Dynamic stability in hovering flight of insects with different sizes

Previous works on the flight dynamic stability of insects have focused on relatively large insects. Here, the longitudinal flight dynamic stability of two hovering miniature insects was computed. With the stability properties of the miniature insects from the present work and those of large insects from previous works, we studied the effects of insect size on the stability properties in the full range of insect sizes. The following results were obtained. Although the insects considered have a 30 000-fold difference in mass, their modal structure of flight stability is the same: an unstable oscillatory mode, a stable fast subsidence mode, and a stable slow subsidence mode; because of the unstable mode, the flight is unstable. An approximate analytical expression on the growth rate of the unstable mode as a function of insect mass $\left(m\right)$ was derived. It shows that the time to double the initial values of disturbances $\left(t_d\right)$ is proportional to the 0.17 power of the insect mass $\left(m\right)$. That is, as $m$ becomes smaller, $t_d$ decreases (i.e., the instability becomes faster). This means that miniature insects need a faster nervous system to control the instability than larger insects. For example, the response time (represented by $t_d$) of a miniature insect, the gall midge $(m\approx 0.05\mathrm{mg})$, needs to be faster by about 7 times than that of a larger insect, the hawk moth $(m\approx 1600\mathrm{mg})$.

Figure 1

The state variables and the reference frames: (a) Side view; (b) back view.

Figure 2

The $u$-series, $w$-series, and $w$-series force and moment data for the gall midge.

Figure 3

The values of $t_d$ for the insects of various size.

Figure 4

The relationship between $t_d$ and $m$.