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Nonlinear Motions and Whipping Loads of High-Speed Crafis in Head Sea S.-K. Chou, F.-C. Chin, Y.-~. Lee (National Taiwan University, China) AssTiACT In this paper, the nonlinear motions and wave loads including whipping effects of high-speed crafts traveling in head sea are investigated theoretically and experimentally. The analysis is performed on an existing large-sized high-s- peed craft, following a modified nonlinear strip method and treating the ship's hull as an elastic beam from the viev-point of hydroelast- icity. The ship's hull is regarded as an Euler beam or Timoshenko beam and the structural response is represented by modal superposition method and finite element method g separately. The elastic backbone model testing technique is adopted to carry out the experiments for measu- ring vertical bending moments acting on ship's hull . Through the comparison with experimental results, the validity of the present calculati- on method is confirmed, and through serial calculations,the influence of structural rigid- ities on wave loads are also clarified. INTRODUCTION Ship motions and wave loads of a displacement type ship in small amplitude waves can be estimated satisfactorily by the linear strip theory t-3. Dynamic behaviors of a displacement type ship suffered serious slamming in rough seas can be investigated by a nonlinear strip theory developed by Yamamoto, Fujino, and Fukasawa4, taking account of nonlinearities caused by hydrodynamic impact, the ship hull's shape and configurations. Dovever, when a high-speed craft travels even in the moderate sea condition, nonlinear characteristics of ship motions and wave loads get significant because of high-speed traveling in waves. Several years ago, Chin, one of the authors, and Fujino5~8 developed a practical method, which is in principal based on the conventional Ordinary Strip Method synthesis but modified to be able to evaluate nonlinear hydrodynamic impact forces as well as dynamic lift in waves, for calculating vertical motions and wave loads of a high-speed craft which travels in regular head sea, and its validity was verified by comparing the computed motion and wave loads ls7 with experimental results performed by using a ship model of hard chine type as well as a ship model of round bilge type. Recently, it was confirmed further that this method also can be applied even to estimated vertical motions of fishing vessels in head sea With accuracy enough for practical user. In the studies on nonlinear motions and wave loads of a high-speed craft mentioned at the above, the ship's hull is treated as a rigid body. However, in general, the size of high-sp- eed craft has increased significantly8-~, and the occurrence of not only local damage due to serious slamming, but also whole structural damage caused by the subsequent whipping of the hull should be possible in case of a high-speed craft of large size. fence, it becomes importa- nt to investigate the influence of hydro-elast- ic interactions on the structural responses of a large-sized high-speed craft in Haves-. In this paper, the authors investigate the nonlinear characteristics of ship motions and whipping loads of high-speed crafts theoretica- lly and experimentally. The analysis is perfor- med on an existing large-sized high-speed craft, following the above-stated modified nonlinear strip method basically, but extended to treat the ship's hull as an elastic beam, from the view-point of hydroelasticity. That is to say, the ship's hull is regarded as an Euler beam or Timoshenko beam and the structural response is represented by modal superposition method and finite element method, separately. The experiments are carried out by using an elastic backbone model. In order to generate pronounced whipping loads acting on model, rigidity of the elastic backbone is selected more flexible than that should be scaled down directly from the actual ship. Comparing the results of serial calculations of different structural representation methods with results obtained in elastic backbone model experiments, the influence of wave length, wave height and advance speed on wave loads are discussed. Furthermore, the influence of hull vibration , which is related to flexural rigidity and shear rigidity of hull structure, are also examined.

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2.THEOlY In the previous formulation of nonlinear vertical rigid-body-motions of high-speed crafts running in head sea, nonlinearities of hydrodynamic forces acting on the ship hull are assumed to be exclusively due to the time- variation of submerged portion of the hull. This approach is followed basically in this paper, except the ship hull's girder is discre- tized into Timoshenko beam for considering both bending and shear deformation, and the formula- tion of finite element method is used to take both low frequency and high frequency vibration into consideration. The formulation of the present method is highly similar to that descr- ibed in detail in the referencess'~'7.Therefore, for convenience sake, the basic concept of the method will be subsequently described briefly. 2.l Coordinate System and Incident Wave The coordinate systems and the sign conventi- on of translational and angular displacements used hereafter are shown in Fig.~. A space-fix- ed Cartesian coordinate system O-lYZ is introd- uced so that the X-Y plane coincides with the still water surface and the Z-axis directs downward. The ship advances in the negative ~ direction at a constant speed V. Another coord- inate system o-xyz is ship-fixed with origin o located at the center of gravity of the ship and the x-axis parallel to the base line of the ship. ri is the initial trim, and rs and (s are the increments of trim and sinkage due to steady running In calm water respectively, While ~ and ~ denote the variation of pitch angle and heave displacement in waves respectively. The counter clockwise rotation around the y-axis and down- ward heave displacement are regarded as positi- ve. The incident wave (w is described as (W=(acos (AX COST+ sinr+wet) in the ship-fixed coordinate system. For small I, the wave profile can be approximated by (W=(acos(~x+~et)' and the sub-wave profile can be similarly approximated by cosh~(h-zal (c=(a cosh~h---= CS(~X+wet) where (a is the wave amplitude, ~ is the wave number, h is the water depth, me is the encoun- ter frequency defined by ~e=~+~V and ~ the wave frequency. za is the instantaneous draft at section x and expressed by z~=d-x~tan(rs+~+~`s+~+w<-(w)/cosr (~) where d is the sectional draft of the ship without forward speed in calm water ~ is defined as r=ri+rs+8 WV is the displacement induced by elastic vibr- at~on. Hi. ~ or ~-,.;^~~-x t . ~ Fig.! Coordinate system 2.2 Sectional External Forces The velocity of wave particles relative to the ship's hull can be divided into two compon- ents, Ur and Vr, which are parallel to x- and z-axes respectively. By assumption of orbital velocity component of wave particle in X-direc- tion is negligible and high order term dropped. Ur and Vr are approximately expressed by U=-Vcosr Van ( (e-() C O S Jinx REV S inT=\Jv (2) Meanwhile, the two components of relative velocity for a ship running in calm water are expressed by UO=-V ~ cos ( [i+Ts) Vo=-V sin(ri+rs) (3) In deriving the hydrodynamic forces, the state of steady running in calm water is consi- dered as the initial reference condition. From this point, the z-direction relative velocity Vr is expressed as follows Vm (fir-Vo) ~Vo =(`e-~)cos7+xt~wv-visinr-sin(ri+rs)) -Ysin(ri+rs) (4) thus Or is separated into one steady term TO associated with running in calm water and the remaining oscillatory term Vr-VO due to waves. 2.2.1 Sectional Force due to Change of Fluid Momentum - Denoting the heave added mass of a tranverse section located at x for oscillatory motion with pSztx,t), and the sectional heave added mass for steady running in wave and in calm water with pSztx,t) and pSzO(x), respectively, then the sectional hydrodynamic force due to the time variation of fluid momentum, when a ship is traveling in waves, can be described as follows is8

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f m=a~{pSz (x, t ) ( Vr-V a) +pS z (x, t ) V O-pS z O (x) V 0} (5 ) In equation (5), pSz(x,t) and pSz(x,t) are both evaluated by the instataneous submerged portion of the ship section while running in Haves. The last term pSzO is evaluated by the submerged port ion of the ship sect ion while running in calm water. The operator ;~ can be expressed as =~U~Vcosr~, thus the sectional force due to change of fluid momentum can be decompo- sed into the following five components ~ T f m=f ma+f mj +f mj +f imp+f i mp ( 6 ) Where f ma =pSz fix, t ~ EVE fmj =-Vcosr-8PS (x,t) . ~Vr-VO~ f no =-Vcos rt dpS`'o (x, t ~ _ Ups z O (x) ~ V f imp= - x ~ (fir-To) f ~mp=P5dx~x~t).Yo (7) The physical meaning of each term in the r.h.s of equation (6) are as follows fma: sectional hydrodynamic inertia force fmj: hydrodynamic force due to longitudinal variation of sectional heave added mass associated with vertical oscillatory velocity * fmj hydrodynamic force due to longitudinal variation of sectional heave added mass associated with vertital steady velocity component of constant forward speed. limp: sectional impact force due to time variation of sectional heave added mass associated with the vertical oscillatory velocity limp sectional impact force due to time variation of sectioal heave added mass associated with vertical steady velocity of constant forward speed. 2.2.2 Sectional Damping Force In a similar manner, the sectional damping force fr is expressed as follows * * f~pNz(x,t) (Vr-Vo)+pNz(x~t)vo-pNzo(x)vo (8) where the sectional heave damping coefficient pNz(x,t), pNz(x,t) and pNzo(x) have the physic- al meanings analogous to those of sectional heave added mass pSz(x,t), pSz(x,t) and pSzO(x) , respectively. 2.2.3 Restoring Force and Froude~rylov Force The sum of sectional restoring force and Froude-Krylov force can be approximatly expres- sed as follows fs=-pg{A(x,t)-Ao(x))cosr (9) where p is the fluid density and 9 is gravitio- nal acceleration AD is the sectional area of the portion under the undisturbed still Hater surface and A(x,t) represents the sectional area of the portion under the undisturbed effective incident wave surface by considering Smith Correction. The sectional external force in total is obtained by summing the force components stated above, ie. fz=fm~fr~fs 2.3 Equation of lotion (10) The displacement and rotation angle of a ship's section, which includes the vibration component as well as the rigid-body-motion component, are denoted by ~ and , respectivel- y. Then the bending strain ex and shear strain can can be expressed by ax= 7=~ '-,~ Fig.2 Displacement representation (11) The kinet ic energy T. strain energy V, and Work done by external forces W can be describ- ed as follow T= ~ lp(~2dX (12-~) Where ,~ denotes the sectional mass of ship's hull V= Vbi Vs (12-2) where Vb, denotes the strain energy due to bending deformation, is expressed by Vb==lEI(~) dx (12-3) Vs. denotes the strain energy due to shear deformation, is expressed by 59

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VS=-~-IGAw(~-~) dx (12-4) EI and GAW are sectional flexural rigidity and shear rigidity , respectively W=-~-liz~dx (12-5) In this manner, variational approach can be introduced to derive the Eamilton's Principle o.s o.o [( T-V+ ~-~0 (13) _o.s where D denotes the dissipation function and [D is described by be= fib ~ ~ Vb+ As ~ VS ~ = ~ W=W ( 13 - 1 ) where fib and As are the structural damping coefficients corresponding to bending and shear deformation, respectively. The total displacement of a ship's section, a, can be expressed by a linear combination of N coordinate functions Wj as follows Mode Shape Function /: // ~_` /~\,,';;% ,//,/,/i it, / \ \~,,~/ `, ~ / / S. S. Fig.3 lode shape functions of 3rd-Sth mode X, X~ X3 _ 1 1 1 > X O t/2 l w = ~ Wj(X)-~(t) (14) ov, ~ ~v2: ~v3 J_ _ we wv2 wv3 where qj's are the generalized coordinates, and j=1,2 denote the rigid body motions correspond- ing to heave and pitch, j>3 are related to vibration components. 2.3.1 Nodal Superposition Method Form_lation Por modal superposition analysis, the mode shape functions wj are obtained by lyklestad'- St3 method for a free-free Euler beam, the first 4 mode shape functions corresponding to vibration deformation, j=3-S are illustrated in Figure 3. Equations of motion derived from Equation (13) by applying Galerkin method can be expres- sed in matrix form as follows tdij]{qj)~tCij]{qj)+tKij]~={fi) (~5) where the assumption =~ is used and detailed expression of generalized mass, damping and stiffness matrices, :~],~C],tK] and generalized force function {f3 are summarized in the Appen- d~x A. 2.3.2 Finite Element Method Formulation In finite element method, the generalized coordinates It's correspond to nodal displacem- ents. The vibration components of w and , denoted by TV and ~v, can be approximated in terms of {qv), ie. |vVll = ~N]{qv~i (16) vJlx=xj-xj+~ {qv), the degrees of freedom at node point Fig.4 Degree of freedom in element within the j-th element, are {qv~j = ~qwvt ~qevJ (16-1) and the corresponding interpolation functions tN] are set to be quadratic forms END = tg: Where ~wv] {qwv~j = Wv21 Nv3] j (16-2) riv; (16-3) {qov}j= lpv2 (16~) [NV]=[1-36+262 46~2-~+262 0 0 0 ] (16-5) [N~]=[O O 0 1-3~+262 46~2 _~+262 ] (164) X-X j = x (16-7) The resultant equations of motion in terms of total degrees of freedom combined by different elements can be expressed similarly in follovi- ng matrix form: [11 ] {q }+ [C ] {q}+ [K ] {q}={f } (17) The detailed expressions of various elements included in the coefficient matrices t11 ], tC ], OK ], and force vector {I } are summariz- ed in the Appendix B. 160

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a. NUdERICAL SOLUTION The numerical values of instantaneous sectio- nal hydrodynamic coefficients are required for various sections of the ship to calculate the dynamic responses of a ship in waves. The evaluations for these coefficients are performed under the following assumptions: Blithe sectional hydrodynamic coefficients for heave motion in the z-direction are assumed to be equivalent to those in the Z-direction. (2)The sectional hydrodynamic coefficients corresponding to oscillatory motion are evalua- ted at encounter frequency He for the part due to rigid body motion, and those at infinite frequency are used for the part due to vibrati- on. (3)The sectional hydrodynamic coefficients corresponding to steady forward velocity are evaluated at infinite frequency under the high-speed condition. (4)The sectional hydrodynamic coefficients are evaluated for the instantaneous submerged portion under the undisturbed wave surface. The nonlinearities of hydrodynamic forces related to the time-varying sectional hydrodyn- amic coefficients and hydrodynamic impact forces are treated in such a manner as describ- ed subsequently. The sectional hydrodynamic coefficients at several different prescribed drafts of a secti- on are computed by Frank close-fit methods for each transverse section. Those hydrodynamic coefficients for different drafts are expressed by a polynomial of n-th order as a function of the instantaneous sectional draft. taking use of such a polynomial expression, the sectional hydrodynamic coefficients are evaluated at each time step during numerical integration of equations of vertical motion. If some section is clear of water at.a certain time step, then during the following re-entry stage, water surface pile-up is consi- dered according to Wagner's wedge impact theor- yt~ that is to say, the instantaneous draft is assumed to be r/2 times of that under undistur- bed Wave surface. This consideration will be disregarded when the pile-up water surface cross over the chine. Furthermore, it is assum- ed that the hydrodynamic impact force is able to be disregarded when the ship section is detaching from the Hater. The validity of this assumption is confirmed by the results of the forced oscillation test performed by Yamamoto, Fujino and Ohtsubotfi. The bottom impact and flare impact are evaluated with different schemes of computing the rate of change of the added mass. The structural damping coefficient fib and ~ are set to be of same value and can be express- ed by pb=ys= r ~ (18) where ~2V is the natural frequency of 2-node vibration and ~ is the corresponding logarithm- ic decrement. For the numerical integration of the equatio- ns of motion, Newmark-4 method with =/4 is used, and the discrete time increment it adopt- ed for time integration is 1/500 of the encoun- ter period. From the view point of the stability of numerical integration, the encountered wave amplitude grows up gradually to steady state during the calculation and the dynamic respons- es are recorded only after stationary state motion is reached. Wave loads can be estimated either by the integration of fz- ~ along the ship's length or by the evaluation from differential formula in terms of W. the discrepancy between these two methods is insignificant and the former method gives more consistent results. Hence, the calculation results presented in this paper are all obtained by applying the integration evaluation, exclusively. 4. EXPERIlENTS AND NDlERICiL PREDICTION 4.1 Elastic Backbone Model In order to investigate the sectional Have loads along the ship's length and verify the validity of the numerical prediction method described in the previous sections, elastic backbone model testing techniques' t7 has been selected for experiments. ~ model in scale 1:14.S of an existing 44.5 meter high-speed craft of hard chine type is used. The principal particulars and body plan are shown in Table ~ and Figure 5, respectively. it square station 1 to 8, the model made of wood, is divided into 9 segments which are connected with a backbone composed of 2 aluminum alloy (6063) beams as shown in Figure 6. The bending and shear rigid Unit mm 3 4 5 1 ~ 1~-o 11 1 2 o o 1<- 3750 - ~ 'A ;~ /~ Fig.5 Body plan of Boat-4450 Length Overal 1 L Breadth ( I) B Depth ( ~ ) D Draft ( 38~) d Displacement W Longitudinal Position of C.G. LOG __ _ Longltudinal Gyradius Yk ~- 44.50 m 7 Fin m . . ~ _ 3.50 m 1.58 m 220.0 ton 2.05 m aft 3E - 27.1 AL . Table ~ Principal particulars of Boat - 450

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_,B EI IHI I IN I IHI I F1 Id IHI IHI I IHI ~ IHI I ~g o 8 ' ~I:~ AS 1 2 A 3 4 5 s 6 78 \ FP 0.7 J Backbone , ~ ~ ~ ~ - 1: r i -my ~ ~ detail of A A ~.2scm2 131 1 g7g Section B-B ton/m \ - ~31 1 Ir1 3?9 6.0 (/ tB OCR for page 157
on modes) while in some comparison cases, 4 mode shape functions calculation called "IODE 4") and even only the first two rigid-body-mot- ions mode shape functions calculation (hence called "RIGID") are also performed. (b)Finite element calculation called "F.E.~." or "EULER") In this kind of calculation, if Timoshenko beam element formulation is used to evaluate the dynamic response of the ship hull's struct- ure, the notation "F.E.~." is adopted. The calculation in terms of Euler beam element formulation by neglecting the &hear deformation is called "EULER" for distinction. 5. COlPARSION BETWEEN NDIERICAl PREDICTION AND EXPERIMENTAL RESULTS Figures 9 to 18 illustrate the nondimensiona- lized peak-to-peak bending moment distribution along the ship's length under various wave and speed conditions. In Figures 9 to Il. the experimental results for the case of Fn=0.35 which may be considered as a typical speed of "non-planning" condition are shown together with the results predicted by the two kinds of numerical computations, namely, DIODES " and "F.E.~.". Both of the predicted values by modal superposition calculation "IODE-6" and finite element calculation "F.E.~." agree well with the experimental results. In the cases of Fn=0.70 and i.0 which may be considered as a typical speed of "semi-planning" and "planning" condition respectively, shown in Figures 12 to 18, the predicted values by "IODE-6'r are satis- factory, except for the cases of relatively short waves, in which it tends to underestimate the wave loads acting on the fore-bodies. Nevertheless, the discrepancy in fore-bodies is improved significantly by the "F.E.~." calcula- tion. As seen in these figures , it can be said that both of "MODE-6" and "F.E.~."calculations give reasonable results, and the agreement between the predicted responses and experiment- al results seems satisfactory enough for the practical point of view. Figure 19 illustrates the forward speed dependence of bending moments at various square stations 4 to 7 in the selected wave condition of A/L=~.5 and Hw/l=l/4o. In Figures l9 (e) and Figure l9(b), the nondimensionalized peak to peak bending moments obtained from experiments are plotted together with predicted results by "NODE-6" and "F.E.~." calculations, respective- ly. As shown in these figure, the discrepancy in the trend between the "Iode-6" prediction and the measured responses tends to be signifi- cant in the speed range of Fn=0.70 to 1.0. However, the predicted values obtained by "F.E.~." calculation and experimental results show qualitatively similar trends in full speed range, and their agreement in values is also remarkable. In order to manifest the validity of the present nonlinear prediction of responses , the time histories of bending moments at square station 1 to 8 as well as C.G. acceleration and bow acceleration obtained by "F.E.~." calculat- ion are shown together with the measured ones 0.06 . A_ 0.05 - 0.04 0.03 0.02 0.01 Fn = 0 .3 5 Hlr/A Esp. Mode-6 F.E.1L A/L = 1.125 1447 D o.ooo. o~z.lo 3.10 4.10 5.10 6.0 7.0 8.0 /,,'' A, /,,'' i,' 8 ff~. /,~' "" ~ "A ho it\\ " \\ " \\ "\\ " \ \ "\ \ ~ ~ \ I I I i-- ~ on inn Fig.9 Longitudinal distribution of vertical bending moment (Fn=0.35,)/~=1.125) 0.06 - $ 0.05 be Q - 0.04 0.03 n no ~ . ~ ~ ~ 0.05 Not 0.04 0.03 c~ 0.02 PA o 0.01 0.000 o 163 Fn = 0.3 5 H~/A EYP. Mode-6 F.E.M. \/L = 1.50 1~40 0 _ 0.02 0.01 o.oo o. O / ~ ~ 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1 0.0 S. S. Fig.iO Longitudinal distribution of vertical bending moment (Fn=0.35,1/L=~.5) Fn = 0 .3 5 | Her/)< Exp.Mode-6 F.E.M. \/L = 1.70 1 1440 a__ _ 1 1.0 2.0 3.0 4.0 5.0 o.Oi.D 8.0 9.0 1 0.0 S. S. Fig.ll longitudinal distribution of vertical bending moment (Fn=0.35,1/L=~.7)

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0.06 0.06 _ _ - m 005 ~ N~ 06 Q 0.04 _ 0.03 _ 0.02 _ (4 ~0.01 ~^,\m Fn = 0 .7 0 Hlr/A Exp. Mode-6 F.E.M. A/L = 1.125 1549 0 ~ ~8 //, ~ \ \ /,' //' //,' I'o ~,,~ _ /.' ~i~\. 0.0 1.0 2.0 3.0 4.0 5.0 6.07.0 8.0 9.0 10.0 S. S. Fig. 12 Longitudinal distribution of vertical bending moment (Fn=0 . 70 ,1/L=1 . 125) 0.07 0.06 ~_ 0.05 0.04 0.03 ~ 0.02 P~ o 0.01 0.00, Fn = 0 . 7 0 Hw/A Esp. Mode-6 F.E.M. \/L= 1.50 1/26 1/31 ~ 1/40 0 -- 1/50 0 - - 0 0 ,P- _ /-~ `,_` \ ~ ~ 1 1 1 d v.u 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 S. S. Fig. 13 Longitudinal distribution of vertical bending moment (Fn=0.70,I/L=~.5) _ - m 0 05 bD 0.04 Fn = 0 .7 0 H'r/A Exp. Mode-B F.E.M. )~/L = 1.70 1551 0 _ - _ ~ 0 0.03 - /~ / / ~0 n '. \o 0.02 a) o 0.01 0.00 0 ~' o - '"~ ~1 1 1 1 1 1 ~ _.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1 0.0 S. S. Fig. 14 Longitudinal distribution of vertical bending moment (Fn=0.70,1/L=~.7) _% m 005 0.04 0.03 0.02 c) o 0.01 P~ ~ ^^ . Fn = 0.70 H'r/A Exp. Mode-B F.E.M. \/L = 2.00 1S45 O .~ /,' 8 :,,~a ,/''B "~\ "\ \ "~4 '`~\\ u.uu 0. ~ 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 .0 S. S. Fig. 15 Longitudinal distribution of vertical bending moment (Fn=0 . 70, )/L=2 . 0) n n7 . _ . 0.06 _ ~_ m 005 ~o 0.04 0.03 0.02 0.01 000 - 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Fn = 1 . O O H'r/~` EYP. Mode-B F.E.M. ~/L = 1.50 1S540 o / 1: ~ _~' / 1 1 1 1 1 1 1 ~ . \ '>\ '\\ 1 s. s. Fig.16 Longitudinal distribution of vertical bending moment (Fn=~.O,I/~=l.S) 0.06 0.05 bO Q 0.04 0.03 0.02 o ~0.01 P~ nnn . . _ Fn = 1.00 H,r/A Exp. Mode-B F.E.~. A/L = 1.70 1/3Gt 0 - - _ _ ~o /-_ \ /'~ ~ 'o\ . '~ \~ I t .-- o. ~t 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 l 0 0 S. S. Fig.17 Longitudinal distribution of vertical bending moment (Fn=~.0,1/~=~.7) 164

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o.o6r Fn = 1. O O H'r/A Esp. Ilode-B F.E.~. A/L = 2.00 1~40 O - ~ 0.05 an 0.04 _ 0.03 _ ~ 0.02 PA 0.01 0.00, i' Pro" ,~/ ~ ,~ ~ 1 1 1 1 ~ 1 1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 S. S. Fig.18 Longitudinal distribution of vertical bending moment (Fn=l.O,}/L=2.0) 0.07 0.06 A_ ~ 0.05 g - 0.04 0.03 0.06 - m by Q 0.04 0.05 0.03 0.02 P. o 0.0 . 0.00 t in both cases of Fn=0.70 and 1.0 in Figures 20 and 21, respectively. Although, the time histories of bending moments obtained by "F.E.~." calculation show slight difference of shape at hogging conditio- n, from that obtained by measurements, as seen in Figures 20(a) and 21(a), the predicted time histories of bending moments and accelerations agree qualitatively well with the measured ones. The plausible reason of the discrepancy in time histories at hogging condition, in which the bow sections emerge from the water surface, between the predicted and the measured bending moments is that the incoming wave surface is assumed to be undisturbed even when the ship travels in waves at a high speed, namely, the effects of spay while planning occurred are not taken into consideration. 6. EFFECTS OF STRUCTURAL RIGIDITIES . A SL 1.5 B~p. IlODB-i, ~ 0.02 4) O.01 /: /// /~/! ire/ o/ . 000 , 1 1 1 , 1 1 1 , ~ , 0. ~ 0.2 0.4 0.6 0.8 1.0 1.2 Fn ~ Vs/~L ~ Fig.19(a) Effect of ship speed on vertical bending moment ("IODE-~" calculation) 0.o7r 1 Hw; 2 ~- 1/40 S S. 4 S.S.6 S.S.7 _ _ :~.' _ . 11 , 1 , 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fn ( Vs/~L ) Fig.l9(b) Effect of ship speed on vertical bending moment ("F.E.~." calculation) Exp. F.E.5I. o . ~ ,/ ,,~ ~ / , ,',/ orb. / ~ try \A A' As clarified in the previous sections, it can be said that vertical wave loads, in which whipping loads are included , acting on a high-speed craft traveling in head sea, can be predicted by the present "F.E.~." calculations with accuracy enough for the practical use. F7 0 70 Exp. F.E.M. Hw/~: 1/40 ~,~ ~ ~ ~;S.S.~ S.S 1 I ~ t Full Scale 2 SEC Fig.20(a) Time histories of bending moment by experiment and calculation (Fn=0.7~/L=1.5 7 Hw/~=1/40) 165

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Fn : 1.00 A/L 1.50 Exp . F.E .M ~ ~ \: V ~ Bow C .G . 2 SEC Full Scale Fig.20(b) Time histories of bow acceleration and acceleration at C.G. (Fn=0 ~ 7, A/L=1 5,Hv/~=1/40) Fn : 1.00 A/L: 1.50 Exp. F.E.M. Hw/~: 1/40 AS Ace: A-; :~ \\ ~ IS.S 6 ~ A 1.,~ ~ if' V; ~ ': ':I'~S.S7~: \/ ~ (\~ is.sV2\/\~ ~S.S~ 0 .,4 p: 0 , ' 1 ~ , 0 up I ~t Full Scale 2 SEC Fig. 21 (a) Time histories of bending moment by experiment and calculation (Fn=~.O,)/L=1.5,Hw/~=~/40) Fn : 0.70 A/L 1 50 Exp. F.E.M ,_ ~ : ~ ~ c ~ ID ~C G. 0 P I ~ t Full Scale 2 SEC Fig.21(b) Time histories of bow accelertion and acceleration at C.G. (Fn=t 0, l/L=1 5 ,Hw/l=1/40) Fn : 0.70 ~ Bow Bending Moment /L 1~50 Acceleration at S.S.7 Hw/~: 1/20 ~ RIGID ~< MODE-4 ~,~ MODE-6 \ < EULER F. E . M. j Actual Ship ,= A o _4 a) m o 1 cd on Fig.22 Time histories of accelertion at C.G. and bending i moment at S.S.7 by various calculations (Fn=0 7, A/L=! 5, Hv/~=~/20) 166

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0.030 ~ 0.040 no30 0.020 _ A_ ~ 0.010 m 06 a O.ooo - -0.010 -0.020 -0.030 -0.040 Fn : u.tu A/L: 1.50 ~ I-I/: 1/20 Ho.~ RIGID MODE-4 )dODE-6 B F.R Y. A.P'~ / '_ 1 \\ ~/ \\ 'a ~ -. it' Fig.23 Longitudinal distribution of vertical bending moment by various calculations (Fn=0.7,1/L=1.5,Hw/l=~/20) In this section , various kinds of calculati- on stated in section 4.3 are applied on the actual high speed craft to investigate the effects of structural rigidities on wave loads. in order to illustrate the effects more clearl- y, the computation performed at a severe wave condition of l/L=~.5 and H~/l=1/20 with forward speed of Fn=0.70, are shown in Figures 22 and 23. Figure 22 shows the time histories of bow acceleration and bending moments at square station 7, and Figure 23 shows the nondimensio- nalized bending moment peak values distribution along ship's length obtained by various calcul- ations. It can be seen in these figures, by comparing with "EULER" calculation, the "RIGID" calculation, in which structural rigidities are considered to be infinite and no vibration can be recognized, may underestimate the sagging moment significantly. Furthermore, by comparing with "F.E.~." calculation, the "EULER" calcula- tion, in which shear rigidity is assumed to be infinite and no shear deformation can be recog- nized, may underestimate the hogging moment remarkably. Therefore, the "F.E.~. calculation by treating ship hull's girder as an Timoshenko beam seems to be necessary for predicting the wave loads acting on it at severe condition. Furthermore, in order to illustrate the rigidities' dependence of wave loads, "F.E.~." calculation is applied on the actual high speed craft with variations of flexural rigidity and shear rigidity separately, at the same conditi- ons which is stated above , namely, I/L=1.5 and Hw/~=1/20 with forward speed of Fn=0.70. Figu- re 24 shows the flexural rigidity's dependence of nondimensionalized bending moment peak values at square stations 5, 7 and 8, while the shear rigidity is kept to be original value of the actual ship. It can be seen in this figure, decreasing the flexural rigidity may reduce the 0.020 ^ 0.010 $ m o0 lo. o.ooo -0.01 0 -0.020 F.E.U | I lain : 0. (0 S S 5 A/L: 1.30 S.S.8 --- lo/): 1/20 -- ~ ~ 0 ~' 1 10 10 1 10 ~ Log(EI/EIs ) _ _ l o Sag \ / / Fig.24 Effect of flexural rigidity on vertical bending moment (Fn=0.7,1/L=1.5,Hw/l=1/20) 0.030 r 0.020 ~ 0.010 $ of Q 0.000 - -0.010 ~ I Say -0.020 -0.030 ' S.S.5 S.S.7 S.S.B - - - F.E.H in 1 I/L 1 50 LIIW/N 1/r O | I , 1O 10 ' 1O ~ Log(GAw/GAws ) Fig.25 Effect of shear rigidity on vertical bending moment (Fn=0.7,1/~=1.5,Hw/l=~/20) sagging moment acting on midship section signi- ficantly. Similarly, Figure 25 shows the shear rigidity's dependence of nondimensionalizied bending moment peak values at square stations 5, 7 and 8, while the flexural rigidity is kept to be original value of the actual ship. It can be seen in this figure that although decreasing the shear rigidity may reduce sagging moment of midship remarkably, the hogging moment of midship may be increased, and the sagging as well as hogging moments acting at sections of fore-bodies may be increased significantly. However, it can be said that for the prediction of wave loads acting on the actual high-speed craft, neglecting shear deformation may undere- stimate the hogging moment, but has no signifi- cant effects on sagging moment. 167

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CONCLUSIONS From the present investigation into nonlinear motions of large-sized high-speed craft in head sea and whipping effects included wave loads acting on it, the following conclusions may be drawn: (1) Through the comparison between numerical prediction and elastic backbone model testing results, the present "F . E. M. " calculation metho- d, which is principally based on a modif fed nonlinear strip method, and following the Timoshenko beam elment formulation, can be applied to estimate nonlinear motions and wave loads including whipping effects of a high-spe- ed craft in head sea with accuracy enough for the practical point of view. (2) Through serial calculations of different structural representation methods, the influe- nces of neglecting the effects of vibration related to flexural deformation or shear defor- mat ion on the accuracy f or predict ing the vertical wave loads of a high speed craft can be summarized as f allows: ( i) The prediction, which neglecting the effects of vibration related to flexural defor- mation, may underestimate the sagging moments along the ship's length significantly. (ii) The prediction ,which neglecting the effects of vibration related to shear deformat- ion, may underestimate the hogging moments along the ship's length significantly. (3) Through serial calculations on various structural rigidity of hull's structure on vertical wave loads acting on a high-speed craft can be summarized as follows: ( i ) By decrees ing the f lexural rigidity, the sagging moment at mid-ship section which depen- ds on impact strongly is reduced. (ii) By decreasing the shear rigidity, saggi- ng moment at mid-ship section is reduced signi- f icantly, but at the same time, the hogging moment of mid-ship section and the sagging as well as hogging moments of forward ship's sections may be increased. ACKNOWLEDGEMENTS The f inancial support from the National Science Council of Republic of China under the grant NSC-77 0403-E002~8 is gratefully acknow- ledged. The authors wish to acknowledge the encouragement and the helpful discussions of Prof. M. Fujino, University of Tokyo. They also would like to express their cordial thanks to fir. C. H. [i of the National Taiwan University for his cooperation in carrying out the experi- ments. The computation was carried out by CDC CYBER 2.3 in the computer center, National Taiwan University. 1. Krovin-Kroukovsky, B. V., "Investigation of Ship [lotions in Regular Waves" Trans., Society of Naval Architects and Marine Engineers, Vol.63 (1955) 2. Tasai, F. and Takagi, M., "A Theory on Ship Dynamic Responses in Regular Waves and its Prediction Method ", The 1st Symposium on Seakeeping, Society of naval Architects of Japan (1969) (in Japanese) 3. Salvesen, N. ,E.O.Tuck and Faltinson, O., "Ship lotions and Sea Loads", Trans., Society of Naval Architects and Marine Engineers, Vol.78 (1970) 4 . Yamamoto , Y ., Fuj ino 11. and Fukasawa, T ., "Iot ion and Longitudinal Strength of a Ship in Read Sea and Effects of Nonlinearities (1st, 2nd, 3rd Reports) ", Journ. Society of Naval Architects of Japan, Vol.143(1978), Vol.144 (1978), Vol.145(1979) (in Japanese) 5. Fuj ino, M. and Chiu, F. C ., "Vertical motions of high-speed Boats in Head Sea and Wave Loads", ~ of Naval Architects of Japan, Vol .154 (1983) ( in Japanese) 6. Chiu, F. C . and Fuj ino , 111., "Nonlinear Prediction of Vertical lotions and Wave Loads of High-speed Crafts in Head Sea", Internation- al Shipbuilding Progress, Vol.36, No.406 (1989) 7. Chiu, F. C. and Fuj ino, 1., "Nonlinear Prediction of Vertical lotions of a Fishing Vessel in Head Sea", Journal of Ship Research, (to be published, Accepted: July, 1989) 8. Kaneko Y. and Baba, E ., Structural Des ign of Large Aluminium Alloy High-speed Craft, London, Royal Institution of Naval Architects (1982) 9. Kaneko Y., Takanashi, T. and Kihara, K. "A Proposal for Design Load on Structural Hull Girder and Bottom Structure of Large High-speed Craft ( 1st , 2nd Reports) " , Trans ., West-Japan Society of Naval Architects, Vol.70 (1985), Vol.72 (1986) (in Japanese) 10. Wang C.T., etc. "Iodel Test on the 4450 Boat", NTU-INA Tech. Rept .213, Institute of Naval Architecture, National Taiwan University (1985) 11. Kanedo, Y. and Takahashi, T., "Comparison between Nonlinear Strip Theory and Nodal Exper- iment on Wave Bending Moment Acting on a Semi- displacement Type High-speed Craft", Trans. West-Japan Society of Naval Architects, Vol.71 (1986) ( in Japanese) 12. Chiu, F . C ., Lee , Y. J . and thou , S . K ., "A Consideration on Vertical Wave Loads Acting on a Large-sized High-speed Craft", Journ., Socie- ty of Naval Architects of Japan, Vol.163 (1988) 13. Frank,W. and Salvesen, N., "The Frank Close-f it Ship lotion Computer Program", NSRDC Report No . 3289 (1970) 14. Wagner, H., "Uber Stoss-und Greitvorgange an der Oberf lache von Flus s igke iten", Z . A . 1 . 11 ., Band 12, Heft 4 (1932) 15. Yamamoto,Y., Fuj ino, 11., and Ohtsubo, H., "Slamming and Whipping of Ships among Rough Seas", Numerical Analysis of the Dynamics of Ship Structures, EUROlECH 122, ATMA, Paris (1979) 16. Bishop, R.E.D.,"Myklestad's Method for Non-uniform Vibration Beam", The Engineer, Dec. 14 (1956) 17. Takahashi, T., and Kaneko, Y., "Experimen- tal Study on Wave Loads Acting on a Semi-displ- acement Type lligh-speed Craft by Means of Elastic Backbone Models, ~iah-sueed Surface Craft Conference '83, London (1983) 68

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7. CONCLUSIONS From the present investigation into nonlinear motions of large-sized high-speed craft in head sea and whipping effects included wave loads acting on it, the following conclusions may be drawn: (~) Through the comparison between numerical prediction and elastic backbone model testing results,the present "F.E.II. " calculation metho- d, which is principally based on a modif fed nonlinear strip method, and f allowing the Timoshenko beam elment formulation, can be applied to estimate nonlinear motions and wave loads including whipping effects of a high-spe- ed craft in head sea With accuracy enough for the practical point of view. (2) Through serial calculations of different structural representation methods, the influe- nces of neglecting the effects of vibration related to flexural deformation or shear defor- mation on the accuracy for predicting the vertical wave loads of a high speed craft can be summarized as follows: (i) The predication, which neglecting the effects of vibration related to flexural defor- mation, may underestimate the sagging moments along the ship's length significantly. ~ ii) The prediction Which neglecting the effects of vibration related to shear deformat- ion, may underestimate the hogging moments along the ship's length significantly. (3) Through serial calculations on various structural rigidity of hull' s structure on vertical wave loads acting on a high-speed craft can be summarized as follows: (i) By decreasing the flexural rigidity, the sagging moment at mid-ship section which depen- ds on impact strongly is reduced. (ii) By decreasing the shear rigidity, saggi- ng moment at mid-ship section is reduced signi- f icantly, but at the same time, the hogging moment of mid-ship section and the sagging as well as hogging moments of forward ship's sections may be increased. ACKNOWLEDGE]lENTS The f inancial support from the National Science Council of Republic of China under the grant NSC-77-0403-E002~8 is gratefully acknow- ledged. The authors wish to acknowledge the encouragement and the helpful discussions of Prof. 11. Fujino, University of Tokyo. They also would like to express their cordial thanks to fir. C. H. Li of the National Taivan University for his cooperation in carrying out the experi- ments. The computation was carried out by CDC CYBER 2.3 in the computer center, National Taiwan University. REFERENCES i. Krovin~roukovsky, B.V., "Investigation of Ship [lotions in Regular Waves" Trans., Society of Naval Architects and Marine Engineers, Vol.63 (1955) 2. Tasai, F. and Takagi, ]1., "A Theory on Ship Dynamic responses in regular Waves and its Prediction Method ", The 1st Svmnosium on Seakeeping, Society of naval Architects of Japan (1969) (in Japanese) 3. Salvesen, N. ,E.O.Tuck and Faltinson, O., "Ship [lotions and Sea Loads", Trans., Society of Naval Architects and Marine Engineers, Vol.78 (1970) 4. Yamamoto , Y ., Fuj ino 11. and Fukasawa, T ., "[lotion and Longitudinal Strength of a Ship in lead Sea and Effects of Nonlinearities (Ist, 2nd, 3rd Reports) ", Journ. Society of Naval Architects of Japan, Vol.143(1978), Vol.144 (1978), Vol . 145 (1979) (in Japanese) 5. Fuj ino , ]1. and Chiu, F . C ., "Vertical motions of high-speed Boats in lead Sea and Wave Loads", Journ. Society of Naval Architects of Japan, Vol.154(1983) (in Japanese) 6. Chiu, F. C . and Fuj ino , 1111., "Nonlinear Prediction of Vertical lotions and Wave Loads of ligh-speed Crafts in Head Sea", Internation- al Shipbuilding Progress, Vol.36, No.406 (1989) 7. Chiu, F . C . and Fuj ino , 11., "Nonlinear Prediction of Vertical [lotions of a Fishing Vessel in Head Sea", Journal of Shin Research, (to be published, Accepted: July, 1989) 8. Kaneko Y. and Baba, E., Structural Design of Large Aluminium Alloy High-speed Craft, London, Royal Institution of Naval Architects (1982) 9 . Kaneko Y., Takanashi , T . and Kihara, K ., "A Proposal for Design Load on Structural Hull Girder and Bottom Structure of Large Bigh-speed Craft (ist , 2nd Reports) " , Trans ., West-Japan Society of Naval Architects, Vol.70 (1985), Vol.72 (1986) (in Japanese) 10. Wang C.T., etc. Model Test on the 4450 Boat", NTU-INA Tech. Kept .213, Institute of Naval Architecture, National Taiwan University (1985) 11. Kanedo, Y. and Takahashi, T., "Comparison between Nonlinear Strip Theory and Model Exper iment on Wave Bending Ioment Acting on a Semi displacement Type ~igh-speed Craft", Trans. West-Japan Society of Naval Architects, Vol.71 (1986) (in Japanese) 12. Chiu, F.C., Lee, Y.J. and thou, S.K., "A Consideration on Vertical Wave Loads Acting on a Large-sized High-speed Craft", Journ.~ Socie- ty of Naval Architects of Japan, Vol.163 (1988] 13. Frank,W. and Salvesen, N., "The Frank Close-fit Ship lotion Computer Program", NSRDC report No. 3289 (1970) 14. Wagner, H., "Uber Stoss-und Greitvorguge an der Oberflache von Flussigkeiten", Z.A.~.! Band 12, Deft 4 (1932) 15. Yamamoto,Y., Fujino, 1., and Ohtsubo, I., "Slamming and Whipping of Ships among Rough Seas", Numerical Analysis of the Dynamics of Ship Structures, EUROlECD 122, ATBA, Paris (1979) 16. Bishop, R.E.D.,"lyklestad's Method for Non-uniform Vibration Beam", The Engineer, Dec 14 (1956) ~ 17. Takahashi, T., and Kaneko, Y., "Experimen- tal Study on Wave Loads Acting on a Semi-displ- acement Type Digh-speed Craft by leans of Elastic Backbone Model", Di~h-sueed Surface Craft Conference '83, London (1983) 169

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APPENDIX A lij=J (~+pszj)Viwjdx Cij=J,iEIWiWjdx+J( ~pSZi IpNzj)WiWjdx ~Vcos~JpSzj(WiWi-WiW~)dx -Vcosr[pSziWiWj]A Kij=JEIWiW~dx-Vcos~J( ~ pNzj)WiW~dx -V2cos 2rJpSzjW,iWJdx +v2cos2~[pszjwi'~]A f i=cosr{pSz~ Widx+Vcos~JpSzieWidx ' + J (~ pNZ) (eWidx-VcOsr [pSzieWi] A -pgJ (A-io)`idx} -Vsin(ri+rs) {Vcos~J (pSZ-pSzo)w/dx +J(pNz-pNzo+ ~ )Widx-Vcos7(pSz-pSzO)Wi]A} (19) (20) (21) (22) APPENDIX B The elemental coefficient matrices [! ]j [C tj [K ]j and force vector {f }j associated vit j-th element are given by * 1J T [l ]j=(~+#Sz)jlO [Nw] [Nw]dg [K ]j=(EI)jlo [Ne]T[Ng]d~ +(0iW)jlO ([Ne]-[Nw]) ([Ne]-[Nw])dg (23) -Vcosr{( ~ pNz)jl i[Nw]T[Nw]d: (24) ~Vcosr(pSz);(lO [Nw]T[Nw]d(-[Nw] [Nw]l~j)} [C ] j= ( ~bEI) j| o [Ne] T [Ne] d~ +(7sGAw)jlo([Ng]-[Nw])T([Ne]-[Nw])d~ +( ~ PIZ)ii i[Nw]T[Nw]d~ (25) +(Vcos~pSz);~{|O([Nw] [Nw]-[Nw] [Nw])d~ n1 -[Nw] [NW]I1;} * {f }j=cosr{[pSz ~ ( ~ pNz) ie-P9(~-~) -Vsin(ri+rs) (