Why this inertial regulator of dragonfly wing pitch reduces self-excited coupled flapping and feathering vibrations and improves gliding flight speed

Question

The pterostigma of insect wings an inertial regulator of wing pitch is linked in this answer to Why do dragonflies have these special little spots on their wings? (open access copy here)

The Summary of the paper includes the following points. I've highlighted some in bold.

Question: Is it possible to explain in simpler terms what "Inertial regulator of dragonfly wing pitch reduces self-excited coupled flapping and feathering vibrations and improves gliding flight speed" means, and why these small weights do it?

1. The pterostigma of insect wings usually is a pigmented spot close to the leading edge far out on the wing, having a greater mass than an equally large wing piece in adjacent wing regions.

2. In several dragonfly (Odonata) species the position of the spanwise torsion axis of the wings, the mass distribution of the wings, and the position of the chordwise centre of mass in chordwise wing strips were determined.

3. In the dragonflies investigated, the torsion axis of the wing lies ahead of the chordwise centre of mass of the wing except at the pterostigma (Fig. 1).

4. A wing having its mass axis behind its torsion axis is very susceptible to self-excited coupled flapping and feathering vibrations, making gliding flight above a critical speed impossible. Due to unfavourable, inertial, wing pitching tendencies, a still lower speed limit is set to active flight.

5. Due to its mass contribution and favourable location, the pterostigma tends to raise these speed limits by causing favourable, inertial, pitching moments during the acceleration phases of wing flapping.

6. The favourable pitching moment of the pterostigma is proportional to the distance from the wing base to the pterostigma, and to the distance of the pterostigma ahead of the wing's spanwise torsion axis. The pterostigma usually has an optimal position at the leading edge of the wing near the wing tip, just where the wing curves backwards. Further optimization of pterostigma mass localization has been obtained in different ways in various insects, involving both pterostigma position (Fig. 4) and form (Fig. 5a).

7. The function of the pterostigma of raising the critical gliding speed, at which self-excited vibrations set in, was demonstrated in dragonflies. Although contributing only 0.1 % (one pterostigma) of the total dragonfly weight, it raised the critical speed by 10–25% in one species.

8. The pterostigma is common among the insect orders Odonata, Neuroptera, Psocoptera, Hemiptera, and Hymenoptera. By passive, inertial, pitch angle control, the pterostigma probably makes the wing beat more efficient in slow and hovering flight of small insects, while its raising of the critical flight speeds probably is of more importance to larger insects. The ability of active pitch angle control in many insects does not detract from the value of the pterostigma, since it contributes passively, without power expenditure, towards an efficient wing stroke.

Figures from the linked paper (click for full size):

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Images from the linked question, click for full size:

Answer 1

Short answer:

Adding a mass ahead of the elastic axis of the wing tends to increase the critical flutter speed of the wing, thus allowing a larger flight envelope. Above this critical speed the wing would experience vibrations of increasing amplitude that could lead to its destruction (or to a limit-cycle flutter mode, which is also bad but less so than rapid unplanned disassembly).

Flutter is a destructive aeroelastic phenomenon that can be described (in a simplified way) as a combination of two natural vibration modes of a wing. These modes are torsion around a spanwise axis and bending with respect to the root. Each mode will have a natural frequency, and the flutter speed will have a minimum (not guaranteed to be a global one, btw) when both of these frequencies are the same, because then they will reinforce one another.

The distance between the center of mass and the elastic axis of the wing is very important for the torsion response of the wing, simply because the wing will twist around this elastic axis, and a center of mass far away from it will give high inertial moments.

By adding weight to the leading edge, the dynamic behavior of the wing is modified in a way that moves the critical flutter speed upwards, extending the safe flight envelope. This technique can be seen most often in helicopter blades, many of which have some trim or balance weight at the leading edge.

Long answer (math ahead):

WIP - The analytical treatment is quite long so I am trying to decide which results are actually worth typing into MathJax