[MUSIC] Well, at this stage we have got a lot of simplification. First we have considered the cases where the velocity of the fluid is very small. And this gave us a condition on the reduced velocity being smaller than the displacement number. Second we are focused on motions of small amplitude in the sense that the displacement number is much smaller than 1. We are in a specific region of our space of parameters. And there we have a set of equations that we hope to be able to solve. The key result we obtained was that the force exerted by the fluid on the solid was made of two terms. The first one was related to the fluid motion caused by the solid and the second one was not. If this later force does not depend on the motion of the fluid, you can compute it without doing any fluid mechanics. Let us do that. So I just look now at this part of the force acting on the solid and disregard the other one. What do we have here? We have a force, f, that depends on the modal displacement q. The other ingredients are the mass number, M, the pressure, P0, the modal shape, Phi, and, of course, the geometry of interface through the local normal and the spatial integration. But we know all of this. In particular the gradient of P0 is given by the momentum balance at the order 0 that I recall here. It is related to the loading by gravity which is in the dynamic Froude number FD. I can substitute the gradient of the pressure P0 and I take the dynamic Froude number of the integral. Finally the force takes a very simple form. It is the product of the displacement, q, of a ratio of dimensionless numbers and of an integral that mixes the shape of the motion phi and the geometry of the interface through the normal, and ds, the elementary area. This is a general result and it seems simple enough. So this force is proportional to the displacement, it is a stiffness force. The coefficient between them the stiffness Kf is a constant that I can compute easily. What do I need? The dimensionless number M is the ratio of the two densities. I know it. The dynamic fluid number is given, once I know the size L, gravity g, and when I choose a time scale T solid. And the integral is just a sum over the interface of a mix of the local modal shape Phi, and the local normal. This is pure geometry. So to summarize, from the point of view of the solid, moving in a fluid with a pressure gradient is strictly equivalent to being connected to an elastic spring. An important result is that I can compute the stiffness of this spring once and for all, for a given problem. I do not have to recompute the force at each position of the solid. [MUSIC] Now let's go to a practical application. Imagine an iceberg, or even simpler, an ice cube in water. Let us consider the vertical oscillations and just small oscillations. We assume that the reduced velocity is small. I want to compute the stiffness force and see how this is going to affect the motion of my ice cube. The mass number M and the dynamic fluid number, I can keep them as they are, and then give their value later when I want to be more specific on the size of the ice cube. The modal shape Phi is here just the unit vector on the vertical axis because I'm considering vertical motion. And if L is the size of the ice cube, I can use it as my reference length. All I have to compute now is the sum of this quantity over the interface. On the sides of my ice cube the normal n to the interface is horizontal. So the product Phi n is equal to 0, the sum is 0. On the bottom of the ice cube, the normal n is opposed to Phi, and so the product Phi n is -1. I sum the local value of -1 over the whole bottom of the ice cube of dimensionless size 1, and I get simply -1 as a result. So the total integral is -1. With the three ingredients, the two dimensionless numbers and the sum of the interface I have my dimensionless stiffness force as a function of the parameters of the problem. It opposes the motion of the ice cube just as a spring. I can use that force on the equation of motion of the ice cube. Actually, it is the only force acting on the ice cube. So, the oscillator equation for the vertical motion of the ice cube is simply here, d square q over d t square equals minus the stiffness times q. I move my fluid induced stiffness force on the left hand side, and this looks now exactly like a classical oscillator with a stiffness and a mass. Now if you remember the time I'm using here is actually t bar, which I noted t for the sake of simplicity. So the true dimensionless equation actually starts with d-square q over d-t-bar-square. I can now go back to the dimensional time by substituting t bar which is t over t solid in the equation. I obtain a simple oscillator equation. Note that the T solid disappears. My solid time T was rather arbitrary, so this is fine. From this oscillator equation, I can extract the frequency of pure oscillation, omega, which depends only on the two densities, the gravity and the size of the ice cube. This is very much like the frequency of a pendulum with length and gravity. This is actually a fluid solid pendulum. You can test this equation for an ice cube of one centimeter you get a period of oscillation of about 0.2 second. Of course, you can arrive at this same result by directly using Archimedes principle, which is simple to use here, but, in the general case, our approach here is much, much simpler. How do we go from an ice cube to an iceberg? We would do exactly the same thing. The sum over the interface is a bit more complex to compute because the shape is complex. But the sum always simplifies whatever the shape and reduces to just the area where the iceberg crosses the surface of water. Of course, because of the dimensionless numbers that depend on L, there is going to be a huge difference in the frequency of oscillation between an ice cube and an iceberg. For a cubic iceberg of 100 meter, the period of free oscillation would be 20 seconds, not 0.2 seconds. [MUSIC] This was for small vertical motions. But actually, I can apply my formula to any kind of slow motion and any shape. Here is a shape of a ship hull. The mode shape, Phi, can be any kind of solid body motion. For instance, pitching, or rolling. For any of these motions, we only have to compute the sum over the interface of the quantities involving the modal shape Phi, and the local normal n. This will give me the stiffness that opposes rolling, pitching, and so on. This is pure geometry. And this can be done very easily at the design stage of a hull, by a basic computer program. Moreover, if you imagine several modes of motion, they may be cross stiffness terms between all the modes. I will not detail this, but the fundamentals are here. All you need to do to have the fluid-induced stiffness is to compute a sum over the interface of geometrical quantities. And remember, no fluid mechanics is needed. Well, actually, some is needed, which is hydrostatics, but we solved it and incorporated the solution into the formula for the fluid-stiffness. So to summarize we see that the second term of my fluid induced force is very important in practice and that it may be very easily computed. But there's also the other term which I've left aside. In the brackets you see that it depends under pressure p and the velocity small u. This is going to be a bit more complex because I will have to solve a problem of linearized fluid dynamics to have it. Here we are. We have been exploring what the interaction looks like when the reduced velocity is small and when the motions are small. Just oscillations of a boat, vibrations of an inflatable damn, or small displacement of a fish. We learned that these motions are opposed by a fluid induced stiffness force easy to compute. Next you are going to compute the other component of the force. [MUSIC]
