downloaded from www.ship-research.com Received: 2016 - 07 - 28 Supported by: National Natural Science Foundation of China (51239007) Author(s):LIU Junfeng, male, born in 1993, master candidate. Research interest: ship collision and stranding. E-mail: [email protected]HU Zhiqiang (Corresponding author), male, born in 1975, Ph. D., associate professor. Research interests: ship collision and stranding, marine renewable energy, dynamics performance of marine engineering structures. E-mail: zhqhu@sjtu. edu.cn CHINESE JOURNAL OF SHIP RESEARCH, VOL.12, NO.2,APR 2017 DOI: 10.3969/j.issn.1673-3185.2017.02.011 Translated from: LIU J F, HU Z Q. 3D analytical method for the external dynamics of ship collisions and investigation of the coefficient of restitution [J ] . Chinese Journal of Ship Research, 2017, 12 (2 ): 84-91. http: // english.ship-research.com 0 Introduction Ship collision is a highly nonlinear process. In or⁃ der to analyze the collision process, the collision mechanism is generally divided into two indepen⁃ dent processes for research: external dynamics and internal dynamics [1] . The external dynamics mainly uses the rigid body motion theory to analyze the mo⁃ tion attitude of the striking ship and the struck ship, as well as the energy dissipation in the collision pro⁃ cess. The internal dynamics mainly uses the elastic⁃ plastic mechanics theory to solve the problems of structural deformation resistance, deformation ener⁃ gy dissipation and structural damage of the striking ship and the struck ship. Ship collision will lead to disastrous consequenc⁃ es, therefore, many scholars have carried out re⁃ search on ship collision. In the study of external dy⁃ namics, Minorsky [2] carried out pioneering research. It is assumed that fully plastic deformation occurs on the struck ship. By the law of conservation of momen⁃ tum, the effect of the surrounding additional water is considered as additional mass and not changed in the collision process, so as to estimate the energy dis⁃ sipation in ship collision. Pedersen et al. [3] proposed the shock dynamics model of ship collision, and ob⁃ tained the energy dissipation and impulse change value in each direction by integrating the contact force and relative displacement in each direction. However, the model can only be applied to two-di⁃ 3D analytical method for the external dynamics of ship collisions and investigation of the coefficient of restitution LIU Junfeng 1,2 , HU Zhiqiang 1,2 1 State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China Abstract: The analytical method for predicting the dynamic responses of a ship in a collision scenario features speed and accuracy, and the external dynamics constitute an important part. A 3D simplified analytical method is implement⁃ ed by MATLAB and used to calculate the energy dissipation of ship-ship collisions. The results obtained by the pro⁃ posed method are then compared with those of a 2D simplified analytical method. The total dissipated energy can be ob⁃ tained through the proposed analytical method, and the influence of the collision heights, angles and locations on the dissipated energy is discussed on that basis. Furthermore, the effects of restitution on the conservative coefficients and the effects of conservative coefficients on energy dissipation are discussed. It is concluded that the proposed 3D analy⁃ sis yields a less energy dissipation than the 2D analysis, and the collision height has a significant influence on the dissi⁃ pated energy. In using the proposed simplified method, it is not safe to simplify the conservative coefficient as zero when the collision angle is greater than 90 degrees. In the future research, to get more accurate energy dissipation, it is a good way to adopt the 3D simplified analytical method instead of the 2D method. Key words: ship collisions; external dynamics; 3D analytical method; energy dissipation; restitution coefficient CLC number: U661.43 61
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Received2016 - 07 - 28Supported byNational Natural Science Foundation of China (51239007)Author(s)LIU Junfeng male born in 1993 master candidate Research interest ship collision and stranding E-mail
liujunfengsjtueducnHU Zhiqiang (Corresponding author) male born in 1975 Ph D associate professor Research interests ship collisionand stranding marine renewable energy dynamics performance of marine engineering structures E-mail zhqhusjtueducn
CHINESE JOURNAL OF SHIP RESEARCHVOL12NO2APR 2017DOI103969jissn1673-3185201702011Translated fromLIU J FHU Z Q 3D analytical method for the external dynamics of ship collisions and investigation of the
coefficient of restitution[J] Chinese Journal of Ship Research201712(2)84-91
http englishship-researchcom
0 Introduction
Ship collision is a highly nonlinear process In order to analyze the collision process the collisionmechanism is generally divided into two independent processes for research external dynamics andinternal dynamics[1] The external dynamics mainlyuses the rigid body motion theory to analyze the motion attitude of the striking ship and the struck shipas well as the energy dissipation in the collision process The internal dynamics mainly uses the elasticplastic mechanics theory to solve the problems ofstructural deformation resistance deformation energy dissipation and structural damage of the strikingship and the struck ship
Ship collision will lead to disastrous consequences therefore many scholars have carried out research on ship collision In the study of external dynamics Minorsky[2] carried out pioneering researchIt is assumed that fully plastic deformation occurs onthe struck ship By the law of conservation of momentum the effect of the surrounding additional water isconsidered as additional mass and not changed inthe collision process so as to estimate the energy dissipation in ship collision Pedersen et al[3] proposedthe shock dynamics model of ship collision and obtained the energy dissipation and impulse changevalue in each direction by integrating the contactforce and relative displacement in each directionHowever the model can only be applied to two-di
3D analytical method for the external dynamics ofship collisions and investigation of the coefficient
of restitution
LIU Junfeng12 HU Zhiqiang12
1 State Key Laboratory of Ocean Engineering Shanghai Jiao Tong University Shanghai 200240 China2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration Shanghai 200240 ChinaAbstract The analytical method for predicting the dynamic responses of a ship in a collision scenario features speedand accuracy and the external dynamics constitute an important part A 3D simplified analytical method is implemented by MATLAB and used to calculate the energy dissipation of ship-ship collisions The results obtained by the proposed method are then compared with those of a 2D simplified analytical method The total dissipated energy can be obtained through the proposed analytical method and the influence of the collision heights angles and locations on thedissipated energy is discussed on that basis Furthermore the effects of restitution on the conservative coefficients andthe effects of conservative coefficients on energy dissipation are discussed It is concluded that the proposed 3D analysis yields a less energy dissipation than the 2D analysis and the collision height has a significant influence on the dissipated energy In using the proposed simplified method it is not safe to simplify the conservative coefficient as zerowhen the collision angle is greater than 90 degrees In the future research to get more accurate energy dissipation it isa good way to adopt the 3D simplified analytical method instead of the 2D methodKey words ship collisions external dynamics 3D analytical method energy dissipation restitution coefficientCLC number U66143
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mensional plane space and since the equation is established in the global coordinate system it is difficult to be extended to three-dimensional space because of its complexity Stronge[4] proposed athree-dimensional collision solution scheme but themodel only deals with the solution of velocity and acceleration of the collision objects On the basis of theresearch of Stronge[4] Liu et al[5] proposed thethree-dimensional dynamic model of ship collisionand obtained the energy dissipation and impulsechange value in each direction The main characteristic of this method is that all the equations are established in the local coordinate system and the closedsolution of the energy dissipation values in the localcoordinate system along each coordinate axis can beobtained On the other hand in order to truly simulate the ship collision process some scholars studythe dissipation process of collision energy in time domain Petersen[6] presented an analytical method fortwo-dimensional external dynamics of ship collisionin time domain Based on the ships horizontal motion equation the side of the struck ship is simplified into 4 nonlinear springs then the ships motionresponse energy dissipation and damage depth inthe ship collision are obtained using numerical integration of time Aiming at the ship collision processand combining the data of actual ship and model experiment Tabri et al[7-9] proposed an analytical method for the ship motion in ship collision based on theship maneuvering model Using the calculation module of the explicit nonlinear finite element softwareLS_DYNA Yu et al[10] presented a method of external and internal dynamics coupling by adding hydrodynamic through subprogram and calculating collision force by LS_DYNA
In the study of internal dynamics Luumltzen et al[11]used the super element method to estimate the deformation resistance and energy dissipation in ship collision which considered the influence of side structure and the shape of the bow Kitamura[12] used FEMto discuss the problems that are neglected in manysimplified analytical methods including the lateralbending of ship hull girder the equivalent failurestrain the forward speed and collision angle of thestruck ship etc Haris et al[13] proposed a simplifiedanalytical method that can quickly estimate the damage and energy dissipation in ship collision whichhas been verified by the numerical results of LS_DYNA A simplified method proposed by Sun et al[14]can estimate the deformation resistance and energy
dissipation of side plates stiffeners and web girdersso as to obtain the deformation resistance and energydissipation of the whole side structure
In the simplified analytical method of external dynamics the restitution coefficient has a direct impacton the value of collision energy but the detail research results are rare at present Pedersen et al[3]proposed a fast estimation method for the energy dissipation and collision impulse of two-dimensionalship collision and this method has closed solution Inthe study restitution coefficient value e ranges in 0leele1 For fully plastic collision the restitution coefficient is 0 and the restitution coefficient is 1 for thefully elastic collision however the method of selecting the restitution coefficient was not explained indetail Liu et al[5] proposed a three-dimensional model of the external dynamics of ship collision andused it to calculate the force of the ship-ice collision but the selection of the restitution coefficientwas also not discussed in depth only set to 0 for calculation Restitution coefficient is an important parameter of external dynamics model which determines the relation of relative velocity between twoships before and after the collision It is generally believed that if the restitution coefficient is 0 the striking ship will not be bounced off by the side of thestruck ship after the collision the strain energy isthe largest at the moment and the energy dissipationis the largest which is conservative If the restitutioncoefficient is 1 the striking ship will be bounced offand the energy dissipation is the least which is dangerous In the case the restitution coefficient is notknown we can simply take the restitution coefficientof 0 But in fact the restitution coefficient of 0 is notconservative For different collision scenarios conservative restitution coefficient (here defined as therestitution coefficient that makes energy dissipationmaximum and ranges in 0leele 1) is different
In this paper we will use the MATLAB program torealize the three-dimensional simplified analyticalmethod for external dynamics of ship collision anddiscuss the impact of the collision height angle andlocation on the collision energy In addition we willalso discuss the influence of collision scenario on therestitution coefficient and compare the new collisionenergy calculated by the conservative restitution coefficient (ie energy dissipation in the collision process) with the collision energy corresponding to restitution coefficient of 0
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1 Three-dimensional analyticalmethod for external dynamics
11 Coordinate system
The definition of the global coordinate system(fixed coordinate system) and the local coordinatesystem is shown in Fig 1 The global coordinate system is an inertial system with the origin located atthe center of gravity of the ship which is aright-handed coordinate system The X axis is alongthe bow direction and the Z axis is upward The origin of the local coordinate system is located at thecollision point C and its 3 mutually perpendicularunit vectors are denoted as n1 n2 and n3 The vectors n1 and n2 are located at the tangent plane ofcollision and the vector n3 is perpendicular to thetangent plane In order to elaborate more conveniently the body-fixed coordinate system is introducedwhich coincides with the global coordinate systeminitially
12 Motion equation of ship collision
In the process of ship collision the ship is regarded as a rigid body so the Newton-Euler equationcan be used to describe the motion of the rigid bodyThe Newton-Euler equation for the general motion ofa rigid body is
M dvdt
+ Mω acute v = F (1)J dω
dt+ω acute Jω = G (2)
where ω is the angular velocity vector of the rigidbody in the local coordinate system v is the velocity vector of the rigid body at the center of gravity inthe local coordinate system M is the mass matrixin the local coordinate system F is the force vectorin the local coordinate system J is the rotational inertia matrix in the local coordinate system and G isthe moment vector
In the collision the influence of hydrodynamicforce on the collision process was simplified by considering the additional mass and additional inertia in
the mass matrix and the rotational inertia matrixFormulas (1) and (2) are nonlinear and inconve
nient to solve It is assumed that the collision timewas very short then Mω acute v and ω acute Jω weresmall quantities and the linear Formulas (3) and (4)are obtained
Mdv = Fdt (3)Jdω = r acute Fdt (4)
In the formulas r represents the location vectorfrom the center of gravity to the collision point
The symbol was used to distinguish between thestriking ship and the struck ship The physical quantity with refers to the striking ships physical quantity otherwise it represents the physical quantity ofthe struck ship In the process of collision the collision force was assumed constant at collision point Cand impulses dPi and dP primei of the two ships are expressed as
dPi = Fidt (5)dP primei = F primeidt (6)
Then the ships equations of motion in the tensorform are
In Formulas (7)-(10) the repeatedly emergent subscripts represent the summation and εijk is the permutation matrix When the subscripts of εijk were arranged clockwise the value was + 1 when arrangedcounterclockwise the value was -1 and the valuewas 0 when there were repeated subscriptsM ijM primeij Iij and I primeij respectively represent massmatrix and inertia matrix of the two ships in the localcoordinate system r j and rprimej respectively representthe location vectors of the two ships from their centerof gravity to the collision point V j and V primej respectively represent velocities at the center of gravity
The above Formulas (7)-(10) are consistent withthe three-dimensional collision model proposed byStronge[4] which can be solved by the same processas Liu et al[5]2 Example analysis
MATLAB was used to compile the program to realize the three-dimensional simplified analytical method for the external dynamics of ship collision andthe energy dissipation in the ship collision was obtained
Fig1 Global and local coordinate systems[5]
X
YZ
COG
Ship
rn2
n1
n3
CF = f1n1 + f2n2 + f3n3
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21 Collision scenario
In this paper we used the example of Ref[15]The forward speed of the two supply ships was both45 ms and the two ships was 825 m in length 188 min breadth 76 m in draft and 4 000 t in displacement The additional mass coefficients in the X andY directions were 005 and 085 respectively Theadditional rotational inertia coefficient of yawing wastaken as 1 The inertia radius was 025 of the corresponding ship length In order to compare with the results of the example the friction coefficient μ0 ofthe two ships was equal to that of the example taking the value of 06
In order to analyze the effect of different collisionlocations on the collision results we selected 24points from the bow to the stern of the struck ship asthe collision points The location coordinates in theXYZ global coordinate system are shown in Table 1where α is waterline angle representing the curvature of waterline of the struck ship The collision location of the striking ship was selected at the bow ofthe striking ship as shown in Fig 2 In addition inorder to consider the influence of different collisionangles θ on the collision results the values of θ
were taken as 30deg 60deg 90deg 120deg and 150deg
In the example of this paper the above two supplyships were selected as the analysis object and theaddition mass inertia coefficient and turning radiuswere obtained by empirical formulas[16] In the two-dimensional collision the influence of vertical direction is neglected that is the vertical coordinates Zc
and Z primec of the collision points of the striking ship Aand the struck ship B are assumed to be 0 Therefore there are only heave sway and yaw motions inthe results of the calculation The influence of vertical height of the collision points needs to be considered in the 3D collision The vertical coordinate Zc
of collision point at struck ship A changes in therange of 0-1Rxx where Rxx is the turning radius of rollmotion of ship A and the range is obtained based onthe assumption that the vertical coordinates of thecenter of gravity of the ship is half of the draft In thecalculation it was assumed that the vertical coordinates Zc of the collision points on the struck ship Awere 0 025Rxx 05Rxx 075Rxx and 1Rxx and the vertical coordinate Z primec of the collision points on thestriking ship B was 05Rprimexx The coordinates Xc andYc of collision points on the struck ship A in theglobal coordinate system and the coordinates X primecand Y primec of collision point on the striking ship B arethe same as those in the two-dimensional collisionscenarios The collision angles of the two ships were30deg 60deg 90deg 120deg and 150deg respectively Therefore there were a total of 120 collision locations and600 collision scenarios22 Results analysis
The energy dissipation rate of collision (the ratioof energy dissipation to initial energy) of the 600 collision scenarios was calculated and the curves forthe change of energy dissipation rate of collision withthe collision location along ship length at specifiedcollision angle θ and in the vertical collision location scenarios were drawn (Fig 3)
It can be seen from Figs 3 (a)-(e) that the collision height has a great influence on the energy dissipation rate When the coordinates Xc and Yc of collision location were determined and the collision an
Table 1 Collision locations and waterline angles[3]
Note L in the table refers to the length of the struck ship
Fig2 Collision locations along ship length
23 21 19 17 11 9 7 53 1 X
α
Y
θ
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gle was given the energy dissipation rate of collisiondecreased with increasing collision height Zc Especially in the mid-ship collision the phenomenonwas more obvious This is because the larger the Zc the larger the induced roll motion and the smallerthe energy dissipation that is the larger the energydissipation rate of collision With the same height ofcollision strong yaw and roll motion would becaused when the collision location is near the bow orstern when the collision location is in mid-ship rollmotion caused by the collision height accounts forthe major part and yaw motion is small Thereforein the mid-ship collision the effect of collisionheight is obvious Figs 3(a)-3(e) also give thetwo-dimensional results of the Ref[15] It can beseen that the collision energy obtained by thethree-dimensional analytical method was less thanthat by the two-dimensional analytical methodWhen the collision height was equal to 0 the resultsobtained by the three-dimensional method were similar to those by the two-dimensional method
As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
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mensional plane space and since the equation is established in the global coordinate system it is difficult to be extended to three-dimensional space because of its complexity Stronge[4] proposed athree-dimensional collision solution scheme but themodel only deals with the solution of velocity and acceleration of the collision objects On the basis of theresearch of Stronge[4] Liu et al[5] proposed thethree-dimensional dynamic model of ship collisionand obtained the energy dissipation and impulsechange value in each direction The main characteristic of this method is that all the equations are established in the local coordinate system and the closedsolution of the energy dissipation values in the localcoordinate system along each coordinate axis can beobtained On the other hand in order to truly simulate the ship collision process some scholars studythe dissipation process of collision energy in time domain Petersen[6] presented an analytical method fortwo-dimensional external dynamics of ship collisionin time domain Based on the ships horizontal motion equation the side of the struck ship is simplified into 4 nonlinear springs then the ships motionresponse energy dissipation and damage depth inthe ship collision are obtained using numerical integration of time Aiming at the ship collision processand combining the data of actual ship and model experiment Tabri et al[7-9] proposed an analytical method for the ship motion in ship collision based on theship maneuvering model Using the calculation module of the explicit nonlinear finite element softwareLS_DYNA Yu et al[10] presented a method of external and internal dynamics coupling by adding hydrodynamic through subprogram and calculating collision force by LS_DYNA
In the study of internal dynamics Luumltzen et al[11]used the super element method to estimate the deformation resistance and energy dissipation in ship collision which considered the influence of side structure and the shape of the bow Kitamura[12] used FEMto discuss the problems that are neglected in manysimplified analytical methods including the lateralbending of ship hull girder the equivalent failurestrain the forward speed and collision angle of thestruck ship etc Haris et al[13] proposed a simplifiedanalytical method that can quickly estimate the damage and energy dissipation in ship collision whichhas been verified by the numerical results of LS_DYNA A simplified method proposed by Sun et al[14]can estimate the deformation resistance and energy
dissipation of side plates stiffeners and web girdersso as to obtain the deformation resistance and energydissipation of the whole side structure
In the simplified analytical method of external dynamics the restitution coefficient has a direct impacton the value of collision energy but the detail research results are rare at present Pedersen et al[3]proposed a fast estimation method for the energy dissipation and collision impulse of two-dimensionalship collision and this method has closed solution Inthe study restitution coefficient value e ranges in 0leele1 For fully plastic collision the restitution coefficient is 0 and the restitution coefficient is 1 for thefully elastic collision however the method of selecting the restitution coefficient was not explained indetail Liu et al[5] proposed a three-dimensional model of the external dynamics of ship collision andused it to calculate the force of the ship-ice collision but the selection of the restitution coefficientwas also not discussed in depth only set to 0 for calculation Restitution coefficient is an important parameter of external dynamics model which determines the relation of relative velocity between twoships before and after the collision It is generally believed that if the restitution coefficient is 0 the striking ship will not be bounced off by the side of thestruck ship after the collision the strain energy isthe largest at the moment and the energy dissipationis the largest which is conservative If the restitutioncoefficient is 1 the striking ship will be bounced offand the energy dissipation is the least which is dangerous In the case the restitution coefficient is notknown we can simply take the restitution coefficientof 0 But in fact the restitution coefficient of 0 is notconservative For different collision scenarios conservative restitution coefficient (here defined as therestitution coefficient that makes energy dissipationmaximum and ranges in 0leele 1) is different
In this paper we will use the MATLAB program torealize the three-dimensional simplified analyticalmethod for external dynamics of ship collision anddiscuss the impact of the collision height angle andlocation on the collision energy In addition we willalso discuss the influence of collision scenario on therestitution coefficient and compare the new collisionenergy calculated by the conservative restitution coefficient (ie energy dissipation in the collision process) with the collision energy corresponding to restitution coefficient of 0
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1 Three-dimensional analyticalmethod for external dynamics
11 Coordinate system
The definition of the global coordinate system(fixed coordinate system) and the local coordinatesystem is shown in Fig 1 The global coordinate system is an inertial system with the origin located atthe center of gravity of the ship which is aright-handed coordinate system The X axis is alongthe bow direction and the Z axis is upward The origin of the local coordinate system is located at thecollision point C and its 3 mutually perpendicularunit vectors are denoted as n1 n2 and n3 The vectors n1 and n2 are located at the tangent plane ofcollision and the vector n3 is perpendicular to thetangent plane In order to elaborate more conveniently the body-fixed coordinate system is introducedwhich coincides with the global coordinate systeminitially
12 Motion equation of ship collision
In the process of ship collision the ship is regarded as a rigid body so the Newton-Euler equationcan be used to describe the motion of the rigid bodyThe Newton-Euler equation for the general motion ofa rigid body is
M dvdt
+ Mω acute v = F (1)J dω
dt+ω acute Jω = G (2)
where ω is the angular velocity vector of the rigidbody in the local coordinate system v is the velocity vector of the rigid body at the center of gravity inthe local coordinate system M is the mass matrixin the local coordinate system F is the force vectorin the local coordinate system J is the rotational inertia matrix in the local coordinate system and G isthe moment vector
In the collision the influence of hydrodynamicforce on the collision process was simplified by considering the additional mass and additional inertia in
the mass matrix and the rotational inertia matrixFormulas (1) and (2) are nonlinear and inconve
nient to solve It is assumed that the collision timewas very short then Mω acute v and ω acute Jω weresmall quantities and the linear Formulas (3) and (4)are obtained
Mdv = Fdt (3)Jdω = r acute Fdt (4)
In the formulas r represents the location vectorfrom the center of gravity to the collision point
The symbol was used to distinguish between thestriking ship and the struck ship The physical quantity with refers to the striking ships physical quantity otherwise it represents the physical quantity ofthe struck ship In the process of collision the collision force was assumed constant at collision point Cand impulses dPi and dP primei of the two ships are expressed as
dPi = Fidt (5)dP primei = F primeidt (6)
Then the ships equations of motion in the tensorform are
In Formulas (7)-(10) the repeatedly emergent subscripts represent the summation and εijk is the permutation matrix When the subscripts of εijk were arranged clockwise the value was + 1 when arrangedcounterclockwise the value was -1 and the valuewas 0 when there were repeated subscriptsM ijM primeij Iij and I primeij respectively represent massmatrix and inertia matrix of the two ships in the localcoordinate system r j and rprimej respectively representthe location vectors of the two ships from their centerof gravity to the collision point V j and V primej respectively represent velocities at the center of gravity
The above Formulas (7)-(10) are consistent withthe three-dimensional collision model proposed byStronge[4] which can be solved by the same processas Liu et al[5]2 Example analysis
MATLAB was used to compile the program to realize the three-dimensional simplified analytical method for the external dynamics of ship collision andthe energy dissipation in the ship collision was obtained
Fig1 Global and local coordinate systems[5]
X
YZ
COG
Ship
rn2
n1
n3
CF = f1n1 + f2n2 + f3n3
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21 Collision scenario
In this paper we used the example of Ref[15]The forward speed of the two supply ships was both45 ms and the two ships was 825 m in length 188 min breadth 76 m in draft and 4 000 t in displacement The additional mass coefficients in the X andY directions were 005 and 085 respectively Theadditional rotational inertia coefficient of yawing wastaken as 1 The inertia radius was 025 of the corresponding ship length In order to compare with the results of the example the friction coefficient μ0 ofthe two ships was equal to that of the example taking the value of 06
In order to analyze the effect of different collisionlocations on the collision results we selected 24points from the bow to the stern of the struck ship asthe collision points The location coordinates in theXYZ global coordinate system are shown in Table 1where α is waterline angle representing the curvature of waterline of the struck ship The collision location of the striking ship was selected at the bow ofthe striking ship as shown in Fig 2 In addition inorder to consider the influence of different collisionangles θ on the collision results the values of θ
were taken as 30deg 60deg 90deg 120deg and 150deg
In the example of this paper the above two supplyships were selected as the analysis object and theaddition mass inertia coefficient and turning radiuswere obtained by empirical formulas[16] In the two-dimensional collision the influence of vertical direction is neglected that is the vertical coordinates Zc
and Z primec of the collision points of the striking ship Aand the struck ship B are assumed to be 0 Therefore there are only heave sway and yaw motions inthe results of the calculation The influence of vertical height of the collision points needs to be considered in the 3D collision The vertical coordinate Zc
of collision point at struck ship A changes in therange of 0-1Rxx where Rxx is the turning radius of rollmotion of ship A and the range is obtained based onthe assumption that the vertical coordinates of thecenter of gravity of the ship is half of the draft In thecalculation it was assumed that the vertical coordinates Zc of the collision points on the struck ship Awere 0 025Rxx 05Rxx 075Rxx and 1Rxx and the vertical coordinate Z primec of the collision points on thestriking ship B was 05Rprimexx The coordinates Xc andYc of collision points on the struck ship A in theglobal coordinate system and the coordinates X primecand Y primec of collision point on the striking ship B arethe same as those in the two-dimensional collisionscenarios The collision angles of the two ships were30deg 60deg 90deg 120deg and 150deg respectively Therefore there were a total of 120 collision locations and600 collision scenarios22 Results analysis
The energy dissipation rate of collision (the ratioof energy dissipation to initial energy) of the 600 collision scenarios was calculated and the curves forthe change of energy dissipation rate of collision withthe collision location along ship length at specifiedcollision angle θ and in the vertical collision location scenarios were drawn (Fig 3)
It can be seen from Figs 3 (a)-(e) that the collision height has a great influence on the energy dissipation rate When the coordinates Xc and Yc of collision location were determined and the collision an
Table 1 Collision locations and waterline angles[3]
Note L in the table refers to the length of the struck ship
Fig2 Collision locations along ship length
23 21 19 17 11 9 7 53 1 X
α
Y
θ
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gle was given the energy dissipation rate of collisiondecreased with increasing collision height Zc Especially in the mid-ship collision the phenomenonwas more obvious This is because the larger the Zc the larger the induced roll motion and the smallerthe energy dissipation that is the larger the energydissipation rate of collision With the same height ofcollision strong yaw and roll motion would becaused when the collision location is near the bow orstern when the collision location is in mid-ship rollmotion caused by the collision height accounts forthe major part and yaw motion is small Thereforein the mid-ship collision the effect of collisionheight is obvious Figs 3(a)-3(e) also give thetwo-dimensional results of the Ref[15] It can beseen that the collision energy obtained by thethree-dimensional analytical method was less thanthat by the two-dimensional analytical methodWhen the collision height was equal to 0 the resultsobtained by the three-dimensional method were similar to those by the two-dimensional method
As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
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1 Three-dimensional analyticalmethod for external dynamics
11 Coordinate system
The definition of the global coordinate system(fixed coordinate system) and the local coordinatesystem is shown in Fig 1 The global coordinate system is an inertial system with the origin located atthe center of gravity of the ship which is aright-handed coordinate system The X axis is alongthe bow direction and the Z axis is upward The origin of the local coordinate system is located at thecollision point C and its 3 mutually perpendicularunit vectors are denoted as n1 n2 and n3 The vectors n1 and n2 are located at the tangent plane ofcollision and the vector n3 is perpendicular to thetangent plane In order to elaborate more conveniently the body-fixed coordinate system is introducedwhich coincides with the global coordinate systeminitially
12 Motion equation of ship collision
In the process of ship collision the ship is regarded as a rigid body so the Newton-Euler equationcan be used to describe the motion of the rigid bodyThe Newton-Euler equation for the general motion ofa rigid body is
M dvdt
+ Mω acute v = F (1)J dω
dt+ω acute Jω = G (2)
where ω is the angular velocity vector of the rigidbody in the local coordinate system v is the velocity vector of the rigid body at the center of gravity inthe local coordinate system M is the mass matrixin the local coordinate system F is the force vectorin the local coordinate system J is the rotational inertia matrix in the local coordinate system and G isthe moment vector
In the collision the influence of hydrodynamicforce on the collision process was simplified by considering the additional mass and additional inertia in
the mass matrix and the rotational inertia matrixFormulas (1) and (2) are nonlinear and inconve
nient to solve It is assumed that the collision timewas very short then Mω acute v and ω acute Jω weresmall quantities and the linear Formulas (3) and (4)are obtained
Mdv = Fdt (3)Jdω = r acute Fdt (4)
In the formulas r represents the location vectorfrom the center of gravity to the collision point
The symbol was used to distinguish between thestriking ship and the struck ship The physical quantity with refers to the striking ships physical quantity otherwise it represents the physical quantity ofthe struck ship In the process of collision the collision force was assumed constant at collision point Cand impulses dPi and dP primei of the two ships are expressed as
dPi = Fidt (5)dP primei = F primeidt (6)
Then the ships equations of motion in the tensorform are
In Formulas (7)-(10) the repeatedly emergent subscripts represent the summation and εijk is the permutation matrix When the subscripts of εijk were arranged clockwise the value was + 1 when arrangedcounterclockwise the value was -1 and the valuewas 0 when there were repeated subscriptsM ijM primeij Iij and I primeij respectively represent massmatrix and inertia matrix of the two ships in the localcoordinate system r j and rprimej respectively representthe location vectors of the two ships from their centerof gravity to the collision point V j and V primej respectively represent velocities at the center of gravity
The above Formulas (7)-(10) are consistent withthe three-dimensional collision model proposed byStronge[4] which can be solved by the same processas Liu et al[5]2 Example analysis
MATLAB was used to compile the program to realize the three-dimensional simplified analytical method for the external dynamics of ship collision andthe energy dissipation in the ship collision was obtained
Fig1 Global and local coordinate systems[5]
X
YZ
COG
Ship
rn2
n1
n3
CF = f1n1 + f2n2 + f3n3
63
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21 Collision scenario
In this paper we used the example of Ref[15]The forward speed of the two supply ships was both45 ms and the two ships was 825 m in length 188 min breadth 76 m in draft and 4 000 t in displacement The additional mass coefficients in the X andY directions were 005 and 085 respectively Theadditional rotational inertia coefficient of yawing wastaken as 1 The inertia radius was 025 of the corresponding ship length In order to compare with the results of the example the friction coefficient μ0 ofthe two ships was equal to that of the example taking the value of 06
In order to analyze the effect of different collisionlocations on the collision results we selected 24points from the bow to the stern of the struck ship asthe collision points The location coordinates in theXYZ global coordinate system are shown in Table 1where α is waterline angle representing the curvature of waterline of the struck ship The collision location of the striking ship was selected at the bow ofthe striking ship as shown in Fig 2 In addition inorder to consider the influence of different collisionangles θ on the collision results the values of θ
were taken as 30deg 60deg 90deg 120deg and 150deg
In the example of this paper the above two supplyships were selected as the analysis object and theaddition mass inertia coefficient and turning radiuswere obtained by empirical formulas[16] In the two-dimensional collision the influence of vertical direction is neglected that is the vertical coordinates Zc
and Z primec of the collision points of the striking ship Aand the struck ship B are assumed to be 0 Therefore there are only heave sway and yaw motions inthe results of the calculation The influence of vertical height of the collision points needs to be considered in the 3D collision The vertical coordinate Zc
of collision point at struck ship A changes in therange of 0-1Rxx where Rxx is the turning radius of rollmotion of ship A and the range is obtained based onthe assumption that the vertical coordinates of thecenter of gravity of the ship is half of the draft In thecalculation it was assumed that the vertical coordinates Zc of the collision points on the struck ship Awere 0 025Rxx 05Rxx 075Rxx and 1Rxx and the vertical coordinate Z primec of the collision points on thestriking ship B was 05Rprimexx The coordinates Xc andYc of collision points on the struck ship A in theglobal coordinate system and the coordinates X primecand Y primec of collision point on the striking ship B arethe same as those in the two-dimensional collisionscenarios The collision angles of the two ships were30deg 60deg 90deg 120deg and 150deg respectively Therefore there were a total of 120 collision locations and600 collision scenarios22 Results analysis
The energy dissipation rate of collision (the ratioof energy dissipation to initial energy) of the 600 collision scenarios was calculated and the curves forthe change of energy dissipation rate of collision withthe collision location along ship length at specifiedcollision angle θ and in the vertical collision location scenarios were drawn (Fig 3)
It can be seen from Figs 3 (a)-(e) that the collision height has a great influence on the energy dissipation rate When the coordinates Xc and Yc of collision location were determined and the collision an
Table 1 Collision locations and waterline angles[3]
Note L in the table refers to the length of the struck ship
Fig2 Collision locations along ship length
23 21 19 17 11 9 7 53 1 X
α
Y
θ
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gle was given the energy dissipation rate of collisiondecreased with increasing collision height Zc Especially in the mid-ship collision the phenomenonwas more obvious This is because the larger the Zc the larger the induced roll motion and the smallerthe energy dissipation that is the larger the energydissipation rate of collision With the same height ofcollision strong yaw and roll motion would becaused when the collision location is near the bow orstern when the collision location is in mid-ship rollmotion caused by the collision height accounts forthe major part and yaw motion is small Thereforein the mid-ship collision the effect of collisionheight is obvious Figs 3(a)-3(e) also give thetwo-dimensional results of the Ref[15] It can beseen that the collision energy obtained by thethree-dimensional analytical method was less thanthat by the two-dimensional analytical methodWhen the collision height was equal to 0 the resultsobtained by the three-dimensional method were similar to those by the two-dimensional method
As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
66
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
67
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
68
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
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21 Collision scenario
In this paper we used the example of Ref[15]The forward speed of the two supply ships was both45 ms and the two ships was 825 m in length 188 min breadth 76 m in draft and 4 000 t in displacement The additional mass coefficients in the X andY directions were 005 and 085 respectively Theadditional rotational inertia coefficient of yawing wastaken as 1 The inertia radius was 025 of the corresponding ship length In order to compare with the results of the example the friction coefficient μ0 ofthe two ships was equal to that of the example taking the value of 06
In order to analyze the effect of different collisionlocations on the collision results we selected 24points from the bow to the stern of the struck ship asthe collision points The location coordinates in theXYZ global coordinate system are shown in Table 1where α is waterline angle representing the curvature of waterline of the struck ship The collision location of the striking ship was selected at the bow ofthe striking ship as shown in Fig 2 In addition inorder to consider the influence of different collisionangles θ on the collision results the values of θ
were taken as 30deg 60deg 90deg 120deg and 150deg
In the example of this paper the above two supplyships were selected as the analysis object and theaddition mass inertia coefficient and turning radiuswere obtained by empirical formulas[16] In the two-dimensional collision the influence of vertical direction is neglected that is the vertical coordinates Zc
and Z primec of the collision points of the striking ship Aand the struck ship B are assumed to be 0 Therefore there are only heave sway and yaw motions inthe results of the calculation The influence of vertical height of the collision points needs to be considered in the 3D collision The vertical coordinate Zc
of collision point at struck ship A changes in therange of 0-1Rxx where Rxx is the turning radius of rollmotion of ship A and the range is obtained based onthe assumption that the vertical coordinates of thecenter of gravity of the ship is half of the draft In thecalculation it was assumed that the vertical coordinates Zc of the collision points on the struck ship Awere 0 025Rxx 05Rxx 075Rxx and 1Rxx and the vertical coordinate Z primec of the collision points on thestriking ship B was 05Rprimexx The coordinates Xc andYc of collision points on the struck ship A in theglobal coordinate system and the coordinates X primecand Y primec of collision point on the striking ship B arethe same as those in the two-dimensional collisionscenarios The collision angles of the two ships were30deg 60deg 90deg 120deg and 150deg respectively Therefore there were a total of 120 collision locations and600 collision scenarios22 Results analysis
The energy dissipation rate of collision (the ratioof energy dissipation to initial energy) of the 600 collision scenarios was calculated and the curves forthe change of energy dissipation rate of collision withthe collision location along ship length at specifiedcollision angle θ and in the vertical collision location scenarios were drawn (Fig 3)
It can be seen from Figs 3 (a)-(e) that the collision height has a great influence on the energy dissipation rate When the coordinates Xc and Yc of collision location were determined and the collision an
Table 1 Collision locations and waterline angles[3]
Note L in the table refers to the length of the struck ship
Fig2 Collision locations along ship length
23 21 19 17 11 9 7 53 1 X
α
Y
θ
64
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gle was given the energy dissipation rate of collisiondecreased with increasing collision height Zc Especially in the mid-ship collision the phenomenonwas more obvious This is because the larger the Zc the larger the induced roll motion and the smallerthe energy dissipation that is the larger the energydissipation rate of collision With the same height ofcollision strong yaw and roll motion would becaused when the collision location is near the bow orstern when the collision location is in mid-ship rollmotion caused by the collision height accounts forthe major part and yaw motion is small Thereforein the mid-ship collision the effect of collisionheight is obvious Figs 3(a)-3(e) also give thetwo-dimensional results of the Ref[15] It can beseen that the collision energy obtained by thethree-dimensional analytical method was less thanthat by the two-dimensional analytical methodWhen the collision height was equal to 0 the resultsobtained by the three-dimensional method were similar to those by the two-dimensional method
As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
66
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
67
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
70
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gle was given the energy dissipation rate of collisiondecreased with increasing collision height Zc Especially in the mid-ship collision the phenomenonwas more obvious This is because the larger the Zc the larger the induced roll motion and the smallerthe energy dissipation that is the larger the energydissipation rate of collision With the same height ofcollision strong yaw and roll motion would becaused when the collision location is near the bow orstern when the collision location is in mid-ship rollmotion caused by the collision height accounts forthe major part and yaw motion is small Thereforein the mid-ship collision the effect of collisionheight is obvious Figs 3(a)-3(e) also give thetwo-dimensional results of the Ref[15] It can beseen that the collision energy obtained by thethree-dimensional analytical method was less thanthat by the two-dimensional analytical methodWhen the collision height was equal to 0 the resultsobtained by the three-dimensional method were similar to those by the two-dimensional method
As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
66
downloaded from wwwship-researchcom
sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
68
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
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As can be seen from Fig 3(f) the influence of collision angle and collision location is also obvious onthe energy dissipation rate of collision When the collision angle was 0-120deg the energy dissipation rateof collision increased with the increase of the collision angle When the collision angle was about120deg-150deg the energy dissipation rate of collisiondecreased with the increase of the collision angleBecause when the collision angle increases to a certain value relative motion occurs between the twoships making the energy dissipation decrease In addition when the collision angles were 120deg and150deg collision energy at the bow of ship A was significantly greater than the corresponding collision energy at the stern of the ship and the greater the collision angle the more obviously bow collision energyincreases The reason is that the curvature of the bowof the ship A is not 0 which makes the collisionclose to the frontal collision3 Effect of restitution coefficient
The value of restitution coefficient may be relatedto the characteristics of the two ships (the side structure length and breadth of ship draft and ship type)and collision scenario parameters (velocities of thestriking ship and the struck ship collision angle andcollision location) In this section we will discussthe impact of collision scenarios and the characteristics of the two ships on the conservative restitutioncoefficient and the influence of the conservative restitution coefficient on the collision energy31 Influence of collision scenario and
characteristics of the two ships on theconservative restitution coefficient
In order to study the influence of collision scenario on the conservative restitution coefficient keepingother values unchanged the velocities of the strikingship and the struck ship collision angle and collision location were adjusted respectively to obtaindifferent energy dissipation-restitution coefficientcurves as shown in Figs 4-7 The point at which thex is marked in the figure represents the highestpoint of the curve and its abscissa corresponds tothe conservative restitution coefficient In the title ofthe figure v_stru represents the velocity of the struckship and v_stri represents the velocity of the strikingship
As can be seen from Fig 4 in the case of the collision angle less than 90deg energy dissipation was thelargest when the restitution coefficient was equal to
0 Because the smaller the restitution coefficient thesmaller the part of the struck ships internal energy (ie strain energy) that was restituted to the kinetic energy the larger the energy dissipation For the caseof collision angle greater than 90deg such as the colli
Fig4 Relationship between energy dissipation and restitutioncoefficient with different collision angles(mid-shipcollisionv_stru=v_stri=45 ms)
Fig6 Relationship between energy dissipation and restitutioncoefficient with different velocities of the struck ship(mid-ship collisionθ=60deg v_stri=45 ms)
0 02 04 06 08 10Restitution coefficient
40353025201510050
times107
Energy
dissip
ationJ
v_stru=2 msv_stru=4 msv_stru=6 msv_stru=8 ms
Velocity of thestruck ship
66
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
67
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
68
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
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sion angles of 120deg and 150deg however the maximumvalue was not obtained when the restitution coefficient was equal to 0 This is due to that the energydissipation is composed of the friction energy (tangential force work) and internal energy (normal forcework) In order to verify this interpretation the corresponding friction energy and internal energy are calculated at 60deg and 120deg respectively
From Figs 8 and 9 it can be seen that at 60deg thefriction energy was significantly smaller than the internal energy Because the collision angle was lessthan 90deg velocity directions of the striking ship andstruck ship were at an acute angle the relative sliding distance of the tangential force along the contactsurface was small and work of friction force wassmall the change of the energy dissipation was thesame as that of the internal energy At 120deg the friction energy was equivalent to the internal energyeven greater than the internal energy Because thecollision angle was greater than 90deg the relative sliding distance of the tangential force along the contactsurface was large and work of friction force waslarge the changes of energy dissipation were different from the change of internal energy It can be seenfrom Figs 5-7 that the collision location the velocity of the struck ship and the velocity of the strikingship also have an influence on the selection of theconservative restitution coefficient
In addition to the collision scenarios the influence of the respective characteristics of the two ships(length breadth draft displacement and block coefficient) on the conservative restitution coefficient canbe obtained by using similar methods It is found inresearch that the influence of the characteristics ofthe two ships is very small on the conservative resti
tution coefficient respectively This is because the influence of the restitution coefficient depends on therelative size of friction energy and internal energyThe friction energy is mainly related to the relativesliding distance while the influence of the characteristics of the two ships on the sliding distance respectively can be ignored
32 Influence of the conservative restitu-tion coefficient on the collision energy
In order to analyze the influence of the conservative restitution coefficient on the collision resultsZxx = 05Rxx and Zc = 05Rc are taken to obtain theenergy dissipation rate of collision at two kinds ofspeed The calculation process is shown in Fig 10The calculation program is divided into the main program and subprogram The main program calculatesthe energy dissipation in the collision scenario andthe subprogram calculates the conservative restitution coefficient in the specific collision scenarioWhen the conservative restitution coefficient was tak
Fig7 Relationship between energy dissipation and restitutioncoefficient with different velocities of the striking ship(mid-ship collisionθ=60degv_stru=45 ms)
0 02 04 06 08 10Restitution coefficient
1816141210080604020
times107
Energy
dissip
ationJ
Velocity of thestriking shipv_stri=2 msv_stri=4 msv_stri=6 msv_stri=8 ms
Fig8 Relationship between energy and restitution coefficient(θ=60deg)
0 02 04 06 08 10Restitution coefficient
12
10
08
06
04
02
0
times107
Energy
dissip
ationJ
Internal energyFriction energyEnergy dissipation
Fig9 Relation between energy and restitution coefficient(θ=120deg)
0 02 04 06 08 10Restitution coefficient
4540353025201510050
times107
Energy
dissip
ationJ Internal energy
Friction energyEnergy dissipation
67
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en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
68
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
70
downloaded from wwwship-researchcom
en as the restitution coefficient it is necessary to callthe subprogram The calculation process of the subprogram is shown in Fig 10 and the calculated results are shown in Figs 11 and 12
From Figs 11 and 12 it can be seen that for thecollision scenario with the collision angle greaterthan 90deg restitution coefficient of 0 is not conservative or not safe In particular the bigger the collision angle is the bigger the gap of the correspondingenergy dissipation rate is between the conservativerestitution coefficient and the restitution coefficientof 0 For the case where the collision angle was lessthan 90deg the curves of energy dissipation rate of collision with the two kinds of restitution coefficientsare nearly the same In Fig 12 the correspondingcurve of 60deg rises in the bow because the velocityvector direction of the struck ship relative to thestriking ship forms a small angle with the tangentialdirection of the bow waterline a larger relative slid
ing is obtained leading to larger friction energy andincrease of the collision energy dissipation
The corresponding conservative restitution coefficients of each collision scenario of Figs 11 and 12were given in Tables 2 and 3 It can be seen thatwhen the collision angle was close to 90deg most of theconservative restitution coefficients were close to 0and most of the conservative restitution coefficientswere close to 1 at about 150deg
Fig10 Flow chart of analysis
Start
Solution oflinear Formulas(7)~(10)
Subprogram for calculatingconservative restitution
Conservative restitution coefficientRestitution coefficient is 0
Fig12 Relation between energy dissipation and collisionlocation when the velocities of the struck ship andthe striking ship are 45 ms and 6 ms separately
Table 2 Selection of the conservative restitution coefficientwhen the velocities of both ships are 45 ms
68
downloaded from wwwship-researchcom
4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
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4 Conclusion
In this paper a set of analytical methods and procedures for three-dimensional ship collision are presented Compared with the two-dimensional analytical method of ship collision 3D analytical method ofship collision proposed in this paper considers theroll pitch and heave motions so the calculated collision energy was less than the result of 2D ship collision
By analyzing the results of different collision scenarios the conclusions are obtained as follows
1) The collision height has obvious impact on thecollision energy When mid-ship collision occursthe impact is particularly evident
2) The collision location and collision angle haveobvious influence on the collision results Especiallywhen the frontal collision occurs bow collision willproduce greater energy dissipation
3) For the scenario when collision angle is greaterthan 90deg it is not safe to simply take 0 as the restitution coefficient The program presented in this papercan be used to calculate the corresponding conserva
tive restitution coefficient for each specific collisionscenarioReferences[1] WANG Z LGU Y N Motion lag of struck ship in colli
sion[J] Shipbuilding of China200142(2)56-64(in Chinese)
[2] MINORSKY V U An analysis of ship collisions withreference to protection of nuclear power plants[R]New YorkSharp(George G)Inc1958
[3] PEDERSEN P TZHANG S M On impact mechanicsin ship collisions[J] Marine Structures199811(10)429-449
[4] STRONGE W J Impact mechanics[M] CambridgeCambridge University Press2004
[5] LIU Z HAMDAHL J A new formulation of the impact mechanics of ship collisions and its application toa ship-iceberg collision[J] Marine Structures201023(3)360-384
[6] PETERSEN M J Dynamics of ship collisions[J]OceanEngineering19829(4)295-329
[7] TABRI KBROEKHUIJSEN JMATUSIAK Jet alAnalytical modelling of ship collision based onfull-scale experiments[J] Marine Structures200922(1)42-61
[8] TABRI KMAumlAumlTTAumlNEN JRANTA J Model-scaleexperiments of symmetric ship collisions[J] Journal ofMarine Science and Technology200813(1)71-84
[9] TABRI KVARSTA PMATUSIAK J Numerical andexperimental motion simulations of nonsymmetric shipcollisions[J] Journal of Marine Science and Technology201015(1)87-101
[10] YU Z LAMDAHL J RSTORHEIM M A new approach for coupling external dynamics and internalmechanics in ship collisions[J] Marine Structures201645110-132
[11] LUumlTZEN MPEDERSEN P T Ship collision damage[D] DenmarkTechnical University of Denmark2002
[12] KITAMURA O FEM approach to the simulation ofcollision and grounding damage[J] Marine Structures200215(45)403-428
[13] HARIS SAMDAHL J Analysis of ship-ship collision damage accounting for bow and side deformationinteraction[J] Marine Structures20133218-48
[14] SUN BHU Z QWANG G An analytical method forpredicting the ship side structure response in rakedbow collisions[J] Marine Structures 2015 41288-311
[15] ZHANG S M The mechanics of ship collisions[D]DenmarkTechnical University of Denmark1999
[16] POPOV Y NFADDEEV O VKHEISIN D Eet alStrength of ships sailing in ice[R] DTIC DocumentWashingtonDCArmy Foreign Science amp Technology Center1969
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905
126-129(in Chinese)[3] ZHU W WWANG W FGUO B Cet al A practical
symmetric aerofoil with high effectiveness[J] Journalof Shanghai Jiaotong University199630(10)35-40(in Chinese)
[4] CHEN W M The effect of rudder forms on ship maneuverability[J] Journal of SSSRI200225(1)40-43(in Chinese)
[5] YU H X The numerical simulation and experimentalresearch of ichthyoid rudder steady lift[D] HarbinHarbin Engineering University2003(in Chinese)
[6] YANG J M Prediction of performance of the shillingrudder behind a propeller[J] Journal of Hydrodynamics199914(2)162-168(in Chinese)
[7] MA Y CLIN J XZHAO Y 敞水舵水动力数值计算
及分析[J] China Water Transport20088(12)
1-3(in Chinese)[8] LI S ZZHAO FYANG L Multi-objective optimiza
tion for airfoil hydrodynamic performance design basedon CFD techniques[J] Journal of Ship Mechanics201014(11)1241-1248(in Chinese)
[9] GIM O SLEE G W Flow characteristics and tip vortexformation around a NACA 0018 foil with an end plate[J] Ocean Engineering20136028-38
[10] XIONG Z YSUN H XZHU X Q 船尾伴流场与螺
旋桨低频噪声相关性研究[C]Proceedings of theFourteenth Symposium on Ship Underwater NoiseChongqingChinese Society of Naval Architects andMarine Engineers2013282-288(in Chinese)
[Continued from page 60]10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905