Analysis of Responses of Floating Dual Pontoon Structure Analysis of Responses of Floating Dual Pontoon Structure

Analysis of Responses of Floating Dual Pontoon Structure

  • 期刊名字:中国海洋工程(英文版)
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  • 论文作者:WENG Wen-Kai,CHOU Chung-Ren
  • 作者单位:Department of Harbor and River Engineering
  • 更新时间:2020-12-06
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论文简介

China Ocean Engineering , Vol.21,No.1 ,pp.91- 104◎2007 China Ocean Press , ISSN 0890-5487Analysis of Responses of Floating Dual Pontoon StructureWENG Wen-Kai(翁文凯) and CHOU Chung-Ren(周宗仁)Department of Harbor and River Engineering , National Taiwan Ocean University ,Keelung 20229 ,Taiwvan , China( Received 18 November 2005 ; received revised form 20 September 2006 ;accepted 29 October 2006 )ABSTRACTA numerical model is developed by use of the boundary integral equation method to investigate the responses of atwo- dimensional floating structure. The structure under consideration consisting of two pontoons , is connected by a rigidframework , and linked to the sea floor by a mooring system。The theoretical conception is based on potential theory withlinear extermal forces , and applied to an arbitrarily shaped body and water depth. The discussion includes the influence ofdraft and space between pontoons on the responses of the floating structure. Finally , the validity of the method is ade-quately verified by experimental results .Key words : boundary element method ; floating structure ; dual ponoon1. IntroductionFloating structures are increasingly used in the nearshore regions to prevent wave energy or controlshoreline erosion. Owing to the various types and the advantage of easy installation , floating structuresoffer engineers another choice to design a suitable or relatively inexpensive structure for local environ-ment( e.g. weak foundation , large tide range ) or special requirements( e.g. aesthetic , water circula-tion and ecological considerations ). Though floating structures have many excellences in environmentalimprovement , they are often preferred in relatively low wave energy regions . .Many discussions concerning the characteristics and efficiency of floating structures have beenmade , not only in various analytic theories but also in the improvement of structure type , by many au-thors in recent years. Some researchers ,e.g. , Leonard et al. ( 1983 ) , discussed the hydrodynamicinterference between floating cylinders in oblique sea by taking advantage of finite element method.McIver( 1986 ) used the method of matched eigenfunction expansions to investigate interaction effectsdue to waves incident upon an adjacent floating bridge. Wang et al. ( 2006 ) showed the dynamic be-havior of a pontoon-separated floating bridge by means of finite element method. Drimer et al. ( 1992 )presented a simplified analytical model for a floating breakwater in finite water depth. Sannasiraj et al.( 1998 ) utilized two- dimensional finite element me中国煤化工; forces and responses ofa single floating pontoon-type breakwater. On the:YHCNMHGweredevotedtothede-velopment of the forms of floating structures and the investigation of the efficiency of structures. Mc-Cartney( 1985 ) classified floating breakwaters into four categories . Discussions including advantagesand disadvantages of structures , mooring system , anchorage methods , etc. , were presented in detail in1 CorresporahT 熱据. E-mail : wkweng @ mail. ntou . edu.twWENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104that paper. Isaacson and Byres ( 1988 ) showed the responses of a floating breakwater and compared theresults with experimental and field data. Murali and Mani ( 1997 ) discussed the eflction and trans-mission characteristics of cage floating breakwaters in an experimental manner. .A numerical model is developed for analyzing the behaviors of a floating structure by use ofboundary element method in this paper. The structure comprising two arbitrarily shaped pontoons isconnected by a rigid framework and linked to the seabed by a linear mooring system . Theoretical analy-sis is based on potential theory with linear conditions , and both scattering and radiating waves are dis-cussed in the analytic process. A model test is carried out in a water tank to verify the numerical re-sults. Finally , the influences of each pontoon' s dimensions and clear distance between pontoons on itsresponses are discussed .2. Theoretical FormulasThe definition sketch of the analyzed region is shown in Fig. 1. A floating structure consists oftwo pontoons and is sited on a sea . Cartesian coordinates are employed and the z-axis is directed verti-cally upwards from its origin on the undisturbed free surface . The structure is located symmetrically atx=0. Each pontoon is linked to the sea floor by an idealized mooring system. The motions of thestructure are assumed to be small and linear when the structure is subjected to a train of small ampli-tude waves of height 5o and frequency σ traveling in the negative x-direction. The fluid in the ana-lyzed region are assumed to be inviscid and incompressible. On the above assumptions , the motion offluid will be irrotational and can be described in terms of velocity potential 中( x ,z ;it )=( gζo/σ )$( x ,z)e'" ,and the potential of the analyzed region must satisfy Laplace' s equation .v°(x ,z)= 0.(1)x=-l2x=1I'Incident wave~x又o。(运)|a(x。,=.)一米ahIRegion 3,(xo,z)Region 2Region 1|中国煤化工Fig. 1. DMHCNM HGThe analyzed region will be further divided into three sub- regions to simplify the problem. Thosesub-regions are termed as region 1( x≥l ),region 2( - h≤x≤l )and region 3( x≤- l2 ), andtheir complex potentials denoted by φf( j=1 ,2 ,3 ). Region 2 includes the floating structure ; the flu-id motion万京数据region will contain scattering and radiating effect. Regions 1 and 3 are set at theWENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104)3positions far away from the structure and are assumed to be beyond the disturbance caused by floatingstructure' s motions .2.1 Potential of Regions 1 and 3The potential of regions 1 and 3 can be described in the following form , respectively ,coshh( h + )φ(x ,z)=x(x-y)+ k,e+]coshk:h(2)φ6τ,z)= K,i*+ty)cosh(h+ 3),(3)cosh khwhere , K, and K, , in complex forms , are cofficient of reflection and transmission , respectively ih isthe incident wave number , which is the root of the linear dispersion relation σ2= gh tanh kh. The po-tential of regions 1 and 3,together with its normal derivative at auxiliary boundaries x = l and x =- l2 can be expressed by :φ(1.z)=(1+ K, )osh(hy+ z);(4)φ(1z)= i(1- K, )cosh(h+ 2);(5)φ(- l2,z)= K,coshk(h+z).(6)φ(- l2 z)=- ikK,(7 )2.2 Boundary Conditions of Region 2Region 2 is enclosed by the free surface Sp , the immersed structure surface S, , the impermeablesea floor surface S, and two fictitious boundaries S; and S2. The boundary conditions on the free sur-face and sea floor are subject to the following equations , respectively ,on z=0 ;(8)8aφ=0(9)JzThe requirements of continuity of mass and energy flux across the fluid interfaces between each re-gion imply the following matching conditions :φ= φ2onx =l;( 10)8φ1 φ2Jx=dxx=l;(11 )中2=中3I、:( 12)中国煤化工Jφ2中3AxdxMYHCNM HG( 13)For the analysis of structure responses ,it is assumed that the structure behaves as a two-dimen-sional rigid body , and the structure will undergo small amplitude surge , heave and pitch motions whenit is in response to the incident and diffracted waves . The displacement for those three mode motionsmay then瓦友執据ed as :94WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104x0-xo=ξe°,zo→zo=(14).in which ,( xo 120 ) is the coordinate of the center of mass of the structure at rest and( xo 1zo)is its in-stantaneous position ; ξ η and w are amplitude of surge , heave and pitch motions , respectively. Thefirst order kinematic boundary on the immersed surface of the structure may then be written as :中_ xx0-0)x, X20-20)三、 δx._;,)_az。月.( 15)an =Jton+Jn + alanz- z- (xThe equations of motion of the structure under fluid pressure and extermal forces may be expressedas :d( xo- x)pgXds +dt2Jnd(zo- 2o)(16).di2In岛的dδPp[(=-20)-驱x-xo)]ds+M,+之M;where , m is the mass of the structure , I, the mass moment of inertia about the center of mass of thestructure , P the dynamic pressure of the fluid , R: and M, are restoring force and moment , and Fx,F: and M, are forces and moments acting upon the structure due to each mooring lines . The foregoingpressure of the fluid and restoring components are given byaφP =- ρ,=- ipgζoφei"、dzR: =-| og( zo- zo)d:.(17)'JnM, =-| ρg8( xo- xo((z-zo)-三(x- x。)dsThe mooring forces and moments ,F。,F: , and M, , are caused by the linear system , likespring ,as shown in Fig. 1. Considering the j-th mooring line AB , with its spring constant R' andpretension Fo ,the coordinates of attachment point on the structure and on the sea floor are( x。iZa )and( x 26 ), respectively. Ignoring the inertia efcts of the mooring line and the viscous forces onthe line , each component of forces and moments due to this mooring line can be expressed as :P =- K(xo- xokj1._ )_ kji_x中国煤化工F! =- KL(xo- xo)YHCNM HG( 18)M; =- K(xo- xo)- K。(zo- 20)- Kδwhere ,K'% =(x- x。尸。, (2-么。尸Fo.品lab'WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104)5K_ =(2-工。尸Ki +(x- x丫FoKx=Ki =(x-x。Xzb-Z。v-品);K。=K.=(x6-xa2[(20-z。Kx。-xo)-(x6-x。x。-2o)] R+ (36-32[(x。-x。Xx。-0)-(z-2美2。-o)] 5o;品K=K%z=(石-石。[(zp-zXx。-x)-(x-x。Xz。-0)k+(x,- xa[(x-x。xx。-xo)+(z_zXz。←zo]L;.K%a=[(xo-x。美z。-2zo)-(z0-z。美x。-xo)Ri1口D[(z-z。美z。- 2)+(x-x。Xx。-x)lab|[(z-石员z。- 2o)+(xp- x。美x。- xo)]Fo;lnb =/(x-x。歹+(z-2).3. Development of the TheoryThe above problem for the fluid potential of region 2 is solved numerically by use of the boundaryintegral equation method. According to Green' s second identity law , the potential of any point on theboundaries of region 2 is subject to the potential on the boundaries together with its first normal deriva-tive. It is written as :(x' ,)= 1f [以x 2n1_收xz)是(In!)]dds( 19)πJr, ldnwhere Inis the solution of Laplace equation. W hen the boundaries enclosing region 2 are parti-tioned into N segments , Eq. ( 19 ) has the following matrix form :地}=[o,了{出}中国煤化工(20)3.1 Cofficients of Reflection and Transmission.MYHCNM HGThe coefficient of reflection and transmission can be acquired by use of continuity of mass and en-ergy flux on the fictitious boundaries . Substituting Eq. (5 ) into Eq.( 11 ) , muliplying the result withcoshk( h + z ), and integrating from z= - h to z=0 ,one has the reflection coefficient , K, ,in termsof the norhar市南振tives of potential 中,as :96WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104K,=1 +φcoshk(h+z)dz,onx=l,(21 )NosinhkhJ - hwhere No=[ 1 + 2hh/sinh2hh ]2.Substituting Eq. (21 )into Eq. (4 ), associating the result with Eq.( 10) , one can write the re-lation between the potential of the auxiliary boundary ,x= l ,and its normal derivative as :收小z)= 2coshK(h+z).+ 2icoshh(h+z)(l ,z )xoshl(h + z)ls.(22)cosh khNosinh2khSimilarly , the coffcient of transmission can be obtained by way of substituting Eq. ( 7 ) into Eq.( 13 ) , multiplying the result with coshl( h+ z ), and integrating from z= - h to z=0. It has thefollowing form :K, =Nosinhhh.中2coshk(h + z)z,on x =- l2.(23 )The relation between potential with its normal derivative on the auxiliary boundary ,x = - l2,can also be obtained through substituting Eq. (23 ) into Eq. ( 6), associating the result with Eq.( 12),and is expressed as :(- 2,z)= 2i coshk 1+z)|(- l2,z)coshK( h + z)dls.(24 )3.2 Motions of StructureResponses of a floating structure are not only affected by hydrodynamic forces of fluid , inducedfrom scattering and radiating waves , but by the effects of the mooring system. For simplicity , the .mooring system is taken to be symmetric fore and aft of the structure . Some terms of mooring forces in-fluencing the motions of the structure will be canceled out , and Eq. ( 16 ) can then be simplified to asimpler form. The motion amplitude of the structure can be obtained after arranging Eq. ( 16 ), interms of the potential 中b on the immersed surface of structure ,as :Jxs/5o-=__i”dids-c2j,[x(=-2o)- n(x-xo)d} (25a)C1C4一C2C3η/50= εJ中,三d.( 25b)dx0/5o0C1C4 - C2C3;{-.an'ds+q[影(=-3(25c)where ,K..ngc1 =2pgPgKxiKox中国煤化工c2 =2MHCNM HG(26)C4 =262+f(xx-- zo)-x- x)ds_上己g+ρgKz +三mσJnWENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104)7Eqs. ( 25a )~( 25c ) are the amplitude of surge , heave , pitch motions of the structure , respec-tively. Substituting Eqs. (25a)~( 25c)into Eq. ( 15), associating the result with Eqs. ( 14a)~( 14c ) , the potential on immersed surface of structure can be written as :_C4x「ax=中。"dsdng [c1C4- C2C3 anJs,'b dnC2帆[z-zo)-≤(x-xo)]dsc1C4- C2C3 JnJJzcax,+中ds-cs JnJ s,C1C4 - C2C3L[NX(=-zo)-(x-川]。”日iC1[2(z-)-n(x- x0,川」。啊路(二30)- x-ola}.CC4 - c2c3Lon(27 )By substituting Eqs. (8),(9),(22),(24 )and Eq. ( 27) into Eq. ( 20 ) , one can obtain thepotential and its normal derivative on the boundaries of region 2. The motion amplitude of each modecan be obtained also by means of substituting the potential on the immersed surface of the structure intoEqs. ( 25a)~( 25c ).3.3 Forces on Mooring LinesFor a mooring line AB in Fig. 1 ,its coordinate of attachment point on the structure is transferredfrom( x。za )to( x。' z。’) when the structure oscillates in response to fluid forces . The forces on themooring line can be estimated easily from lengthening ε and spring constant Kab of mooring line AB ,when mooring lines are treated as a linear system , and is expressed as :Fau =二1(xo-x0),(( zo- zo),- x。+(z6-z。)Kabζo. lab6oζ+[(x-x。美z。-zo)-(z-zXx0-x)]g}.(28)S0」4. Results and Discussions4.1 Comparison of Numerical and Experimental ResultsA wave tank of50 m in length and 1.2 m in width was used for a series of experiments for verifi-cation of the numerical results. Water depth h was maintained at a constant 0.5 meter during the mod-el test. A floating structure model with each rectangular pontoon width a = 25 cm ,draft d = 15 cmand clear distance between pontoons l = 50 cm was sited in the wave tank for the experiment. Thelength of test model was 117 cm in its axis directiMH中国煤化工gap belween the modeledge and tank wall under consideration was aboutCNMH.Peristics of the test modelwere : weight= 76.05 kg , mass moment of inertia I, =9.82 kg m2 . ,and position of the center of massunder free surface zo= - 12.1 cm. For simulation of the mooring system , four taut springs were ap-plied to moor the model ; each spring had a spring siffness K = 0.275 kg/ cm in the linear region , andan angle of 12黎据n 54. 5° to the horizontal axis. All the above parameters of the model were normal-WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104ized for the necessary input data of numerical computation , and were given as stiffness K/( ogh )=0.094 , pretension of each mooring line FoM pgh2 )= 0.0246 , mass of structure m/( pgh2 )=0.26,mass moment of inertia I,( pght )= 0.157 ,and dimensions of structure a/h=0.5 ,d/h=0.3 ,l/h= 1.0 ,and zo/h= - 0.242 , respectively .Three mode displacements of the floating structure and the reflection coefficients induced by thestructure were investigated during the model test. The reflection cofficients were obtained by means ofthe analytic method of Goda and Suzuki ( 1976 ). Acceleration transducers and angle sensors weremounded on the structure for measuring displacements of the structure. The position of sensors is gen-erally unable to consist with the center of mass of the model . Therefore , the data from the sensors needto be arranged again through coordinate transformation .Figs. 2~4 ilustratle the amplitude of three- mode motions of the structure modulating in dimen-sionless wave period σ h/ g from the experiment and numerical calculation , respectively. Essentially ,the dimensionless amplitude of surge motion decreases monotonously when the wave period shortens ;pitch motion has a peak value at the higher frequency σ2h/g≈2.45. As regards the heave motion , itsamplitude decreases gradually when the incoming wave period σ°h/g is less than 1.3 ,but it has apeak value at σ2 h/ g≈2.3. The results of rflction cofficient are shown in Fig. 5 , and it generallyhas an excellent performance in wave attenuation when the structure is afected by incoming waves witha shorter period .On the whole , most experimental results are in sufficient correlation with numerical analysis re-sults , as observed from the comparisons. The prominent capability of dispersing wave energies andstructure motions is therefore confirmed here as being able to be validly predicted. However , the pre-dicted results of heave motion are apparently larger than the experimental results at its peak frequency .It is believed that those differences may be induced by viscosity efcts of fluid and energy losses fromthe gap between the model and water tank. .2.0Experimental resultsNumericul rpultsNumerical resultsExperioeotul rooults |1.5了1.00.50.01.03.04.0oh/g中国煤化工oh/g .Fig.2. Comparison between numerical andMYHC N M H Gen numerial andexperimental results for surge motion .experimental results for heave motion .( a/h=0.5 ;d/h=0.3 ;l/h=1.0冶=54.5° )( a/h=0.5 ;d/h=0.3 ;l/h=1.0冶=54.5°)WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104)92.0Experimental results1.00Numerical results1.50.75:' ,了1.0必0.50 .音0.250.00.001.03.04.0ofh/goh/g .Fig.4. Comparison between numerical andFig.5. Comparison between numerical andexperimental results for pitch motion .experimental results for releetion cofficient.( a/h=0.5 ;d/h=0.3;l/h=1.0;θ=54.5° )( a/h=0.5 ;d/h=0.3 ;l/h=1.00=54.5° )4.2 Influence of Space Between Pontoons on Motions of StructureThe influences of space ,l ,between pontoons on the motions of a floating structure are discussedhere. A structure composed of two pontoons and having its center of mass Zo= - 0.5d is symmetrical-ly sited on the sea. Each pontoon of width a =0.25h and draft d =0.25h is ideally assumed to behomogeneous ; the mass and mass moment of inertia of the structure will therefore be calculated by a1preumed form epressed as m=2upad ,and 1,=号v2pud[ +d2++a2+(a+1月],respective-ly. Ceofficents U1 and U2 are dependent on the density and shape of the structure ,and are simply giv-en the same constant value 0.9 in here. Having the stffness K/( pgh )= 0.03 and the angle of incli-nation θ = 60° , each spring is moored on the edge of pontoon bottom. Pretension of each spring will begivenas Fo/( pgh2 )=[ 2ad/h2- m/( ph2 )]( 2sin0 ).Fig. 6 illustrates the variations of surge motion with a dimensionless period σ2 h/ g for various ,spaces of pontoons . The figure demonstrates that the amplitude of surge motion decreases monotonouslywith shortening wave period , until the amplitude reaches a small value .Fig. 7 shows the results for heave motion. The dimensionless amplitude of heave motion will ap-proach to a constant value when the structure is affected by a longer period wave , and then will have amonotonic decrease with the shortening wavelength. Besides , heave motion in the ranges analyzed hasa peak response which appears in a higher frequency range except for the case of small space betweenpontoons l =0.1h. It has a tendency that a longer space between pontoons will lead to a lower fre-quency of peak response . Those phenomena are di-ingle floating bodies and中国煤化工it is believed that those variations are caused by:MYHC NMH G,The influences of space between pontoons on pu mouou ie SIUWII i川rig. 8. As can be seen ,the peak frequency of pitch motion trends down as the space between pontoons increases gradually , andthe value of peak response also decays. The main reason is that a longer space will give the structure a .larger mass m娃of inertia and a larger restoring moment. .万万数据100WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104//h= 1.0i/h= 0.25l/h- 0.1心2F0324oh/gFig.6. Influence of space between pontoonsFig.7. Influence of space between pontoonson surge motion .on heave motion.( a/h= d/h=0.25 ; K( pgh )=0.03 ;θ=60 )8/h= 0.25l/h= 0.5:作最61/h= 1.02}1A0.51.01.52.02.5cb/gkFig.8. Influence of space between pontoonsFig.9. Relation between peak responses ofon pitch motion.heave motion and space .( a/h= d/h=0.25 ;K/( pgh )=0.03 ;=60° ) ( a/h=0.25 ;d/h=0.5 ;K/( pgh )=0.05 ;旧=30* )According to the above results , the space has a great effect upon the structure' s motions ; it notonly changes the peak frequency of pitch motion , but induces a peak response of heave motion at ahigh frequency . Those peak frequencies of heave motion will have a correlation between each other ,iffloating structures are under the same conditions but with different spaces between pontoons. Fig. 9 il-lustrates the relations between heave motions and parameter hl for three cases l/h =0.25 ,0.5 and1.0. The conditions of the structure are :width a =0.25h ,draft d =0.5h of each pontoon and cen-ter of mass zo= -0. 5d. The stiffness of the mooring line and its angle of inclination are K/( pgh )=0.05 and θ = 30° , respectively. As can be seen中国煤化reporse localed at khl=2.0472 ,1 .0282 and0.514 for l/h=1.0 ,0.5MYHCN MH Geponse is found to havelinear relation to space of pontoons l/h. For the case l/h=1.U ,the kl ot peak response is twice ofcase l/h =0.5,and is four times of the case l/h = 0.25. The results for another case , a= d=0.5h , K/( ogh )=0.06 and θ= 90( i.e. tension leg type), are shown in Fig. 10. It also displaysthe same明哮数据S described above notwithstanding the frequencies of peak responses are differentWENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104101 .from each other. Obviously , parameter kl is a very important factor for the peak response of heave mo-tion , and the understanding of its influence will be useful for the applications of this kind of floatingstructures .l/h= 0.25l/h= 0.5l/h= 1.0了三00.51.01.52.0kiFig.10. Relation between peak responses of heave motion and space .( a/h= d/h=0.5 ; K/ pgh )=0.06;θ=90° )4.3 Influences of Pontoon Dimension on Motions of StructureThe influences of pontoon dimension on the motions of the structure are shown in Figs. 11 ~ 13for pontoon draft d/h =0.25 ,0.375 and 0.5. In those cases , all structure parameters , includingpontoon width , distance between pontoons , sifness and angle of inclination of mooring lines , are keptconstant expect for the center of mass of the structure. The coordinate of the center of mass in theabove cases is similarly taken as Zzo= -0.5d ,and it changes with the pontoon draft .Figs. 11 ~ 13 ilustrate the normalized ampltude of three mode motions , surge , pitch and heave,as a function of dimensionless period σh/ g,respectively. The variations of surge motion , as can beseen from Fig. 11 ,do not have a large difference when the pontoon draft is changed. The peak fre-quency of pitch motion , as shown in Fig. 12 , obviously has a tendency towards longer period as pon-toon draft is increased. It can be understood that the floating structure with a larger draft will cause alarger mass moment of inertia and will induce a lower frequency of peak response . Although the char-acteristic of structure' s motions in surge and pitch mode is obvious and comprehensible ,it is still diffi-cult to distinguish between the variations of a dua中国煤化工ingle foaing body whenone observes those two mode motions from Figs.YHCNM HGThe results for heave motion ,as shown in Fig. 13 , are similar to the behavior of a single floatingbody in low frequency range , but they often have a peak response in high frequency range. The fre-quency of peak response also has a tendency to decrease when the pontoon draft increases ,implyingthat the peat智循se of heave motion at high frequency is subject not only to the space between pon-102WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104toons ,but to the forces of the mooring system , hydrodynamic buoyancy , mass of the structure , etc.4gd/h= 0.25d/h=0.25.d/h = 0.375d/h= 0.375d/h= 0.5....... d/h= 0.50234σh/goh/gFig.11. The infuence of pontoon draftFig. 12. The infuence of pontoon drafton surge motion.on pitch motion.( a/h= d/h=0.25 ;K/ pgh )= 0.05 ;( a/h= d/h=0.25 ;K( pgh )=0.05 ;l/h=0.5 ;θ=450 )l/h=0.5 ;θ=45° )d/h = 0.25... d/h= 0.50.0了E0.40.81.62.0orh/gkFig.13. The influence of pontoon draftFig.14. Effect of siffness on surge response .on heave motion.( a/h= d/h=0.25 ;K/( pgh )= 0.05 ;l/h=0.5 ;8=45° )4.4 Influences of Stiffmness of Mooring Cable on Structure' s ResponsesA floating dual pontoon structure with each pontoon width a = 0.25h , draft d = 0.25h , andspace between pontoons l = 0.5h , is investigated for understanding the influences of the stiffness ofthe mooring system on structure' s responses. The results are shown in Figs. 14~ 16 for four stiffnessesof the mooring lines K( ogh )=0.12 ,0.08 ,0.04 and 0.0. The angle inclination of each mooringline is kept to θ = 60° during numerical analysis .Fig. 14 demonstrates the relationship betwee:MYH中国煤化工of surge moio and theparameter kl. As can be seen , the influences ofCNMH G,aare not very obvious forsurge motion of the structure. The amplitude of heave motion is described in Fig. 15 ,and all the re-sults have an identical zero value at the frequency of kl≈1.1 when the sifness of the cable changesfrom 0.0 to 0.12. The overall modulation of heave motion has not large differences from that for a sin-gle floating互黎振, except for the regionof hl≥1.1. As can be seen from Fig. 15 ,the peak re-WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104103sponse of heave in the region kl≥ 1.1 tends to have higher frequency with the increasing stifness ofthe mooring system. Pitch motion as ilustrated in Fig. 16 , also changes its peak frequency to a higherfrequency region when the structure is linked by a cable of larger siffness .5rK/(pgb) 0.12KI (0g)-0.083上K/ (pgh)-0.043t2F0.40.81.21.62.0kIkdrig. 15. Effect of siffness on heave response.Fig. 16. Effct of siffness on pitch response .5. ConclusionA numerical model has been developed by means of the boundary element method for the analysisof the two-dimensional linearized hydrodynamic problem of a floating dual pontoon structure . Compar-isons between numerical results and experimental ones show good agreement for a wide range of param-eters , and the validity of this method is therefore confirmed .The results obtained here indicate that the clear space between pontoons is an important parameterfor the behaviors of a floating dual pontoon structure. The clear space has a great effect upon responsesof the structure ; it not only changes the natural frequency of the structure , but causes heave motion tohave a peak response in high frequency range. However , the responses of the floating dual pontoonstructure gradually become similar to a single floating body' s with the shortening clear distance .ReferencesDrimer ,N. , Agnon , Y. and Stiassnie ,M.,1992. A simplified analytical model for a floating breakwater in water of fi-nite depth , Applied Oeean Research ,14( 1 ):33 ~41.Goda,Y. and Suzuki , Y. ,1976. Estimation of incident and rllected waves in random wave experiments , Coastal En-gineering , Vol. 1 ,ch. 48 , 828 ~ 845.Isaacson ,M. and Byres , R. , 1988. Floating breakwater response to wave action , Coastal Engineering , 162 ,2189 ~2200 .Leonard J W. ,Huang ,M. C. and Hudspeth ,R. T.中国煤化工between floating cylinders inoblique seas , Applied Ocean Research ,5( 3 ):158 ~MeCartney , B. L. , 1985. Floating breakwater design ,MHCNMH Gw! and Ocean Engineering ,ASCE ,11(2):304~317.Mclver ,P. , 1986. Wave forces on adjacent floating bridges , Aptied Ocean Research , 82):67~ 75.Murali ,K. and Mani ,J S.,1997. Performance of cage floating breakwater , Journal of Watervay , Port , Coastal andOcean Enginserirg ,ASCE ,1234):172~ 179 .Sannasiraj , 5页数循ndar , v. and Sundarvadivelu , R. , 1998. Mooring forces and motion responses of ponto-type104WENG Wen-hai and CHOU Chung ren/ China Ocean Engineering ,21( I ),2007 ,91- 104floating breakwaters , Ocean Engineering ,25( 1 ):27 ~48.WANG Cong ,FU shi-xiao ,LI Ning , CUI Wei-cheng and LIN Zhu-ming , 2006. Dynamic Analysis of a pontoon- Separat-ed Floating Bridge Subjected to A Moving Load , China Ocean Eng. ,20(3 ):419 ~ 430.中国煤化工MYHCNM HG

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