Prediction of Ship Manoeuvrability

Transcription

1 epinted: Website: epot 58S, Mch 97, ethelnds Ship esech Cente TO, Shipbilding Deptment, Schoemkestt 97, 68 VK Delft The ethelnds. Pediction of Ship Mnoebility G. n Leewen nd J.M.J. Jonée Delft niesity of Technology Smmy Sttic swy- nd oscilltoy ywing tests with :55 model of the DWT tnke "Bitish Bombdie" e discssed. The pincipl ppose of these tests ws to detemine the coefficients of non-line mthemticl model to pedict nmbe of stndd mnoees, which wee elie pefomed with the fll-scle ship befoe. Clke (965) descibes the eslts of these fllscle mnoees. The mthemticl model chosen is bsed on the Abkowitz Tylo-expnsion of the hydodynmic foces nd moments; see Abkowitz (964). Howee, thee is pincipl diffeence with espect to the ibles inoled, which enbles moe coect desciption of some non-line phenomen. Compison of the pedicted mnoees with the coesponding fll-scle dt shows the good geement. Fo compison pposes some expeiments he been pefomed s well with smll model of the sme tnke ( α = 00 ). Howee it is fond tht scle effects, de to the ey low eynolds nmbe, he consideble inflence on the hydodynmic deities. Some inteesting dditionl figes e gien showing the contibtions of ech tem of the mthemticl model ding tning cicle mnoee while lso the chnge of the stbility oots ding this mnoee is plotted. Intodction At the Delft Shipbilding Lbotoy model tests wee pefomed to detemine the coefficients of non-line mthemticl model, which descibes the still wte mnoebility popeties of ship. Two types of tests e pefomed. Fist the sttic towing tests with constnt dift nd dde ngle nd second oscilltion tests to detemine the dded mss effect in the swying motion nd the hydodynmic deities of the ywing

2 motion. The min pticls of both ship nd models e smmised in Tble. Tble Min Pticls of Ship nd Models The pincipl ppose of the tests is to obtin infomtion on the possibility to pedict the pincipl mnoees of ship, sing n deqte mthemticl model nd model expeiments to detemine the coefficients of this model. Fo compison pposes, Clke (965) pefomed fllscle tils. Fige Ship Model nde Oscillto in Towing Tnk The model expeiments he been pefomed t fo diffeent initil speed conditions, coesponding with constnt popelle powe ech. Oiginlly the eslts of these fo sets of tests he been kept septed, becse it ws ssmed tht some of the non-dimensionl coefficients cold chnge with the Fode nmbe, bsed on the initil speed. In tht cse set of coefficients which wold be diffeent fo ech initil speed wold he been fond. Clke (965) gies the eslts of this tenttie nlysis. Ding the nlysis of the expeimentl dt, it ws fond tht the pincipl diffeences between some of the nondimensionl coefficients cold be descibed effectiely by consideing the locl wte elocity ne the dde. In this wy the ppent Fode-effect in these coefficients the cold be clled "powe-effect" while the diffeences in the othe coefficients wee not consideed significnt, in iew of both the ilble infomtion nd the cccy of the mesements. It is not to be expected howee, tht this method of descibing the phenomen mentioned boe will hold fo othe ships. Especilly when the Fode nmbe becomes high e.g. 0.0 o highe, it is not ey likely tht thee will be no el Fode effect in some hydodynmic deities if they e comped with the coesponding les fond t Fode nmbe of 0.0 e.g. Poided cetin mthemticl model hs been dopted, thee e seel methods to detemine the coefficients of sch model by model tests. The min poblem is to find ot how f ncopling of the thee motions in the hoizontl plne is llowed. Of cose, ding n ctl mnoee the motions e lwys copled nd een if the mthemticl model contins tems, which e ment to descibe the coss-copling effects one is not se bot the wy these effects he to be detemined. The most conenient method might be to pefom fee nning model tests nd find the

3 coefficients of the mthemticl model by nlysing the dt concening position nd cose of nmbe of epesenttie mnoees. In tht cse ll ibles emin copled in the ntl wy. In genel howee too less oom is ilble fo these kinds of mnoees, so tht one is pessed to find n cceptble ltentie. In this espect the foced hoizontl oscilltion test poides n ltentie soltion. On the othe hnd, the poblem of ncopling the motions s mentioned boe is intodced. Anothe pcticl poblem is to choose the ight combintions of oscillto feqencies nd mplitdes. Consideing n ctl oscilltoy motion, de to the hmonic motion of the dde, it is fond tht the combintions of feqencies nd mplitdes inoled in these motions cnnot esily be simlted by hoizontl oscilltion tests. This is becse the ctl nge of mplitdes is of the mgnitde of one hlf to mny ship lengths. In most towing tnks sfficient width is not ilble in this espect. These poblems e consideed in gete detil in by Vn Leewen (969). The conclsion is tht most hoizontl oscilltion tests inole n nntl eltion of mplitdes nd feqencies. In othe wods the tios of elocity nd cceletion mplitdes e qite diffeent fom the ctl les. Appently this is no poblem, becse most of the mthemticl models which e now in se do not contin ny coss-copling tems between elocities nd cceletions. This does not imply, howee, tht sch cosscopling effects cold not be intodced, if the nge of tios of these ibles is extended too f. Some finl emks on the mthemticl model. In the cose of the yes lot of stdies wee deoted to this sbject, stting with Didson nd Schiff (946), who descibed model bsed on the line eqtions of motion. This set of eqtions, which oiginlly inoles thee eqtions, descibing the sging, swying nd ywing motion, hs been sed by seel thos, thogh nfotntely omitting the sge eqtion. Abkowitz (964) hs poposed one of the most extended nonline mthemticl models. In this model the hydodynmic pt of the foces is expnded into Tylo seies of the ibles concened. This pinciple is ey sefl, pticlly if the constnts of the model e to be detemined by the nlysis of foced model tests, becse ll imginble hydodynmic effects in pinciple cn be descibed in this wy. An impotnt qestion inoled in this Tylo expnsion is p to wht degee it hs to be extended to be se tht the pincipl nonline hydodynmic effects e descibed coectly. On the othe hnd it is qestioned to wht extend it is necessy to etin get nmbe of tems in sch model fo esonble ccte desciption of mnoees, een if the septe hydodynmic effects inoled cn be mesed with foced model tests. In othe wods it is sggested tht, on the gond tht ding n ctl mnoee the tios of the ibles stisfy jst one eltion, it might be possible to descibe the joint effect of nmbe of tems by one tem only. In this wy mch simple mthemticl model wold ise, the coefficients of which hd to be consideed fnctions of the coefficients of the oiginl model. Vn Leewen (970) descibed some simple non-line models bsed on these gonds. A disdntge of sch simplified mthemticl models is tht its coefficients cnnot be detemined by ncopling the thee motions which mens tht they cn only be deied fom fee nning tests, eithe fll-scle o model tests. Fo pcticl pposes howee, sch s simltion stdies nd tomtic piloting, these simplified non-line models cn be pplied sccessflly. The pinciple of the mthemticl model sed fo the pesent model-tests is, pt

4 fom some detils, the sme s hs been sed by Abkowitz. The wy in which Abkowitz tets the inflence of chnge of the fowd speed, howee, bings bot tht no insight is gined into the physicl bckgond of this inflence. In this ppe, the hypothesis is sed tht if the motions e simil, egding elocities nd cceletions, the pincipl hydodynmic foces on the hll e popotionl with the sqe of the instntneos fowd speed. Fo the foces, which minly depend on the effectie ngle of ttck of the dde, popotionlity with the sqe of the locl wte elocity is ssmed. The genel concept of this hypothesis is confimed by the model expeiments. In the following chpte this will be discssed in moe detil. Eqtions of Motion. Intodction If we fom mthemticl model nd we stt fom the fct tht the hydodynmic foces e fnctions of the elocities nd cceletions inoled in motion, we cn expnd these foces, s hs been done by Abkowitz (964), in Tylo seies of these elocities nd cceletions. On the gond of considetions of mgnitde we cn ignoe the tems whose ode is highe thn e.g. the thid. Thee e some objections to this pocede, howee. Consideing tem popotionl to the thid powe of e.g. the ngl elocity the omission of the foth ode tems mens tht the contibtion of this tem, egdless the fowd speed, emins popotionl with the thid powe of the ngl elocity. Fom model expeiments it is known tht this - nd simil tems - e eesed popotionl with the fowd speed. eglecting this speed dependence conseqently coesponds with ndeestimting the non-line effects descibed by these thid ode tems in the cse of speed edction. Fo speed edction of 50 pe cent, sch non-line effect is ndeestimted by fcto two. Anothe objection, thogh of less impotnce, is tht if the septe elocities e consideed s ltel, fowd nd ngl elocity, the pticl ole plyed by the fowd speed becomes hdly ppent. In section., diffeent bsis hs been chosen fo the mthemticl model, the hypothesis mentioned in chpte being sed. The effects of the foth degee tems, mentioned boe, e inoled in the thid degee Tylo expnsion of the foces, if they e consideed to be fnctions of chcteistic ibles, descibing the simility of the motions. It is emphsised howee, tht the concept of this is not new, becse lso Didson nd Schiff (946), omoto (957) nd Ed nd Cne (96) ledy pid ttention to the impotnce of these ibles. Both elie wok nd the pesent inestigtion jstify the doption of the hypothesis concening the foces.. Components of Hydodynmic Foces The eqtions of motion descibing the blnce of foces nd moments ding still wte mnoee cn be witten s follows (see lso Fige ): m m ( & ) I = & = ( & ) = X x zz x Eqtion -,b,c whee epesents the component of the hydodynmic foces pependicl to the ship nd the coesponding moment, while X epesents the component of 4

5 these foces cting in longitdinl diection. components e consideed to be the eslt of the fct tht the sm of the hydodynmic foces is not obtined by the speposition of the foces cting on the hll nd those on the dde, which my be ppoximtely te fo the side foces on siling ychts. Concening the longitdinl foce blnce, foth gop hs to be consideed, which inoles the foces de to the esistnce nd the chnge of thst csed by speed loss ding mnoeing. This gop detemines the diffeence between the fowd speed of the cente of gity nd the speed of the wte ne the dde.. Hypothesis Concening the Hydodynmic Foces Acting on the Mnoeing Ship Fige Co-odinte System nd Definition of Vibles The sm of the hydodynmic foces cn be diided into thee gops:. The fist gop contins the components, which depend on the condition of motion of the ship withot popelle nd dde. The ibles inoled in this cse will be discssed in section... The second gop contins the foces, which ct on the dde. They depend on the effectie ngle of ttck of the dde nd s this qntity depends on the ships condition of motion, these components will depend on the ibles of the fist gop s well s on the dde ngle itself.. The thid gop contins the foce components, which e csed, mong othe things, by the chnge of cicltion ond the ship, de to the dde deflection. In genel, these The hypothesis mentioned in the peceding chpte, concening the fist gop foces, is fomlted s follows: If two simil motions of cetin ship (o model) e comped, thn these foces will be popotionl to the sqe of the fowd speeds inoled, poided these speeds do not diffe too mch. This hypothesis is minly bsed on model expeiments, thogh the eslts of fllscle mnoees, exected t diffeent fowd speeds, poide n indiction fo it s well. The explntion fo the seflness of this hypothesis cn be deied fom the pincipl impotnce of the ineti foces nd fthe fom the smll ole plyed by the geneted wes in cetin speed nge. If comping the simil motions of ship nd he model, the we pttens will only be simil if the Fode nmbes in this cse e eql, de to the constnt le of the cceletion of gity. On the gond of the smll inflence mentioned of the geneted wes in cetin speed-nge, the hypothesis is to be pplied in this cse within esticted nge of Fode 5

6 nmbes, while the foces e popotionl to the fcto L. Concening the fowd speed, the ppe limit of the seflness of the hypothesis logiclly follows fom the incesing impotnce of the wes, when the speed is highe. The lowe limit, howee, cnnot be deied only fom the decesing we genetion, s this, on the conty, the is eson to expect its seflness. Appently thee e othe esons fo this, the pincipl of which pobbly is the incesing impotnce of the fictionl foces, nd in genel, of the iscos effects, comped to the ineti foces. The eqtions of motion of floting body, moing in hoizontl plne, cn be witten in sch fom, tht the hypothesis is expessed by it. Accoding to the hypothesis the foces, cting on the body, depend linely on the foce nit 0.5ρ L, so tht the foces, diided by this nit, cn only be fnctions of nondimensionl pmetes, which descibe the (esticted) simility of the instntneos conditions of motion to be comped. Assming only the elocities (,, ) nd the cceletions (&, &, & ) to ply ole in the eqilibim of foces, the eqtions of motion cn be witten s follows: m m ( & ) I zz x = & = ρ ( & ) x = ρ X L L L L & L & L &,,,, L L & L & L &,,,, ρ L L L & L & L &,,,, Eqtion -,b,c If the pinciple of the hypothesis is lso pplied to the foce components of the second gop, then the locl simility is chcteised by the ctl ngle of ttck of the dde, while the foces e then popotionl to the sqe of the locl wte elocity. As the ctl ngle of ttck nd the mgnitde nd the diection of the locl wte elocity, these foces my be ppoximted s follows: = ρ L ( ct ) whee with: is the locl wte elocity nd = p p p ct = nd L = Concening the components of the thid gop, it is ssmed tht they will minly depend on the ibles of both the fist nd second gop. They will ptly be popotionl to the sqe of the fowd speed of the cente of gity nd fo the est to the sqe of the locl (dde) speed. On this bsis the mthemticl model cn be bild p, consideing the thee gops to be fnctions of the non-dimensionl ibles, concened. The expnsion of the thee gops in thid degee Tylo seies leds to nmbe of tems pt of which being popotionl with the sqe of the fowd speed, while the emining tems will be popotionl to the sqe of the locl dde speed. Consideing e.g. the line tem in, then the following expession is fond: 6

7 ρl { ( ) } st gop nd gop d gop (foces on dde de to effectie ngle of ttck) (foces on hll withot dde nd popelle) (intection dde-hll; foces de to filing of speposition pinciple) The expessions fo the othe tems will he simil foms. On this gond the mthemticl model is to be witten in the following fom: m m ( & ) I ( & ) x zz = ρ & = ρ = ρ X x L L L X ρ ρ ρ L L L X Eqtion -,b,c In these eqtions the components mked with n steisk contin the tems oiginted fom the Tylo expnsions of the fist nd second gop. The dshed components contin the coesponding tems of the second nd thid gop. Both steisk nd dsh mked components e fnctions of the dde ngle nd the fie ibles detemine the second ode simility. The longitdinl foce component X descibes the diffeence between the ship s stight-line esistnce nd the chnge of thst de to the speed edction. On the gond of theoeticl considetions nd the expeience fom elie inestigtions nmbe of ssmptions he been mde, which simplify the expessions fo the ios components. Some of these ssmptions he been inestigted pticlly, while othes e not contdicted by the mesements. The ssmptions concened e smmised s follows:. The fowd speed, s ible of Eqtion, cn be eplced by its longitdinl component x. Conseqently, the ibles nd e defined s / x nd L / x espectiely, while the nit ds is defined s x dt / L.. The hydodynmic ltel foces e independent of the longitdinl cceletion, nd the hydodynmic longitdinl foces e independent of the swy nd yw cceletions.. on-line cceletion effects do not occ in the nge of inteest. 4. If the ship is on stight cose, the foces de to cetin dde deflection e popotionl to the sqe o the locl dde speed. (This ssmption my be consideed the definition of the qntity fo the pesent inestigtions). 5. The inflence of the dde te on dded mss effects is negligible fo pcticl pposes..5 Set of Eqtions of Motion Execting the Tylo expnsions of the thee gops of foces nd moments nd pplying the ssmptions indicted boe, the eqtions of motion cn be witten s follows: 7

8 8 ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } { } m m = & & & & & & Eqtion 4- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } { } zz I = & & & & & & Eqtion 4-b

9 m X X X X & X m X ( ) ( ) ( ) X & ( ) ( ) X { X } { X ( ) X } X T & X = X X ( ) X Eqtion 4-c In these eqtions ll ibles he been mde non-dimensionl with the initil speed 0 so tht e.g.: x = = 0 ( ) ( ) = Some dditionl emks concening these eqtions:. The side foce nd moment eqtions contin some tems to descibe the symmeticl behio of the ship. With zeo dde deflection these tems e ( ) ( ) nd while the symmeticl dde effectieness is descibed by the tems nd espectiely. b. The longitdinl foce component X is diided into two components, the fist of tht descibes the blnce between the oiginl thst nd the stight-line esistnce while the second descibes the (lineised) incement of the thst ding mnoee. c. The foth degee tems, which wee mentioned in the intodction, e de to the fctos / ( ), which my be lineised in the nge 0.60 < < 0.00 to d. If the chnge of speed ding mnoee is lge thn the nge in which the hypothesis is lid, thn the st nd dsh coefficients cn be consideed line fnctions of the speed. In tht cse dditionl coefficients s nd shold be dded. Fo the pesent inestigtion, this ppeed not necessy howee. Exection of Tests. Mesing Eqipment The pincipl popety of the mesing eqipment is tht only hmonic components of the foces e detemined. Fo the pesent inestigtion only the fist hmonic components wee needed. This detemintion is chieed by mltiplying the foces by sin ωt nd cosωt espectiely, while these podcts e integted ding one o moe peiods of the oscilltion. The non-oscilltoy components of the foces e detemined by integtion of the foce ding nmbe of peiods. 9

10 Zndedop nd Bitenhek (96) gie in moe detiled discssion on the oscillto nd the mesing eqipment. The sttic dift ngle djstment ding the oscilltion tests is chieed by tning the model with espect to the connecting line of the oscillto stts (the pe ywing line ). This is sketched in Fige. Tble Tim s Fnction of Speed A pticl inestigtion might gie the nswe, bt it is not expected tht ndesible coss-copling effects wold distb the side foce mesements, de to the ncopling of the swying nd ywing motion. This is lso expected with espect to the olling motion. As it ws not the ppose of this inestigtion to find n nswe to this qestion, it ws consideed sefl ppoximtion to djst the constnt dghts, deied fom stightline tests nd to estin the model lso fom olling. Fige Dift Angle Adjstment of Oscillto. Detemintion of Dght nd Tim As the model ws estined fom heing nd pitching the dght nd tim, s dependent on the fowd speed nd the nmbe of popelle eoltions hd to be detemined. The eoltions djsted fo these tests wee estimted fom the ilble fllscle dt. A chnge of the pm did inflence neithe the men dght no the tim howee. In Tble the ios dghts foe nd ft fo model nd ship e gien. These dghts he been djsted fo the tests concened. It is qestioned howee if it is coect to estin the model in eticl diection ding oscilltion nd othe tests.. Detemintion of esistnce nd Poplsion Coefficients The desciption of the blnce between longitdinl esistnce nd the popelle thst is ptly bsed on some fll-scle of pm t 5.5 knots nd ptly on the popelle chcteistics. Fom these dt the wke fction ws deied while fo the lowe speeds, csed by mnoeing. This wke fction ws consideed constnt. Fom the fct tht ding mnoee the powe does not chnge, the incement of thst nd the decement of pm cold be clclted sing the popelle chcteistics. Fo compison pposes lso the fll-scle mesements of these qntities e gien in Fige 4 nd Fige 5. Concening the othe initil speeds,.4, 9. nd 6. knots, line eltion between these speeds nd the pm ws dopted (see Fige 6). The incese of thst nd the decese of the pm ding mnoees with these 0

11 initil speeds wee detemined s this ws done fo the 5.5 knots initil speed. Fige 4 Vition of Thst with Speed ding Tning Cicles t 00 ominl PM t t = 0 Fige 7 Mesed nd Compted Vles of Thst Fige 5 Vition of PM with Speed ding Tning Cicles t 00 ominl PM t t = 0 Fige 8 Mesed nd Compted Vles of Toqe Fige 6 Adopted Initil Speed-PM eltion In Fige 7 nd Fige 8 the clclted thst nd toqe e plotted fo the initil conditions. It cn be shown tht on stight cose pbolic eltion between the thst nd the fowd speed exists, poided the thst ce of the popelle digm is lineised in the nge of inteest. Conseqently the ssmption of pbolic eltion between the longitdinl esistnce nd the speed is eqilent to the ssmption of speed-independent thst dedction fction. This nmbe ws estimted to be 0.0. On this bsis the esistnce coefficient X ws clclted: X =

12 while fo the effectie thst incement coefficient X ws fond: T X T = In Tble the compted ps, concening the ios conditions, e smmised. Tble Compted Popelle te Vles It is noted howee tht these dt e coected fo the diffeences between the fll-scle nd model popelle ( 8 - B 5.60 espectiely). The ps les, djsted ding the model tests, e not the sme s gien in Tble, howee, s oiginlly these les wee bsed on the fll-scle eltion between thst nd speed s gien by Clke (965). Ding the nlysis of the test dt this eltion did not ppe to coespond with the fll-scle dt of pm nd the popelle digm, conseqently no did the pm- eltion. This is shown in Fige 9. As, in ddition, the coefficients, which descibe the blnce between esistnce nd thst ding mnoee bsed on this eltion, eslted in ey lge diffeences between the mnoees compted nd those exected on fll-scle, the line eltion between pm nd speed ws dopted. The coefficients of the foce nd moment components, nd X he been coected s f s necessy, which ws possible becse thei eltion with the pm ws known fom the model expeiments..4 Detemintion of dde Speed Fige 9 Discepncies between the Mesed Fowd Speed nd the Speed Deied fom Thst nd Toqe Mesements (Fll-Scle) Fom this fige it is ssmed tht the fllscle speed mesements concened e not coect. The qntity hs been deied fom the stight-line tests with constnt dde ngle. These tests wee exected fo the 0 combintions of speed nd ps, while in ech cse the dde ngle ws ied fom 6 degees pot to 6 degees stbod with steps of 9 degees. A foml desciption of the wte elocity ne the dde hs been bsed on the implse theoy with espect to the popelle. Accoding to this theoy the wte elocity t cetin distnce fom the popelle disk cn be witten s (see Fige 0): = V µ C e Eqtion 5

13 sing this le of µ, the les of wee clclted nd plotted in Fige ess the eltie speed loss. Fige 0 Chnge of Wte Velocity ne the Popelle In both cses the pime denotes mking it non-dimensionl with the initil speed 0. If the ces of the popelle toqe nd thst e lineised in the speed nge of inteest the expession fo C cn be witten s follows: C V e = Λ Λ Eqtion 6 whee the coefficients nd descibe the lineised toqe nd thst. sing the wke fction ψ (see chpte.), the les of C cn be clclted fo ech of the 0 speed-ps combintions gien in. If these les e pplied to Eqtion 6, the nknown speed is eplced by the qntity µ, which oiginlly indicted the distnce fom the popelle disk. It mst be noted howee tht in this cse the deitions de to the ssmptions sed clminte t the compttion of µ, so tht this qntity the hs to be consideed clcltion qntity thn n indiction fo the distnce between dde nd popelle. The sme is to be pplied to the qntity. Sbstittion of the expession fo into the foml desciption of the side foce nd moment mesements t = = 0, fo ech of the dde ngles pplied, le of µ ws obtined. In Fige the podcts µ hs been plotted ess the dde ngle fom which the optiml le of 0.76 ws deied. Fige Expeimentl Detemintion of Fcto µ fom Eqtion 5 Fige Lineistion of the Qntity In Fige A (Appendix A), fo the 0 combintions of speed nd pm the mesed dde foces nd moments e plotted on the bsis of while the dde ngle is pmete. The les of nd wee too smll to distingish between st nd dsh components. A men le, deied fom the swying tests nd the pesent tests is obtined.

14 Tble 4 Sttic Swy Test Pogm Fo the compttion of the dde coefficients the mesements concened wee coected with these men les..5 Detemintion of emining Coefficients.5. Sttic Swy Tests These tests, exected t the 0 combintions of speed nd pm, poided the coefficients of the following ibles: nd eqtion:,,, X eqtion:, while lso the nd coefficients wee detemined. In Tble 4 scheme of the test pogm concened is gien. sing the test eslts t = 0 the coefficients of, nd e detemined while compting the, nd coefficients; the coefficients fist mentioned wee consideed to be known qntities. In Fige A, Fige A nd Fige A 4, the side foce nd moment mesements concened e plotted, while the longitdinl foce mesements e shown in Fige A, Fige A nd Fige A 4. As no infomtion ws ilble inoling the inflence of the dde speed, the dsh coefficients concened cold not be detemined..5. Oscilltoy Swying Tests These tests wee minly exected to detemine the ltel dded mss effect. Only two initil speed conditions e consideed while the inflence of chnge of pm ppeed negligible. Conseqently & nd & e set to zeo. The nge in which the non-dimensionl cceletion L & / x ws chnged; it ws extended to 0.5 thogh n estimtion of the mximm fll-scle le is bot 0.5. eetheless the mesed foce ppeed line with the cceletion in the whole nge. In Fige A 5 the dt concened e plotted whee the speeds e consideed pmete..5. Oscilltoy wing Tests with Constnt Dift nd dde Angle In genel these tests wee lso exected fo the ten conditions gien in Tble. The mplitde of the chcteistic ible ws ied between 0.05 nd 0.70 coesponding with tning dii of ppoximtely 0 nd ship lengths espectiely. 4

15 The choice of the oscillto feqency hs been bsed on the following considetions:. Bsed on the tnk width ilble nd the popeties of the oscillto the nondimensionl feqency γ = ω x / g, which is the leding fcto detemining the oscilltoy pt of the we ptten, ws kept s smll s possible. Concening the inflence of this ible the ede is efeed to Vn Leewen (964). b. The foces inoled in the lowest speed, coesponding with Fode nmbe 0.07, hd to be esonbly mesble. In ode to jdge the feqency nge sed t these oscilltoy motions, the qntity ω of ship lengths siled ding one peiod. The esticted tnk width nd oscillto mplitde inoles disgeement between the foced motions of the model nd fll-scle mnoees e.g. sinsesponse tests. These fll-scle mnoees inole the lge mplitdes in the nge of pcticl fllscle feqencies. It is not known, howee, how f this discepncy between model-scle nd fll-scle mnoees inflences the hydodynmic deities. In the efeences Vn Leewen (969) nd Vn Leewen (969b) some moe detils concening this mtte e discssed. π / cn be sed, indicting the nmbe In Tble 5 scheme of the complete ywing pogm is gien. Tble 5 w Test Pogm 5

16 The mesed foces nd moments concening these tests e to be diided into thee components:. popotionl to the ngl elocity (sine component) b. popotionl to the ngl cceletion (cosine component) c. the constnt component. In Tble 6 the ios components e smmised while the ibles concened e mentioned in the seqence of detemintion. inoled the eslts e consideed not ey tstwothy. The test pogm consisted of:. sttic swy tests b. oscilltoy swying tests c. oscilltoy ywing tests (withot dde ngle nd dift ngle) The inflence of the speed edction nd conseqently of the thst incement ws not exmined in pticl. It ws ssmed tht the hypothesis mentioned in Chpte.4 wold hold. This mens tht the tests only hd to be exected fo the initil speed conditions. In Tble 7 the eslts of these tests e smmised nd e comped with those of the lge model. In Fige A 8 thogh Fige A 7 the mesements e shown. Tble 6 Vibles nd Components De to the citei gien in the peceding chpte, concening the nges of oscillto feqencies nd mplitdes, the mximm le of the ngl cceletion mplitde exceeds the coesponding le ee occing t the fll-scle ship, the ltte being estimted bot.0. Conseqently the coefficients concened he been detemined in this fll-scle nge the moe so s otside this nge the model expeiments showed consideble non-line effect. In Fige A 6 thogh Fige A the side foce nd moment mesements e plotted, while in Fige A 4, Fige A 6 nd Fige A 7 the coesponding longitdinl foce mesements e gien. Tble 7 Coefficients of Smll Model (:00) As follows fom Tble 7 fo the impotnt coefficients, the mgnitde of the diffeences between the coefficients of the lge nd the smll model is bot 0 pecent while this pecentge fo the less impotnt coefficients is bot 5..6 Some Expeiments with Smll Model (α = 00) Fo compison ppose the eslts of esticted nmbe of tests with smll model e gien. These tests he been exected befoe those with the lge model. Becse of the ey low speeds Tble 8 Compison of 9 0 Stbod Cicle fo Scle :00, :55 nd : 6

17 The impotnce of these diffeences is ptly shown in Tble 8 in which the eslts of tning cicle mnoee e gien, compted with the :00 model coefficients, completed with some of the lge model. Concening the nge of ibles pplied with this model it is noted tht nely the sme mximm les of dde ngle, dift ngle nd ngl elocity wee djsted. The diffeence between the feqency nges of the two models is expessed by the two qntities π / ω, denoting the nmbe of ship lengths coeed ding one peiod of the oscilltion, nd γ which qntity goens the we ptten ding the oscilltoy motion. The fist qntity ies fom.0 t 0 = 0.0 to. t 0 = 0. 50, which is nely the sme nge s is pplied fo the lge model. The mximm le of γ t 0 = 0.50 (0.7) is bot two times the coesponding le of the lge model howee. eetheless this le is consideed sfficiently low s to oid distbing inflences of this pmete. Ptting the eslts of the two models in the light of the hypothesis, mentioned in chpte.4, it ppes tht the hypothesis holds fo both models septely, bt not if comping both models. As in both cses the Fode nmbes hd the sme les the diffeences between the coesponding coefficients cn be tced to the smll eynolds nmbe of the smll model. It is fond tht the pm effect on the line tems is ey smll, comped to the mgnitde of these tems. Concening the non-line tems, it ws not possible to distingish between the noml sctte of the dt nd this pm effect, de to the esticted cccy of the mesements nd the eltiely smll les of this nonline tem. This does not pply to the pe dde ngle dependnt tems of cose, the chnge of which with pm is consideble. Anothe impotnt eslt is the seflness of the hypothesis, concening the popotionlity of the foces with the sqe of the instntneos speed nd the chcteistic ibles nd. This is clely shown e.g. in Fige A, Fige A nd Fige A 6. Thogh it hs lso been tied to distingish between the eslts concening the fo initil speeds by Vn Leewen (969c), it follows fom the figes jst mentioned tht the diffeences, which cold be consideed Fode nmbe effect, e not significnt howee. Tble 9 Ltel Foce Coefficients.7 Some emks Concening Compted Coefficients Tble 0 w Moment Coefficients 7

18 Tble Longitdinl Foce Coefficients In Tble 9, Tble 0 nd Tble in the left colmns the coefficients of the set of Eqtion e gien while in the ight colmns the coesponding coefficients of Eqtion 4 e smmised, the ltte being deied fom the peceding ones. This deition is ptly bsed on the following ppoximtions: ( ) = = = The cccy of these ppoximtions is shown in Fige, Fige nd Fige 4 in which these qntities e plotted. Concening the signs of the coefficients in these tbles the le holds tht ll tems e tnspoted to the ight hnd sides of the eqtions. Fige Lineistion of Speed Dependent Fcto ( ) Fige 4 Lineistion of Speed / Dependent Fcto ( ) 4 Compte Pogms 4. Lest Sqes Anlysis of Mesed Dt This IBM 60/65 compte pogm SBSL#M0 hs been deeloped to compte the coefficients of foth ode polynomil of fo ibles: F ( x, x, x, x4 ) = p t= C x t k x l x m x n 4 sing the lest sqe citeion. Heein is p 70 nd k l m n 4. If some of the coefficients e ledy known they cn be gien nd the coesponding tems e sbtcted fom the mesed le of the fnction. If the distnce between mesement nd the compted le of the polynomil exceeds two times the MS le the dt concened e dopped while the coefficients e compted gin. This dt point selection pocede pimily 8

19 sees to locte mesing, witing o typing eos. The fcto sed in this citeion hs been fond expeimentlly. In sttistics, slly fcto is pplied, thogh it hs been fond tht if the nmbe of mesement is eltiely smll een lge eos e not locted in tht cse. This dt point selection pocede comes into opetion gin, nless:. the nmbe of selected dt exceeds ten pecent of the totl nmbe, b. the MS le is ledy smlle thn bondy le gien befoehnd. In the next stge the mximm le of ech tem is compted. If these mximm contibtions of nmbe of tems is smlle thn the bondy le jst mentioned the pocede is epeted, thogh withot the coefficient the mximm contibtion of which ws the smllest. Also this coefficients selection pocede is epeted ntil the contibtions of ll tems emined e sfficient. Fthe the stndd deition nd the coeltion coefficient of ech coefficient e compted. The fist qntity is plotted in Fige 5 nd Fige 6 concening the lge nd the smll model espectiely. A lte PC esion of this compte pogm cn be fond t the Intenet: Fige 5 Stndd Deitions of Coefficients Fige 6 Stndd Deition of Coefficients (Smll Model :00) 9

20 4. Soltion of Diffeentil Eqtions In this compte pogm (IBM 60/65 pogm SBSL#M0) the diffeentil eqtions e soled fo gien time depending dde signls, whee the nge-ktt pocede is pplied. Two cses e consideed. In the fist the dde te is constnt o zeo, while in the second sinsoidl dde inpt cn be gien to detemine the feqency chcteistics of ship. Fige 7 Time Histoies of Septe Tems of Side Foce Eqtion of Tning Cicle D Fige 8 Time Histoies of Septe Tems of Moment Eqtion of Tning Cicle D 0

21 Fige 9 Time Histoies of Septe Tems of Longitdinl Foce Eqtion of Tning Cicle D Fige 0 Time Histoies of Stbility oots of Tning Cicle D The otpt qntities cn lso be eqied to obtin the inpt fo the coefficients pogm M0, to detemine the coefficients of mthemticl model with edced nmbe of coefficients. Fthe fo ech step the le of ech tem of the set of eqtions of motion is compted which enbles to get n insight in the impotnce of the ios components. An illsttion of this is gien in Fige 7, Fige 8 nd Fige 9. Anothe wy to obsee the pocess of mnoee is to lineise the eqtions of motion t ech step to the set:

22 & = & = & = 4 sing the coefficients of this set of eqtions the stbility of the system cn be obseed. The time constnts of the system cn be fond fom the oots of the set: λ λ λ 4 = 0 An exmple of the chnge of these constnts, indicting the chnge of stbility, ding tning cicle mnoee, is gien in Fige 0. Concening the time intel between two steps of the compttion, fo ll mnoees ten steps pe ship length, bsed on the initil speed, ws pplied. The time intels following fom this e gien in Tble. 4 The eslts of the compted tning cicles e shown in Fige B thogh Fige B 4. As follows fom these figes, the (finl) tes of tn e somewht smlle thn the fll-scle les thogh, combined with the finl speed, the tning dimetes gee ey well howee. Fo compison with the spil mnoee eslts some dditionl tning cicles he been compted. Only the finl les of the ios qntities he been sed to obtin complete ces. The compttion of the 7 degees pot dde tning cicle hs been epeted omitting cetin coefficients, which significnce ws ey little, ccoding to the model tests. The eslts e shown in Fige B. As ppes fom this fige these coefficients he lso little impotnce in the mthemticl model. 5. Zig-Zg Tils Tble Time Intels Applied fo Compttions 5 Compison of Compted nd Fll-Scle Mnoees 5. Tning Cicles The pincipl dt of the tning cicles e smmised in Tble. In Tble 4, the pincipl dt of the zigzg tils e smmised. Concening the initil conditions, only the cose ψ ws consideed while no othe dt wee ilble. The les of the dde te of tn wee deied fom the dt nd figes gien by Clke (965). The compted zig-zg mnoees e plotted in Fige B 5 thogh Fige B nd comped with the fll-scle mesements. The oeshoot ngles e somewht smlle, bt the men oscilltion peiods gee ey well. These two qntities e plotted in Fige. Concening the initil speed conditions of these mnoees the dopted line eltion: 0 =.5 x pm hs been pplied, while the pm les fo both fllscle tils nd compttions e the sme. Tble Tning Cicle Dt

23 6 Finl emks Tble 4 Zig-Zg Til Dt To jdge the eslt of the model expeiments discssed in this ppe, Fige nd Fige my see in the fist plce. They poide n oell picte of the pincipl pmetes of tning cicle, spil nd zig-zg tils: Fige, c ginst 0: eltion between tning cicle dimete / c ) nd dde ngle. Fige b, c ginst 0: finl speed edction of tning cicle mnoees. Fige, t p ginst / 0: eltion between oscilltion peiods of zig-zg tils nd initil speed. Fige b, ψ / ginst mx 0 / 0: eltion between oeshoot ngle nd nominl dde ngle. Fom these figes, in which the compted qntities e comped with those mesed ding the fll-scle tils, it ppes tht it is possible to pedict the mnoeing popeties of the ship concened by mens of oscilltion tests with esonble cccy. It mst be noted howee tht both fll-scle dt nd compted dt he thei ncetinties. In pticl this my be impotnt if x, y 0 0 plots e comped, becse these plots e obtined ey indiectly. Concening the zig-zg tils it is fond tht little chnge of the exection cose hs eltiely lge effect on the mximm cose deition nd the oscilltion peiod. Finlly we he to keep in mind tht the detemintion of the mnoeing popeties of ship i hoizontl oscilltion tests is the indiect, t lest while the motions of the model ding these tests e the nelistic. Fom the fo figes boe mentioned, it my then be conclded tht if scle effects ply ole in the pesent inestigtion thei impotnce is not ey lge nd of the sme mgnitde s the cccy of both the fll-scle nd modelscle mesements.

24 Fige Pincipl eslts of Pedicted Cicles t 00 ominl PM Fige Pincipl eslts of Pedicted Zig-Zg Tests 7 ecommendtions The mthemticl desciption of the mnoeing popeties is bsed on hypothesis, which descibes the eltion between the foces cting on the ship nd the fowd speed. The ppliction of this hypothesis is flly jstified by the pesent model expeiments. This does not men howee tht ll hydodynmic effects, which ply pt in this mthemticl model, e elly necessy fo sfficient desciption of the hoizontl motions. Theefoe, it might be inteesting to find simple mthemticl model, which popeties e not to gie nd ccte desciption of the hydodynmic phenomen bt the of the motions. 4

25 8 List of Symbols m I ρ zz Mss of the ship Moment of ineti of the ship Density of wte x, y 0 0 Co-odintes in spce bonded co-odinte system Ltel foce, positie to stbod Moment ond Z0-xis, positie to the ight X Longitdinl foce & Added mss in ltel diection & Added moment of ineti Added mss in longitdinl diection X &,,,,,, X, X X, X,, Components of, nd X espectiely Components popotionl to Thst incement coefficient Longitdinl esistnce coefficient = / ρ L / ρ x L = / ρ L / ρ x L X = X / ρ L X / ρ x L 0 Initil speed, & Instntneos fowd speed nd cceletion (ecto) x, & x Instntneos longitdinl speed nd cceletion Locl elocity ne the dde, & Swy (dift) elocity nd cceletion, & w ngl elocity nd cceletion dde ngle Speed edction: x 0 / = sin β / = tn β & = d / ds / ds & ψ β φ φ & s s = x = d ψ / ds = L / L / L / = d Heding ngle Dift ngle Deition ngle = d φ / ds = L / = L / L / Distnce coeed by the ship Distnce coeed by the ship in ship lengths dis of cte x 5

26 Dift dis of cte ( w dis of cte ( m I zz = m / 0.5ρL = I zz / 0.5ρL = /0 = L / 0 = /0 & = L / & 0 & & = L & / = L & / 0 = / L / β ds = d / ) L/ ψ ds = d / ) 9 efeences Abkowitz (964) M.A. Abkowitz, Lectes on Ship Hydodynmics - Steeing nd Mnoebility, HyA epot H-5, Decembe 964. Clke (965) D. Clke, Mnoeing Tils with the Ton Ded-weight Tnke Bitish Bombdie, BSA epot, Decembe 965. Didson nd Schiff (946) K.S.M. Didson nd L.J. Schiff, Tning nd Cose Keeping Qlities, SAME 946. Ed nd Cne (96) H. Ed, nd C.L. Cne, esech on Ship Contollbility, D.L. epot, Steens Institte of Technology, 96. omoto (967) K. omoto, On the Steeing Qlities of Ships, I.S.P. Jly 957. Vn Leewen (964) G. n Leewen, The Ltel Dmping nd Added Mss of Hoizontlly Oscillting Shipmodel, T..O. epot 65S, Decembe 964. Vn Leewen (969) G. n Leewen, Enkele poblemen bij het ontwepen n een hoizontle oscillto (in Dtch), T.H. Shipbilding Lbotoy, epot 5, Jny 969. Vn Leewen (969b) Vn Leewen, Some otes on the Discepncies between the Ltel Motions of Oscilltion Tests nd Fll- Scle Mnoees, T.H. Shipbilding Lbotoy epot, Apil 969. Vn Leewen (969c) G. n Leewen, Tenttie eslts of the Hoizontl Oscilltion Tests with 4- Mete Model of the Bitish Bombdie, T.H. Shipbilding Lbotoy, epot 9, Jne 969. Vn Leewen (970) G. n Leewen, A Simplified on-line Model of Mnoeing Ship, T.H. Shipbilding Lbotoy, epot 6, Mch 970. Zndedop nd Bitenhek (96) H.J. Zndedop nd M. Bitenhek, Oscillto Techniqes t the Shipbilding Lbotoy, T.H. Shipbilding Lbotoy, epot, oembe 96. 6

27 0 Appendix A Fige A Ltel Swy Dmping Moment Fige A Ltel Foce nd Moment de to dde Deflection Fige A Ltel Swy Dmping Foce Fige A 4 dde Angle Dift Coss Copling Effect in Ltel Foce nd Moment 7

28 Fige A 5 Swying Foce nd Moment de to Ltel Mss Fige A 7 w te Dift Coss Copling Effects in Ltel Foce nd Moment Fige A 6 Ltel w Dmping Foce nd Moment Fige A 8 dde Angle w te ( ) Coss Copling Effect in Ltel Foce nd Moment 8

29 Fige A 9 w te dde Angle ( ) Coss Copling Effect in Ltel Foce nd Moment Fige A wing Foce nd Moment de to Moment of Ineti Fige A 0 Dift w te dde Angle Coss Copling Effect in Ltel Foce nd Moment 9

30 Fige A Longitdinl Swy Dmping Foce Fige A Longitdinl Foce de to dde Deflection Fige A 4 Longitdinl w Dmping Foce 0

31 Fige A 5 dde Angle Dift Coss Copling Effect in Longitdinl Foce Fige A 8 Swying Foce de to Ltel Mss (Smll Model :00) Fige A 9 Swying Moment de to Ltel Mss (Smll Model :00) Fige A 6 dde ngle w te Coss Copling Effect in Longitdinl Foce Fige A 0 wing Foce de to Moment of Ineti (Smll Model :00) Fige A 7 Longitdinl Foce de to Centifgl Acceletion Fige A wing Moment de to Moment of Ineti (Smll Model :00)

32 Fige A Ltel Foce de to dde deflection (Smll Model :00) Fige A 5 Ltel Swy Dmping Moment (Smll Model :00) Fige A Ltel Moment de to dde Deflection (Smll Model :00) Fige A 6 Ltel w Dmping Foce (Smll Model :00) Fige A 4 Ltel swy Dmping Foce (Smll Model :00) Fige A 7 Ltel w Dmping Moment (Smll Model :00)

33 Appendix B Fige B Time Histoies nd X- Plot of Tning Cicle A t 4 0 to Stbod nd 00 ominl PM Fige B Time Histoies nd X- Plot of Tning Cicle B t 7 0 to Pot nd 00 ominl PM

34 Fige B Time Histoies nd X- Plot of Tning Cicle D t 9 0 to Stbod nd 00 ominl PM Fige B 4 Tning Cicle Chcteistics t 00 ominl PM 4

35 Fige B 5 Time Histoies nd X- Plot of Zig-Zg Test A (0/0) t 00 ominl PM Fige B 6 Time Histoies nd X- Plot of Zig-Zg Test C (0/0) t 00 ominl PM 5

36 Fige B 7 Time Histoies nd X- Plot of Zig-Zg Test D (0/0) t 00 ominl PM Fige B 8 Time Histoies nd X- Plot of Zig-Zg Test E (0/0) t 85 ominl PM 6

37 Fige B 9 Time Histoies nd X- Plot of Zig-Zg Test F (0/0) t 85 ominl PM Fige B 0 Time Histoies nd X- Plot of Zig-Zg Test A (0/0) t 85 ominl PM 7

38 Fige B Time Histoies nd X- Plot of Zig-Zg Test J (0/0) t 70 ominl PM Fige B Time Histoies nd X- Plot of Zig-Zg Test L (0/0) t 60 ominl PM 8

39 9