[MUSIC] So, we are now trying to develop models in an intermediate range of reduced velocities where the dynamics of the solid is coupled to a flow that is not that fast. We have defined a new framework, the assumption of pseudo-static aeroelasticity. Under this approximation, can we say anything in general on the effect of the fluid loading on the dynamics of the solid? Again let us use the simplest frameworks for the dynamics of the solid. That of a single mode approximation. In a single mode approximation the position of interface depends upon a single parameter, q. So considering that interface is fixed at the position defined by q, and with a velocity q dot. We have in the fluid domain a steady state problem for the boundary condition dependent on q and q dot. The pressure and velocity in the fluid will depend on these values, q and q dot, and so will the fluid loading projected on the mode F sub FS. We may state that the fluid loading on the solid at a given time, will depend on the position and velocity of the interface at the same time. [MUSIC] Here is the dimensionless form of the force exerted by the fluid on the solid projected on the mode. I have used this form several times. You see that the force is actually proportional to the Cauchy number CY, as noted before. So I write the force as the Cauchy number times the function of the other parameters, such as the Reynolds number and my interface displacement q, and my velocity q dot. For small motion, D much smaller than 1, we may now expand the fluid loading as a function of the small parameter D. The first term is the permanent loading corresponding to the reference position of q equals 0. The next term is D times the fluctuation of the fluid loading force, small f sub FS. This is the flow-induced force resulting from the motion of the interface. It contains a term proportional to the modal displacement q. This is the flow-induced stiffness force that we already obtained, but it also contains a term proportional to the velocity q-dot. This is a new flow-induced damping force. From the point of view of the dynamics of the solid the coupling to the flow is equivalent to the coupling with the spring plus a damper and the characteristics of this spring and damper depend on the fluid mechanics in two aspects. First, they increase in magnitude through the Cauchy number which is proportional to the square of the fluid velocity. Then the spring and damper coefficients depend on the sensitivity of the fluid loading to the deformation and velocity of the interface. These are the dF/dq and dq dot terms. What is the consequence of such a fluid induced damping on the dynamics of our solid? Let us go back to the dynamics of our mode which is governed by the demensionless oscillator equation that we have been using. Now, we have a model for the fluid force that applies on it. Let us simply incorporate this force in the dimension-less form of the oscillator equation. If we are interested in the dynamics of the solid, we should now change the reference time and use the dimensionless time t bar based on t solid. The equation keeps the same form, but the original oscillator part is in generic form, and you may note that the coefficient of the damping term has changed, because the time differentiation has changed. Here is the equation again. As the fluid force contains a stiffness term and a damping term, we may just incorporate them in the classical oscillator equation. So, in the presence of flow, we have a new oscillator with the damping and the stiffness that vary with the flow velocity. We know the effect of the fluid stiffness that we discussed before. So I shall not comment on this. What about the damping term? First of all there is no reason that the damping cf should be positive, or negative. If it is positive, then the flow is going to damp the free motion of the solid. Conversely, if it is negative any perturbation will be exponentially amplified in time. This is a dynamic instability again. We already found dynamic instabilities in the previous chapter for fast flows, but there is a major difference of mechanism here. In the instability by coincidence and non-symmetry coupling, the first effect of the flow is to bring the frequencies of two modes together, and then a combined mode with negative damping appeared. But the dynamic insatiblity we just obtained, the mode is unstable itself as soon as the fluid force acts. Of course in practice this instability will set on only when the total damping of the mode will reach zero and becomes negative. So if there is an initial mechanical damping C nought, then the total damping will decrease under the effect of flow and reach zero at the critical value of the reduced velocity UR critical. This value maybe predicted as a function of the initial damping c nought, of the mass number M, and of the sensibility of the fluid loading to a velocity at the interface. Let us summarize. We found that if the velocity of the solid at the interface cannot be neglected, there is a possibility that the vibration of the solid is negatively damped and the negative damping induced by the flow would then increase with the flow velocity. The dynamic instability caused by this mechanism seems much more dangerous than the one you had before. Why? Because it does not require a coincidence of two modes and actually does not even require that the solid has a kinematic that is based on a two mode approximation. Any simple single mode motion may be unstable. Next we shall explore how this might happen on a system that we have been considering several times, the oscillating airfoil in a steady flow [MUSIC]