎢ ⎥ + + + + ⎣ δ p˙ + ⎦ = ⎢ I + L + + I + N + Iz L p + I x z N p ⎥ z v x z v ⎢ ⎥ 0 + + +2 δ γ˙ ⎣ I x+ Iz+ − I x+2 ⎦ I x Iz − I x z z 0 1 0 ⎡ +⎤ δv × ⎣ δp + ⎦ , (23) δγ ⎡

⎤

the characteristic equation of which is a cubic equation and the exact expression of its three roots can be obtained, providing the expressions for λ1 and λ2,3 . It should be noted that for mode 1, δv + , δp + and δγ + are at least 4 times as large as δr + , but for mode 2, δv + and δp + are only about 2 times as large as δr + . Thus the above simplification might only give a crude approximation to mode 2 (see Appendix A). For mode 3 (stable fast subsidence mode), δv + is negligibly small (Table 4), we can neglect the “Y -force equation” (the first equation in Eq. 18) and set δv + in other equations zero. The equations of motion (Eq. 18) are simplified to the following ⎡ + + ⎤ Iz L p + I x+z N p+ Iz+ L r+ + I x+z Nr+ 0⎥ ⎡ +⎤ ⎢ + + + + +2 ⎢ I x Iz − I x+2 ⎥ I I − I δ p˙ z x z x z ⎢ ⎥ ⎥ ⎣ δr˙ + ⎦ = ⎢ I + L + + I + N + I + L + + I + N + ⎢ xz p ⎥ x p xz r x r ⎢ ⎥ 0 δ γ˙ + + +2 ⎣ I x+ Iz+ − I x+2 ⎦ I I − I z x z xz 1 0 0 ⎡ +⎤ δp × ⎣ δr + ⎦ . (24) δγ Since δγ does not appear in the first and second equations of Eq. 24, the third equation is de-coupled from the first two, and we have ⎡ + + ⎤ Iz L p + I x+z N p+ Iz+ L r+ + I x+z Nr+ + ⎢ + + ⎥ +2 I x+ Iz+ − I x+2 δ p˙ ⎢ I x Iz − I x z ⎥ δp z , =⎢ + + ⎥ δr˙ + ⎣ I x z L p + I x+ N p+ I x+z L r+ + I x+ Nr+ ⎦ δr I x+ Iz+ − I x+2 z

I x+ Iz+ − I x+2 z

(25)

the characteristic equation of which is a quadratic equation and the exact expressions of its two roots can be easily obtained; one of them provides the expression for λ4 . The roots of the characteristic equations of Eq. 23 (mode 1 and mode 2) and Eq. 25 (mode 3) are obtained in Appendix A. As discussed there, Appendix A, the exact expressions of the roots, especially these of the cubic equation, are rather complex and it is difficult to see how a parameter influence the roots. Relatively simple expressions of the roots are obtained by neglecting the small terms (see Appendix A)

L+ L + g+ p , (26) λ1 ≈ 3 + v +2 + + + I x − I x z /Iz 3(I x+ − I x+2 z /I z )

183

ˆ λ2,3 = nˆ ± iω, where −1 nˆ ≈ 2

3

(27a)

+ L+ vg

+ I x+ − I x+2 z /I z

+

L+ p + 3(I x+ − I x+2 z /I z )

√ + 33 L+ vg ωˆ ≈ , + 2 I x+ − I x+2 z /I z λ4 ≈

1 2

L+ p + + + + I x − I x+2 Iz z /I z

,

(27b)

(27c) Nr+ + − I x+2 z /I x

+

+ I x+z L + p Nr

. I x+ Iz+ − I x+2 z (28)

As discussed in Appendix A, the above approximate expressions of λ1 , λ4 , and the real part of λ2,3 represent the major contributions to the roots. 3.3.2 Physical interpretations of the modes With the approximate expressions for the eigenvalues, we can give physical interpretation of the natural modes of motion. First, we examine the unstable slow divergence mode. The expression of λ1 (Eq. 26) shows that L v , g and L p play major role in this mode. L v and g are both positive, hence √ 3 L v g is positive, giving a destabilizing contribution (note + that (I x+ − I x+2 z /I z ) is positive); L p is negative, giving a stabilizing contribution. The physical reason of the desta√ bilizing effects of 3 L v g is explained as following. L v is the “roll moment” produced by side motion velocity, and g represents the weight or vertical aerodynamic force. As described above, in this mode, the insect conducts horizontal side motion with its body tilted to the same direction as that of the side motion. Suppose that the insect moves and tilts to right, the side motion (v > 0) will produce a moment L v v(> 0) which will enhance the roll and at the same time, the roll will tilt the vertical force (equal to mg) to the right, which will enhance the side motion, producing the destabilizing effects (the case of translating and rolling to the left can be similarly explained). The physical reason of L p ’s stabilizing effect is clear: L p gives a damping moment. From the above discussion, we see that because of the roll moment produced by the horizontal side motion and the side force produced by the roll rotation, the motion is unstable, and because of the roll damping moment, the instability becomes relatively weak (i.e. the divergence is slow). Next, we examine stable slow oscillatory mode. The expression of the real part of λ2,3 (Eq. 27b) is similar to that of λ1 , except that the term involving L v and g is negative (multiplied by−1/2). Thus, in this mode, L v and g, as well as L p , give stabilizing contribution. Unlike mode 1, in this mode the insect conducts horizontal side motion with its body rolled to the opposite direction of the side motion.

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Thus the moment produced by the side velocity is in the opposite direction of the roll and the side force produced by the roll is in the opposite direction of the side motion, stabilizing the motion. Finally, we look at the stable fast subsidence mode. δp and δr are the main variables in this mode. As seen from the expression of λ4 (Eq. 28), L p and Nr are both negative, hence each term in the expression of λ4 produces stabilizing effect (note that I x z is negative). The physical reason is clear: L p and Nr give damping moments. Now, let us explain why this mode is relatively fast. As above mentioned, in this mode the insect rotates about its long axis of body (δp and δr approximately same magnitude but opposite sign, and x-axis is about 45◦ from the long body axis or the x -axis, hence the resultant angular velocity is about the x -axis). The moment of inertia about the long axis of body is relatively small (as shown above, I x is almost by one order of magnitude smaller than I y and Iz ), whilst the damping moments, representing by + L+ p and Nr , are relatively large. This results in large decelerations in δp + and δr + , causing δp + and δr + to die out very fast. 3.4 How the aerodynamic derivatives are produced The stability properties are determined by the aerodynamic derivatives (together with the mass and the moments and products of inertia of the insect). As seen above, the magnitude of L v , Yv , L p and Nr are relatively large and play a major role in the disturbance motion, but other derivatives are relatively small. Let us examine how the derivatives are produced. 3.4.1 Yv , L v and Nv First, we consider the derivatives with respect to v. As see in Table 1, the magnitudes of L v and Yv are relatively large and Nv is almost zero. Figure 6 shows the time courses of the coefficients of lift (CL ), drag (CD ) and the y-axis component of the drag (CY ) of each wing in one cycle, when the insect has a horizontal side motion (moving right: v + = 0.15, p + = r + = 0), and their counterparts at reference flight (hovering) are included for comparison. The drag of a wing is the component of the aerodynamic force on the wing that is parallel to the translational velocity of the wing and its direction is opposite to that of the translational velocity, hence, it is in general positive. For convenience, we define a non-dimensional time, tˆ, such that tˆ = 0 at the start of the downstroke and tˆ = 1 at the end of subsequent upstroke. As seen in Fig. 6, at hovering (reference flight), CL and CD of one wing are the same as that of the other wing (in Fig. 6, the dashdot-curves for the left and right wings at reference flight overlap); but when the insect moves right, the lift and drag of the left wing are larger than that of the right wing in the latter part of a half-

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Fig. 6 Time courses of the coefficients of lift (CL ), drag (CD ), and the y-component of drag (CY ) of wing in one cycle at v + = 0.15( p + = r + = 0) and their counterparts at reference flight

stroke, i.e. when tˆ ≈ 0.25–0.5 and tˆ ≈ 0.75–1 (in the rest part of the halfstroke, the differences in C L and CD between the left and right wing are relatively small). The difference in lift between the left and right wing will produce a positive moment about the x-axis (similarly, a negative moment about the x-axis will be produced by the lift when the insect moving left). This gives the relatively large, positive, L v derivative (note that the drag of the wings is in the stroke plane and has relatively small distance to the x-axis, hence its contribution to the moment about the x-axis is relatively small).


The drag of a wing is parallel to the x–y plane and its y-axis component, CY , contributes to the side force (note that the lift, being perpendicular to the stroke plane, hence perpendicular to the x–y plane, does not give any side force). At reference flight, CY of one wing has same magnitude as and opposite sign to that of the other wing, resulting in a zero side force (Fig. 6c). When moving right, the magnitude of CY of the left wing is larger than that of the right wing in the latter part of a halfstroke (Fig. 6c; tˆ ≈ 0.25–0.5, tˆ ≈ 0.75–1) because, as mentioned above, the drag of the left wing is larger that of the right wing there, and CY of the left wing is negative but that of the right wing is positive (Fig. 6c), resulting in a negative net side-force (similarly, when the insect moving left, v < 0, a positive net side-force will be produced by the drag). This gives the relatively large, negative, Yv derivative. The drag of each wing in the downstroke is approximately the same as that in the upstroke (Fig. 6b), but they have opposite direction. The moment about the z-axis produced by the drag of each wing in its downstroke is canceled by that in the upstroke. As a result, the drag will produce little wingbeatcycle-average moment about the z-axis, and the lift, being parallel to the z-axis, will not produce any moment about the z-axis either, resulting in the almost zero Nv derivative. Now, the question is, why CL and CD of the left wing are larger than that of the right wing when the insect moves right (and why CL and CD of the right wing are larger than that of the left wing when the insect moves left). This is explained as follows. Figure 7 gives sectional vorticity plots of the wings for the above case of the insect moving to the right, showing the leading-edge-vortex (LEV) of the wings. The LEV of the left wing (the left columns in Fig. 7) is more concentrated than that of the right wing (the right columns in Fig. 7). When the insect moves to the right, the left wing sees a relative air velocity that is approximately directed from wing-root to wing-tip, but the right wing sees a relative velocity that is approximately directed from wing-tip to wing-root. It is believed that the velocity form wing-root to wing-tip had made the LEV of the left wing more concentrated, increasing its aerodynamic force (the aerodynamic force is generally normal to the wing surface, hence both lift and drag are increased), and the velocity from wing-tip to wing-root had made the LEV of the right wing less concentrated, decreasing its aerodynamic force. (The case of the insect moving to the left can be similarly explained.) 3.4.2 Y p , L p and N p Next, we examine the derivatives with respect to p. The magnitude of L p is large and that of Y p and N p are very small (Table 1). Figure 8 shows the time courses of CL , CD and CY of each wing in one cycle, when the insect rotates about the x-axis (rolling to the right: p + = 0.15, v + = r + = 0), and

185

Fig. 7 Vorticity plots at wing sections 0.6R–0.9R from wing root during the downstroke. Left column, results of the left wing; right column, results of the right wing. The insect is moving to the right at v + = 0.15. Solid and broken lines indicate positive and negative vorticity, respectively. The magnitude of the non-dimensional vorticity at the outer contour is 2 and the contour interval is 3. a tˆ = 0.25, b tˆ = 0.30, c tˆ = 0.35

their counterparts at reference flight are included for comparison. As seen in Fig. 8, in the mid-portion of a halfstroke (tˆ ≈ 0.1–0.4 and tˆ = 0.6–0.9), the lift and drag of the right wing is larger than that of the left wing. The difference in lift between the left and right wings will produce a negative moment about the x-axis (similarly, rolling to the left will produce a positive moments about the x-axis), giving the relatively large, negative, L p derivative (as mentioned above, the drag’s contribution to the moment about the x-axis is small). Similar to the above case of v-derivatives, the drag of each wing in the downstroke is approximately the same as that in the upstroke (Fig. 8b) and can produce little wingbeat-cycleaverage moment about the z-axis, and as mentioned above, the lift could not produce any moment about the z-axis, resulting in the very small N p derivative. As mentioned above, the difference in drag between the left and right wings occur in the mid-portion of a halfstroke. In this period, a wing is approximately perpendicular to the

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Y. Zhang, M. Sun

Fig. 7 continued Fig. 7 continued

longitudinal symmetrical plane and the drag is approximately parallel to the x-axis. Thus the difference in drag between the left and right wings can produce little side force and as mentioned above, the lift does not contribute to the side force, resulting in the very small Y p derivative. The physical reason for the lift and drag of the right wing being larger than that of the left wing (when rolling to the right) is clear: during the rolling, the outer part of the right wing moves downwards and sees a upward “incoming flows”, and the effective angle of attack is increased; whilst for the left wing , the opposite is true. 3.4.3 Yr , L r and Nr Finally, we examine the derivatives with respect to r . As seen in Table 1, the magnitude of Nr is large and that of Yr and L r are almost zero. Figure 9 shows the time courses of CL , CD and CY of each wing in one cycle, when the insect rotates about the z-axis (yawing to the right: r + = 0.15, v + = p + = 0), and their counterparts at reference flight are included in the figure. As seen in Fig. 9, when yawing to the right, the lift and drag of left wing is larger than that of the right wing in the downstroke (tˆ = 0–0.5), but the opposite is true in the upstroke (tˆ = 0.5–1). Since the difference in

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lift between the wings in the downstoke is opposite to that in the upstroke, little wingbeat-cycle-average moment about the x-axis can be produced (this is also true when yawing to the left), resulting in the almost zero L r derivative. Similarly, the difference in drag between the wings in the downstroke is opposite to that in the unstroke and little wingbeat-cycle-average side force can be produced, resulting in the almost zero Yr derivative. In the downstroke (Fig. 9b; tˆ = 0–0.5), the drag of each wing points to the negative x-direction, but the drag of the left wing is larger than that of the right wing, producing a negative net moment about the z-axis. In the upstroke (Fig. 9b; tˆ = 0.5–1), the drag of each wing points to the positive x-direction, and the drag of the right wing is larger than that of the left wing, also producing a negative moment about the z-axis. That is, when yawing to the right, negative moment about the z-axis is produced in both the downstrokes and upstrokes (similarly, when yawing left, positive moment will be produced in both the downstrokes and upstrokes), giving the relatively large, negative, Nr derivative. The physical reason for the differences in CL and CD between the left and right wings in this case is also clear: when the insect rotates to the right about the z-axis, during the downstroke, the left wing sees a larger relative velocity, but the right wing sees a smaller one, thus the CL and CD


187

Fig. 8 Time courses of the coefficients of lift (CL ), drag (CD ), and the y-component of drag (CY ) of wing in one cycle at p + = 0.15(v + = r + = 0) and their counterparts at reference flight

Fig. 9 Time courses of the coefficients of lift (CL ), drag (CD ), and the y-component of drag (CY ) of wing in one cycle at r + = 0.15(v + = p + = 0) and their counterparts at reference flight

of the left wing are larger than that of the right wing; in the upstroke, the opposite is true.

motion has three natural modes of motion: an unstable slow oscillatory mode, a stable fast subsidence mode and a stable slow subsidence mode. The longitudinal aerodynamic derivatives of the present model dronefly have been calculated by Wu and Sun in a previous study on flight control at hovering [15]. Using these derivatives, the longitudinal natural modes of motion are computed (see Appendix B). The eigenvalues are: 0.048 ± 0.098i, −0.12 and −0.012, giving the unstable slow oscillatory mode, the stable fast subsidence mode and the stable slow subsidence mode, respectively. The corresponding time constants (growth or decay

3.5 The longitudinal modes of motion The above analysis has shown the stability property of the lateral disturbance motion. As mentioned above, the disturbed flight of the model insect includes both longitudinal and lateral motions. The longitudinal disturbance motion of several other hovering model insects has been studied by our group [3,5]. It was shown that the longitudinal disturbance

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rate) are (computed using Eq. 22): tdobule = 14.4, thalf = 5.8 and thalf = 56.5, respectively. We see that both the lateral and longitudinal disturbance motions are unstable because of the unstable slow divergence mode of the lateral motion and the unstable slow oscillatory mode of the longitudinal motion. 3.6 Some discussion on the generalization of the present results In the present study, morphological and kinematical data of hovering droneflies are used for the model insect. Here, we have some discussion on whether the results obtained for the droneflies are applicable to other insects. In the above discussion on the production of aerodynamic derivatives (see Sects. 3.1 and 3.4), it is shown that when the stroke plane is approximately horizontal and the center of mass is below the stroke plane, there are mainly 3 moment derivatives, i.e., L v , L p and Nr , and L v is positive and L p & Nr are negative (the magnitudes of other moment derivatives are relatively small). In the discussion on the physical reasons for the genesis of the natural modes of motion (see Sect. 3.3), it is shown that mode 1 (unstable slow divergence mode) and mode 2 (stable slow oscillatory mode) are mainly produced by the L v derivative and the tilting of the stroke plane, and mode 3 is mainly produced by the L p and Nr derivatives. These discussions indicate that if an insect hovers with a horizontal stroke plane and the center of mass is below the stroke plane, similar aerodynamic derivatives would be produced (i.e., the magnitudes of L v , L p and Nr are large and L v is positive and L p & Nr are negative, and the magnitudes of other moment derivatives are relatively small) and similar natural modes of motion would result. Many flies, bees and moths hover with a horizontal stroke plane and the center of mass of the insect is under the stroke plane. Thus, it is reasonable to expect that they would have similar natural modes of motion (we will test this speculation in our future work).

Y. Zhang, M. Sun

modes of motion, are derived. The expressions clearly show the sources of the unstable mode: tilting of the vertical force due to rolling about the x-axis would enhance the side translation, and the side translation would produce roll moment that enhances the rolling, resulting in the unstable mode. For the unstable divergence mode, td , the time for initial disturbances to double, is about 9 times the wingbeat period (the longitudinal motion of the model insect was shown to be also unstable and td of the longitudinal unstable mode is about 14 times the wingbeat period; the growth rate of the instability of the model dronefly is determined by the faster one of the two unstable modes). Thus, although the flight is not dynamically stable, the instability does not grow very fast and the insect has enough time to control its wing motion to suppress the disturbances. Appendix A: Derivation of approximate roots The characteristic equation of Eq. 23 is λ3 − (c1 + c2 + a1 )λ2 + (−b1 a1 − b2 a2 + a1 c1 + a1 c2 )λ − (b1 + b2 )g + = 0, where a1 = Yv+ /m + = −0.0094,

a2 = Y p+ /m + = −0.0011,

+ + +2 b1 = Iz+ L + v /(I x I z − I x z ) = 0.1797,

b2 = I x+z Nv+ /(I x+ Iz+ − I x+2 z ) = −0.0056, + + +2 c1 = Iz+ L + p /(I x I z − I x z ) = −0.2671,

c2 = I x+z N p+ /(I x+ Iz+ − I x+2 z ) = 0.0175, g + = 0.0158. Because the magnitudes of c2 and a1 are much less than that of c1 , the magnitudes of b1 a1 , b2 a2 and a1 c2 are much less than that of a1 c1 , and the magnitudes of b2 is much less than b1 , Eq. 29 can be approximated as λ3 − c1 λ2 + a1 c1 λ − b1 g + = 0.

4 Conclusion For the lateral disturbed motion of the model insect, three natural modes of motion were identified: one unstable slow divergence mode, one stable slow oscillatory mode, and one stable fast subsidence mode. The unstable slow divergence mode and stable slow oscillatory mode mainly consist of a rotation about the x-axis (horizontal, pointing forward) and a side translation (the rotation about the vertical axis is relatively small); the stable fast subsidence mode mainly consists of a rotation about the horizontal x-axis and a rotation about the vertical axis, with negligible side translation. Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural

123

(29)

(30)

Let c 3 b g + a1 c12 1 1 A=− − + , 2 6 ⎧ 3 2 ⎨ c 3 b g + a1 c12 1 1 + − B= − ⎩ 3 2 6 3 ⎫1/2 a1 c1 c1 2 ⎬ − + . ⎭ 3 3 The roots of Eq. 29 are [16] √ √ c1 3 3 λ1 = −A + B + −A − B + , 3

(31a)

(31b)

(32)


λ2,3 = nˆ ± iω, ˆ

(33a)

c √ 1 √ 1 3 3 nˆ = − −A + B + −A − B + , 2 3 √ √ 3 √ 3 3 ωˆ = −A + B − −A − B . 2

(33b) (33c)

Noting that in Eq. 31 the magnitude of b1 g + /2 is much larger than that of [a1 c12 /6 − (c1 /3)3 ], A and B can be simplified as b1 g + , A≈− 2 1/2 2 b1 g + a1 c1 c1 2 3 B≈ + . − 2 3 3

(35)

λ4 =

Using Eqs. 34 and 36, the roots given in Eqs. 32 and 33 become c1 (37) λ1 = 3 b1 g + + , 3 (38a)

where 1 nˆ = − 3 b1 g + , √2 3 3 ωˆ = b1 g + . 2

f1 = f2 = h1 = h2 =

I x+z Nr+ /(I x+ Iz+ − I x+2 z ) = 0.2924, + + + + +2 Iz L r /(I x Iz − I x z ) = 5.80 × 10−4 , + + +2 I x+z L + p /(I x I z − I x z ) = 0.1760, I x+ N p+ /(I x+ Iz+ − I x+2 z ) = −0.0209, + + + + +2 I x Nr /(I x Iz − I x z ) = −0.3497, −4 I x+z L r+ /(I x+ Iz+ − I x+2 z ) = −3.82 × 10 .

1 2

L+ p + I x+ − I x+2 z /I z

+

Nr+ + + Iz − I x+2 z /I x

+

(43a) (43b)

(43c)

+ I x+z L + p Nr

. I x+ Iz+ − I x+2 z (44)

Values of λ1 , λ2,3 and λ4 computed using Eqs. 42–44 are

λ2 − (c1 + c2 + h 1 + h 2 )λ + (c2 h 1 + c2 h 2 + c1 h 1

where

(41)

λ2,3 = nˆ ± iω, ˆ

+ L+ L+ −1 3 p vg , + nˆ ≈ + +2 + + + 2 I x − I x z /Iz 3(I x − I x+2 z /I z ) √ + 33 L+ vg ωˆ ≈ , + 2 I x+ − I x+2 z /I z λ4 ≈

(38c)

1 (c1 + h 1 ) − d1 f 1 , 2 1 or λ4 = (c1 − h 1 ) + d1 f 1 . 2

The second choice of λ4 gives a value that is very different from the real value and is discarded. Substituting the definitions of a1 , a2 , b1 , b2 , . . . etc. into Eqs. 37, 38 and 41, the expressions of the roots are

L+ L + g+ p , (42) λ1 ≈ 3 + v +2 + + + I x − I x z /Iz 3(I x+ − I x+2 z /I z )

(38b)

The characteristic equation of Eq. 25 is + c1 h 2 − d2 f 1 − d2 f 2 − d1 f 1 − d1 f 2 ) = 0,

The roots of which are 1 2 (c1 + h 1 ) ± (c1 + h 1 ) − 4(c1 h 1 − d1 f 1 ) 2 1 2 (c1 + h 1 ) ± (c1 − h 1 ) + 4d1 f 1 = 2 1 ≈ (c1 + h 1 ) ± d1 f 1 , 2 where (c1 − h 1 )2 is smaller than 4d1 f 1 by one order of magnitude and is neglected. Thus

is smaller by two orders of magnitude than (b1 g + /2)2 , B can be further simplified as

2 b1 g + b1 g + . (36) =− B≈ 2 2

λ2,3 = nˆ ± iω, ˆ

(40)

(34)

Further, noting that in Eq. 35, a1 c1 c1 2 3 − 3 3

d2 =

Neglecting the higher-order small terms in the coefficient of λ and in the constant-term, Eq. 39 becomes λ2 − (c1 + h 1 )λ + c1 h 1 − d1 f 1 = 0.

where

d1 =

189

(39)

λ1 = 0.053, λ2,3 = −0.071 ± 0.12i, λ4 = −0.54. Comparing this values with the “exact values” given in Table 2 shows that the approximate values of λ1 and the real part of λ2,3 are about 70 and 80% of the exact values, respectively, and the approximate value of λ4 is very close to the exact value. This shows that the above approximate expressions of λ1 , λ4 and the real part of λ2,3 represent the major contributions.

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Y. Zhang, M. Sun

Appendix B: The longitudinal natural modes of motion The variables that define the longitudinal motion (see Fig. 1 of Ref. [15], also of Ref. [3]), are the forward (u) and dorsal-ventral (w) components of velocity along the x- and z-axes, respectively, the pitching angular-velocity (q), and the pitch angle between the x-axis and the horizontal (θ ). Let X , Z and M denote the x- and z-components of the wingbeatcycle-averaged total aerodynamic force and the aerodynamic +, Z + pitching moment, respectively. Let X u+ , Z u+ & Mu+ , X w w + + + + & Mw and X q , Z q & Mq denote the non-dimensional aerodynamic derivatives with respect to u, w and q, respectively. Wu and Sun computed the derivative when studying the control problem of the longitudinal motion [15]. The values + = 0.10, are: X u+ = −1.24, Z u+ = −0.06, Mu+ = 1.78; X w + + + + Z w = −1.13, Mw = −0.12; X q = −0.23, Z q = −0.03, Mq+ = −0.20. The system matrix of the longitudinal motion [3,15] is ⎡ + + + /m + X + /m + −g + ⎤ X u /m Xw q + /m + Z + /m + 0 ⎥ ⎢ Z u+ /m + Z w q ⎥, (45) A =⎢ ⎦ ⎣ M + /I + M + /I + M + /I + 0 u y w y q y 0 0 1 0 with the above derivatives and the known values of m + , g + and I y+ , elements in A can be computed ⎡ ⎤ −0.0134 0.0011 −0.0025 −0.0158 ⎢ −0.0006 −0.0122 −0.0003 0 ⎥ ⎥ , (46) A =⎢ ⎣ 0.0895 ⎦ −0.0062 −0.0098 0 0 0 1 0 the eigenvalues of A is computed as λ1,2 = 0.048 + 0.098i,

(47a)

λ3 = −0.12,

(47b)

λ4 = −0.012,

(47c)

123

representing an unstable slow oscillatory mode, a stable fast subsidence mode and a stable slow subsidence mode, respectively.

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