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Turbines). The effect of the surrounding water on the vibration of a body is (almost) completely represented by these coefficients. Roughly speaking, the body submerged in water vibrates like a body in air with increased inertia. 22 In the following section the physical problem is described. The liquid is modeled by a potential flow. The hydrodynamic coefficients are expressed in terms of certain potentials. Section 3 contains a short analysis of the resulting mathematical problem, namely the Laplacian equation with von Neumann boundary conditions.
The velocity of the rigid body at some point x and time t is given by U(t) + w(t) V(x,t) where U(t) d dt R(t) x (x-R(t)), is the velocity of the origin R(t). The motion of the rigid body is governed by Newton's law: d dt P(t) F(t) d dt L(t) with N(t) P(t) Ps V(x,t)dx J G2 (t) linear momentum L(t) Ps J G2 (t) angular momentum F (t), N (t) X X V(x,t)dx total force and total torque about the origin 0 Ps constant density of the rigid body 24 For further reasoning, it is convenient to express the equations of motion in terms of the kinetic energy of the rigid body: ( 2.
18), respectively. 18) involving the velocity potential ¢ can be written in terms of the kinetic energy by using Gauss' Theorem: So, finally, we have Fl Nl = d aT 1 - dt au - w X d aT 1 - dt aw- X - w aT 1 au 1 - p 1 J nn dS aT aiT - u aG 2 X aT~ au- p 1 J n(x oG 2 X n) dS Using this representation for F 1 and N1 we obtain for the equations of motion of the rigid body: 32 with T (E identity matrix) These equations differ from the corresponding equations of motion of the rigid body in air (better in vacuum) a.