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Reduction of hydroelastic responses of a very-long floating structure

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Ocean Engineering 33 (2006) 610–634 www.elsevier.com/locate/oceaneng Reduction of hydroelastic responses of a very-long ﬂoating structure by a ﬂoating oscillatingwater-column breakwater system D.C. Honga,*, S.Y. Hongb, S.W. Hongb aCenter for Advanced Transportation Vehicles, College of Engineering, Chungnam National University, 220 goong-dong, Yusung, Daejon 305-764, Republic of Korea bMaritime & Ocean Engineering Research Institute, KORDI, P.O. Box 23, Yusung, Daejon 305-600, Republic of Korea Received 23 November 2004; accepted 30 June 2005 Available online 10 October 2005 Abstract The hydroelastic responses of a very-long ﬂoating structure (VLFS) placed behind a reverse T- shape freely ﬂoating breakwater with a built-in oscillating water column (OWC) chamber are analyzed in two dimensions. The Bernoulli–Euler beam equation is coupled with the equations of rigid and elastic motions of the breakwater and the VLFS. The interaction of waves between the ﬂoating rigid breakwater and the elastic VLFS is formulated in a consistent manner. It has been shown numerically that the structural deﬂections of the VLFS can be reduced signiﬁcantly by a suitably designed reverse T-shape ﬂoating breakwater. q 2005 Elsevier Ltd. All rights reserved. Keywords: Floating reverse T-shape breakwater; Reduction of hydroelastic responses; Very-long ﬂoating structure (VLFS); Equations of rigid and elastic motions; Oscillating water column (OWC) 1. Introduction The conventional bottom-mounted breakwaters have been found to be inadequate to a very-long ﬂoating structure (VLFS) since they prevent the circulation of seawater around * Corresponding author. Tel.: C82 42 868 7588; fax: C82 42 868 7519. E-mail address: mmedchong@kornet.net (D.C. Hong). 0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.06.005 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 611 Nomenclature D(M) displacement vector of a point M on the ﬂoating breakwater E Young’s modulus of the VLFS FkB wave exciting force coefﬁcients of the ﬂoating breakwater FkC excitation coefﬁcients of the ﬂoating breakwater due to the air pressure in the OWC chamber FkV generalized wave exciting force coefﬁcients of the VLFS G0 two-dimensional Rankine-type Green function G two-dimensional Kelvin-type Green function in a ﬁnite-depth water H water depth hl(x)(lZ2,3,.N) generalized mode functions of the VLFS I moment of inertia of the VLFS Kkl m0 M(x) stiffness matrix of the VLFS wave number bending moment of the VLFS m(x) mass distribution of the VLFS MkBl inertia coefﬁcients of the ﬂoating breakwater MBBklB hydrodynamic coefﬁcients of the ﬂoating breakwater due to its own motions MBBklV hydrodynamic coefﬁcients of the ﬂoating breakwater due to the motions of the VLFS MBCkl coefﬁcients of the ﬂoating breakwater due to the air pressure in the OWC chamber MkVl generalized inertia coefﬁcients of the VLFS MBVklB hydrodynamic coefﬁcients of the VLFS due to the motion of the ﬂoating breakwater MBVklV hydrodynamic coefﬁcients of the VLFS due to its own motions qBl ðlZ 1; 2; 3Þ complex amplitude of rigid sway, heave, and roll motions of the ﬂoating breakwater qV1 complex amplitude of the rigid sway motion of the VLFS qVl ðlZ 2; 3; .; NÞ complex amplitude of generalized modes of the VLFS Q(x) shear force of the VLFS RBkl hydrostatic restoring coefﬁcients of the ﬂoating breakwater RVkl hydrostatic stiffness matrix of the VLFS SB wetted surface of the ﬂoating breakwater SV wetted surface of the VLFS SBW waterplane of the ﬂoating breakwater SVW waterplane of the VLFS Ud complex-valued relative vertical mean velocity of the airﬂow through the duct w(x) vertical displacement of the VLFS J0 incident wave potential JS scattering wave potential 612 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 JR radiation wave potential jBl ðlZ 1; 2; 3Þ unit-amplitude radiation potentials due to the motions of ﬂoating breakwater jVl ðlZ 1; 2; .; NÞ unit-amplitude radiation potentials due to the motions of the VLFS g equivalent linear damping parameter the VLFS, which may cause serious environmental damage. Thus, various ﬂoating breakwaters and anti-motion devices of oscillating water column (OWC)-type, L-shape, reverse L-shape, horizontal plates and multiple vertical plates have been proposed as alternatives (Takaki et al., 2000; Ohmatsu et al., 2001; Maeda et al., 2001; Hong et al., 2002). The wave energy absorbing ability of a submerged body with a ﬂat horizontal top has been studied by Guevel et al. (1981). In their study, it has been shown numerically that the submerged body with rolling period tuned to the period of the incident wave can totally reﬂect the waves. They also proposed a horizontal plate ﬁxed under the free surface as a breakwater. The anti-wave performance of a submerged horizontal plate attached to the leeside end of a VLFS has been studied experimentally by Ohta et al. (1999) and analyzed by Watanabe et al. (2003) and Takagi et al. (2000) by making use of the linear potential theory. In this paper, hydroelastic responses of a mat-like VLFS of which the breadth is assumed to be inﬁnite, placed behind a reverse T-shape freely ﬂoating breakwater with a built-in OWC chamber have been analyzed in two dimensions by making use of the linear potential theory. Numerical results have been presented to show the hydroelastic response reduction performance of the ﬂoating breakwater. 2. Theoretical analyses A VLFS is freely ﬂoating on the free surface of ﬁnite-depth water under gravity. Cartesian coordinates (x, y, z) attached to the mean position of the VLFS are employed with the origin in the waterplane, and the x-axis parallel to the lengthwise direction of the VLFS and the y-axis vertically upwards. There are no motions in the direction of z-axis. This two-dimensional VLFS can be modeled as a Bernoulli–Euler beam (Wu et al., 1995). A reverse T-shape freely ﬂoating breakwater with a built-in OWC chamber is placed in front of the VLFS as shown in Fig. 1. The ﬂoating breakwater performs simple harmonic rigid body oscillations of small amplitude about its mean position with circular frequency u of plane progressive linear waves incident from xZKN. The vertical displacement w(x) of the VLFS can be found by solving the Bernoulli– Euler beam equation. ðEIwðxÞ00Þ00eKiut Z f ðxÞeKiut (1) where E is Young’s modulus, I the moment of inertia and f(x) the external force ﬁeld. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 613 Fig. 1. Coordinate system and conﬁguration of ﬂoating breakwater and VLFS. According to the Rayleigh–Ritz method, w(x) can be expressed as XN wðxÞeKiut Z ðqVl eKiutÞhlðxÞ (2) lZ2 where hl(x) are the base functions or the generalized mode functions and qVl ðlZ 2; 3; .; NÞ the corresponding complex amplitude. The boundary conditions of the freely ﬂoating beam are wðxÞ00 Z 0 and ðEIwðxÞ00Þ0 Z 0 at x ZGLV=2 (3) where LV is the length of the VLFS. The following characteristic functions representing the natural modes of the beam in air and satisfying the above free end boundary conditions, can be used as the mode functions (Newman, 1994). pﬃﬃ h2ðxÞ Z 1=2 and h3ðxÞ Z 3x=LV (4) hlðxÞ Z cosh 2mlK2x=LV cosh mlK2 C cos 2mlK2x=LV cos mlK2 for l Z 4; 6; 8; . (5) hlðxÞ Z sinh 2mlK2x=LV sinh mlK2 C sin 2mlK2x=LV sin mlK2 for l Z 5; 7; 9; . (6) where mj are the positive real roots of tan mj C ðK1Þjtanh mj Z 0; jZ 2; 3; 4; . (7) The above base functions or mode functions are mutually orthogonal as follows. LðV =2 hlðxÞhkðxÞdx Z dlkLV=4 (8) KLV =2 614 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 As indicated by Eq. (4), the second and third modes represent rigid heave and roll motions of the VLFS. We assume that the VLFS performs also rigid sway motion of complex amplitude qV1 independent of the motions corresponding to qVl ðlZ 2; 3; .; NÞ. With the usual assumptions of an incompressible ﬂuid and irrotational ﬂow without capillarity, the ﬂuid velocity is given by the gradient of a velocity potential Fðx; y; tÞZ RefJðx; yÞeKiutg. vðx; y; zÞ Z VJðx; y; zÞ (9) where the complex-valued velocity potential J satisﬁes Laplace’s equation. V2J Z 0 (10) J can be expressed as follows. J Z J0 C JS C JR (11) Here, J0 is the incident wave potential J0 a0u ZK m0 cosh m0ðy C HÞ eim0x sinh m0h (12) with m0H tanh m0H Z k0; k0 Z u2 g ; (13) where a0 denotes the amplitude of the incident wave, g the gravitational acceleration, m0 the wave number and H the water depth. JS is the scattering potential representing the disturbance of the incident wave potential by the ﬂoating breakwater and the VLFS ﬁxed at their mean position. The radiation potential JR due to both the rigid and elastic motions of the ﬂoating breakwater and the VLFS can be expressed as follows. " # X3 XN JR ZKiu qBl jBl C qVl jVl (14) lZ1 lZ1 where qBl ðlZ 1; 2; 3Þ are the complex amplitude of rigid sway, heave, and roll motions of the ﬂoating breakwater and jBl , jVl are unit-amplitude radiation potentials due to the motions of the ﬂoating breakwater and the VLFS, respectively. On the wetted surfaces SV and SB of the ﬂoating breakwater and the VLFS at their mean position, the unit-amplitude radiation potentials satisfy vjV1 vn Z n1; vjVl vn Z n2hl for l Z 2; 3; .; N on SV (15) vjVl vn Z 0 for l Z 1; 2; .; N on SB (16) D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 615 vjBl vn Z nl on SB; vjBl vn Z 0 on SV for l Z 1; 2 (17) vjB3 vn Z ðe3 !rÞ$n on SB; vjB3 vn Z 0 on SV (18) where r denotes position vector measured from the center of rotation O of the ﬂoating breakwater and n a normal vector directed into the ﬂuid region from the body surface. The boundary condition for JS on SVgSB is vJS vn vJ0 ZK vn on SV g SB (19) The OWC motion relative to the ﬂoating breakwater generates airﬂow through the duct as shown in Fig. 1. The formulation of the radiation–diffraction problem concerning the ﬂoating OWC chamber has been presented by Hong et al. (2004a,b) in detail where the free surface boundary conditions have been presented as follows. 8 vJ >>< 0 on Fe Kk0J C vz Z >>: i gu rg Ud on Fi (20) where Fe and Fi represent the free surface outside and inside the OWC chamber, respectively, g the equivalent linear damping and Ud the complex-valued vertical mean velocity of the airﬂow through the duct relative to the ﬂoating body as follows. ð 1 Ud Z ld vJ vy C iuDðMÞ$e2 dl; M on Fi (21) Fi where D(M) denotes the displacement vector of a point M on the oscillating body, ld the length of the duct for air ﬂow. The pressure drop across the duct is Re½pCeKiut Z g !Re½UdeKiut (22) where pc is the air pressure inside the chamber. Substituting Eq. (11) into Eq. (20), we have Kk0jPl C vjPl vy gu Z i rg 1 ld ð vjPl vy KdBP kPl dl on Fi for P Z V or B (23) Fi and Kk0JS C vJS vy gu Z i rg 1 ld ð vðJS C vy J0 Þ dl on Fi (24) Fi 616 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 where kB1 Z 0; kB2 Z 1; kB3 Z ðxKx0Þ (25) At zZKH, we have vJ vz Z 0 (26) The radiation and scattering potentials must also satisfy the radiation condition at inﬁnities. Since all the boundary conditions are linearized and prescribed, the unit-amplitude radiation potentials can be found as solutions of the following Green integral equations by applying Green’s theorem to the potentials and the Green functions in De and in Di, respectively, divided by the common boundary Sc as shown in Fig. 1. jPl ð 2K jPl vG0 vni dl C ð vjPl vni G0dl ð ZK vjPl vni G0dl FigSBi gSc FigSc SBi on Fi g SBi g Sc (27) for P Z V or B jPl 2C ð jPl vG vn ð dlK vjPl vn Gdl Z ð vjPl vn Gdl on SV g SBe g Sc for SVgSBe gSc Sc SVgSBe (28) P Z V or B where SBi and SBe are the internal and external wetted surfaces of the ﬂoating breakwater inside and outside of Sc, respectively, ni a normal vector directed into the body from SBi , G0 the Rankine-type Green function and G the Kelvin-type Green function in a ﬁnite-depth water (Wehausen and Laitone, 1960). The conditions on the matching boundary Sc are jPl ðMCÞ Z jPl ðMKÞ and vjPl ðMCÞ vn Z vjPl ðMKÞ vn ; M on Sc (29) where the superscripts C and K denotes the inside and outside surfaces of Sc, respectively. Analogously, we have the following integral equations and matching boundary conditions for JS. ð ð ð JS 2K JS vG0 vni dl C vJS vni G0dl ZK vJS vni G0dl on Fi g SBi g Sc (30) FigSBi gSc FigSc SBi ð ð ð JS 2C vG JS vn dlK vJS vn Gdl Z vJS vn Gdl on SV g SBe g Sc (31) SVgSBe gSc Sc SVgSBe D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 617 JSðMCÞ Z JSðMKÞ and vJSðMCÞ vn Z vJSðMKÞ vn ; M on SC (32) Here, the following condition should be combined with each system of integral equations in order to obtain the solution free of the irregular frequencies (Hong, 1987). ð vjðMÞ vn GðP; MÞKjðMÞ vGðP; vn MÞ d1 ¼ 0; P 2SBW g SVW (33) SegSc where SBW and SVW is the waterplane of the ﬂoating breakwater and the VLFS, respectively. The technique to solve the systems of integral Eqs. (27), (28) and (33) as well as Eqs. (30), (31) and (33), is analogous to that presented by Hong et al. (2004a,b). 3. Hydroelastic and rigid motion responses Applying the principle of virtual work to the Bernoulli–Euler beam and substituting Eq. (2) into Eq. (1), we have XN LðV =2 KklqVl Z f ðxÞhkðxÞdx for k Z 2; 3; .; N (34) lZ2 KLV =2 where Kkl is the stiffness matrix LðV =2 Kkl Z ðEIhlðxÞ00Þ00hkðxÞdx: (35) KLV =2 Decomposing f(x) into its components due to the hydrodynamic pressure ru Re{iJ eKiut}, the hydrostatic pressure, the body force and the inertial force, and moving terms involving the unknown complex amplitude of elastic and rigid motions to the lefthand side of Eq. (34), we have the following generalized equation of motions of the VLFS and the ﬂoating breakwater. XN X3 ½ð1Kdk1ÞðKkl Ku2MkVl C RVklÞKu2ðMBVklVÞqVl Ku2 ðMBVklBÞqBl lZ1 lZ1 (36) Z FkV; k Z 2; 3; .; N The equation of rigid sway motion of the VLFS coupled with the other motions of the VLFS and the ﬂoating breakwater can be expressed as follows. ( ) XN X3 Ku2 ½d1lM1Vl C MBV1lVqVl C ðMBV1lBÞqBl Z F1V (37) lZ1 lZ1 618 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 The coefﬁcients in Eqs. (36) and (37) are as follows. LðV =2 M1V1 Z mðxÞdx KLV =2 (38) LðV =2 MkVl Z mðxÞhlðxÞhkðxÞdx; l Z 2; 3; .; N; k Z 2; 3; .; N (39) KLV =2 ð RVkl ZKrg hlðxÞhkðxÞn2dl; l Z 2; 3; .; N; k Z 2; 3; .; N (40) SV ð MBV1lV Z jVl n1dl; l Z 1; 2; .; N (41) SV ð MBVklV Z jVl hkðxÞn2dl; l Z 1; 2; .; N; k Z 2; 3; .; N (42) SV ð MBV1lB Z jBl n1dl; l Z 1; 2; 3 (43) SV ð MBVklB Z jBl hkðxÞn2dl; l Z 1; 2; 3; k Z 2; 3; .; N (44) SV ð F1V ZKiur ðJ0 C JSÞn1dl (45) SV ð FkV ZKiur ðJ0 C JSÞhkðxÞn2dl; k Z 2; 3 (46) SV where m(x) is the distribution of mass of the VLFS, MBVklV and MBVklB the generalized hydrodynamic coefﬁcients—the sum of the generalized added mass and wave damping coefﬁcients—due to the motions of the VLFS and the ﬂoating breakwater, respectively, RVkl the hydrostatic stiffness matrix (Huang and Riggs, 2000) and FkV the generalized wave exciting force coefﬁcients. The equations of rigid motions of the ﬂoating breakwater with an OWC chamber, coupled with the motions of the VLFS, are as follows. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 619 XN X3 Ku2 MBBklVqVl C ½Ku2ðMkBl C RBklÞKu2ðMBBklB C MBCklÞqBl lZ1 lZ1 (47) Z FkB C FkC; k Z 1; 2; 3 where MkBl, RBkl, MBBklB, FkB are the coefﬁcients of inertia, restoring, hydrodynamic and wave exciting forces of rigid ﬂoating body, respectively, MBBklV the generalized hydrodynamic coefﬁcients due to the motions of the VLFS and MBCkl, FkC are the coefﬁcients due to the air pressure in the OWC chamber (Hong et al., 2004a,b). Solving simultaneously Eqs. (36), (37) and (47), the complex amplitude of rigid and elastic motions qVl ðlZ 1; 2; .; NÞ of VLFS and rigid motions qBl ðlZ 1; 2; 3Þ of ﬂoating breakwater can be found. We can also calculate the bending moment M(x) and the shear force Q(x) according to the Bernoulli–Euler beam theory as follows. XN MðxÞ Z qVl EIhlðxÞ00 (48) lZ2 XN QðxÞ Z qVl ½EIhlðxÞ000 (49) lZ2 4. Numerical results and discussions The longitudinal section of the VLFS is a rectangular of which the horizontal length LV is 1000 m and the draft dv is 2 m. It has a uniform mass distribution with Young’s modulus of 2.06!1011 Pa and its moment of inertia per unit breadth is 0.666 m3. In order to study the hydroelastic response reduction performance of the reverse T-shape freely ﬂoating breakwater with a built-in OWC chamber, two numerical models, RTOWC-1 and RTOWC-2, are chosen. Their principal dimensions in meters are shown in Table 1 where yG and RG denote the y coordinates of the center of gravity and the roll radius of gyration, respectively. The thickness of vertical and horizontal plates of the ﬂoating breakwater is 2 m everywhere. Numerical tests have been carried out for waves of which the period ranges from 6 to 20 s (wavelength range: 56–326 m) in a water of 30 m deep. Table 1 Principal dimensions and inertial properties of the reverse T-shape ﬂoating breakwaters with OWC a b c d lF lD dB yG RG RTOWC-1 40 10 5 3 20 1 10 K8 20 RTOWC-2 60 10 5 3 32 1 10 K8 28 620 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Fig. 2. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ6 s. dsZ2.5 m. The two bodies are freely ﬂoating without any constraints. Two values—2.5 and 50 m—of ds, the distance between them, are chosen. For each value of ds, the amplitude of vertical displacements of the VLFS is presented for three periods—6, 13 and 20 s in Figs. 2–13. The vertical displacement of the VLFS without breakwater is also presented in each ﬁgure to show Fig. 3. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ13 s. dsZ2.5 m. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 621 Fig. 4. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ20 s. dsZ2.5 m. Fig. 5. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ6 s. dsZ50 m. 622 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Fig. 6. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ13 s. dsZ50 m. Fig. 7. Vertical displacement of VLFS without and with RTOWC-1 for different g TZ20 s. dsZ50 m. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 623 Fig. 8. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ6 s. dsZ2.5 m. Fig. 9. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ13 s. dsZ2.5 m. 624 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Fig. 10. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ20 s. dsZ2.5 m. the differences in hydroelastic responses of the VLFS. Four values of the equivalent linear damping parameter, gZ0, 500, 1000, 1500 kg-mass/m2 s, were chosen to see its inﬂuence on the motion responses. The complex-valued non-dimensional motion responses of the ﬂoating breakwater and rigid sway motion responses of the VLFS, for each case mentioned above, are presented in Fig. 11. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ6 s. dsZ50 m. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 625 Fig. 12. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ13 s. dsZ50 m. Tables 2–7. The rigid sway motion response of the VLFS without breakwater is also presented in each table. The effect of the present freely ﬂoating breakwaters on the hydroelastic responses for long waves of TZ20 s are very poor as shown in Figs. 4, 7, 10 and 13. It might Fig. 13. Vertical displacement of VLFS without and with RTOWC-2 for different g TZ20 s. dsZ50 m. Table 2 Complex-valued non-dimensional motion responses, dsZ2.5 m, TZ20 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 K1.81C0.397i K1.85C0.434i K1.880.477i K1.89C0.523i K1.709C0.465i K1.78C0.54i K1.828C0.626i K1.852C0.71i qB2 =a0 0.427C0.632i 0.338C0.689i 0.267C0.761i 0.218C0.84i 0.543C0.617i 0.46C0.668i 0.397C0.733i 0.355C0.804i qB3 H=a0 K0.255C0.184i K0.413C0.302i K0.535C0.447i K0.617C0.602i K0.134C0.354i K0.272C0.468i K0.37C0.603i K0.429C0.742i qV1 =a0 K0.0449C0.186i K0.047K0.182i K0.048K0.178i K0.048K0.174i K0.036K0.183i K0.039K0.176i K0.04K0.169i K0.039K0.163i qV1 =a0 (without breakwater)0.016K0.009i D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 626 Table 3 Complex-valued non-dimensional motion responses, dsZ2.5 m, TZ13 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 K1.27C0.238i K1.179C0.254i K1.1080.291i K1.06C0.33i K1.155K0.024i K1.016C0.125i K0.95C0.24i K0.922C0.327I qB2 =a0 0.525C0.61i 0.329C0.626i 0.175C0.59i 0.063C0.534i 0.546C0.644i 0.322C0.546i 0.196C0.448i 0.122C0.368i qB3 H=a0 K0.235C0.52i K0.49C0.617i K0.709C0.63i K0.88C0.597i 0.049C0.931i K0.331C0.94i K0.571C0.885i K0.729C0.821i qV1 =a0 K0.037K0.014i K0.038K0.01i K0.04K0.007i K0.042K0.005i K0.024K0.015i K0.029K0.006i K0.034K0.0016i K0.038K0.001i qV1 =a0 (without breakwater)K0.025C0.06i 627 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Table 4 Complex-valued non-dimensional motion responses, dsZ2.5 m, TZ6 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 0.001K0.006i 0.002C0.004i 0.004C0.006i 0.005C0.002i K0.189K0.153i K0.166K0.145i K0.147K0.145i K0.133K0.15i qB2 =a0 K0.071C0.074i K0.073C0.068i K0.076C0.064i K0.08C0.062i 0.052C0.049i 0.048C0.044i 0.044C0.04i 0.04C0.039i qB3 H=a0 0.228K0.239i 0.197K0.246i 0.17K0.24i 0.15K0.23i 0.048C0.064i 0.033C0.069i 0.024C0.078i 0.019C0.085i qV1 =a0 0.001K0.0004i 0.001C0.0002i 0.002C0.0006i 0.003C0.0008i K0.006K0.014i K0.006K0.013i K0.005K0.012i K0.004K0.012i qV1 =a0 (without breakwater)0.014K0.026i D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 628 Table 5 Complex-valued non-dimensional motion responses, dsZ50 m, TZ20 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 K0.536C1.65i K0.543C1.71i K0.535C1.76i K0.517C1.81i K0.418C1.629i K0.42C1.735i K0.394C1.831i K0.35C1.901i qB2 =a0 0.762C0.126i 0.741C0.225i 0.744C0.323i 0.768C0.41i 0.795C0.025i 0.777C0.115i 0.783C0.199i 0.807C0.272i qB3 H=a0 0.073C0.293i 0.045C0.485i 0.064C0.669i 0.119C0.831i 0.247C0.347i 0.234C0.517i 0.265C0.672i 0.326C0.801i qV1 =a0 K0.047K0.189i K0.046K0.187i K0.046K0.186i K0.044K0.184i K0.043K0.189i K0.042K0.186i K0.04K0.184i K0.038K0.182i qV1 =a0 (without breakwater)0.016K0.009i 629 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Table 6 Complex-valued non-dimensional motion responses, dsZ50 m, TZ13 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 0.286C1.244i 0.286C1.15i 0.31C1.076i 0.345C1.024i 0.028C1.187i 0.134C1.02i 0.235C0.934i 0.314C0.889i qB2 =a0 0.638K0.518i 0.652K0.342i 0.617K0.2i 0.56K0.096i 0.66K0.569i 0.58K0.341i 0.484K0.211i 0.4K0.136i qB3 H=a0 0.561C0.209i 0.634C0.438i 0.632C0.637i 0.588C0.792i 0.916K0.037i 0.932C0.324i 0.871C0.557i 0.797C0.708i qV1 =a0 K0.043K0.018i K0.04K0.015i K0.039K0.014i K0.038K0.012i K0.036K0.026i K0.032K0.019i K0.031K0.013i K0.031K0.009i qV1 =a0 (without breakwater)K0.025C0.06i D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 630 Table 7 Complex-valued non-dimensional motion responses, dsZ50 m, TZ6 s RTOWC-1 RTOWC-2 g 0 500 1000 1500 0 500 1000 1500 qB1 =a0 0.041C0.008i 0.043C0.046i 0.059C0.073i 0.078C0.089i K0.363K0.991i K0.282K0.957i K0.209K0.949i K0.15K0.958i qB2 =a0 K0.1K0.014i K0.097K0.018i K0.096K0.022i K0.097K0.026i K0.039C0.109i 0.032C0.104i 0.026C0.102i 0.021C0.101i qB3 H=a0 0.359C0.056i 0.346C0.044i 0.331C0.039i 0.32C0.04i K0.0006C0.013i K0.012C0.018i K0.02C0.025i K0.025C0.032i qV1 =a0 0.0005C0.0003i 0.0004C0.0006i 0.0005C0.0009i 0.0006C0.001i K0.001K0.01i K0.0009K0.01i K0.0002K0.01i K0.0003K0.01i qV1 =a0 (without breakwater) 0.014K0.026i 631 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 632 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 be due to the fact that the dimensions of the present ﬂoating breakwaters are not sufﬁciently large so that they may play a signiﬁcant role. For waves of TZ13 s, the effects of the present breakwaters are noticeable. In particular, RTOWC-2 ﬂoating 2.5 m before the VLFS reduces the hydroelastic responses of the VLFS signiﬁcantly as shown in Fig. 9 where the amplitude is inversely proportional to the value of g. For relatively short waves of TZ6 s, the effects of the RTOWC-2 are very poor. On the other hand, the effects of the RTOWC-1 are excellent especially for dsZ50 m. For the waves of TZ6 s, the amplitude of hydroelastic responses with the RTOWC-1 is proportional to the value of g while that with the RTOWC-2 is inversely proportional to g. These phenomena might be due to the characteristics of the whole vibrating system in waves. So, the principal dimensions of the ﬂoating breakwater should be determined in accordance with the eventual wave energy spectrum so as to maximize the global reduction efﬁciency of the hydroelastic responses of the VLFS. The curves of bending moments and shear forces of the VLFS ﬂoating 2.5 m behind the RTOWC-2 in waves of TZ13 s, have also been presented in Figs. 14 and 15. The amplitude of bending moments and shear forces has been shown to be reduced signiﬁcantly as g grows. 5. Conclusions An analytical method to evaluate the reduction efﬁciency of the hydroelastic responses of a VLFS by making use of a freely ﬂoating breakwaters with a built-in OWC chamber has been proposed. Fig. 14. Bending moment of VLFS without and with RTOWC-2 for different g TZ13 s. dsZ2.5 m. D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 633 Fig. 15. Shear force of VLFS without and with RTOWC-2 for different g TZ13 s. dsZ2.5 m. Although the differences between the RTOWC-1 and 2 are the length a of the submerged horizontal plate and lF,, the length of the OWC chamber, the present numerical results will be useful to design a reverse T-shape ﬂoating breakwater. More systematic studies can be done by varying each of the principal dimensions shown in Table 1, with or without a suitable mooring system which are not considered here. If the reverse T-shape ﬂoating breakwater is connected to the leeside end of the VLFS by a pin, for example, it will probably act as an excellent anti-motion device. Acknowledgements The present work is a part of the research program for the development of design technology of VLFS funded by the Ministry of Maritime Affairs and Fisheries of Korea. References Guevel, P., Hong, D.C., Marti, J., Delhommeau, G., 1981. Considerations sur le fonctionnement des houlomoteur du type oscillant. Bulletin de l’ATMA, Paris, Paper No. 1877. Hong, D.C., 1987. On the improved Green integral equation applied to the water-wave radiation-diffraction problem. Journal Society of Naval Architect of Korea 24 (1), 1–8. Hong, S.Y., Choi, Yoon R., Hong, S.W., 2002. Analysis of hydroelastic responses of pontoon-type VLFS coupled with ﬂoating breakwaters using a higher-order boundary element method, Proc 12th Int Offshore Polar Eng Conf., Kitakyushu, Japan 2002 pp. 313–318. Hong, D.C., Hong, S.Y., Hong, S.W., 2004a. Numerical study of the motions and drift force of a ﬂoating OWC device. Ocean Engineering 31 (2), 139–164. 634 D.C. Hong et al. / Ocean Engineering 33 (2006) 610–634 Hong, D.C., Hong, S.Y., Hong, S.W., 2004b. Numerical study on the reverse drift force of ﬂoating BBDB wave energy absorbers. Ocean Engineering 31 (10), 1257–1294. Huang, L.L., Riggs, H.R., 2000. The hydrostatic stiffness of ﬂexible ﬂoating structures for linear hydroelasticity. Marine Structures 13, 91–106. Maeda H., Rheem, C.K., Washio, Y., Osawa, H., Nagata, Y., Ikoma, T., Fujita, N., Arita, M., 2001. Reduction effects of hydroelastic reponses on a VLFS with wave energy apsorption devices using OWC system. Proc. 20th OMAE Conf., Rio de Janeiro, Brazil, OMAE01/OSU-5013. Newman, J.N., 1994. Wave effects on deformable bodies. Applied Ocean Research 16, 47–59. Ohmatsu S., Kato, S., Namba, Y., Maeda, K., Kobayashi, M., Nakagawa, H., 2001. Study on ﬂoating breakwater for eco-ﬂoat mooring system. Proc. 20th OMAE Conf., Rio de Janeiro, Brazil, OMAE01/OSU-5012. Ohta, H., Torii, T., Hayashi, N., Watanabe, E., Utsunomiya, T., Sekita, K., Sunahara, S., 1999. Effect of attachment of a horizontal/vertical plate on the wave response of a VLFS. Proc. 3rd Int. Workshop on VLFS, Honolulu, USA, pp. 265–274. Takagi, K., Shimada, K., Ikebuchi, T., 2000. An anti-motion device for a very large ﬂoating structure. Marine Structures 13, 421–436. Watanabe, E., Utsunomiya, Ohta, H., Hayashi, N., 2003. Wave response analysis of VLFS with an attached submerged plate - Veriﬁacation with 2-D model and some 3-D numerical examples, Proc. Int. Symp. On Ocean Space Utilization Technol., Tokyo, Japan 2003 pp. 147–154. Wehausen, J.V., Laitone, E.V., 1960. Surface Waves. Encyclopedia of Physics 9, 446–778. Wu, C., Watanabe, E., Utsunomiya, T., 1995. An eigenfunction expansion-matching method for analyzing the wave-induced responses of an elastic ﬂoating plate. Applied Ocean Research 17, 301–310.

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