Hydroelastic behavior of a floating ring-shaped plate

4 A.I. Andrianov & A.J. Hermansfor z < 0 in the case of infinite water depth (IWD) or −h < z < 0 for finite waterdepth (FWD), supplemented by the boundary conditions at the free surface and atthe bottom for FWD. The linearized kinematic condition in the plate and waterregions, at the surface z = 0, has the form∂Φ∂z = ∂w∂t , (2)where w(ρ, ϕ, t) denotes either the free-surface elevation or the deflection of the plate;t is the time. Later, to distinguish the vertical displacements, we denote the freesurfaceelevation in the open water region F by ζ, while the plate deflection in Pis still denoted by w. The dynamic condition, derived from the linearized Bernoulliequation, has the formP − P atmρ w= − ∂Φ∂t− gw, (3)at z = 0, where ρ w is the density of the water, g is the gravitational acceleration,P (ρ, ϕ, t) is the pressure in the fluid, and P atm is the atmospheric pressure. Relations(2–3) are the kinematic and dynamic conditions at the free surface. The linearizedfree-surface condition in the water region F takes the form∂Φ∂z = −1 ∂ 2 Φg ∂t 2 (4)at z = 0. The other boundary condition at the bottom z = −h, applying for the caseof finite depth, is written in the following form∂Φ∂z= 0. (5)The edges of the ring are free of vertical shear forces, bending and twistingmoments. Therefore, the free-edge conditions at ρ = r 1 and ρ = r 0 are:{(∇ 2 (1 − ν) ∂−ρ ∂ρ + 1 )}∂ 2ρ ∂ϕ 2 w = 0, (6){ ( ∂ (1 − ν) ∂∂ρ ∇2 +ρ 2 ∂ρ − 1 ) }∂2ρ ∂ϕ 2 w = 0, (7)where ν is the Poisson ratio. Here and in the sequel the Laplacian is two-dimensional,except when the Green’s function is derived.3. Integro-differential equationIn this section the main integro-differential equation (IDE) is derived along the platecontour, consisting of the inner and outer edges of the ring. We use the approachdescribed and applied in [7–8] for plates of rectangular shape in Cartesian coordinates.The derivation of IDE in polar coordinates has recently been published by the authors[6].AndrianovHermans.tex; 15/03/2005; 17:37; p.4