[MUSIC] Using our single mode approximation of the fluid-conveying pipe, we have obtained some models of the fluid structure interaction in different ranges of velocities. As I said before,it is a bit frustrating to have patches of models. Actually, for this particular system we can say much more. To derive the torque exerted by the flow on the pendulum, I used the Angular Momentum Theorem. In the torque, the first term was the time dependence of the total fluid angular momentum, and because I was under the approximation of a pseudo static aeroelasicity, the angular velocity was frozen in time, and so this term was equal to zero. If now the velocity is time dependent, which is the general case, then this term reads simply d/dT of M/3 theta dot, which is M/3, theta double dot. I now have the most general form of the torque exerted by the solid on the fluid, which reads M/3 theta double dot + MUR theta dot. If I include this in the equation of the fluid pendulum I recover added mass and added damping at the same time. One plus M/3, Theta double dot plus MUR Theta single dot plus Theta equals zero. And this is valid for all reduced velocities UR. Actually, I have simultaneously added mass and added damping. This means that added mass exists at all reduced velocities, not only at very small ones. And that added damping exists at all reduced velocities not only at intermediate ones. Yes, schematically, here's what happens. At all reduced velocity, I have added mass. Added Coriolis damping always exists, but becomes larger and larger as the reduced velocity is increased. Eventually, it becomes the dominant effect. This gives some deeper meaning to our patches of models. For our fluid conveying pendulum, the simple models of each range of velocity were right, but they could not give us more than the dominant effect. The full model combines all this : Effects add up. [MUSIC] The fluid solid interactions effects do not just add up like this in the most general case, but what is true that they do not replace one by the other. You remember that at low reduced velocity, all effects were independent of the reduced velocity. That in pseudo static aeroelasticity, the fluid induced damping effects that gave us galloping were proportional to the reduced velocity UR. And that in quasi static aeroelasticiy , the fluid induced stiffness effect that gave us divergence or flutter were proportional to the Cauchy number that varies like the square of the reduced velocity. So, the general feature is more like this. Depending on the range of reduced velocity, each type of effect dominates. At low reduced velocities, added mass, at intermediate velocities flow-induced damping and at higher reduced velocities, flow-induced stiffness. When we build up the classical models of added mass, fluid damping and so on we went to limit cases so that we could neglect some terms in comparison to others and focus on the dominant effect. But we can say now that the root causes of all the mechanisms we have found are always there but that depending on the range of reduced velocity, one or the other dominates. [MUSIC] Let us go back to our fluid-conveying pipe instability. We have found that the fluid conveying pendulum, the single mode approximation was enough to give us the added mass and the fluid induced Coriolis damping, but that does not explain the famous garden horse instability. In the fluid conveying pendulum, the higher the fluid velocity, the higher the damping. The effect is only stabilizing. We have to enrich the model somehow. And this is not a question of too much simplification in the description of the interaction, because we have solved the model for any flow velocity. The only direction of improvement is to have a better approximation of the dynamics of the pipe. Look at this motion here and here. This does not look like a single mode oscillating. It seems a combination of two modes. Let us go in that direction. [MUSIC]
