SUMMARY The longitudinal dynamic flight stability of a hovering bumblebee was studied using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis for solving the equations of motion. For the longitudinal disturbed motion, three natural modes were identified:one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. The unstable oscillatory mode consists of pitching and horizontal moving oscillations with negligible vertical motion. The period of the oscillations is 0.32 s (approx. 50 times the wingbeat period of the bumblebee). The oscillations double in amplitude in 0.1 s; coupling of nose-up pitching with forward horizontal motion (and nose-down pitching with backward horizontal motion) in this mode causes the instability. The stable fast subsidence mode consists of monotonic pitching and horizontal motions, which decay to half of the starting values in 0.024 s. The stable slow subsidence mode is mainly a monotonic descending (or ascending) motion, which decays to half of its starting value in 0.37 s. Due to the unstable oscillatory mode, the hovering flight of the bumblebee is dynamically unstable. However, the instability might not be a great problem to a bumblebee that tries to stay hovering: the time for the initial disturbances to double (0.1 s) is more than 15 times the wingbeat period (6.4 ms), and the bumblebee has plenty of time to adjust its wing motion before the disturbances grow large.

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the "rigid body" assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the "rigid body" assumption is tested and how differences in size and wing kinematics influence the applicability of the "rigid body" assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the "rigid body" assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the "rigid body" assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode.

Dragonflies are capable of long-time hovering, fast forward flight and quick manoeuvres. Scientists have always been fascinated by their flight. Kinematic data such as stroke amplitude, inclination of the stroke-planes, wing beat frequency and phase-relation between the fore-and hindwings were measured by taking high-speed pictures of dragonflies in free-flight (e.g. Norberg, 1975;Wakeling and Ellington, 1997b) and tethered dragonflies (e.g. Alexander, 1984). Using these data in quasi-steady analyses (not including the interaction effects between forewing and hindwing), it was shown that the lift coefficient required for flight was much greater than the steady-state values measured from dragonfly wings (Wakeling and Ellington, 1997a). This suggested that unsteady wing motion and/or flow interaction between the fore-and hindwings must play important roles in the flight of dragonflies (Norberg, 1975;Wakeling and Ellington, 1997c).Force measurement on a tethered dragonfly was conducted by Somps and Luttges (1985). It was shown that over some part of a stroke cycle, lift force was many times larger than that measured from dragonfly wings under steady-state conditions. This clearly showed that the effect of unsteady flow and/or wing interaction was important. Flow visualization studies on flapping model dragonfly wings were conducted by Luttges (1988, 1989), and it was shown that constructive or destructive wing/flow interactions might occur, depending on the kinematic parameters of the flapping motion. In these studies, only the total force of the fore-and hindwings was measured and, moreover, force measurements and flow visualizations were conducted in separated works.In order to further understand the dragonfly aerodynamics, it was desirable to determine the aerodynamic force and flow structure simultaneously and also to know the force on the individual forewing and hindwing during their flapping motions. Freymuth (1990) conducted force measurement and flow visualization on an airfoil in hover modes. One of the hover modes was for hovering dragonflies. Only mean vertical force was measured. It was shown that large mean vertical force coefficient could be obtained and the force was When the midstroke angles of attack in the downstroke and the upstroke are set to 52°and 8°, respectively (these values are close to those observed), the mean vertical force equals the insect weight, and the mean thrust is approximately zero. There are two large vertical force peaks in one flapping cycle. One is in the first half of the cycle, which is mainly due to the hindwings in their downstroke; the other is in the second half of the cycle, which is mainly due to the forewings in their downstroke. Hovering with a large stroke plane angle (52°), the dragonfly uses drag as a major source for its weightsupporting force (approximately 65% of the total vertical force is contributed by the drag and 35% by the lift of the wings).The vertical force coefficient of a wing is twice as large as the quasi-steady value. The interaction between t...

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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, t d , 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 t d 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.

It has been shown that conventional aerodynamic theory, which was based on steady flow conditions, cannot explain the generation of large lift by the wings of small insects (for reviews, see Ellington, 1984a;Spedding, 1992). In the past few years, much progress has been made in revealing the unsteady high-lift mechanisms of flapping insect wings. Dickinson and Götz (1993) measured the aerodynamic forces of an airfoil started rapidly at high angles of attack in the Reynolds number (Re) range of the fruit fly wing (Re=75-225; for a flapping wing, Re is based on the mean chord length and the mean translation velocity at radius of the second moment of wing area). They showed that lift was enhanced by the presence of a dynamic stall vortex, or leading edge vortex (LEV). After the initial start, lift coefficient (CL) of approximately 2 was maintained within 2-3 chord lengths of travel. Afterwards, CL started to decrease due to the shedding of the LEV. But the decrease was not rapid, possibly because the shedding of the LEV was slow at such low Re; and The unsteady aerodynamic forces of a model fruit fly wing in flapping motion were investigated by numerically solving the Navier-Stokes equations. The flapping motion consisted of translation and rotation [the translation velocity (ut) varied according to the simple harmonic function (SHF), and the rotation was confined to a short period around stroke reversal]. First, it was shown that for a wing of given geometry with ut varying as the SHF, the aerodynamic force coefficients depended only on five non-dimensional parameters, i.e. Reynolds number (Re), stroke amplitude (Φ), mid-stroke angle of attack (αm), non-dimensional duration of wing rotation (∆τr) and rotation timing [the mean translation velocity at radius of the second moment of wing area (U), the mean chord length (c) and c/U were used as reference velocity, length and time, respectively]. Next, the force coefficients were investigated for a case in which typical values of these parameters were used (Re=200; Φ=150°; αm=40°; ∆τr was 20% of wingbeat period; rotation was symmetrical). Finally, the effects of varying these parameters on the force coefficients were investigated.In the Re range considered (20-1800), when Re was above ~100, the lift (CL) and drag (CD) coefficients were large and varied only slightly with Re (in agreement with results previously published for revolving wings); the large force coefficients were mainly due to the delayed stall mechanism. However, when Re was below ~100, CL decreased and CD increased greatly. At such low Re, similar to the case of higher Re, the leading edge vortex existed and attached to the wing in the translatory phase of a half-stroke; but it was very weak and its vorticity rather diffused, resulting in the small CL and large CD. Comparison of the calculated results with available hovering flight data in eight species (Re ranging from 13 to 1500) showed that when Re was above ~100, lift equal to insect weight could be produced but when Re was lower than ~100, additiona...

SUMMARYThe time courses of wing and body kinematics of three freely hovering droneflies (Eristalis tenax) were measured using 3D highspeed video, and the morphological parameters of the wings and body of the insects were also measured. The measured wing kinematics was used in a Navier-Stokes solver to compute the aerodynamic forces and moments acting on the insects. The time courses of the geometrical angle of attack and the deviation angle of the wing are considerably different from that of fruit flies recently measured using the same approach. The angle of attack is approximately constant in the mid portions of a half-stroke (a downstroke or upstroke) and varies rapidly during the stroke reversal. The deviation angle is relatively small and is higher at the beginning and the end of a half-stroke and lower at the middle of the half-stroke, giving a shallow U-shaped wing-tip trajectory. For all three insects considered, the computed vertical force is approximately equal to the insect weight (the difference is less than 6% of the weight) and the computed horizontal force and pitching moment about the center of mass of the insect are approximately zero. The computed results satisfying the equilibrium flight conditions, especially the moment balance condition, validate the computation model. The lift principle is mainly used to produce the weight-supporting vertical force, unlike the fruit flies who use both lift and drag principles to generate the vertical force; the vertical force is mainly due to the delayed stall mechanism. The magnitude of the inertia power is larger than that of the aerodynamic power, and the largest possible effect of elastic storage amounts to a reduction of flight power by around 40%, much larger than in the case of the fruit fly. Supplementary material available online at

SUMMARY The aerodynamics and forewing-hindwing interaction of a model dragonfly in forward flight are studied, using the method of numerically solving the Navier-Stokes equations. Available morphological and stroke-kinematic parameters of dragonfly (Aeshna juncea) are used for the model dragonfly. Six advance ratios (J; ranging from 0 to 0.75) and, at each J, four forewing-hindwing phase angle differences(γd; 180°, 90°, 60° and 0°) are considered. The mean vertical force and thrust are made to balance the weight and body-drag, respectively, by adjusting the angles of attack of the wings, so that the flight could better approximate the real flight. At hovering and low J (J=0, 0.15), the model dragonfly uses separated flows or leading-edge vortices (LEV) on both the fore- and hindwing downstrokes; at medium J (J=0.30, 0.45), it uses the LEV on the forewing downstroke and attached flow on the hindwing downstroke; at high J (J=0.6, 0.75), it uses attached flows on both fore- and hindwing downstrokes. (The upstrokes are very lightly loaded and, in general, the flows are attached.) At a given J, at γd=180°, there are two vertical force peaks in a cycle, one in the first half of the cycle, produced mainly by the hindwing downstroke, and the other in the second half of the cycle, produced mainly by the forewing downstroke; atγ d=90°, 60° and 0°, the two force peaks merge into one peak. The vertical force is close to the resultant aerodynamic force[because the thrust (or body-drag) is much smaller than vertical force (or the weight)]. 55-65% of the vertical force is contributed by the drag of the wings. The forewing-hindwing interaction is detrimental to the vertical force (and resultant force) generation. At hovering, the interaction reduces the mean vertical force (and resultant force) by 8-15%, compared with that without interaction; as J increases, the reduction generally decreases (e.g. at J=0.6 and γd=90°, it becomes 1.6%). A possible reason for the detrimental interaction is as follows: each of the wings produces a mean vertical force coefficient close to half that needed for weight support, and a downward flow is generated in producing the vertical force; thus, in general, a wing moves in the downwash-velocity field induced by the other wing, reducing its aerodynamic forces.