Dynamic flight stability of a hovering model insect:lateral motion

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.

Dynamic flight stability of hovering insects

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 157Hz）. 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 Mu （pitching moment produced by unit horizontal speed） as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Zw （vertical force produced by unit vertical speed） as the primary source of the stable slow subsidence mode.

Dynamic flight stability of a bumblebee in forward flight

The longitudinal dynamic flight stability of a bumblebee in forward flight is studied. The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are employed for solving the equations of motion. The primary findings are as the following. The forward flight of the bumblebee is not dynamically stable due to the existence of one （or two） unstable or approximately neutrally stable natural modes of motion. At hovering to medium flight speed [flight speed Ue = （0-3.5）m s^-1; advance ratio J = 0-0.44], the flight is weakly unstable or approximately neutrally stable; at high speed （Ue = 4.5 m s^-1; J = 0.57）, the flight becomes strongly unstable （initial disturbance double its value in only 3.5 wingbeats）.

Dynamic flight stability of a model dronefly in vertical flight

The dynamic flight stability of a model dronefly in hovering and upward flight is studied.The method of computational fluid dynamics is used to compute the stability derivatives and the techniques of eigenvalue and eigenvector used to solve the equations of motion.The major finding is as following.Hovering flight of the model dronefly is unstable because of the existence of an unstable longitudinal and an unstable lateral natural mode of motion.Upward flight of the insect is also unstable,and the instability increases as the upward flight speed increases.Inertial force generated by the upward flight velocity coupled with the disturbance in pitching angular velocity is responsible for the enhancement of the instability.

A Computational Study on Lateral Flight Stability of the Cranefly in Hover

The dynamic flight stability of hovering insects includes the longitudinal and lateral motion.Research results have shown that for the majority of hovering insects the same longitudinal natural modes are identified and the hovering flight in longitudinal is unstable.However,in lateral,the modal structure for hovering insects could be different and the stability property of lateral disturbance motion is not as robust as that of longitudinal motion.The cranefly possesses larger aspect ratio and lower Reynolds number,and such differences in morphology and kinematics may make the lateral dynamic stability different.In this paper,the lateral flight stability of the cranefly in hover is investigated by numerical simulation.Firstly,the stability derivatives are acquired by solving the incompressible Navier–Stokes equations.Subsequently,the dynamic stability characteristics are checked by analyzing the eigenvalues and eigenvectors of the linearized system.Computational results indicate that the lateral dynamic modal structure of cranefly is different from most other insects,consisting of three natural modes,and the weakly oscillatory mode illustrates the hovering lateral flight is nearly neutral.This neutral stability is mainly caused by the negative derivative of roll-moment vs.sideslip-velocity,which can be attributed to the weaker‘changingLEV-axial-velocity’effect.These results suggest that insects in nature may exhibit different dynamic stabilities with different morphological and kinematic parameters,which should be considered in the designs of flapping wing air vehicles.

Dynamic Flight Stability of a Model Hoverfly in Inclined-Stroke-Plane Hovering

Most hovering insects flap their wings in a horizontal plane, called ＇normal hovering＇. But some of the best hoverers, e.g. true hoverflies, hover with an inclined stroke plane. In the present paper, the longitudinal dynamic flight stability of a model hoverfly in inclined-stroke-plane hovering was studied. Computational fluid dynamics was used to compute the aerodynamic derivatives and the eigenvalue and eigenvector analysis was used to solve the equations of motion. The primary findings are as follows. （1） For inclined-stroke-plane hovering, the same three natural modes of motion as those for normal hovering were identified： one unstable oscillatory mode, one stable fast subsidence mode, and one stable slow subsidence mode. The unstable oscillatory mode and the fast subsidence mode mainly have horizontal translation and pitch rotation, and the slow subsidence mode mainly has vertical translation. （2） Because of the existence of the unstable oscillatory mode, inclined-stroke-plane hov- ering flight is not stable. （3） Although there are large differences in stroke plane and body orientations between the in- clined-stroke-plane hovering and normal hovering, the relative position between the mean center of pressure and center of mass for these two cases is not very different, resulting in similar stability derivatives, hence similar dynamic stability properties for these two types of hovering.