Dynamic flight stability of a hovering bumblebee

In last twenty years, much work has been done to study the aerodynamics and energetics of insect flight, and considerable progress has been made in these areas (e.g. Dudley and Ellington, 1990a,b; Dickinson and Götz, 1993; Ellington et al., 1996; Dickinson et al., 1999; Wang, 2000; Sun and Tang, 2002; Usherwood and Ellington, 2002a,b). The area of insect flight stability has received much less consideration,however. Recently, with the current understanding of the aerodynamic force mechanisms of insect flapping wings, researchers are beginning to devote more effort to understanding this area.

Thomas and Taylor (2001)and Taylor and Thomas (2002)studied static stability (an initial directional tendency to return to equilibrium after a disturbance) of gliding animals and flapping flight,respectively. They found that flapping did not have any inherently destabilizing effect: beating the wing faster simply amplified the existing stability or instability, and that flapping could even enhance stability compared to gliding flight at a given speed.

Taylor and Thomas (2003)studied dynamic flight stability in the desert locust Schistocerca gregaria, providing the first formal quantitative analysis of dynamic stability in a flying animal (the dynamic stability of a flying body deals with the oscillation of the body about its equilibrium position following a disturbance). A very important assumption in their analysis was 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 degrees of freedom (termed rigid body approximation). In the rigid body approximation, 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. In addition, the gyroscopic forces of the wings were assumed to be negligible. 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. The authors first measured the aerodynamic force and moment variations of the tethered locust by varying the wind-tunnel speed and the attitude of the insect, obtaining the aerodynamic derivatives. Then they studied the longitudinal dynamic flight stability of the insect using the techniques of eigenvalue and eigenvector analysis.

In the study of Taylor and Thomas(2003), the dynamic stability of forward flight at high flight speed (the advance ratio was around 0.9) was studied. Many insects often hover. In hovering, unlike in forward flight, the stroke plane is generally approximately horizontal, the body angel is relatively large and the wing in the downstroke and in the upstroke operates under approximately the same conditions. As a result, the aerodynamic derivatives, hence the dynamic stability properties of hovering, must be different from those of forward flight. It is of great interest to investigate the dynamic flight stability of hovering.

In the present paper, we study the longitudinal dynamic flight stability in a hovering bumblebee. The bumblebee was chosen because previous studies on bumblebees provide the most complete morphological data and wing-motion descriptions. In the study by Taylor and Thomas(2003), due to the limits of the experimental conditions, the insect had to be tethered and the reference flight might not have been in equilibrium, so some derivatives could not be measured directly. If a computational method were used to obtain the aerodynamic derivatives, the above difficulties could be solved. More importantly, the computational approach allows simulation of the inherent stability of a flapping motion in the absence of active control. This is very difficult or impossible to achieve in experiments using real insects, as was done by Taylor and Thomas(2003). In the present study,we used the method of computational fluid dynamics (CFD) to compute the flows and to obtain the aerodynamic derivatives. First, conditions for force and moment equilibrium were determined. Then, the aerodynamic derivatives at equilibrium flight were computed. Finally, the longitudinal dynamic flight stability of the hovering bumblebee was studied using the techniques of eigenvalue and eigenvector analysis.

Similar to Taylor and Thomas(2003), we make the rigid body approximation: the insect is treated as a rigid body of 6 degrees of freedom(in the present case of symmetric longitudinal motion, only three degrees of freedom) and the action of the flapping wing is represented by the wingbeat-cycle average forces and moment (in addition, the gyroscopic effects of the wing are assumed negligible). This model of the hovering bumblebee is sketched in Fig. 1A.

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 the x-axis is horizontal and points forward at equilibrium. The variables that define the motion(Fig. 1B) are the forward(u) and dorso-ventral (w) components of velocity along x- and z-axes, respectively, the pitching angular-velocity around the center of mass (q), and the pitch angle between the x-axis and the horizontal (θ). o_Ex_Ey_Ez_Eis a coordinate system fixed on the earth; the x_E-axis is horizontal and points forward.

In deriving the linearized equations (Equations 1, 2, 3, 4, 5, 6), the aerodynamic forces and moment (X, Z and M) are represented as analytical functions of the disturbed motion variables (δu, δw andδ q) and their derivatives(Etkin, 1972; Taylor and Thomas, 2003), e.g. X is represented as X=X_e+X_uδu+X_wδw+X_qδq,where the subscript e (for equilibrium) denotes the reference flight condition. In so doing, the effects of the whole body motion on the aerodynamic forces and moment are assumed to be quasisteady (terms that include δu̇,δ ẇ, etc. are not included). The whole body motion is assumed to be slow enough for its unsteady effects to be negligible.

Flight data for the bumblebee are taken from Dudley and Ellington(1990a,b). The general morphological data are as follows: m=175 mg; wing length R=13.2 mm; c=4.01 mm, r₂=0.55R;area of one wing (S) is 53 mm²; free body angle(χ₀) is 57.5°; body length (l_b) is 1.41R; distance from anterior tip of body to center of mass divided by body length (l) is 0.48l_b, distance from wing base axis to center of mass divided by body length (l_l) is 0.21 l_b; pitching moment of inertia of the body about wing-root axis (I_b) is 0.48×10-8 kg m². Assuming that the contribution of the wing mass to the pitching moment of inertia is negligible (the added-mass on the wings has been included in the CFD model), I_y, the pitching moment of the bumblebee about y-axis, can be computed as

The wing-kinematic data are as follows: Φ=116°; n=155 Hz;β=6°; χ(body angle)=46.8°. U is computed as U=4.59 m s^-1.

In determining the equilibrium conditions of the flight, we need to calculate the flows around the wings (at equilibrium, the body does not move);to obtain the aerodynamic derivatives, we need to compute the flows around the wing and around the body. In the present CFD model, it is assumed that the wings and body do not interact aerodynamically, neither do the contralateral wings, and the flows around the wings and body are computed separately. It is also assumed that the wing is inflexible. The wing planform used(Fig. 2A) is approximately the same as that of a bumblebee (Ellington,1984a). The wing section is a flat plate with rounded leading and trailing edges, the thickness of which is 3% of the mean chord length of the wing. The body of the insect is idealized as a body of revolution; the outline of the idealized body (Fig. 2B)is approximately the same as that of a bumblebee. Neglecting the axial asymmetry of the bumblebee can cause some differences in the computed body aerodynamic force. However, near hovering, the body aerodynamic force is much smaller than that of the wings (i.e. the aerodynamic force of the insect is mainly from the wings), and a small difference in the body aerodynamic force may not affect the aerodynamic derivatives greatly (see below).

In the flapping motion described above, the mid-stroke angles of attack(α_d and α_u), the flip duration(Δτ_r), the flip timing (τ_r), the period of flapping cycle (τ_c), the mean positional angle(⁠

U has been computed above. Re and τ_c are computed as Re=1326 and τ_c=7.12. On the basis of the flight data in Dudley and Ellington(1990a), the flip duration(Δτ_r) is set to 0.22τ_c and the flip rotation is assumed to be symmetrical (thus the flip timing τ_ris determined in terms of Δτ_r). α_d,α _u and

The flow equations (the Navier-Stokes equations) and the solution method used in the present study are the same as those described in Sun and Tang(2002). Once the flow equations are numerically solved, the fluid velocity components and pressure at discretized grid points for each time step are available. The aerodynamic forces and moments acting on the wing (or the body) are calculated from the pressure and the viscous stress on the wing (or the body) surface.

The lift (L_b) and drag (D_b) of the body are the vertical (z₁ direction) and horizontal(x₁ direction) components of the resultant aerodynamic force of the body, respectively. The pitching moment of the body(m_y,b) is the moment about the mass center due to the aerodynamic force of the body. The above forces and moment are non-dimensionalized by 0.5ρU²S_tand 0.5ρU²S_tc,respectively. The coefficients of L_w, T_w, m_y,w, L_b, D_b and m_y,b are denoted a C_L,w, C_T,w, C_M,w, C_L,b, C_D,b and C_M,b, respectively.

As seen above, the kinematic parameters of the wing left undetermined are the mid-stroke angles of attack (α_d, α_u) and the mean positional angle of the wing(⁠

Conditions in equilibrium flight are taken as the reference conditions in the aerodynamic derivative calculations. In order to estimate the partial derivatives X_u, X_w, X_q, Z_u, Z_w, Z_q, M_u, M_w and M_q, we make three consecutive flow computations for the wing: a u-series, in which u⁺ is varied whilst w⁺, q⁺ and θ are fixed at the reference values (i.e. w⁺, q⁺ andθ are zero), a w-series, in which w⁺ is varied whilst w⁺, q⁺ and θ are fixed at zero and a q-series, in which w⁺, q⁺ and θ are fixed at zero (in all the three series,wing kinematical parameters are fixed at the reference values); similar flow computations are conducted for the body. Using the computed data, curves representing the variation of the aerodynamic forces and moments with each of the w⁺, q⁺ and θ variables are fitted. The partial derivatives are then estimated by taking the local tangent(at equilibrium) of the fitted curves.

After the aerodynamic derivatives are determined, the elements of the system matrix A would be known. Equation 7 can be solved to yield insights into the dynamic flight stability of the hovering bumblebee.

The solution process of the present problem is summarized as follows. The eigenvalues and eigenvectors of A in Equation 7 are calculated, giving the natural modes; analyzing the motions of the natural modes gives the dynamic stability properties of the hovering bumblebee.

The code used for the flow computations is the same as that in Sun and Tang(2002) and Sun and Wu(2003). It was tested in Sun and Wu (2003) using measured unsteady aerodynamic forces on a flapping model fruitfly wing. The calculated drag coefficient agreed well with the measured value. For the lift coefficient, the computed value agreed well with the measured value, except at the beginning of a half-stroke, where the computed peak value was smaller than the measured value. The discrepancy might be because the CFD code does not resolve the complex flow at stroke reversal satisfactorily. There is also the possibility that it is due to variations in the precise kinematic patterns,especially at the stroke reversal. Wu and Sun(2004) further tested the code using the recent experimental data by Usherwood and Ellington(2002b) on a revolving model bumblebee wing. In the whole α range (from -20° to 100°), the computed lift coefficient agreed well with the measured values. The computed drag coefficient also agreed well with the measured values except when αis larger than approximately 60°.

In the above computations, the computational grid was of the O-H type and had 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. The outer boundary was set at 20 chord lengths from the wing. The time step was 0.02. 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.

In the present study for the wing, we used similar grid dimensions as used in the test calculations (Wu and Sun,2004); for the body, the grid dimensions were 71×73×96 in the normal direction, along the body axis and in the azimuthal direction,respectively (tests have been conducted to show that these grid dimensions are appropriate for the present computations).

For different set of values of α_d, α_u and

As defined above, X⁺ and Z⁺ are x- and z-components of the non-dimensional total aerodynamic force due to the wing and the body and M⁺ is the corresponding non-dimensional pitching moment. After the equilibrium flight conditions have been determined, aerodynamic forces and moments on the wing and on the body for each of u, w and q varying independently from the equilibrium value are computed. The corresponding X⁺, Z⁺ and M⁺ are obtained. In Fig. 4A-C, the u-series, w-series and q-series data, respectively,are plotted (in the figure the equilibrium value has been subtracted from each quantity). X⁺, Z⁺ and M⁺ vary approximately linearly with u⁺w⁺ and q⁺ in a range of-0.1≤Δu⁺, Δw⁺ andΔ q⁺≤0.1, showing that the linearization of the equations of motion is only justified for small disturbances. (In the computations, we found that the aerodynamic forces and moment of the body are negligibly small compared to those of the wing; this is because the relative velocity that the body sees is very small.) The aerodynamic derivatives

Let us examine the aerodynamic derivatives and discuss how they are produced. First, we consider the derivatives with respect to u⁺. As seen in Table 1,

Next, we examine the derivatives with respect to w⁺.

Finally, we examine the derivatives with respect to q⁺. As seen in Table 1,

With the aerodynamic derivatives computed, the elements in the system matrix A are now known. The eigenvalues and the corresponding eigenvectors can then be computed, and the results are shown in Tables 2 and 3.

As seen in Table 2, there are a pair of complex eigenvalues with a positive real part and two negative real eigenvalues, representing an unstable oscillatory motion (mode 1) and two stable subsidence motions (mode 2 and mode 3), respectively. The period for the oscillatory mode and the t_double or t_half for the three modes, computed using Equations 14, 15, 16, 17, are shown in Table 4. Hereafter, we call modes 1, 2 and 3 unstable oscillatory mode, fast subsidence mode and slow subsidence mode, respectively.

As mentioned in Materials and methods, the eigenvector determines the magnitudes and phases of the disturbance quantities relative to each other. These properties can be clearly displayed by expressing the eigenvector in polar form (Table 5): since the actual magnitude of an eigenvector is arbitrary, only its direction is unique,and we have scaled them to make δθ=1.

The non-dimensional period of the oscillatory mode is T=48.7 and the non-dimensional time of doubling the amplitude is T_double=15.4 (Table 4). Note that the reference time used in non-dimensionalization of the equations of motion is the period of wingbeat cycle. Thus the period of the insect oscillation is about 49 times of the wingbeat period (the wingbeat period is t_w=1/n=6.4 ms), and the starting value of the oscillation will double in 15 wingbeats.

As seen in Table 5, the unstable oscillatory mode is a motion in which δq⁺,δθ and δu⁺ are the main variables(δw⁺ is smaller than δq⁺and δu⁺ by two orders of magnitude; δθis seen to be very large, but it is the result ofδ q⁺ and the long period).δ q⁺ represents pitching motion and

For the fast subsidence mode, t_half is 3.5(Table 4); disturbances decrease to half of the starting values in about four wingbeats. As seen in Table 5, in this modeδ q⁺, θ and δu⁺ are also the main variables (δw⁺ is smaller by 3 orders of magnitude). δq⁺ and θ are out of phase(they have opposite signs); δu⁺ and θ are in phase. That is, when δθ has a positive initial value, so doesδ u⁺, but δq⁺ has a negative initial value. The insect would pitch down (back to the reference attitude) and at the same time moves forward (see the sketch in Fig. 9B). Note that this is different from the case of the unstable oscillatory mode, in which the insect pitches down while moving backwards and pitches up while moving forward (see Fig. 8B). The characteristic transients of δu⁺, δq⁺ andθ are plotted in Fig. 9A.

For this mode, t_half is 57.8(Table 4); it takes about 58 wingbeats for the disturbance to decrease to half of its initial value. Unlike the above two modes, in which δw⁺ is negligibly small, this mode is a motion in which δw⁺ is the main variable (Table 5); other variables are one order of magnitudes or more smaller. Since

The eigenvalue and eigenvector analysis of the longitudinal small disturbance equations of the bumblebee has identified one unstable oscillatory mode and two (stable) monotonic subsidence modes. It is desirable to examine the physical processes of the motions of the natural modes and interpret the motions physically.

Let us examine the first cycle of the motion. For clarity, the first half-cycle of the characteristic transients of δu⁺(⁠

At the beginning of the cycle (t⁺=0; Fig. 11), δθ is at its local minimum value, δu⁺(⁠

When δθ has increased to zero at t⁺≈10(the bumblebee has moved to the configuration shown in Fig. 12B),δ q⁺ does not reach its local maximum value, but continues to increase (see Fig. 11). This is because at this time δu⁺ is still large and so is the nose-up pitching moment(ΔM⁺). As a result, δθ would increase with time at a faster rate than when it is smaller than zero, which would cause the amplitude of δθ to become larger than that in the preceding quarter cycle. We thus see that the combination of forward motion and nose-up pitching causes the instability.

Now δθ has become positive and F₀ is tilted backwards, which would slow the forward motion. At t⁺≈18, δu⁺ decreases to zero and changes sign (the bumblebee has moved to the configuration of Fig. 12C). Then, the bumblebee moves backward. The backward motion would produce a nose-down pitching moment,reducing the nose-up pitching rate (δq⁺). At t⁺≈24, δq⁺ changes sign,δθ reaches its local maximum value (the bumblebee has moved to the configuration of Fig. 12D).

In the next half-cycle, the above process repeats in an opposite direction(Fig. 12D-A); here it is the combination of backward motion and nose-down pitching that produces the destabilizing effect.

In this mode, as seen in Table 5 and Fig. 9A, whenδθ and δu⁺ have positive initial values,δ q⁺ has a negative initial value. The positiveδθ tilts F₀ backwards, the horizontal component of which tends to reduce δu⁺; the forward motion (δu⁺) produces a nose-up pitching moment(⁠

In this mode, when the bumblebee descends initially due to some disturbance(i.e. δw⁺ has a positive initial value), the descending rate will decrease with time(Fig. 10). A positiveδ w⁺ produce a upward force(⁠

In the preceding discussion, the effects of the rate derivatives(⁠

Taylor and Thomas (2002)have discussed the constraints on the rigid body approximation in detail. In general, the rigid body approximation only works well if the wingbeat frequency is at least an order of magnitude (10 times) higher than the highest frequency of the natural modes. They reasoned, using reduced order approximations to the natural modes of motion, that this could be expected to be true in animal flight. In the present study on the disturbed longitudinal motion of the hovering bumblebee, the period of the oscillatory mode is about 50 times the wingbeat period (see Table 4), which is much more than 10 times larger than the wingbeat period.

It should be noted that in the fast subsidence mode, a disturbance quantity varies from its initial (maximum) value to half of the value in 3.5 wingbeats. Is this too short to apply the rigid body approximation? To answer this question, let us look at an oscillatory mode, in which a variable would usually vary from a peak value to half of the value in a time 16% of its period. For the rigid body approximation to be appropriate, as stated above,the period should be at least 10 times as long as the wingbeat period; then 16% of the period is at least 1.6 wingbeat period. We thus see that in the fast subsidence mode, a variable decreasing from its initial value to half of that value in 3.5 wingbeat periods is slow enough for the rigid body approximation.

The above discussion shows that application of the rigid body approximation in the present analysis is appropriate. The result here provides an example that supports the reasoning of Taylor and Thomas(2002) on the applicability of the rigid body approximation to animal flight.

The present model simulates the inherent dynamic stability of the bumblebee in the absence of active control. That is, in the disturbed motion, the model bumblebee uses the same wing kinematics as in the reference flight. A real bumblebee, if the motion is dynamically stable and the disturbances die out fast, might not make any adjustment to its wing kinematics, and could return to the equilibrium `automatically'. In this case, the disturbed motion history predicted by the model represents that of the real bumblebee. In general,however, a real bumblebee makes continuous adjustments to its wing kinematics in order to keep to the reference flight, and the disturbed motion predicted by the model would be altered at an early stage.

In the present study, some of wing kinematic parameters (n, Φ,etc.) at reference flight are taken from or determined from the experimental data of Dudley and Ellington(1990a) and Ellington(1984b), and the others(α_u, α_d and

As mentioned above, in general the disturbed motion is a linear superposition of the simple motions represented by the natural modes. For the hovering bumblebee, when disturbed from its reference flight, the disturbed motion is a linear combination of an unstable oscillatory mode and two stable subsidence modes. The growth of the disturbed motion is determined by the unsteady oscillatory mode. The unstable oscillatory mode doubles its amplitude in about 15 wingbeats (Table 4), which is about 0.1 s (the wingbeat period is 6.4 ms). To a person or a man-made machine, this growth rate is fast. But to a bumblebee,which can change its wing motions within a fraction of a wingbeat period, this growth rate might not be fast; the insect, if wishing to keep to the reference flight, has plenty of time to adjust its wing motion before the disturbances have grown large. For example, in the forward moving phase(Fig. 12A-C), the bumblebee might slightly decrease and increase the angle of attack of the wings during the downstroke and during the upstroke, respectively, from the equilibrium value of the angle of attack, and in the backward moving phase(Fig. 12D-F), the bumblebee might do the opposite. This would produce effects on the lift and the drag opposite to those produced by δu⁺, thus the destabilizing δM⁺ could be eliminated.

List of symbols

 
• A

  system matrix

 
• c

  mean chord length

 
• C_L,w

  vertical force coefficient of wing

 
• C̄ _L,w

  mean vertical force coefficient of wing

 
• C_M,w

  pitching moment coefficient of wing

 
• C̄ _M,w

  mean pitching moment coefficient of wing

 
• C_T,w

  thrust coefficient of wing

 
• C̄ _T,w

  mean thrust coefficient of wing

 
• D_b

  body drag

 
• e

  reference flight condition

 
• E

  earth

 
• F₀

  aerodynamic force of reference flight

 
• g

  the gravitational acceleration

 
• i

  imaginary number,

  \(i=\sqrt{-1}\)

 
• I_b

  pitching moment of inertia of the body about wing-root axis

 
• I_y

  pitching moment of inertia about the y-axis of insect body

 
• l

  length

 
• λ

  distance from anterior tip of body to center of mass divided by body length

 
• l_b

  body length

 
• λ_l

  distance from wing base axis to center of mass divided by body length

 
• λ_b

  body length divided by R

 
• L_b

  body lift

 
• L_w

  vertical force of wing

 
• m

  mass of the insect

 
• M

  total aerodynamic pitching moment about center of mass

 
• M⁺

  non-dimensional total aerodynamic pitching moment about center of mass

 
• \(M_{\mathrm{q}}^{+}\)

  derivative of M⁺ with respect to q⁺

 
• \(M_{\mathrm{u}}^{+}\)

  derivative of M⁺ with respect to u⁺

 
• \(M_{\mathrm{w}}^{+}\)

  derivative of M⁺ with respect to w⁺

 
• n

  stroke frequency

 
• o, o′, o₁, o_E

  origins of the frames of reference

 
• q

  pitching angular-velocity about the center of mass

 
• q⁺

  non-dimensional pitching angular-velocity about the center of mass

 
• r₂

  radius of the second moment of wing area

 
• R

  wing length

 
• Re

  Reynolds number

 
• S

  area of one wing

 
• S_t

  area of two wings

 
• t

  time

 
• t_double

  time for a divergent motion to double in amplitude

 
• t_half

  time for a divergent motion to half in amplitude

 
• t_w

  period of the wingbeat cycle

 
• t̂

  non-dimensional time (t̂=0 at the start of a downstroke and t̂=1 at the end of the subsequent upstroke)

 
• T

  period of the insect motion

 
• T_w

  horizontal force of wing

 
• u

  component of velocity along x-axis

 
• u⁺

  component of non-dimensional velocity along x-axis

 
• w

  component of velocity along z-axis

 
• w⁺

  component of non-dimensional velocity along z-axis

 
• u_t

  translational velocity of the wing

 
• u_t⁺

  non-dimensional translation velocity of the wing

 
• U

  reference velocity

 
• x, y, z

  coordinates in the body-fixed frame of reference (with origin at center of mass)

 
• x′, y′, z′

  coordinates in the frame of reference with origin at wing root and z′ perpendicular to stroke plane

 
• x₁, y₁, z₁

  coordinates in the frame of reference with origin at wing root and z₁ in vertical direction

 
• x_E, y_E, z_E

  coordinates in a system fixed on the earth

 
• ẋ _E

  x_E-component of the velocity of the mass center of the insect

 
• X

  x-component of the total aerodynamic force

 
• X⁺

  non-dimensional x-component of the total aerodynamic force

 
• \(X_{\mathrm{q}}^{+}\)

  derivative of X⁺ with respect to q⁺

 
• \(X_{\mathrm{u}}^{+}\)

  derivative of X⁺ with respect to u⁺

 
• \(X_{\mathrm{w}}^{+}\)

  derivative of X⁺ with respect to w⁺

 
• ż_E

  z_E-component of the velocity of the mass center of the insect

 
• Z

  z-component of the total aerodynamic force

 
• Z⁺

  non-dimensional z-component of the total aerodynamic force

 
• \(Z_{\mathrm{q}}^{+}\)

  derivative of Z⁺ with respect to q⁺

 
• \(Z_{\mathrm{u}}^{+}\)

  derivative of Z⁺ with respect to u⁺

 
• \(Z_{\mathrm{w}}^{+}\)

  derivative of Z⁺ with respect to w⁺

 
• α

  geometric angle of attack of wing

 
• \({\dot{{\alpha}}}\)

  angular velocity of pitching rotation

 
• \({\dot{{\alpha}}}^{+}\)

  non-dimensional angular velocity of pitching rotation

 
• \({\dot{{\alpha}}}_{0}^{+}\)

  a constant

 
• αd

  midstroke geometric angle of attack in downstroke

 
• αu

  midstroke geometric angle of attack in upstroke

 
• β

  stroke plane angle

 
• δ

  small disturbance notation (prefixed to a perturbed state variable)

 
• Δ

  increment notation

 
• θ

  pitch angle between the x-axis and the horizontal

 
• λ

  generic notation for an eigenvalue

 
• Δλr

  duration of wing rotation or flip duration (non-dimensional)

 
• ρ

  density of fluid

 
• τ

  non-dimensional time

 
• τc

  non-dimensional period of one flapping cycle

 
• τr

  non-dimensional time when pitching rotation starts

 
• ν

  kinematic viscosity

 
• φ

  azimuthal or positional angle

 
• \({\bar{{\phi}}}\)

  mean positional angle

 
• \({\dot{{\phi}}}\)

  angular velocity of azimuthal rotation

 
• \({\dot{{\phi}}}^{+}\)

  non-dimensional angular velocity of azimuthal rotation

 
• Φ

  stroke amplitude

 
• χ

  body angle

 
• χ0

  free body angle

We thank the two referees whose helpful comments and valuable suggestions greatly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (10232010, 10472008).