Parabola Model with the Application to the Single-Point Mooring System

Transcription

1 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 Parabola Model wth the Applcaton to the Sngle-Pont Moorng System Chunle Sun, Yuheng Rong, Xn Wang, Mengjao Tan Faculty of Scence Jangsu Unversty, Jangsu 3 Zhenjang, Chna @qq.com Abstract Set n the CUMCA Problem A, from the two aspects of marne envronmental load and system s parameters, the study descrbes the desgn of sngle-pont moorng system. It s manly dvded nto three parts, ncludng the upper buoy, the chan and the lower anchor. Also, the chan lne s smplfed as a quadratc parabola. Furthermore, statcs model where there are global mechanc analyss and local force analyss s set up. Keywords Sngle-pont moorng system; Parabola; Statc equlbrum I. INTRODUCTION Snce developed by the U.S. navy n the 9s, moorng system has been wdely used n ocean observaton, engneerng, aquaculture and other felds. A sngle-pont moorng system, consstng of a cylndrcal buoy, cylndrcal steel ppes, a cylndrcal steel drum, a heavy ball, a chan and a specal anchor []. The cylndrcal buoy s meters wde, meters hgh and weghs kg. Every ppe s.5 meters wde, meters long and weghs kg. There s a underwater acoustc communcaton set n the steel drum that s meters hgh and.3 meters wde. They wegh kg totally. The model and parameters of the chan are shown n TABLEI. It provdes horzontal recovery force, under the acton of external force n the envronment, to keep the moorng floatng body postonng. In fact, t has a seres of advantages of smple structure, good drecton and low cost, so t s one of the most common ways of postonng the feld of mltary and cvlan n Chna. TABLE I. THE MODEL AND PARAMETERS OF THE CHAIN The mass of the unt Model length(n(kg/m)) I 3. II 7 III.5 IV 9.5 V 8. In the process of desgnng a moorng system, the followng matters should be pad attenton to. Frstly, the tlt angle of the tangent at the lnk of the anchor and chan does not exceed degrees, otherwse the anchor wll be dragged, causng the node to shft. Secondly, when the steel drum s vertcal, the operaton of the underwater acoustc communcaton set n t s the best. If the drum s tlt angle s more than 5 degrees, the equpment works poorly. To reduce the angle, a heavy ball wll hang at the lnk of the drum and chan. It s ultmately requred that we determne the model, length of the chan and the weght of the heavy ball to make the draft of the buoy and steel drum s tlt a n g l e a s s m a l l a s p o s s b l e. Fg.. The sngle pont moorng system The remander of ths paper s organzed as follows. Secton II focuses on statc analyss of the moorng system. Secton III shows system parameters at two JMESTN359 99

2 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 wnd speeds of m/s and m/s. Secton IV descrbes the change of the heavy ball s weght when wnd speed ncreases. Secton V desgns the optmal soluton to the system n a certan sea area and senstvty analyss n ths case. Concluson and p r o s p e c t a r e p r e s e n t e d n S e c t o n VI. B. The analyss of forces appled on ppes B T F II. STATIC ANALYSIS [] T G A. The analyss of forces appled on the buoy Fg.3. The analyss of forces that appled on ppe B Fg.. The analyss of forces appled on the buoy The buoyancy appled on the buoy s D B g ( ) h D gh () where s the densty of seawater, D s the cylndrcal buoy s dameter, h s ts draft. From sea wnd load F=.5 Sv (N), where S s the projecton area of the plane of the wnd speed, v s wnd speed, and water flow F=37 Sv (N), where S s the projecton area of the plane of the water speed, v s flow rate, we can get that the marne envronmental load on t s v F T G F.5 D ( l h) v 37 D hv () The buoyancy appled on Secton (,3,, 5 ) of steel ppes s D B g ( ) l D gl (5) Where D s ts dameter, l s ts length. The marne envronmental load on t s F 37 D l v cos () Secton of ppes remans n a state of rest,so ts statc equlbrum equatons are F T sn T sn (7) T cos B T cos G (8) WhereG s ts gravty, T s the tenson at the bottom of Secton of the steel ppe, s the tlt angle of the steel ppe. C. The analyss of forces appled on the steel drum and the heavy ball where v s wnd speed, v s flow rate, l s the cylndrcal buoy s heght. The buoy remans n a state of rest,so ts statc equlbrum equatons are B G T cos (3) F T sn () WhereG s ts gravty, T s the tenson at the top of the frst ppe, s the tlt angle of the buoy. Fg. 3. The analyss of forces appled on the drum and ball The buoyancy appled on them s B D 7 B7 g ( ) l g (9) The marne envronmental load on them s 3m7 3 F F7 37v [ Dl cos ( ) ] () Ther statc equlbrum equatons are m F () F7 T5 sn5 T sn JMESTN359 9

3 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 T5 cos 5 B B7 T cos G G7 () where G,G7 are ther gravty, B, B7 are the buoyancy acted on them, T s the tenson at the top of the chan, 5 s the drum s tlt angle, s the tlt angle of the chan lne s tangent at the juncton wth the drum. D. The analyss of forces appled on the anchor Fg.. The analyss of forces that appled on the anchor Snce the tlt angle of the chan lne s tangent at the lnk wth the anchor s very small and even none, t s assumed that the tenson that the anchor take, has no component force n the vertcal drecton. As the anchor s volume s very small, ts buoyancy s neglgble. The statc equlbrum equatons of the anchor are approxmately f T (3) N G 9 () Where f s the statc frcton force exerted on t, N s the seabed touchdown, G9 s the gravty of the anchor. E. The shape of the chan lne [3]-[] Catenary and quadratc parabola model are two ways of statc analyss of the chan lne. Snce the catenary equaton s transcendental and the tlt angle of the lne s tangent at the lnk wth the anchor s not equal to, the solutons are very complex and dffcult to calculate. When the angle s small, quadratc parabola can be used nstead of catenary. In ths paper, quadratc parabolc s used to smulate the shape of the chan lne. a) When the chan lne s tangent to the seafloor, we pck the orgn O to be the pont where the anchor s, put the seafloor lne as the x axs and vertcal lne as the y axs, and set up the plane rectangular coordnates system xoy. f It assumed that w s the self-gravty per unt length of the chan, F (F=f) s horzontal force of ts tenson and x s the length of ts part on the seafloor. When the anchor lne s tangent to the seabed, based on the geometrc relatonshp between the curve and force, the parabolc curve equaton s N G 9 T w ( x x), x x y f (5), x x The length of ts part on the seafloor s x l 7 fh w h () where h s ts vertcal projecton s heght, l7 s ts length. b) When the chan lne and the seabed have a certan angle, we pck the orgn to be the pont O, put the tangent as the x axs and the perpendcular lne of the x axs as the y axs to set up System x oy. Because the angle that named between ox and ox s very small, t can be assumed that the y axs and y axs are approxmately concdent and f s approxmately parallel to the x axs. Fg. 5. The model of the system In x oy, the equaton s w f x' y ', so n System xoy, the equaton turns nto On the bass of L and w y x x tan (7) f w tan L tan f L h tan, we can get L h wl arctan( ) (8) L f The horzontal projecton length of the chan lne s l7 h L (9) JMESTN359 9

4 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 F. The analyss of forces of the whole system f B G N Fg.. The analyss of forces of the whole system The whole system statc equlbrum equatons are G F F f B N B 7 9 F B 9 For the whole system, we can see that l G () H h cos h () where G s gravty of the whole system, B s ts buoyancy, H s the of the sea. III. SYSTEM PARAMETERS AT TWO WIND SPEEDS From (9) and (), we establsh nonlnear multvarable equaton systems. The buoy s draft s 9 9 h ( G B) /( gd ) () If we select the chan n model II wth length of.5m, the heavy ball wth weght of kg and the densty of seawater s 5 kg/m3, the densty of steel s 785 kg/m3, we can calculate the self-gravty per unt length of the chan s w 59.59N / m. If seawater s stll and sea s 8 meters deep, when v m / s and v m / s, the chan lne s equaton s (5). In ths case, the solutons can be solved by Newton-teratve method. ) When v m / s,tlt angles ( / ) of steel ppes and the steel drum from top to bottom are.3,.7,.9,.33,.398. The length of ts part on the seafloor ( x ) s 9.7 meters long. ) When v m / s,tlt angles ( / ) of steel ppes and the steel drum from top to bottom are.33,.7,.98,.53,.55. The length of ts part on the seafloor ( x ) s.38 meters long. IV. THE CHANGE OF THE HEAVY BALL S WEIGHT Under the assumpton of Secton III, wnd speed ncreases to 3m/s. In ths case, the lne s equaton s (7) and the other equatons are the same to the front ones. The tlt angle of the steel drum s 8.5 degrees and the tlt angle of the tangent at the lnk of the chan and anchor s 9.93 degrees. In ths case, the anchor wll be dragged and acoustc communcaton sets work poorly. It s necessary that the heavy ball s weght should be ncreased to adjust the system. As the weght of the ball gradually ncreases, the angles are all decreasng and the tlt angle of the drum frstly decreases to 5 degrees. When the weght of the ball contnues to ncrease to kg, the tangent s tlt angle s just 5.93 degrees and the drum s tlt angle s.98 degrees. In ths condton, the system works properly. V. SYSTEM DESIGN AND SENSITIVITY ANALYSIS A. System Desgn [7] [8] Due to some factors such as tdal, sea envronment ncludng water, flow speed and wnd speed, s changng. Based on the analyss of forces of the system, we establsh a mult-objectve optmzaton model mn[ 5, h, R] 5 5 (3) where the model, length of the chan and the ball s weght are decson varables. We make data dscrete among the range of varables and enumerate all the combnaton plans to seek for the optmal soluton. When water changes between meters and meters, sea water velocty can reach.5m/s and wnd speed can reach 3m/s, the better soluton s that the chan s model s V, ts length s.3m and the ball weghs 39kg. B. Senstvty Analyss [9] Under the optmal soluton, senstvty analyss s conducted to assess the nfluence of sea water velocty, wnd speed and water on the system parameters[8]. JMESTN359 9

5 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 a) When v 3m / s, v.5m / s,under dfferent water s, the parameters change as follows. TABLE III. PARAMETERS UNDER DIFFERENT WATER VELOCITIES TABLE II. PARAMETERS UNDER DIFFERENT WATER DEPTHS Water (H/m) Ttle angles ( / )./.8/.75/.8 /.89./.8/.75/.8/.89./.8/.75/.8/.89./.8/.75/.8/.89 Partal length ( x / m ) Draft ( h /m ) Water velocty ( v / m / s ) Ttle angles ( / )././././..5/./.8/.9/.5./.3/./.8/.3 3./ 3.3/ 3.35/ 3.39/3../.8/.75/.8/.89 Partal length ) ( x / m Draft (h/m) /.8/.75/.8/ Fg. 7. The shapes of chan under dfferent water s From TABLE II and Fg. 7, t s clear that water manly affects the tlt angle of the tangent. The deeper the water, the bgger the tlt angle and the smaller the area nstead. But t has lttle effect on the tlt angles of ppes and the drum. Fg. 8. The shapes of chan under dfferent water veloctes From TABLE III and Fg. 8, we can fnd that water velocty has a great mpact on the shape of the chan. And t s postvely correlated wth the tlt angles of ppes and the drums, but negatvely correlated wth the tlt angle of the tangent. c) When v.5m / s, H 8m, under dfferent wnd speeds, the parameters change as follows. b) When v 3m / s, H 8m, under dfferent water veloctes, the parameters change as follows. JMESTN359 93

6 Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: Vol. Issue 3, March - 7 TABLEIV. PARAMETERS UNDER DIFFERENT WIND SPEEDS wnd speed ( v / m/ s ) Ttle angles ( / ) 3.87/ 3.93/././ /.5/./.8/.5.5/./.8/.35/..3/.3/.9/.5/.3./.8/.75/.8/.89 Partal length ( x / m ) Draft (h/m) Fg. 9. The shapes of chan under dfferent wnd speeds Comparng TABLE III and TABLE IV, we can see that the nfluence of wnd speed on the tlt angles of ppes and the drums s smlar to water velocty s. However, t has less nfluence on the chan s shape. VI. CONCLUSIONS AND PROSPECT Based on the CUMCA Problem A, the Sngle-Pont moorng system s studed by settng up statcs model and parabola Model n ths paper. Wthn the allowable range of errors, the complex calculaton s replaced wth the approxmaton to smplfy the model and the Newton-teratve method s used to seek exact solutons to the equatons. After a seres of studes and calculatons, t s clear that water velocty, wnd speed and water are three man varables that affect the system parameters. Water velocty and wnd speed manly affect tlt angles of ppes and the drums. Water manly affects the tlt angle of the tangent and swmmng area of the buoy. Besdes, they all has a great effect on the chan s shape. The statcs and parabola model of the system needs to be mproved. In theory, the dynamc analyss and the non-lnearty of the moorng lne wll be the focus of research n the future. ACKNOWLEDGE Research s supported by the Natonal Natural Scence Foundaton of Chna (No. 773). REFERENCE [] the CUMCA Problem A, [] L. Wang, Study on Dynamcs of Sngle-Pont Moorng systems, Shandong: Ocean Unversty of Chna, pp. 8-3,. [3] W. Han and H. Lu, A revew of the study on Sngle-Pont Moorng systems, Salvage Proceedngs, pp. 7-9, 9. [] Y. Xa and H. Lu, Studes on ant-typhoon buoy SPM and ts applcaton, Shandong: Ocean Unversty of Chna, pp. -,. [5] D. Tang, R. Ju, Q. Yue and S. Wang, Modal dampng rato analyss of dynamcal system wth non-statonary responses, Appled Ocean Research, vol. 59, pp. 38-,. [] W. L, G. Sh, W. L and J.Yang, Force Calculaton of Turret FPSO Sngle Pont Moorng System, Journal of Chongqng Jaotong Unversty, vol. 3, pp ,. [7] K. Xue, W. Wang and W. Wang, A Dynamc Postonng Method for Sngle Pont Moorng System of a Catamaran, Key Engneerng Materals, vol. 5, pp.7-5,. [8] Y. Rho, K. Km, C. Jo and D. Km, Statc and dynamc moorng analyss-stablty of floatng producton storage and offloadng (FPSO) rsers for extreme envronmental condtons, Internatonal Journal of Naval Archtecture and Ocean Engneerng, vol.5,pp , 3. [9] D. DU, L. Fang, M. Cheng and Z.W., Effect of wnd and tdal current on moton stablty of sngle pont moorng system, Journal of Naval of Engneerng, vol. 9, pp. 79-9, 7. JMESTN359 9