The linearization of mooring load distribution problem

Transcription

1 Scientific Journas of the aritime University of Szczecin Zeszyty Naukowe Akademii orskiej w Szczecinie 16, 46 (118), 9 16 ISSN (Printed) Received: ISSN (Onine) Accepted: DOI: 1.174/111 Pubished: The inearization of mooring oad distribution probem Jarosław Artyszuk aritime University of Szczecin 1 Wały Chrobrego St., 7-5 Szczecin, Poand, e-mai: j.artyszuk@am.szczecin.p Key words: ship manoeuvring, mooring, berthing, hydrodynamics, oads, inear mode, inearization Abstract This paper presents a nove step forward in finding the oads of particuar mooring ropes that baance the steady environmenta excitations for a ship staying at berth. The industria static equiibrium method for a rough assessment of ship mooring safety is considered to be we-estabished. The static oads are directy reated to the rope s Ls (minimum breaking oads) whie appying a certain safety margin (usuay 5%). The probem is reduced to a set of inear equations that may be soved anayticay. The generaity in terms of arbitrary horizonta and vertica anges of mooring ropes is preserved. A derivations are provided to enhance trust in the very simpe yet absoutey accurate and fast inear soution. The accuracy is studied both anayticay, throughout a the deveopment stages, and finay by comparison to the exact numerica soution of the origina noninear equiibrium equations for an exempary mooring pattern. A discussion of seected effects in oad distribution is aso given. Using the approach presented, for instance, we can efficienty test mooring safety when any mooring rope of the set is accidentay broken. Introduction The ship navigators and berth/termina operators are often faced with the probem of excessive oads in mooring ropes in both common and extreme weather conditions. Guidance is needed, for both the design of new technica systems and the operation of a currenty existing situation. In both cases, various risk/ faiure options must be considered. One method of practica importance is the static equiibrium anaysis of a moored ship. The geometry of mooring ines with no catenary effect (i.e. the straight ine) is generay assumed here for merchant ships. The ship s static equiibrium can be examined by means of ship manoeuvring simuation, e.g. (Artyszuk, 4), or through purey static methods (Chernjawski, 198; OCI, 13). In both cases, we ony and directy determine a ship s inear and anguar offset from a berth, which resuts in stretching mooring ropes so as to baance the environmenta oad. The subsequent computation of particuar rope tensions is straightforward. In the both the mentioned approaches the use of time-consuming numerica methods is required. In the first case, they dea with a set of ordinary differentia equations in time-domain; in the second one, a set of agebraic equations is soved. Athough the adopted formuation of the mooring probem is not expicity caed inear in the studies of (Chernjawski, 198), and in those reproduced by (Natarajan & Ganapathy, 1995), the matrix equations used (which inherenty present some inearizations) and the simpe input data uniquey ead to a fuy inear probem that can be soved anayticay. However, for reasons that are not expained, a numericay iterative (or incrementa) technique is undertaken, in threference cited works, to sove the probem. The anaytica methods for inear probems are very fast and exact, aowing for various efficient parametric studies. If possibe, they shoud be aways encouraged. oreover, they are readiy avaiabe for everyone, even for an inexperienced user. On the other hand, the consequences and vaidity range of the inearizations made to the mooring probem, Zeszyty Naukowe Akademii orskiej w Szczecinie 46 (118) 9

2 Jarosław Artyszuk which shoud be thoroughy understood, have not been fuy appraised so far in iterature. That is why in the present paper a carefu, systematic deveopment (through successive approximations) of the inear mooring mode is chaenged, starting from a noninear mode. The evauation is made anayticay (whie discussing the simpifications) and numericay, for a specific case study. ooring force mode for a ship s arbitrary movement In order to deveop a fu noninear mode of the mooring rope effect on a ship, we introduce two D reference systems and OXY, see igures 1 and. oth are paced in the horizonta, top-view pane of water. The former is ship-fixed and used for defining ocations of the ship s faireads and for expressing the exciting forces acting on it. The atter coordinate system is earth-fixed and used for motion description, but ony in terms of the ship s position and attitude geometric variations. y defaut, the OXY system is the primary one used for performing the beow vector cacuations. The use of the oca system (whenever necessary) wi be expicity marked in the formuas. It is assumed that and OXY initiay coincide and are aigned with each other, i.e. OXY is naturay positioned at the ship s origin whie the ship is aongside the berth in cam weather. In this case, the ship is sighty touching the Δψ > ship x X O γ rope y initiay =O Y shore igure 1. Reference systems and basic definitions Δx > Δψ < x' ' ' x Δy <, γ H, γ H fenders and has margina tension in mooring ropes. The X-axis corresponds to the direction of the berth ine. The anguar dispacement of the ship to starboard (Δψ) and the horizonta ange of the rope (γ H ) if made fast on starboard side are both given a positive vaue. The mentioned noninearity of the mode derives from the geometric reations invoved in the system consisting of the ship s fairead (), shore-based board (), and mooring rope connecting the two. The noninearity is especiay evident in the case of a fairy arge movement of the ship away from the berth igure. In mechanica aspects, however, the inear aw of rope easticity (strain-stress characteristics) is usuay adopted, and aso used hereafter. In igure, an exampe of the new inear and anguar positions of the ship is denoted by the primesign ('). Determination of the possibe causes of such extreme and sudden departure from the origina situation ( = O, x OX) is beyond the scope of the present work. Athough the ship starts to drift due to wind, and its track initiay foows roughy the direction of the wind (with some heading ateration based on wind moment), the mooring rope immediatey begins stretching and restrains further movement of the ship. The improper, hypothetica scae of the ship s movement, depicted in igure, is uncommon in mooring practice but is nevertheess convenient for defining appropriate reationships y' y environment (e.g. wind) igure. Geometry of ship s departure from origin genera case 1 Scientific Journas of the aritime University of Szczecin 46 (118)

3 The inearization of mooring oad distribution probem within the mode and is intended to constitute a firm background for reasonabe inearization. In particuar, igure shows variations of the horizonta ange (see igure 1), depicted as γ H (actua) and γ H (initia), and in the horizonta pane projection of the 3D rope, represented by its projection engths and. The figure aso shows the inear (Δx, Δy) and anguar dispacements of the ship in this situation, to faciitate the interpretation of the provided numerica resuts. The evoution of the fairead coordinates on earth can be written using the foowing vectoria notation (see igure 3): r r r (1) xo x r, r y, o y cos sin x r y () sin cos where r, r, r stand for the vectors of the ship s origin shift; fairead ocation versus ship s origin (with fixed coordinates of the fairead in x, y ); fairead ocation in OXY after the appied deviation in ship s position and heading, respectivey. The reative change of fairead ocation, required for rope eongation cacuations, reads: Δr r r (3) where r r x, y, r 1 r x 1 y r x y r r (4) Hence x cos sin x x Δr (5) y sin cos y y or x cos 1 sin x Δr (6) y sin cos 1 y The vector r, symboising the rope s orthogona projection (of the fairead-board 3D ine) to the horizonta pane, is expressed by cos H cos H r cos, r (7) sin H sin H where the actua ength,, serves as the basic intermediate unknown in our study of the ship s horizonta motion. oreover, the initia horizonta estimate of is usuay given on input i.e. based on the combined ship-termina top view. The expicit knowedge of shore board coordinates is not needed at a. Note: the horizonta ange γ H is counted against the earth-fixed axis OX. As shown in igure 4, this ength can easiy be turned into the tota (3D) ength, necessary in further evauations of the rope s eastic behaviour. Here, the key input is the fairead-board vertica distance (height), Δh, which is aways directy given. The boards ashore, as we as the berth itsef, are ocated much further beow the ship s faireads. Nevertheess, instead of the square root, it is often beneficia to introduce a proportionaity coefficient between both engths and, and the resuting corresponding forces. This coefficient is the cosine of the vertica ange γ. The vertica ange changes in parae with the tota ength whie the ship undergoes a horizonta movement. oth and γ are aso incuded in (7). r Δ r ' γ > γ Δh > igure 4. Situation in vertica pane of the rope igure 3. ector anaysis Simiary to igure and the reated comments reported earier, igure 4 aso iustrates a arger scae Zeszyty Naukowe Akademii orskiej w Szczecinie 46 (118) 11

4 Jarosław Artyszuk of the ship s movement than the one seen during typica mooring. The initia state of r is marked by the use of the sub-index in (7): cos H cos H r cos, sin H sin H r (8) inay r r Δr (9) where the sign at Δr arises from the fact that we move the fairead (not the board), as seen in igure 4. Introducing the increase in horizonta and tota ength: and taking advantage of:, h (1) (11) we can write: h (1) cos (13) h The reative eongation of the rope (responsibe for deveoping the stress response) is thus: 1 (14) 1 (15) The expression of the rope oad (and the force acting on the ship), according to the assumed inear easticity, reads: L (16) e% and its most interesting horizonta component (the mooring restraint force) takes the form: L cos cos (17) e% where L and e % are the basic rope parameters minimum breaking oad (strength) and dimensioness specific easticity, respectivey. y definition, for e % =.1. (typica for wire), we experience 1 % eongation of rope at L. As can be seen in (17) on the right-hand side, the horizonta force is proportiona to the product of two unknown terms, which can be rewritten as foows: (18) cos 1 1 (19) from which we obtain a simpe function of the variabe Δ : cos () Δ is governed by, see (1) and (7), but according to (9), one gets: X r Y cos H x cos 1 sin H y sin x x sin y cos 1y (1) X Y () We do not intend here to sette the fina expression for (). The geometricay derived non-inear, yet fu, mode of the singe rope-excited mooring force, (17,, 1), expicity provides the force vaue for a given ship position and heading change, as a function of three parameters (dispacements) Δx, Δy, Δψ. This force, through the horizonta ange and appication point (i.e. the fairead), contributes to ongitudina, transverse, and rotationa restraints of the ship in windy conditions. Athough impractica, the static equiibrium in case of a singe rope is theoreticay possibe and sovabe. The inverse probem consists here in computing the ship s three dispacement components based on the provided environmenta (externa) force, which consists of two force and one moment components. In the case 1 Scientific Journas of the aritime University of Szczecin 46 (118)

5 The inearization of mooring oad distribution probem of our noninear mode of mutipe ropes, this can ony be accompished numericay. However, the presence of a unique soution may not be guaranteed. inay, the desired mooring oad is a direct outcome of the resoved dispacements. Approximations towards inearization The approximations introduced from here on mosty consist of appying vaues of some functiona sequence imits to the equations in the preceding chapter. To simpify notation, we deiberatey use = instead of, uness required for carity and consistency, especiay for expressions within the same ine. Step 1 or a genera ange α approaching zero, we can state that: cos 1 whie sin (3) One can numericay estimate, that at east within the range of 1 for the ange, the eft-hand side of the first expression in (3) eads to vaues roughy 1 3 times ower than the second expression, so it may be disregarded. ore accuratey: 1 cos (4) which can be proved by L'Hopita s rue and even found in some eementary textbooks, e.g. (ronsztejn & Siemiendiajev, 1996). Expression (4) is an infinitesima of higher order (exacty second) to the second expression in (3). Incorporating (3) to (1, ), we gain: X cos H x y r (5) Y H y x sin x y cos H x y y x sin y x H (6) Step Let us now remove in (6) the underined infinitesima quantities of higher/second order to the inear combinations of ship s dispacements (in brackets). Hence, after some minor rearrangements, one gets: cos H x sin H y cos y sin x and foowing (1): H H (7) cos H sin H 1 x y 1 cos H y sin H x (8) Step 3 or sma inear and anguar dispacements, represented by a genera variabe x, equation (8) converges to: x 1 x 1 (9) which can be numericay and even anayticay (exacty) proved, or found in textbooks, once again using L'Hopita s rue. Hence cos H sin H x y cos H y sin H x (3) and finay cos H cos x sin H y H sin y H x (31) Step 4 Incorporating (31) into (), and adopting a simiar operation as in the previous step, one comes to the foowing expression cos 1 1 (3) 1 1 Step 5 Disregarding the underined term in (3) as higher order infinitesima, we have cos (33) Zeszyty Naukowe Akademii orskiej w Szczecinie 46 (118) 13

6 Jarosław Artyszuk Step 6 In (33), the denominator in parenthesis of the ast expression can be made equa to unity: cos 3 (34) Step 7 (fina) Negecting the higher order infinitesima, as in step 5, we finaise with: cos cos 3 (35) Athough many steps have been performed so far, they are absoutey justifiabe and do not introduce appreciabe inaccuracies. This wi be evidenced numericay in the ast chapter. The isted steps, sometimes repeatedy appied, coud have been integrated to some extent, but the adopted approach keeps short-ength formuas and ensures carity. The horizonta mooring force utimatey reads: cos and, after impementing (31): 3 cos L e % L e cos H x sin H y cos H y sin H x % (36) (37) Let s remember that Δψ in (31) is expressed in radians. or sma dispacements, the vertica ange γ converges to its initia vaue γ. The same can be said with regards to the behaviour of the horizonta ange γ H, which wi be expicity used in the next chapter. In other words, both anges are practicay preserved. The horizonta ange has not been focused on in the above transformations, where attention has been paid to the vertica pane of the rope, but an anaogy between the two exists. Static equiibrium for fu mooring ayout Let s assume that the static equiibrium of a ship is expressed in its coordinate system. This is natura, since the environmenta oads of current, wind, and wave are determined and pubished in such a system. In view of the definitions introduced in this paper, where both ship- and berth-fixed reference systems initiay coincide, this statement is purey forma. The horizonta restraint force for a singe rope, now marked with the i subscript, is expressed by: i x A y C (38) i where A i, i, C i are rope-specific (more precisey mooring ayout-specific) constants, see (37) for detais. This force can be decomposed into the foowing components: ongitudina restraint xi i cos Hi i cos H i transverse restraint i (39) yi i sin Hi i sin H i (4) (yaw) moment restraint zi i i x x Embedding (38): xi yi zi i i fxi sin sin y cos Hi i Hi H i yi cos H i fxi i fxi (41) x A y C (4a) x A y C (4b) fyi mzi fyi x A y C (4c) where A x, x, C x, x { fx, fy, mz }, are nine newy defined mooring ayout-specific constants. If the tota environmenta oad to be withstood by n number of mooring ropes is depicted by its ongitudina force, Ex, transverse force, Ey, and yaw moment, Ez, then the inear static equiibrium equations take the matrix form: Ex Ey Ez A A A fx fy mz fx fy mz mzi C C C fx fy mz fyi mzi x y (43) where the goba constants (for the whoe mooring ayout) in the first matrix on the right-hand side of the equation, are summations of particuar contributions from each rope, as exempified by A fx A fxi etc. (44) n The minus-sign on the right-hand side of (43) comes from the fact that we need to have opposite mooring forces that baance the environmenta forces, i.e. the sum of a forces must equa zero. The equations in (43) sha be soved against the ship s inear and anguar dispacements as the 14 Scientific Journas of the aritime University of Szczecin 46 (118)

7 The inearization of mooring oad distribution probem unknowns Δx, Δy, Δψ. The set of inear equations is anayticay and easiy sovabe. There are penty of agorithms in iterature and ready computer toos to do so. or practica assessment or appication in navigation practice, the very sma radian vaue obtained for Δψ sha be directy converted to degrees. As stated above, the degree vaue of heading deviation is aso sma. When the dispacements from (43) are substituted into (38) we acquire the mooring oad in a particuar rope. Since in many cases we are mosty interested in the mooring oad reated to (divided by) L, this dimensioness quantity, ', taken as the safety factor, coud be directy gained if we omit L in computations of equation constants. Therefore, we can base the whoe agorithm on the right-hand expression of the beow equation: 3 cos 1 (45) L e% The dimensioness oad ' is thus ineary dependent on the ratio of actua dimensioness eongation Δ' = Δ' / in the horizonta pane to the specific easticity of the rope. However, the environmenta force components in the eft-hand side of (43) shoud aso be divided by L. This approach is thus not practica if different Ls are present in the mooring ayout. One can aso notice that the vaue of rope easticity e %, higher for fibre ropes and ower for wire ropes, does not infuence the oad distribution. The higher the vaue of easticity, the higher are the ship s absoute dispacements, but the rope force is preserved. Numerica exampe and method assessment Let us consider the practica mooring ayout presented in igure 5. It consists of 8 ropes. The ship s faireads are marked with sma back soid dots. The shore boards (hooks) are indicated by big whitefied circes. A horizonta geometric properties (with the ship s ength of approx. 3 m) can be easiy determined. Together with vertica ange data, they are aso more accuratey summarised in Tabe 1 in order to document the foowing numerica cacuations. The L of 4 kn and e % =. are assumed. The former vaue seems to be arge in ight of the mentioned ship s size, but it represents the usua navigationa practice of doubing ropes sent from the same fairead. In this way, having nominay 16 ropes (of actua, physica L = 1 kn 1 t), we can essentiay restrict our agorithm to 8 different ropes (n = 8). The directy obtained oad in kn, and its reation to L, is reevant to each singe rope (tota of 16). Tabe 1. Input data Rope ID. 5 [m] Rope type -5 5 [m] 1-5 x [m] y [m] ship shore [m] γ H [ ] γ [ ] #1 head ine # forward breast #3 forward breast #4 forward spring #5 aft spring #6 aft breast #7 aft breast #8 stern ine The input environmenta force chosen as an exampe, corresponding to a m/s wind bowing from behind the beam, is taken as: Ex = +114 kn (to forward), Ey = 357 kn (to port), Ez = knm (to starboard). In the case of a fuy noninear mode, numerica computation has been attempted. Having a set of three noninear equations with three unknowns, this was essentiay an iterative process of manua (tria and error) tuning. Independent and couped #4 #5 #1 #8 igure 5. Exempary mooring ayout # #3 #6 #7 Zeszyty Naukowe Akademii orskiej w Szczecinie 46 (118) 15

8 Jarosław Artyszuk caibrations of Δx, Δy, and Δψ were utiized to evauate the equiibrium force and moment with the proper degree of accuracy (±.5 kn in force, ±.5 knm in moment) as fast as possibe. In particuar, the heading ateration (Δψ) is seected in reation with the transverse departure (Δy) off the berth as to preserve a positive eongation both in the forward and aft part of the ship. or this purpose, one coud define and automaticay cacuate intermediate, oca transverse dispacements at the fore and aft perpendicuars of the ship. The resuts of numerica soution of the fu mode are as foows (a digits are significant for the required accuracy): Δx = m (to forward), Δy =.878 m (to port), Δψ = º (to starboard). which ead to mooring oads presented in Tabe. According to mooring practice and recommendations of (OCI, 13), safe working oads in mooring ropes shoud be beow approx. 5% of L. These criteria are aso met in our exampe. Tabe. Distribution of mooring oads for fu mode Rope ID. [kn] /L [%] # # # # #5 7.3 # # # The anaytica resuts of the inearized mode, derived in the paper, with one more significant digit are quite identica: Δx = m (to forward), Δy =.8857 m (to port), Δψ = º (to starboard). If such are introduced to the fu mode, we get (in absoute magnitudes) the sum of ongitudina mooring forces ower by 1 kn, transverse mooring forces ower by 6 kn, and mooring moments ower by 335 knm, as compared to the numerica noninear soution. However, the individua oad of each rope ony differs by as much as 1 kn (and.1% L). One can aso notice that in this quite rea exampe the heading variation (ess than.1º) is amost unnoticeabe to navigators and termina operators, maybe due to the assumed ow environmenta moment. Concusions The anaytica and numerica investigations performed in the present work have evidenced the great potentia of a feasibe inear anaytica soution to the probem of mooring oad distribution o among particuar ropes for a ship at berth. A sma excursion of a ship from the reference position and aignment produces a sma eongation, and a sma variation of both the horizonta and vertica anges of the mooring rope. The rope eongation itsef, however, may not be negected as it eads to significant forces acting on the mooring ropes, and thus to identica reactions on the ship. The change of the other two quantities (ange-reated ones) can safey be disregarded. Hence, we arrive at the very efficient (quick, accurate, and compact) method of anaytica inear soution, which can even be impemented in an eectronic spreadsheet. This essentiay 3DO anaysis concentrated on the horizonta panar motions (dispacements) of a ship is typica for current and wind environmenta concerns ony. In the case of 6DO osciatory ship motions, ike in heavy weather/wavy conditions at the site, the presented approach yieds somewhat average (steady) oads, which are not the maximum ones. Therefore, future research wi be directed towards the dynamic anaysis of the mooring oads and the generaization of the agorithm to account for the omitted ship dispacements, such as those pertaining to heave, ro, and pitch motions. urther improvements of the agorithm shoud incude the consideration of pretension of moorings and fender effects, usuay appied but negected in the present anaysis. References 1. Artyszuk, J. (4) athematica ode of the ooring Rope Assisted Ship anoeuvring. Zeszyty Naukowe Akademii orskiej w Szczecinie 3 (75), Studies of Navigation acuty. aritime University of Szczecin.. ronsztejn, I.N. & Siemiendiajev, K.A. (1996) athematics an Encycopedic Guide. 1th Ed. Warsaw: PWN (in Poish). 3. Chernjawski,. (198) ooring of Surface esses to Piers. arine Technoogy 17, 1 (Jan). 4. Natarajan, R. & Ganapathy, C. (1995) Anaysis of oorings of a erthed Ship. arine Structures 8, OCI (13) ooring equipment guideines (EG3). 3 rd Ed. (impression of the origina 8 edition), OCI. Livingston: Witherby Seamanship Int. 16 Scientific Journas of the aritime University of Szczecin 46 (118)