Relationship between wingbeat frequency and resonant frequency of the wing in insects

Abstract

In this study, we experimentally studied the relationship between wingbeat frequency and resonant frequency of 30 individuals of eight insect species from five orders: Odonata (Sympetrum flaveolum), Lepidoptera (Pieris rapae, Plusia gamma and Ochlodes), Hymenoptera (Xylocopa pubescens and Bombus rupestric), Hemiptera (Tibicen linnei) and Coleoptera (Allomyrina dichotoma). The wingbeat frequency of free-flying insects was measured using a high-speed camera while the natural frequency was determined using a laser displacement sensor along with a Bruel and Kjaer fast Fourier transform analyzer based on the base excitation method. The results showed that the wingbeat frequency was related to body mass (m) and forewing area (Af), following the proportionality f ~ m(1/2)/Af, while the natural frequency was significantly correlated with area density (f0 ~ mw/Af, mw is the wing mass). In addition, from the comparison of wingbeat frequency to natural frequency, the ratio between wingbeat frequency and natural frequency was found to be, in general, between 0.13 and 0.67 for the insects flapping at a lower wingbeat frequency (less than 100 Hz) and higher than 1.22 for the insects flapping at a higher wingbeat frequency (higher than 100 Hz). These results suggest that wingbeat frequency does not have a strong relation with resonance frequency: in other words, insects have not been evolved sufficiently to flap at their wings' structural resonant frequency. This contradicts the general conclusion of other reports--that insects flap at their wings' resonant frequency to take advantage of passive deformation to save energy.

Resonance frequencies of honeybee (Apis mellifera) wings

“…Until these alternatives are explored further, we suggest that the simplest interpretation of our data is support for the stiff element hypothesis. This conclusion is in general agreement with the conclusion of Ha et al (2013), that the fundamental frequency of the wings is significantly different from the wingbeat frequency. Ha et al (2013) found fundamental frequency (lowest mode) exceeded wingbeat frequency for all insect groups tested except Hymenopterans, while we found the lowest mode exceeded wingbeat frequency for A. mellifera (Hymenoptera).…”

“…This conclusion is in general agreement with the conclusion of Ha et al (2013), that the fundamental frequency of the wings is significantly different from the wingbeat frequency. Ha et al (2013) found fundamental frequency (lowest mode) exceeded wingbeat frequency for all insect groups tested except Hymenopterans, while we found the lowest mode exceeded wingbeat frequency for A. mellifera (Hymenoptera). Ha et al (2013) measured the first longitudinal mode for the wings of several species of insect, but on excised wings, which are stiffer than in vivo wings (Mengesha et al, 2011).…”

“…Ha et al (2013) found fundamental frequency (lowest mode) exceeded wingbeat frequency for all insect groups tested except Hymenopterans, while we found the lowest mode exceeded wingbeat frequency for A. mellifera (Hymenoptera). Ha et al (2013) measured the first longitudinal mode for the wings of several species of insect, but on excised wings, which are stiffer than in vivo wings (Mengesha et al, 2011). Our preliminary trials of excised wings suggested that once the wings had dried, resonance frequencies of this mode increased by 32% (from 340 Hz to 450 Hz), 75 min after wing removal from the animal (n=1), which is why we conducted our primary experiment on wings attached to the animal.…”

“…An alternative to modeling the resonance spectrum is to measure it empirically, as has been done for bird feathers (Clark et al, 2013) and insect wings. The lowest resonance frequency (first longitudinal mode) was investigated by Ha et al (2013) for eight species of insect using a laser displacement sensor on a single point near the wingtip. However, their method did not allow them to examine higher normal modes.…”

“…Once removed from the animal, insect wings dry (lose mass) and stiffen (Mengesha et al, 2011), both of which should increase resonance frequency relative to the in vivo condition. Ha et al (2013) measured either isolated forewings or hindwings (depending on species), but in most of the species, the forewing/hindwing pair is mechanically coupled during flight. Separating mechanically linked wings may alter the resonance frequency relative to the in vivo condition.…”