Hydroelastic behavior of a floating ring-shaped plate

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10 A.I. Andrianov & A.J. Hermanswhere we have introduced the function c (q)mn as a combination of Bessel and Hankelfunctions in the formc (q)mn = r 0[−r 1[kJ n+1 (kr 1 )H (q)kJ n+1 (kr 0 )H (q)nn(κ mr 0 ) − κ m J n (kr 0 )H (q) ]n+1 (κ mr 0 )], q = 1, 2. (27)(κ mr 1 ) − κ m J n (kr 1 )H (q)n+1 (κ mr 1 )From Equation (26) we can derive 2(M −2)(N +1) relations for the determinationof 2M(N +1) unknown amplitudes. The set of equations for both cases of water depthis completed by the free-edge conditions (6–7) when the deflection function is insertedthere. These conditions at the edges ρ = r 1 and ρ = r 0 give us 4(N + 1) equationsM∑m=1a (1)mn d(1) mn,ij + a(2) mn d(2) mn,ij = 0, (28)where n = 0, ..., N, i = 0, 1, j = 1, 2, and the functions d (q)[()d (q)mn,i1 = H (q)n (κ mr i ) −κ 2 (1 − ν)n(n − 1)m +r 2 imn,ij are[(d (q)mn,i2 = H (q)n (κ m r i ) − n κ 2 m + (1 − ν)n2 (1 − n)r i(+ H (q)n+1 (κ mr i ) κ 3 m +]+ H (q)n+1 (κ (1 − ν)mr i )κ m ,r i)r 3 i)](1 − ν)n2ri2 κ m , (29)where q = 1, 2 is the Hankel-function index and r i is either inner-ring contour r 1 orouter-ring contour r 0 .From now on we consider the IWD and FWD cases separately.6. Ring on water of infinite depthHere the analytic study is given for the case of infinite depth. In the IWD case we onlyconsider part of the solution, namely that related to the water dispersion relation.The analysis given in this section is a good basis from which to derive a solution forthe general and most complicated case of water of finite depth. To demonstrate themethod we wish to use for the FWD case, we consider the contribution of the polek = k 0 in the IWD case. Here we ignore the contribution of the integral along theimaginary axis after closing the contour in (20). This means that local disturbancesare disregarded.The deflection of the plate, the Green’s function and the function F (k) are representedby expressions (21), (20) and (18a), respectively. For deep-water wave numberk 0 equals K, so k 0 = ω 2 /g.Now we analyze the k-integral in IDE (26). The integrand has the poles k = ±κ m ,shown in Figure 3, and can be represented as a sum of four integrals∫∞k 00k(k − k 0 )(k 2 − κ 2 m)[]a (1)mn c(1) mn + a(2) mn c(2) mn J n (kρ) dk= a (1)mn (I 01 − I 11 ) + a (2)mn (I 02 − I 12 ) , (30)AndrianovHermans.tex; 15/03/2005; 17:37; p.10
Hydroelastic Behavior of a Floating Ring-Shaped Plate 11where the path of integration in (31) is below the singularities k = k 0 and k = κ 1 ,respectively, and the integrals I iq for i = 0, 1, q = 1, 2 are[×kJ n+1 (kr i )H (q)n∫∞I iq = k 0 r i0k(k − k 0 )(k 2 − κ 2 m)(κ m r i ) − κ m J n (kr i )H (q)n+1 (κ mr i )]J n (kρ) dk, (31)with r i as r 1 or r 0 , m = 1, ..., M and n = 0, ..., N. The contour of integration may beclosed in the complex k-plane. In the integral I iq we split up the Bessel function oflargest argument to be able to close the contour of integration and, furthermore, useJordan’s lemma.Let us consider the integral I 01 . For further manipulation we use the fact that ρ r 1 and, therefore, to close the contours of integration we represent the Besselfunctions J n (kρ) as the half-sums of corresponding Hankel functions of the first andsecond kind, as was done in (32).When the contours are closed for all integrals, we apply the Cauchy residue lemmaat the poles k = κ m for the integrals with the contour closed in the upper half-planeand at the poles k = −κ m for the integrals with the contour closed in the lowerhalf-plane. The Wronskian W {H (1)n (κ m r i ), H (2)n (κ m r i ) applies for the combination ofHankel functionsH (1)n+1 (κ mr i )H (2)n (κ mr i ) − H (1)n (κ mr i )H (2)n+1 (κ mr i ) = − 4i . (33)πκ m r iThen, adding up the resulting terms and considering the coefficients of J n (κr i ), wederive the plate dispersion relation for deep water (22), where the plus sign in theright-hand side corresponds to the integrals in the upper half-plane, and the minussign to the integrals in the lower half-plane. The derivation of the plate dispersionrelation from IDE is also a justification of our approach.In the upper half-plane we also obtain a contribution of the pole k = k 0 of theintegrand, as seen in Figure 2. This contribution has to cancel the term in the righthandside of IDE (26). If we neglect the integral along the imaginary axis, applicationAndrianovHermans.tex; 15/03/2005; 17:37; p.11
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Hydroelastic behavior of a floating ring-shaped plate

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