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This video shows a Hawker jet with the wing fluttering up and down like it's about to break. What can cause flutter like that? Can it actually cause a wing or stabilizer failure? How can flutter be prevented? What should be done if something like this happens?

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Imbalanced or loose (and, in extreme cases, structurally weak) control surfaces can cause flutter, a type of harmonic motion. Flutter is a very dangerous condition; if it is not stopped, it can cause structural failure and potentially lead to a fatal accident. V-tail Bonanzas had a series of such accidents related to flutter of the butterfly ruddervator tail.

I understand that prevention is a combination of good maintenance and keeping the aircraft below Vne (never exceed speed), or possibly other speeds in some configurations.

The appropriate reaction would be to slow down, first and foremost. Descending is probably also a wise choice.

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    $\begingroup$ Poor maintenance can definitely be a cause of aileron flutter in light aircraft (worn aileron rod end bearings that allow enough play for resonant vibrations to set up are a pretty classic example, and part of why we check the ailerons for free play during preflight). $\endgroup$ – voretaq7 Dec 22 '13 at 8:46
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Flutter is essentially an aerodynamic and structural interaction that has gone unstable. At certain speeds and flight conditions, the aero forces interact with the structural dynamics of the aircraft. This aero and structural dynamic instability can be seen in other places other than aircraft as well, the most famous that comes to mind is the one with the bridge rocking back and forth before collapsing

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