[MUSIC] We have just seen that the approximation of quasistatic aeroelasticity where the fluid and solid respective dynamics are well separated in time scales, allows building very simple models for instabilities due to the flow for instance the torsion divergence of an airfoil, of a bridge can be reasonably modeled by this. But there are some flow induced instabilities that behave quite differently as an oscillation of growing amplitude not just an increase in displacement. And the simple model we used only led to the possibility of increasing displacement by buckling. Let us build now a more sophisticated model. [MUSIC] By looking at the movie of flutter of a wing for instance the motion seems to be on more than one mode. In other terms it is not just a shape in space times a function of time. It seems more complex with a lot of bending of the wing, but at the same time some torsion. [MUSIC] Let us therefore base our new model on a richer approximation of the kinematics of a solid using a two modes approximation for the motion. By this I mean that the displacement Ksi is the sum of the contribution of one mode q1Phi1, and as a second mode q2Phi2. This two modes kinematics may be of several kinds. For instance it may be a combination of a rigid body motions in two directions. Mode one is the translation along X, mode two along Y, and the combination gives you any position in space. We may also imagine the combination of translation and rotation as here, or an even more complex case with the two modes of bending of a beam like this. To compute the dynamics of the solid we have to solve the modal equations of motions here with the fluid loading on each mode. This is identical to what we just did on one mode except that we have two modes and because we have two modes we have different model masses and stiffness, and the same fluid loading was to be projected on the two modes. We have enriched the model of the kinematics of the solid, but in terms of time scales we keep the approximation that these motions are much slower than that of the fluid. We are still in quasistatic aeroelasticity. [MUSIC] The general framework I used previously was the following for one mode. The pressure and velocity fields in the fluid depended only on the variable q, and therefore the fluid force also only dependent on q. We could then expand the field force in terms of q and obtain a flow-induced stiffness force. Now, if we have two modes, then the steady state fluid mechanics problems depends on the instantenous values of both q1 and q2. The resulting fluid loading needs to be projected on the two modes. The two modal forces will both depend on q1 and q2 and , as in the previous case they are proportional to the Cauchy number. Let us now expand each of them in terms of the small quantity D, the amplitude of displacements, which is assumed to be small. The first term is the permanent loading. As before we are only interested in the load that depends on the motion of the solid, and more particularly the first coefficients of the expansion, these two terms. Let us define k sub ij the partial derivatives of the modal forces with respect to each modal variable. The modal equations now take this simple form keeping only the q1 and q2 dependent terms. We now have a set of two equations that are coupled by flow induced stiffness term. [MUSIC] These terms vary with the fluis velocity through the Cauchy number. Note that the diagonal terms of the matrix correspond to the previous analysis where fluid forces resulted in a modified stiffness of a mode, but what is new here is that we have off-diagonal terms and these induce a coupling between the two modes. Now, what happens for the dynamics of a system when we couple modes with such stiffness terms? In general we'll have to recompute new modes of the system. This can be done of course and we will do it, but let us try and anticipate what the result will be. Consider first the effect of the diagonal terms I just mentioned. Each of the two modes will see its frequency change when the Cauchy is varied as discussed previously. So schematically depending on the values of flow induced stiffness K11 and K22 these two frequencies may part form each other or may come closer and closer as you can see here. In that case we may have something called a coincidence of frequencies of the two modes. Another question becomes : what happens when two modes of a system come to the same frequencies and at the same time are coupled by stiffness terms? [MUSIC] There is a simple way to predict what happens for this dynamical system at coincidence. Here I moved on the left hand side of the equation the diagonal terms. As the frequencies are now equal it'll have the two modal equations identical and scaled at a frequency of 1. Now there remains only the coupling forces. For the sake of simplicity let us assume that the coupling stiffness is very small in comparison to the stiffness of the modes. All the results can now be obtained by considering first the symmetric case when the stiffness are equal, and then the antisymmetric case where the opposite. Symmetric coupling first. We look for the modes of this coupled system in the form of the real part of q1, q2 naught e to the i of omega t. By inserting this in the modal equations the frequency then needs to satisfy this condition which implies that the determinant is equal to 0. There are two solutions to this, and because epsilon is small we have simple solutions. Omega equal 1 plus or minus epsilon over 2. [MUSIC] The two mode of the systems, which I call A and B have real frequencies and the eigenvectors are combinations of the original modes one and two and they are also real. This is just weak coupling that makes the system slightly different with a slight difference of frequencies between motions along the two eigenvectors. Now the case of antisymmetric coupling. Let us do exactly the same derivation, but for antisymmetric coupling. I also look at the frequencies. I have a determinant equal to 0, but the big difference is that now the two frequencies read omega equal 1 plus or minus i epsilon over 2. The frequencies have an imaginary part. [MUSIC] So the two modes of this systems are the following with complex frequencies. Even the eigenvectors have an imaginary part as you can see here. What does this mean? Let us start with the first of the two modes. We need to go back to the values of q 1 of t and q 2 of t, which are the real part of the eigenvector times the exponential to the i times the modal frequency times t. Because both the eigenvectors and the frequency are complex quantities we obtain q1 and q2 that oscillate as cosine and sine of t and that decrease exponentially in time. This is a very different case from the previous one. If I start with an arbitrary condition along the mode, the amplitude of q1 and q2 decrease with time. In the q1 q2 plane we have a decreasing spiral. Let us call this the damped mode. Now, the second mode. We obtain q1 and q2, that also oscillate as cosine and minus sin of t, but now they increase exponential in time. This is an unstable mode in the q1 q2 plane. If we start with an initial condition on this mode we shall have a spiral growing in time. These modes correspond to what we shall call a dynamic instability. Dynamic instability as opposed to static instability because there is here an exponential growth and oscillation. In the static instability there was only exponential growth no oscillation. So, to summarize here is what we learned with the simple case of weak coupling. When the coupling is symmetric we have two neutral modes, but when it is antisymmetric we have a damped mode and most important an unstable mode. Why do we have a stable and an unstable mode when the coupling is anti symmetric? In the case of symmetric coupling there is a potential phi equal to epsilon q1 q2 such that the right hand side of the equation reads dphi over a dq1 and dphi over dq2 respectively. This is called a conservative forcing. Any cycle in the q1 q2 plane is going to produce a net zero energy transfer in the modes. Conversely for the antisymmetric coupling there is no potential. This is a nonconservative forcing with energy input or output at each cycle. [MUSIC] Let us go back to our original case. Is the coupling symmetric, or antisymmetric ? In general it is neither of them because the coefficients are CyK12 and CyK21, and remember that KIJ tells the dependence of the projected fluid force on mode i to the displacement of mode j. There is absolutely no reason that they be equal or opposite, but if the coupling is not exactly symmetric it contains a nonzero antisymmetric part and is therefore nonconservative. So, as soon as they are not equal the system is nonconservative, and there will be an unstable mode. Now what can be computed even for nonsmall coupling, and you can expect to have an unstable mode and another mode. We shall have simple example of this. [MUSIC] So, to summarize. If we have a two mode approximation of the dynamics, the modal equations are coupled by flow induced stiffness terms. There is a risk of a dynamic instability when the coupling matrix is such that, as the Cauchy numbers increased, there is a coincidental frequency, and when the coupling is not symmetric. This flow induced instability is often called coupled mode flutter. But coupling is not sufficient, we need coincidence and nonsymmetric coupling. This is an important result. The approximation of quasistatic aeroelasticity allows to predict not only static instabilities, buckling, but also dynamic instabilities. Is this applicable to the oscillating instability of a wing? This is what we shall do next. [MUSIC]