Dynamic Instability of Taut Mooring Lines Subjected to Bi-frequency Parametric Excitation

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1 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. Dyamic Istability of Taut Moorig Lies Subjected to Bi-frequecy Parametric Excitatio Aiju Wag, Uiversity of Strathclyde Hezhe Yag, State Key Laboratory of Ocea Egieerig, Shaghai Jiao Tog Uiversity Nigel Barltrop, Uiversity of Strathclyde Sha Huag, Uiversity of Strathclyde ABSTRACT Parametric excitatio or parametric resoace occurs whe the offshore structure system parameter varies with time ad meets a certai coditio. Moored structure durig the swell sea states causes the taut moorig lie tesio uctuatio which may iduce very large dyamic motio of moorig lies. I this work, the taut moorig lies subjected to bi-frequecy parametric excitatio are studied. The parametric excitatio equatio of moorig lies is derived. The Bubov- Galerki approach is employed to obtai stability chart whe cosider the bi-frequecy excitatio. The resposes of the moorig lies subjected to sigle- ad bi-frequecy excitatio are discussed. Keywords: dyamic istability; moorig lie; parametric excitatiomathieu; bi-frequecy 1. INTRODUCTION The dyamic istability is the oscillatory motio of dyamic system due to timedepedet variatio of structure parameters e.g. iertia or stiffess due to the ifluece of exterally applied force. Differet the forced excitatio, the parametric excitatio is oautoomous system. It ca have catastrophic effects o the structures. The study idicated parametric excitatio would cause very large icrease i lateral dyamic motio of moorig lies due to the variatio of axial tesio. (Röquist et al., 1). It is beeficial to avoid the moorig lie desig to locate i the ustable zoe. Recetly, some researchers have studied the dyamic istability due to parametric excitatio such as parametric rollig of ships, spar ad risers (Falzarao et. al 3, Yag et al. 15, ad Zhag et. al 1). However, previous work all focused o the sigle frequecy excitatio. Actually, the offshore structure is exposed to radom waves which are multi-frequecy excitatio. So, it is ecessary ad beeficial to study the taut moorig lies subjected to bi-frequecy parametric excitatio.. THEORY AND MATHEMATICAL MODEL The geeral dyamic equatio of Beroulli- Euler beam ca be expressed as follow. 4 y y y EI T m f ( x, t) (1) 4 x x t Where EI is the bedig stiffess, T is the axial tesio, m is mass per uit legth, f ( xt, ) is exteral force o the beam. For the moorig lies, the bedig stiffess ofte ca be eglected ad the axial tesio ca be expressed by T T TA () t () 93

2 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. Where() t Acos( t). T is the mea tesio of the lie. T A is the amplitude of lies dyamic tesio. is the tesio variatio frequecy of the lie. is the radom phase. The hydrodyamic o the lies are calculated by Moriso equatio. It is oliear ad ca be liearized as the follows. 1 y y y f( x, t) DC D CDD t t t (3) d y( ) dy( ) c d d a q Acos( k ) y( ) where t, is the basic frequecy. 1 TA k, a ( ), q ( ), T c 1 CD D m. T l m is the atural frequecy of the lie. =1,, (7) Combig equatios (1)-(3) leads to This is the secod order homogeous equatio which amed Hill equatio. y y m C DD t t y T TA Acos( t) x (4) Assumed that the eds are pied, the lateral motio of lies ca be writte as x yxt (, ) y( t)si (5) l 1 Submit Eq. () ito Eq. (1), it follows that d y() t dy() t [ m C DD 1 dt dt x T T A t y t A cos( ) ( ) ( )]si l l (6) The above equatio ca be rewritte ito the geeral form, 3. SINGLE-FREQUENCY EXCITATION CASE I this sectio, oly sigle-frequecy excitatio is take ito accout, ad the the Eq. (7) ca be expressed as follow. It is the well-kow Mathieu equatio. ycy aqcos( ) y (8) For the sigle-frequecy excitatio, the top ed motio of moorig lies correspods to the floatig structure durig the regular sea. 3.1 Stability chart of sigle-frequecy excitatio The stability chart is ofte used to idetify the ustable ad stable zoes of dyamic istability due to parametric excitatio. The stability chart ca show the chage istability as the parameters are varied ad are very useful for the desig guidace. It ca be solved by Floquet theory or perturbatio method. The stability chart of sigle-frequecy excitatio for 94

3 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. the taut moorig lies is show i the Fig.1. It ca be see that the ustable zoe will shrik as the dampig icreases. The resoace will occur whe excitatio is itegral or sub multiple of fudametal frequecy. Fig.1 Stability chart of sigle-frequecy excitatio (Blue lie--c=; red lie--c=.1; gree lie-- c=.; black lie--c=.3) 3. Dyamic respose due to siglefrequecy excitatio i time domai Fig. preset two case for the lateral dyamic respose at the midpoit of the moorig lie (Case I locate at stable zoe ad case II locate at ustable zoe). The directio of respose is orthogoal to the directio of the excitatio. For the case II, It ca be see that the respose will icrease expoetially. This case should be removed for the desig. Respose (Case I) Phase plae trajectory Respose Phase plae trajectory (Case II) Fig. Dyamic respose due to sigle-frequecy excitatio i time domai (Case I: a=5; q=.4; c=.1; Case II: a=5; q=.5; c=.1) 95

4 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. 4.1 Effects of differet d 4. BI-FREQUENCY EXCITATION CASE For real sea coditios, the top ed of moorig lie is subjected to multi-frequecy excitatio from floatig motios i the radom waves. The parametric istability property of moorig lie has great effects o the safety of the desig case ad it ca cause the lateral motio expoetial icrease i the oscillatio. Here, the parametric istability due to bifrequecy excitatio is studied which is coducive to uderstad the mechaism of the dyamic istability. ycy a q(cos( ) dcos(4 ) y (9) The dyamic istability for the bifrequecy excitatio is obtaied by the Bubov-Galerki approach (Perdese, 198). Fig.3 shows the stability chart for the bifrequecy excitatio. It ca be see that the ustable zoe for the bi-frequecy excitatio is obviously differet from the sigle-frequecy excitatio. Fig.4 ad 5 give the stability charts for differet d =.5 ad -.5 respectively. The sig of d mea the differet phase betwee the two excitatios. The ustable zoe chages whe d turs ito egative ad the shape is also differet. It is iterestig to fid that the closed zoe exists. Fig.4 Stability chart for bi-frequecy excitatio for differet dampig (d=-.5) (Blue lie--c=; red lie--c=.1; gree lie-- c=.; black lie--c=.3) Fig.3 Stability chart for sigle- ad bifrequecy excitatio (d=-.5) (Blue lie- sigle-frequecy excitatio; red lie- bi-frequecy excitatio) Fig.5 Stability chart for bi-frequecy excitatio for differet dampig (d=.5) (Blue lie--c=; red lie--c=.1; gree lie-- c=.; black lie--c=.3) 96

5 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. 4. Effects of differet dampig for positive d The effects of differet dampig for positive d are compared i the Fig.6-8. The dampig varies from to.3. It ca be see that the ustable zoe of bi-frequecy excitatio is more sesitive tha the siglefrequecy excitatio. The ustable zoe chages more as the dampig varies. Fig.8 Stability chart for bi-frequecy excitatio for differet d (c=.3) (Blue lie--d=-.3; red lie--d=-.5; gree lie- -d=-.7) 4.3 Effects of differet dampig for egative d Fig.6 Stability chart for bi-frequecy excitatio for differet dampig (c=.) (Blue lie--d=-.3; red lie--d=-.5; gree lie- -d=-.7) Fig.9-11 preset the effects of differet dampig for egative d o the dyamic istability zoe. The dampig varies from to.3. It ca be see that the ustable zoe of bifrequecy excitatio is completely differet from the sigle-frequecy excitatio. The ustable zoe chages more as the dampig varies. The safety case will tur ito usafely whe dampig varies or the sig of d chages. Fig.7 Stability chart for bi-frequecy excitatio for differet d (c=.1) (Blue lie--d=-.3; red lie--d=-.5; gree lie- -d=-.7) Fig.9 Stability chart for bi-frequecy excitatio for differet d (c=.) 97

6 Proceedigs of the 1 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles, Jue 15, Glasgow, UK. (Blue lie--d=.3; red lie--d=.5; gree lie-- d=.7) excitatio. The safety case i the siglefrequecy excitatio may become usafely i the bi-frequecy excitatio. The effects of differet parameters o the stability chart were discussed. The results idicate that multifrequecy should be give cosideratio i a more accurate predictio of parametric excitatio. ACKNOWLEDGEMENTS This work was fiacially supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ). Fig.1 Stability chart for bi-frequecy excitatio for differet d (c=.1) (Blue lie--d=.3; red lie--d=.5; gree lie-- d=.7) 6. REFERENCES Röquist, A. Remseth, S. ad Udahl, G. 1. Ustable No-Liear Dyamic Respose Ivestigatio of Submerged Tuel Taut Moorig Elemets Due to Parametric Excitatio. 3rd Nordic Semiar o Computatioal Mechaics. Jeffrey Falzarao, Ju Cheg, ad Samrat Das. 3. Parametric excitatio of floatig offshore platforms. 8 th Iteratioal Coferece o the Stability of Ships ad Ocea Vehicles. Fig.11 Stability chart for bi-frequecy excitatio for differet d (c=.3) (Blue lie--d=.3; red lie--d=.5; gree lie-- d=.7) 5. CONCLUSIONS I this work, the taut moorig lies subjected to sigle ad bi-frequecy parametric excitatio were studied. The resposes of the moorig lies subjected to sigle- ad bifrequecy excitatio were discussed. The ustable zoe of bi-frequecy excitatio is obviously differet from the sigle-frequecy Yag H.Z., Xu P., 15. Effect of hull geometry o parametric resoaces of spar i irregular waves. Ocea Egieerig. 99:14-. Libag Zhag, Ayma Eltaher, Paul Jukes. 1. Mathieu istability of TTR due to VIV. Proceedigs of the 5 th Iteratioal Offshore Pipelie Forum. Pederse, P Stability of the solutios to Mathieu-Hill equatios with dampig. Igeieur-Archiv, 149 (1),