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.
The effects of corrugation and wing planform （shape and aspect ratio） on the aerodynamic force production of model insect wings in sweeping （rotating after an initial start） motion at Reynolds number 200 and 3500 at angle of attack 40℃ are investigated, using the method of computational fluid dynamics. A representative wing corrugation is considered. Wing-shape and aspect ratio （AR） of ten representative insect wings are considered; they are the wings of fruit fly, cranefly, dronefly, hoverfly, ladybird, bumblebee, honeybee, lacewing （forewing）, hawkmoth and dragon- fly （forewing）, respectively （AR of these wings varies greatly, from 2.84 to 5.45）. The following facts are shown. （1） The corrugated and flat-plate wings produce approximately the same aerodynamic forces. This is because for a sweeping wing at large angle of attack, the length scale of the corrugation is much smaller than the size of the separated flow region or the size of the leading edge vortex （LEV）. （2） The variation in wing shape can have considerable effects on the aerodynamic force; but it has only minor effects on the force coefficients when the velocity at r2 （the radius of the second ：moment of wing area） is used as the reference velocity; i.e. the force coefficients are almost unaffected by the variation in wing shape. （3） The effects of AR are remarkably small： whenAR increases from 2.8 to 5.5, the force coefficients vary only slightly; flowfield results show that when AR is relatively large, the part of the LEV on the outer part of the wings sheds during the sweeping motion. As AR is increased, on one hand, the force coefficients will be increased due to the reduction of 3-dimensional flow effects; on the other hand, they will be decreased due to the shedding of part of the LEV; these two effects approximately cancel each other, resulting in only minor change of the force coefficients.
