Dynamic flight stability of hovering insects

Abstract

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.

The need for higher-order averaging in the stability analysis of hovering, flapping-wing flight

“…In addition, we use the documented x h of the hovering hawkmoth ( = x R 0.22 h [15]). Then, we seek the flapping parameters α d , α u , ϕ 0 , and Δx and the operating θ eqm to ensure trim of the second-order averaged dynamics at hover; that is, to satisfy equation (13 which indicates instability of the hovering hawkmoth.…”

Controlling roll perturbations in fruit flies

“…Moreover, they fly at Reynolds numbers Re ¼ 10 2 -10 4 , in which flows are unsteady [5,6]. Most importantly, recent analytical and numerical analyses, as well as mechanical models, indicate that flapping flight is aerodynamically unstable, on a time scale of a few wing-beats [7][8][9][10][11][12][13][14][15][16][17][18][19]. It is, therefore, intriguing how insects overcome such control challenges and manage to fly with impressive stability, manoeuvrability and robustness, outmanoeuvring any man-made flying device.…”

Control Effectiveness Analysis of the hawkmoth Manduca sexta: a Multibody Dynamics Approach

“…An increase of the symmetric bias of the stroke positional angle ( 0 ) induces increases both in F x and M y (in-phase), which means that the hawkmoth pitches up (down), while going forward (backward) at the same time. This mode of motion is known to be highly unstable [5,19], hence further attitude stabilization control is necessary for using this control strategy. On the other hand, increase of the symmetric bias of the feathering angle (α 0 ) induces a decrease in F x , and an increase in M y (out-of-phase).…”

“…On the other hand, increase of the symmetric bias of the feathering angle (α 0 ) induces a decrease in F x , and an increase in M y (out-of-phase). This means that the hawkmoth pitches down (up), while going forward (backward), and this mode of motion is known to be stable [5,19]. By taking advantage of this coupling effect, it seems that the symmetric bias of the feathering angle (α 0 ) is appropriate for the attitude control to the directions of forward/backward and pitching motions.…”

“…Fig. 7 shows that three control inputs have an effect on F y and M x , which are the asymmetric amplitude of the stroke positional angle (Δ amp ), the asymmetric amplitude of the feathering angle (Δα amp ), and the asymmetric bias of the deviation angle ( 11 forward (backward), and this mode of motion is known to be stable [5,19]. By taking advan this coupling effect, it seems that the symmetric bias of the feathering angle (α 0 ) is appropriate attitude control to the directions of forward/backward and pitching motions.…”