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Hydroelastic responses and drift forces of a very-long ﬂoating structure

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ARTICLE IN PRESS Ocean Engineering 34 (2007) 696–708 www.elsevier.com/locate/oceaneng Hydroelastic responses and drift forces of a very-long ﬂoating structure equipped with a pin-connected oscillating-water-column breakwater system D.C. Honga,Ã, S.Y. Hongb aCenter for Advanced Transportation Vehicles, College of Engineering, Chungnam National University 220 goong-dong, Yusung, Daejon 305-764, Republic of Korea bMaritime and Ocean Engineering Research Institute, KORDI, P.O. Box 23 Yusung, Daejon 305-600, Republic of Korea Received 27 December 2005; accepted 30 May 2006 Available online 25 September 2006 Abstract The hydroelastic responses of a very-long ﬂoating structure (VLFS) in waves connected to a ﬂoating oscillating-water-column (OWC) breakwater system by a pin are analyzed by making use of the modal expansion method in two dimensions. The Bernoulli–Euler beam equation for the VLFS is coupled with the equations of motions of the breakwater taking account of the geometric and dynamic boundary conditions at the pin. The Legendre polynomials are employed as admissible functions representing the assumed modes of the VLFS with pinned-free-boundary conditions. It has been shown numerically that the deﬂections, bending moments and shear forces of the VLFS in waves can be reduced signiﬁcantly by a pin-connected OWC breakwater. The time-mean horizontal drift forces of the VLFS equipped with the breakwater calculated by the near-ﬁeld method are also presented. r 2006 Elsevier Ltd. All rights reserved. Keywords: VLFS; Hydroelastic responses; Pinned-free-boundary conditions of a ﬂoating Bernoulli–Euler beam; Pin-connected ﬂoating breakwater; Drift force; Relative motions; OWC 1. Introduction In order to reduce the hydroelastic responses of a verylong ﬂoating structure (VLFS) in waves, a box-shaped antimotion device rigidly attached to the weather-side edge of a VLFS has been proposed by Takagi et al. (2000). The antiwave performance of a submerged horizontal plate attached to the weather-side end of a VLFS has been studied experimentally by Ohta et al. (1999) and analyzed by Watanabe et al. (2003). The hydroelastic response reduction performance of a oscillating-water-column (OWC) breakwater system freely ﬂoating in front of a VLFS has been presented by Hong et al. (2006) where the boundary-value problem for the velocity potential have been solved by making use of a Green integral equation with Kelvine-type Green function in a ﬁnite-depth water. In this paper, the hydroelastic responses of a twodimensional VLFS connected to the ﬂoating-OWC breakwater system, by a pin have been analyzed by making use of the modal expansion method. The two-dimensional VLFS is modeled as a Bernoulli–Euler beam. The relative motions between the ﬂexible VLFS and the rigid ﬂoating breakwater have been formulated taking account of the geometric and dynamic boundary conditions at the pin. The Legendre polynomials are employed as admissible functions representing the assumed modes of the VLFS with pinned-free-boundary conditions of the Bernoulli– Euler beam. Numerical results have been presented to show the hydroelastic response reduction performance of the pin-connected OWC breakwater system. 2. Theoretical analyses ÃCorresponding author. Tel.: +82 42 868 7588; fax: +82 42 868 7519. E-mail address: mmedchong@kornet.net (D.C. Hong). A VLFS is freely ﬂoating on the free surface of ﬁnitedepth water under gravity. Cartesian coordinates (x, y, z) 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.05.004 ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 697 Nomenclature DB(M) displacement vector of a point M on the breakwater DV(M) displacement vector of a point M on the VLFS E Young’s modulus of the VLFS Fx time-mean horizontal drift force FBk wave exciting force coefﬁcients of the break- water FCk excitation coefﬁcients of the breakwater due to the air pressure in the OWC chamber FVk generalized wave exciting force coefﬁcients of the VLFS H water depth hlðxÞ ðl ¼ 2; 3; . . . ; NÞ admissible functions 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) M B kl MBBklB MBBklV MBCkl M V kl MBVklB MBVklV mass distribution of the VLFS inertia coefﬁcients of the breakwater hydrodynamic coefﬁcients of the breakwater due to its own motions hydrodynamic coefﬁcients of the breakwater due to the motions of the VLFS coefﬁcients of the breakwater due to the air pressure in the OWC chamber generalized inertia coefﬁcients of the VLFS hydrodynamic coefﬁcients of the VLFS due to the motion of the breakwater hydrodynamic coefﬁcients of the VLFS due to its own motions PnðxÞ; n ¼ l À 2; l ¼ 2; 3; 4; . . . Legendre polynomials of order n qBl ðl ¼ 1; 2; 3Þ complex amplitude of rigid sway, heave, and roll motions of the breakwater qV1 complex amplitude of the rigid sway motion of the VLFS qVl ðl ¼ 2; 3; . . . ; NÞ complex amplitude of generalized modes of the VLFS Q ¼ Qxe1 þ Qye2 reaction force at the pin Q(x) shear force of the VLFS RBkl hydrostatic restoring coefﬁcients of the break- water RVkl hydrostatic stiffness matrix of the VLFS SB wetted surface of the breakwater SV wetted surface of the VLFS SBW waterplane of the breakwater SVW waterplane of the VLFS w(x) vertical displacement of the VLFS (xa, 0) coordinates of the pin ðxBo ; yBo Þ coordinates of the center of rotation of the breakwater C0 incident wave potential CS scattering wave potential CR radiation wave potential cBl ðl ¼ 1; 2; 3Þ unit-amplitude radiation potentials due to the motions of breakwater cVl ðl ¼ 1; 2; . . . ; NÞ unit-amplitude radiation poten- tials due to the motions of the VLFS g equivalent linear damping parameter attached to the mean position of the VLFS are employed with the origin at the center of 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. A ﬂoating OWC breakwater system is connected to the left end of the VLFS by a pin as shown in Fig. 1. The breakwater performs simple harmonic rigidbody oscillations of small amplitude about its mean position with circular frequency o of plane progressive linear waves incident from x ¼ À1. The displacement of a point M(x, y) of the breakwater can be expressed as follows: DBðMÞ ¼ qB1 e1 þ qB3 e3 Â ½ðx À xBo Þe1 þ ðy À yBo Þe2Þ, (1) where qBl ðl ¼ 1; 2; 3Þ are the complex amplitude of rigid sway, heave, and roll motions of the breakwater and (xBo, yBo) the coordinates of the center of rotation. The horizontal motion of the VLFS is assumed to be a rigid sway motion with complex amplitude qV1 . The vertical displacement w(x) of the VLFS can be found by solving the Bernoulli–Euler beam equation ðEIwðxÞ00Þ00eÀiot ¼ f ðxÞ eÀiot, (2) where E is Young’s modulus, I the moment of inertia and f(x) the external force ﬁeld. According to the Rayleigh–Ritz method, w(x) can be expressed as wðxÞ eÀiot ¼ X N ÀqVl eÀiot Á hl ðxÞ, (3) l¼2 where hl(x) are the admissible functions and qVl ðl ¼ 2; 3; . . . ; NÞ the corresponding complex amplitude. It should be noted that the pin is moving in accordance with the relative motions between the VLFS and the pinconnected breakwater. The displacement of a point M(x, y) of the VLFS can be expressed as follows: DVðMÞ ¼ qV1 e1 þ wðxÞe2. (4) The pinned-free-boundary conditions of the VLFS modeled as a Bernoulli–Euler beam are wðxÞ00 ¼ 0 at x ¼ ÀLV=2, (5) ARTICLE IN PRESS 698 D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 OWC breakwater Wave lD duct Fe d dB Sc Fi lF c Di a ds b Pin De VLFS y x H Bottom Fig. 1. Coordinate system and conﬁguration of a pin-connected breakwater and the VLFS. wðxÞ00 ¼ 0 and ðEIwðxÞ00Þ0 ¼ 0 at x ¼ LV=2, (6) where LV is the length of the VLFS. Since the characteristic functions satisfying the free-free- boundary conditions of a beam cannot satisfy the pinnedfree-boundary conditions, the Legendre polynomials Pn(x) are introduced here. hlðxÞ ¼ PnðxÞ; n ¼ l À 2; l ¼ 2; 3; 4; . . . , (7) where x is the normalized coordinate of x as follows: x ¼ 2x=LV. (8) The above admissible functions are mutually orthogonal as follows: Z1 hlðxÞhkðxÞ dx ¼ 2=½2ðl À 2Þ þ 1; l ¼ 2; 3; 4; . . . , À1 k ¼ 2; 3; 4; . . . ð9Þ With the usual assumptions of an incompressible ﬂuid and irrotational ﬂow without capillarity, the ﬂuid velocity v iRs egÈivCeðnx;byyÞetÀhieotgÉradient of a velocity potential Fðx; y; tÞ ¼ vðx; y; zÞ ¼ rCðx; y; zÞ, (10) where the complex-valued velocity potential C satisﬁes Laplace’s equation r2C ¼ 0. (11) C can be expressed as follows: C ¼ C0 þ CS þ CR. (12) Here, C0 is the incident wave potential in a ﬁnite-depth water C0 ¼ À a0o m0 cosh m0ðy þ sinh m0h HÞ eim0x (13) with o2 m0 tanh m0H ¼ k0; k0 ¼ g , (14) where a0 denotes the amplitude of the incident wave, g the gravitational acceleration, m0 the wave number and H the water depth. CS is the scattering potential representing the disturbance of the incident wave potential by the breakwater and the VLFS at their mean positions. The radiation potential CR due to both the rigid and elastic motions of the breakwater and the VLFS can be expressed as follows: " # X3 X N CR ¼ Àio qBl cBl þ qVl cVl , (15) l¼1 l¼1 where cBl ; cVl are unit-amplitude radiation potentials due to the motions of the ﬂoating breakwater and the VLFS, respectively. The boundary-value problem for the velocity potential is analogous to the one presented by Hong et al. (2006) where the radiation–diffraction potentials have been obtained as solutions of a set of Green integral equations using a Kelvine-type Green function in a ﬁnite-depth water as well as a Rankine-type Green function. 3. Hydroelastic and rigid-motion responses Applying the principle of virtual work to the Bernoulli– Euler beam and substituting Eq. (3) into Eq. (2), we have X N Z LV=2 K kl qVl ¼ f ðxÞhkðxÞ dx þ QyhkðxaÞ l¼2 ÀLV =2 for k ¼ 2; 3; . . . N ð16Þ where Kkl is the stiffness matrix Z LV=2 Kkl ¼ EI hlðxÞ00hkðxÞ00 dx (17) ÀLV =2 and Qy is the vertical component of the reaction force Q at the pin in accordance with the dynamic boundary condition at the pin of which the coordinates is ðxa; 0Þ. Q ¼ Qxe1 þ Qye2. (18) Decomposing f(x) into its components due to the hydrodynamic pressure roRefiCeÀiotg, the hydrostatic pressure, the body force and the inertial force and moving terms involving the unknown complex amplitude of elastic and rigid motions as well as the unknown reaction force to the left-hand side of Eq. (16), we have the following generalized equations of coupled motions of the VLFS and the pin-connected breakwater: X N Â ð1 À À dk1Þ K kl À o2 M V kl þ RVkl Á À o2MBVklVÃ l¼1 X3 ÂqVl À o2 MBVklBqBl À Qy hk ðxa Þ ¼ F V k l¼1 k ¼ 2; 3 . . . ; N, ð19Þ ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 Table 1 Principal dimensions and inertial properties of the pinned breakwaters 1.5 with OWC a b c d lF lD dB xG RG tW tB OWCBW1 15 2.5 3 4 7.5 1 9 À509 5.8 2 2 OWCBW2 30 5 3 4 15 2 9 À515 11.7 3 2 1 OWCBW3 40 5 3 4 20 1 10 À521 15 4 3 699 γ=0 γ=50 γ=100 γ=150 Sway RAO of OWCBW 1.2 0.5 1 Vertical amplitude/a0 0.8 0.6 γ=0 γ=50 0.4 γ=100 γ=150 Without OWCBW 0.2 0 10 15 20 T (sec.) Fig. 2. Displacements at the weather-side end of the VLFS with pinconnected OWCBW1. γ=0 0.3 γ=50 γ=100 γ=150 Without OWCBW 0.2 0.1 Heave RAO of OWCBW 0 10 15 20 T (sec.) Fig. 4. Sway amplitude of the pin-connected OWCBW1. 1.2 1 0.8 0.6 γ=0 0.4 γ=50 γ=100 γ=150 0.2 0 10 15 20 T (sec.) Fig. 5. Heave amplitude of the pin-connected OWCBW1. Sway RAO of VLFS 0 10 15 20 T (sec.) Fig. 3. Sway amplitude of the VLFS with pin-connected OWCBW1. " # Ào2 X N À d1l M V 1l þ MBV1lVÁqVl þ X3 MBV1lBqBl À Qx ¼ F V1 , l¼1 l¼1 (20) X N À o2 MBBklVqVl l¼1 þ X3 ÂÀo2 M B kl þ RBkl À o2ÀMBBklB þ MBCkl ÁÃqBl þ l¼1 dk1Qx þ dk2Qy þ hÀ dk3 xa À xBo ÁQy À ðya À i yBo ÞQx ¼ F B k þ F C k ; k ¼ 1; 2; 3. ð21Þ Eq. (20) is the equation of rigid sway motion of the VLFS coupled with the other motions of the VLFS and the breakwater. Eq. (21) is the equation of motions of the pinconnected OWC breakwater system, coupled with the motions of the VLFS. 700 7 6 5 4 ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 0.2 γ=0 γ=50 γ=100 γ=150 Without OWCBW Roll RAO of OWCBW IWI/a0 3 γ=0 γ=50 0.1 2 γ=100 γ=150 1 0 10 15 20 T (sec.) Fig. 6. Roll amplitude of the pin-connected OWCBW1. γ=0 γ=250 γ=750 γ=2000 1 Without OWCBW 0 -250 0 250 500 X (M) Fig. 8. Vertical displacement of the VLFS with pin-connected OWCBW1, T ¼ 6 s. γ=0 γ=50 0.3 γ=100 γ=150 Without OWCBW Fx/ρ ga02 IMI/ρ a0 ω2 H3 B 0.2 0 0.1 10 15 20 T (sec.) Fig. 7. Horizontal drift forces of the VLFS and the pin-connected OWCBW1. The coefﬁcients in Eqs. (19) and (20) are as follows: Z LV=2 M V 11 ¼ mðxÞ dx ÀLV =2 (22) Z LV=2 M V kl ¼ mðxÞhlðxÞhkðxÞ dx; ÀLV =2 l ¼ 2; 3 . . . N; k ¼ 2; 3; . . . ; N, ð23Þ Z RVkl ¼ À rg hlðxÞhkðxÞn2 dl; l ¼ 2; 3; . . . ; N; SV k ¼ 2; 3; . . . ; N, ð24Þ 0 -250 0 250 500 X (M) Fig. 9. Bending moment of VLFS with pin-connected OWCBW1, T ¼ 6 s. Z MBV1lV ¼ cVl n1 dl; l ¼ 1; 2; . . . ; N, (25) SV Z MBVklV ¼ cVl hkðxÞn2 dl; l ¼ 1; 2; . . . ; N; SV k ¼ 2; 3; . . . ; N, ð26Þ Z MBV1lB ¼ cBl n1 dl; l ¼ 1; 2; 3, (27) SV ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 701 γ=0 γ=0 γ=50 0.25 γ=100 γ=250 0.8 γ=750 γ=150 γ=2000 Without OWCBW 0.2 Without OWCBW 0.6 0.15 0.4 0.1 IMI/ρ a0 ω2 H3 B 0.05 0.2 IQI/ρ a0 ω2 H2 B 0 -250 0 250 500 X (M) Fig. 10. Shear force of the VLFS with pin-connected OWCBW1, T ¼ 6 s. γ=0 γ=250 0.5 γ=750 γ=2000 Without OWCBW 0.4 0 -250 0 250 500 X (M) Fig. 12. Bending moment of VLFS with pin-connected OWCBW1, T ¼ 8 s. γ=0 γ=250 0.6 γ=750 γ=2000 0.5 Without OWCBW 0.3 0.4 IQI/ρ a0 ω2 H2 B IWI/a0 0.3 0.2 0.1 0 -250 0 250 500 X (M) Fig. 11. Vertical displacement of the VLFS with pin-connected OWCBW1, T ¼ 8 s. Z MBVklB ¼ cBl hkðxÞn2 dl; SV l ¼ 1; 2; 3; k ¼ 2; 3; . . . ; N, (28) Z F V 1 ¼ Àior ðC0 þ CSÞn1 dl, (29) SV Z F V k ¼ Àior ðC0 þ CSÞhkðxÞn2 dl; k ¼ 2; 3 (30) SV where m(x) is the distribution of mass of the VLFS, MBVklV and MBVklB the generalized hydrodynamic coefﬁcients due to the motions of the VLFS and the breakwater, 0.2 0.1 0 -250 0 250 500 X (M) Fig. 13. Shear force of the VLFS with pin-connected OWCBW1, T ¼ 8 s. respectively, RVkl the hydrostatic stiffness matrix and F V k the generalized wave exciting force coefﬁcients. In Eq. (21), M B kl , RBkl , MBBklB, F B k are the coefﬁcients of inertia, restoring, hydrodynamic and wave exciting forces of the breakwater, respectively, MBBklV the generalized hydrodynamic coefﬁcients due to the motions of the VLFS and MBCkl, FCk are the coefﬁcients due to the air pressure in the OWC chamber (Hong et al., 2004). The geometric boundary conditions at the pin are as follows: qV1 ¼ qB1 þ ½qB3 e3 Â ðxBo e1 þ yBo e2Þ Á e1, (31) 702 1.2 1 0.8 ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 1.5 1 γ=0 γ=50 γ=100 γ=150 Sway RAO of OWCBW Vertical amplitude/a0 0.6 0.4 γ=0 γ=50 γ=100 0.2 γ=150 Without OWCBW 0 10 15 20 T (sec.) Fig. 14. Displacements at the weather-side end of the VLFS with pinconnected OWCBW2. 0.5 0 10 15 20 T (sec.) Fig. 16. Sway amplitude of the pin-connected OWCBW2. 1.2 γ=0 γ=50 0.3 γ=100 γ=150 Without OWCBW 0.2 Sway RAO of VLFS 0.1 0 10 15 20 T (sec.) Fig. 15. Sway amplitude of the VLFS with pin-connected OWCBW2. X N wðxaÞ ¼ qVl hlðxaÞ ¼ qB2 þ ½qB3 e3 Â ðxBo e1 þ yBo e2Þ Á e2. l¼2 (32) Solving simultaneously Eqs. (19)–(21), (31) and (32), qVl ðl ¼ 1; 2; . . . ; NÞ and qBl ðl ¼ 1; 2; 3Þ can be found as well as Qx and Qy, the horizontal and vertical components of the reaction force at the pin. The integrals of Legendre polynomials appearing in Eqs. (17), (23) and (24) are obtained by making use of the Heave RAO of OWCBW 1 0.8 0.6 γ=0 0.4 γ=50 γ=100 0.2 γ=150 0 10 15 20 T (sec.) Fig. 17. Heave amplitude of the pin-connected OWCBW2. formulae as follows (Prudnikov et al., 1986): Z1 PðmpÞPðnqÞ dx ¼ 0 for m À n À p þ q ¼ 1; 3; 5; . . . , À1 (33) Z1 PðmpÞPðnqÞ dx ¼ 22ÀpÀq À1 Xp Â ðÀ1ÞkÀ1ðp À kÞ!ðq þ k À 1Þ! k¼1 ! ! m nþqþkÀ1 ! mþpÀk pÀk ! n Â pÀk qþkÀ1 qþkÀ1 for m À n À p þ q ¼ 2; 4; 6; . . . ð34Þ Roll RAO of OWCBW ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 703 6 γ=0 1 γ=50 γ=0 γ=50 5 γ=100 γ=150 0.8 γ=100 γ=150 Without OWCBW 4 0.6 IWI/a0 3 0.4 2 0.2 1 0 10 15 20 T (sec.) Fig. 18. Roll amplitude of the pin-connected OWCBW2. 0 -250 0 250 500 X (M) Fig. 20. Vertical displacement of the VLFS with pin-connected OWCBW2, T ¼ 9 s. γ=0 γ=0 γ=250 γ=50 2 γ=100 γ=750 γ=150 γ=2000 Without OWCBW 3 Without OWCBW 1.5 IMI/ρ a0 ω2 H3 B Fx/ρga20 2 1 1 0.5 0 10 15 20 T (sec.) Fig. 19. Horizontal drift forces of the VLFS and the pin-connected OWCBW2. 0 -250 0 250 500 X (M) Fig. 21. Bending moment of VLFS with pin-connected OWCBW2, T ¼ 9 s. We can also calculate the bending moment M(x) and the shear force Q(x) according to the Bernoulli–Euler beam theory as follows: X N MðxÞ ¼ qVl EI hlðxÞ00, (35) l¼2 X N QðxÞ ¼ qVl ½EI hlðxÞ000. (36) l¼2 4. Time-mean horizontal drift force By making use of the near-ﬁeld method presented by Pinkster and van Oortmerssen (1977), the time-mean horizontal drift force can be obtained as Fx ¼ F B x þ F Vx , F B x ¼ À rg 4 I W B zBr ðM Þ2 n Á e1 dl þ r 2 ZZ SB & 1 2 rCT ðM Þ2 704 1.25 1 0.75 ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 γ=0 γ=50 γ=100 3 γ=150 Without OWCBW 2 γ=0 γ=250 γ=750 γ=2000 Without OWCBW IMI/ρ a0 ω2 H3 B IQI/ρ a0 ω2 H2 B 0.5 1 0.25 0 -250 0 250 500 X (M) Fig. 22. Shear force of the VLFS with pin-connected OWCBW2, T ¼ 9 s. γ=0 γ=250 1 γ=750 γ=2000 Without OWCBW 0.8 0 -250 0 250 500 X (M) Fig. 24. Bending moment of VLFS with pin-connected OWCBW2, T ¼ 12 s. γ=0 γ=250 γ=750 γ=2000 1.5 Without OWCBW IQI/ρ a0 ω2 H2 B IWI/a0 0.6 1 0.4 0.5 0.2 0 -250 0 250 500 X (M) Fig. 23. Vertical displacement of the VLFS with pin-connected OWCBW2, T ¼ 12 s. h i' þ Re DBðMÞ Á rðÀioCTðMÞÞ n Á e1 ds þ 1 2 nh Re qB3 e3 Â i FBm Á o e1 , ð37Þ F V x ¼ À rg 4 I W V zVr ðM Þ2 n h Á e1 dl þ r ZZ 2i' SV & 1 2 rCT ðM Þ2 þ Re DVðMÞ Á rðÀioCTðMÞÞ n Á e1 ds 0 -250 0 250 500 X (M) Fig. 25. Shear force of the VLFS with pin-connected OWCBW2, T ¼ 12 s. ZZ r þ Re À2 ½gDVðMÞ À ioCTðMÞÞ SV Á Â YðMÞ e3 Â n Á e1 ds, ð38Þ where the super-bar indicates the complex conjugate, FBm the inertia force of the breakwater, zBr and zVr the relative wave height along the waterline of the breakwater and the VLFS, respectively, and YðMÞe3 the rotation vector of the Bernoulli–Euler beam YðMÞ ¼ tanÀ1ðdwðxMÞ=dxÞ. (39) ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 705 γ=0 γ=50 1.5 γ=100 1.2 γ=150 Without OWCBW 1 1 0.8 γ=0 γ=50 γ=100 γ=150 Sway RAO of OWCBW Vertical amplitude/a0 0.6 0.5 0.4 0.2 0 10 15 20 T (sec.) Fig. 26. Displacements at the weather-side end of the VLFS with pinconnected OWCBW3. γ=0 γ=50 0.5 γ=100 γ=150 Without OWCBW 0.4 0.3 0.2 Heave RAO of OWCBW 0 10 15 20 T (sec.) Fig. 28. Sway amplitude of the pin-connected OWCBW3. γ=0 1.2 γ=50 γ=100 γ=150 1 0.8 0.6 0.4 0.2 Sway RAO of VLFS 0.1 0 10 15 20 T (sec.) Fig. 27. Sway amplitude of the VLFS with pin-connected OWCBW3. Since the problem is two-dimensional, the integrals along the waterlines and those over the wetted surfaces can be done for unit breadth. 5. Numerical results and discussions The horizontal length LV of the VLFS is 1000 m and the draft dV is 2 m. It has a uniform mass distribution with Young’s modulus of 2.06 Â 1011 Pa and its moment of 0 10 15 20 T (sec.) Fig. 29. Heave amplitude of the pin-connected OWCBW3. inertia per unit breadth is 0.666 m3. In order to study the hydroelastic response reduction performance of the pinconnected breakwater, three numerical models, OWCBW1, OWCBW2 and OWCBW3, are tested. Their principal dimensions in meters are shown in Table 1 where yG denotes the y coordinates of the center of gravity of the breakwater, RG the roll radius of gyration measured from the pin and tW and tB the thickness of vertical and horizontal plates, respectively. The rotational motion and the moment of the breakwater are calculated with respect to the pin position. The distance between the VLFS and the breakwater ds is 2.5 m. The period of incident waves T ARTICLE IN PRESS 706 D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 4 1 γ=0 γ=50 0.8 γ=100 3 γ=150 Without OWCBW 0.6 IWI/a0 Roll RAO of OWCBW 2 γ=0 1 γ=50 γ=100 γ=150 0 10 15 20 T (sec.) Fig. 30. Roll amplitude of the pin-connected OWCBW3. γ=0 γ=250 γ=750 γ=2000 4 Without OWCBW 3 0.4 0.2 0 -250 0 250 500 X (M) Fig. 32. Vertical displacement of the VLFS with pin-connected OWCBW3, T ¼ 13 s. γ=0 γ=50 4 γ=100 γ=150 Without OWCBW 3 2 2 IMI/ρ a0 ω2 H3 B Fx/ρ g a02 1 1 0 10 15 20 T (sec.) Fig. 31. Horizontal drift forces of the VLFS and the pin-connected OWCBW3. in a water of 30 m deep ranges from 6–20 s with a computing interval of 1 s. The length of the Bernoulli–Euler beam is discretized into 1200 line segments. It should be noted that the numerical results, in particular the shear forces which must be equal to the vertical reaction forces at the pinned end, converge when the number of modes of the Legendre polynomials reaches 22–40 for the present study. The amplitude of vertical displacements of the VLFS at the weather-side end where the breakwater is connected by the pin is presented for the three pin-connected break- 0 -250 0 250 500 X (M) Fig. 33. Bending moment of VLFS with pin-connected OWCBW3, T ¼ 13 s. waters, respectively. The vertical displacement of the VLFS without breakwater is also presented in each ﬁgure to show how much the hydroelastic responses of the VLFS can be reduced. The value of the dimensional damping parameter g (kg-mass/m2 s) in ﬁgures is related with the air pressure in the OWC chamber which can absorb the incident wave energy (Hong et al., 2004). Thus additional reduction of the hydroelastic responses of the VLFS can be achieved with an appropriate value of g. The response amplitude ratios (RAO) of vertical displacements of the VLFS at the pin with OWCBW1 is ARTICLE IN PRESS D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 707 γ=0 γ=0 γ=50 2 γ=100 γ=250 5 γ=750 γ=150 γ=2000 Without OWCBW Without OWCBW 4 1.5 IMI/ρ a0 ω2 H3 B 3 1 2 0.5 1 IQI/ρ a0 ω2 H2 B 0 -250 0 250 500 X (M) Fig. 34. Shear force of the VLFS with pin-connected OWCBW3, T ¼ 13 s. γ=0 γ=250 1 γ=750 γ=2000 Without OWCBW 0.8 0.6 IWI/a0 0.4 0.2 0 -250 0 250 500 X (M) Fig. 35. Vertical displacement of the VLFS with pin-connected OWCBW3, T ¼ 16 s. presented in Fig. 2. It shows that OWCBW1 works well for T ¼ 6 and 7 s. Sway RAO of VLFS is presented in Fig. 3. There are small differences between the sway RAOs of VLFS with and without OWCBW1 for T ¼ 6 and 7 s. In Figs. 4–6, the RAOs of breakwater motions are presented. The time-mean horizontal drift forces are presented in Fig. 7. The drift force with pin-connected OWCBW1 for g ¼ 0 is shown to be more than three times greater than the drift force without breakwater when T ¼ 8 s but, it becomes smaller for g ¼ 750. When T48, the drift forces for gX750 become negative. The reverse drift forces shown in this 0 -250 0 250 500 X (M) Fig. 36. Bending moment of VLFS with pin-connected OWCBW3, T ¼ 16 s. ﬁgure are due mainly to the contribution of F B x , the drift force of the OWCBW1, given by Eq. (37). The lengthwise distribution of vertical displacements, bending moments and shear forces of the VLFS for different values of g when T ¼ 6 s are presented in Figs. 8–10, respectively. The role of g is shown to be negligible at this period. In Figs. 11–13, the vertical displacements, bending moments and shear forces of the VLFS for gX250 when T ¼ 8 s are presented. Here, the role of g is shown to be important. However, when TX9 s, the role of g is hardly signiﬁcant and the related numerical results are not presented here. The RAO of vertical displacements of the VLFS at the pin with OWCBW2 is presented in Fig. 14. It shows that OWCBW2 works well for T ¼ 6, 8, 9, 10 and 11 s. Sway RAO of VLFS is presented in Fig. 15. There are small differences between the sway RAOs of VLFS with and without OWCBW2 for Tp10 s but the sway RAOs of VLFS with OWCBW2 grows rapidly for T410 s. In Figs. 16–18, the RAOs of breakwater motions are presented. The time-mean horizontal drift forces are presented in Fig. 19. The drift force with pin-connected OWCBW2 for g ¼ 0 is shown to be more than 10 times greater than the drift force without breakwater when 10pTp12 s but, it becomes smaller for g ¼ 2000. The reverse drift forces are also shown for longer periods. The distribution of vertical displacements, bending moments and shear forces of the VLFS for different values of g when T ¼ 9 s are presented in Figs. 20–22, respectively. The role of g is shown to be negligible at this period. In Figs. 23–25, the vertical displacements, bending moments ARTICLE IN PRESS 708 D.C. Hong, S.Y. Hong / Ocean Engineering 34 (2007) 696–708 γ=0 γ=250 2.5 γ=750 γ=2000 Without OWCBW 2 IQI/ρ a0 ω2 H2 B 1.5 1 0.5 0 -250 0 250 500 X (M) Fig. 37. Shear force of the VLFS with pin-connected OWCBW3, T ¼ 16 s. and shear forces for gX250 when T ¼ 12 s are presented. Here, the role of g is shown to be important. However, when TX13 s, the role of g is not remarkable and the related numerical results are not presented here. The vertical displacements of the VLFS at the pin with OWCBW3 is presented in Fig. 26. It shows that OWCBW3 works well for T ¼ 6, 8, 9, 11–14 s. The role of g is shown to be negligible for Tp11 s but it becomes important for TX12 s. The sway RAO of VLFS is presented in Fig. 27. There are small differences between the sway RAOs of VLFS with and without OWCBW3 for Tp12 s but the sway RAOs of VLFS with OWCBW2 grow rapidly for TX13 s. In Figs. 28–30, the RAOs of breakwater motions are presented. The time-mean horizontal drift forces are presented in Fig. 31. The drift force with pin-connected OWCBW3 for g ¼ 0 becomes more than 10 times greater than the drift force without breakwater when T ¼ 12 s. It reaches its peak value when T ¼ 14 s. But, the drift force with pin-connected OWCBW3 for g ¼ 750 is always smaller than the drift force without breakwater and it becomes negative when TX12. The vertical displacements, bending moments and shear forces of the VLFS for different values of g when T ¼ 13 s are presented in Figs. 32–34, respectively. In Figs. 35–37, the vertical displacements, bending moments and shear forces for gX250 when T ¼ 16 s are presented. Here, the role of g is shown to be important. However, when TX17 s, the role of g is not remarkable and the related numerical results are not presented here. The reverse drift forces of backward-bent duct buoy (BBDB) have been shown by McCormick and Sheehan (1992) experimentally. Hong et al. (2004) have shown it numerically where the reverse drift force occurs for g ¼ 0 and its absolute value becomes smaller as g grows. Here, the shape of the OWC chamber is rather forward bent and the drift forces for g ¼ 0 are always positive for the three OWC breakwaters. For the present OWC chambers, the reverse drift forces occur for g6¼0 and their absolute values become greater as g grows. 6. Conclusions An analytical method to calculate the hydroelastic responses of a VLFS equipped with a pin-connected breakwater with an OWC chamber, taking account of the geometric and dynamic boundary conditions at the pin, has been proposed. Numerical results for three pin-connected breakwaters show that each of them reduces signiﬁcantly the hydroelastic responses of the VLFS in a frequency range pertinent to its size. Additional reduction of the hydroelastic responses has been achieved by the absorption of the incident wave energy by the OWC chamber with an appropriate value of g. The time-mean horizontal drift forces of the VLFS equipped with the OWC breakwater calculated by the nearﬁeld method are also presented. The present analytical method and numerical results will be useful to design a pin-connected breakwater with or without an OWC chamber. Acknowledgments 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 Hong, D.C., Hong, S.Y., Hong, S.W., 2004. Numerical study on the reverse drift force of ﬂoating BBDB wave energy absorbers. Ocean Engineering 31 (10), 1257–1294. Hong, D.C., Hong, S.Y., Hong, S.W., 2006. Reduction of hydroelastic responses of a very-long ﬂoating structure by a ﬂoating oscillatingwater-column breakwater system. Ocean Engineering 33 (5–6), 610–634. McCormick, M.E., Sheehan, W.E., 1992. Positive drift of backward-bent duct barge. Journal of Waterway, Port, Coastal and Ocean Engineering (ASCE) 118 (1), 106–111. 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. In: Proceedings of the Third Internatuinal Workshop on VLFS, Honolulu, USA, pp. 265–274. Pinkster, J.A., van Oortmerssen, G., 1977. Computation of the ﬁrst and second order wave forces on oscillating bodies in regular waves. In: Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, pp. 136–159. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I., 1986. Integrals and Series, vol. 2: Special Functions. Gordon and Breach Science Publishers, p. 448. 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, A., Ohta, H., Hayashi, N., 2003. Wave response analysis of VLFS with an attached submerged plate. International Journal of Offshore and Polar Engineering 13 (3), 190–197.

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