# Why can not two modes coincide perfectly for some cases in Flutter phenomena?

I have a question about flutter phenomena. As you know, flutter occurs when two modes (for example, first mode and second mode) coincide (that is, after a specific velocity, the frequencies of two modes will be the same and equal) and the damping term will change into positive which causes failure.

However, for some cases, frequencies of two modes do not coincide perfectly (they are not equal) and damping parameter negative sign will change into positive, which mean flutter is happening.

Why is this happening? Why can't the frequencies of two modes coincide perfectly?

• Not two modes of the same degree of freedom converge (1st and 2nd mode are harmonics, so their frequency ratio is fixed), but flutter occurs when the frequencies of two different oscillations come close. They do not need to become identical, just close enough that they build up each other's amplitude. May 30, 2014 at 10:25
• @ Peter Kämpf: Thank you for your great help. I want to know why in some cases these 2 modes come close, and in some cases they become exactly identical. May 30, 2014 at 14:40
• Wouldn't this be, perhaps, more likely to be answered over at physics.stackexchange.com? May 30, 2014 at 18:26
• @Jan: Sure, but this goes for all questions about flight mechanics, aerodynamics and dynamics. This question is in the field of aeroelastics, and appropriately enough, it carries this tag. May 30, 2014 at 20:44
• @Shellp in your second graph have you performed a mode tracking to be sure they didn't cross at the flutter point? Oct 7, 2017 at 1:16

Flutter happens when the frequency of two modes coincide. These modes must be of different nature, so their frequencies can move in different directions. Typical examples are elastic modes (with eigenfrequencies independent of speed) and aerodynamic modes (with eigenfrequencies proportional to speed). When flight speed increases, the aerodynamic mode becomes faster (think of dynamic pressure as of the stiffness of the spring in a spring-mass system) while the elastic mode stays constant. At some point both have the same frequency, but flutter will already appear when they are close enough so each builds up the amplitude of the other. This will become more efficient when the frequencies are identical, but this is academic: When flutter starts, it is a bad idea to accelerate further.

Example for an elastic mode: Wing bending. The first mode is just the tips moving up and down, the second mode is one tip up, the other down, which will rotate the fuselage, the third mode (or for pedants: The second symmetric mode) is again both tips up, but now with the midspan section moving down and so on. They are harmonics (like the oscillations of guitar strings), so their frequencies are at a fixed ratio, with the first mode having the lowest frequency. See below, from top to bottom.

Example for an aerodynamic mode: Aileron flutter, fast-period mode (especially when coupling to the bending mode in sweptback flying wings).

For me, your diagrams don't make sense. They should look more like that:

EDIT:

With Shellp's comments I realized that the answer above is too simple. The lines in the graph are for the static case of the elastic vibration. Once this wing moves through the air, it's movement will change the local aerodynamic forces, which will modify the flutter modes. The motion-induced aerodynamic forces counteract the elastic motion at slow speeds, adding damping (negative, because they reduce the motion). With damping, the frequency of the mode is reduced.

At higher speeds, delays in the buildup of pressures mean that now the aerodynamic forces lag behind the motion, and once this lag exceeds a quarter of a period, they produce positive damping (this is an unfortunate wording, so stay with me. The damping term is a negative number when it really damps the motion, but when the term becomes positive at higher speeds, it is still called damping, but now the motion is excited instead of damped in the common sense. So let's call positive damping "excitation" from now on).

This excitation now increases the flutter frequency, so in reality the two flutter phenomena now get locked together. The first graph in the question above shows the frequency going down with speed, which looks unusual to me. The second graph shows this locking-in with an increase in frequency over speed, but at very high frequencies (it must be a small structure). If you want, the excitation is pushing the elastic frequency up, so both lock together without converging completely. But that cannot last forever - at higher speeds they should diverge, but in reality the structure is overloaded way before that happens.

• Thank you Peter! My diagrams are correct[ If you still have doubt, I can send you several papers about flutter topic! ;) ]. The behavior of the V(Flight Speed)-w( Frequency) diagram, depends on which method you use to calculate the Flutter(p,k,p-k,g and...methods). But all in all, the point is that 2 frequencies become identical and damping parameter's sign will change into positive sign. May 31, 2014 at 8:01
• I have read and seen in many papers the diagrams like my second figure that I mentioned above, but none of them mentioned that why for some cases these two frequencies do not become identical and keep distance! I would be grateful if you could help me! :) May 31, 2014 at 8:09
• @Shellp: This could be due to nonlinear effects. If the amplitude is big enough, the simple assumptions in my answer do not hold, and both the elastic and the aerodynamic behavior change. See the SB-9 flutter movie (youtube.com/watch?v=RenbFgLZBNA) - normally the flutter would grow bigger and quickly destroy the plane. Nonlinearity helped to limit the flutter amplitude. Concerning damping: Positive damping means that now the oscillation will build up instead of dying down. That is another way of saying that flutter happens. May 31, 2014 at 8:43