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2102008,20(2):210-215A MULTI-BODY SPACE-COUPLED MOTION SIMULATION FOR A DEEP-SEA TETHERED REMOTELY OPERATED VEHICLE *ZHU Ke-qiang, ZHU Hai-yang, ZHANG Yu-song, GAO JieMaritime College, Ningbo University, Ningbo 315211, China, E-mail: zhukeqiang@ MIAO Guo-pingSchool of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200030, China(Received May 8, 2006, Revised February 28, 2008)Abstract: A multi-body coupled dynamic model is developed to simulate the motion of a Tethered Remotely Operated Vehicle (TROV) system. A strong nonlinear coupling motion between umbilical tether and ROV is discussed .The movement of ROV is considered as six-degrees of freedom. The lumped mass model is applied and an averaged tangential vector technique is included in the three-dimensional dynamic response equations of the cable-segments. The model can simulate the three-dimensional transient coupled motion of the complex multi-body system in typical ship maneuvering conditions and can be used in either a towing problem or a tethered underwater vehicle problem. Simulation results are seen to fit well with the experiment. Key words: cables, ROV, hydrodynamics, coupled motion1. IntroductionDynamic analysis of marine cables has been of interest owing to their wide applications in ocean engineering [1-5]. Experimental study will be the most reliable method to predict the dynamic behavior of marine cable systems. Grosenbaugh [6] measured the motion and tension of a deeply moored buoy. Yoerger et al.[7] and Hover et al.[8] measured the motion of a deeply towed underwater vehicle system.However, the experimental study is time consuming and costly. There exist also some limitations and difficulties in the experiment [9, 10]. The depth limitations of the existing basin laboratories pose questions to the validity of the testing depth. Small scales introduce uncertainties related to viscous effects owing to the nature of the scaling laws. In system design, coupled*Project supported by the National Natural Science Foundation of China (Grant Nos. 10572063 and 50639020), the State Key Laboratory of Ocean Engineering (Shanghai Jiaotong University) (Grant No. 0502) and the Program for Changjiang Scholars and Innovative Research Team in University ( Grant No. IRT0734).Biography : Zhu Ke-qian g (1956-) , Male, Ph. D., Professornumerical analysis will still be an important and practical tool in combination with model tests.Two kinds of modeling techniques are available to predict the response of tethered systems: the continuous (closed-form, analytical) and the discrete models. Continuous models can provide quick estimates of several dynamic characteristics of tethered systems, including motion and tension spectra, transfer functions, the natural frequencies, and the onset of slack in the tether. Both Niedzwecki et al.[11]and Huang et al.[12] used a model of single-degree freedom to characterize the response of a vertically suspended system subject to sinusoidal forces. These simple models provide a first-order approximation to a vertically tethered system. Niedzwecki and Thampi [13]also derived a set of closed equations that represent a multi-segment drill string. Continuous models are effective tools but they also have several limitations. For example, equivalent linearization techniques have to be used to approximate the quadratic drag [14, 8], and the coupled motion equations of ROV are rarely included in these models. Moreover, these analytical models are invalid for a slack tether, which is the precursor of snap-loads.211Discrete models are not constrained by these limitations. They are valid for several non-linear properties like quadratic drag and spatially varying properties of cables. Various components such as ROV cages or in-line clumps can be easily added to the models. Niedzwecki and Thampi [11] developed a multi-segment lumped-parameter model of a drill string connecting a ship to a sub-sea package and used it to investigate snap load. Driscoll and Biggins [15]performed a preliminary study of a strictly vertical caged ROV system using a similar discrete method.The article presents a general multi-body model of space coupled motion for a deep-sea Tethered Remotely Operated Vehicle (TROV) system. A strong nonlinear coupling motion principle between the umbilical tether and the vehicle is discussed.The movement of ROV is considered as six-degrees of freedom. The lumped mass model is applied and an average tangential vector technique is included in the three dimensional equations of the cable-segment. Snap loads that can damage the armour and internal conductors of the tether can be predicted by computing the dynamic tension and the instant configuration. The investigation has an important meaning in avoiding resonance, surviving in resonance, extending the cable life, broadening the operational area of the ROV, or insuring docking and calling-back ROV safely, as well as saving the trial expense.2. Six-degrees freedom motion equations of TROV 2.1 Coordinate systemsTwo orthogonal coordinate systems are used to establish the present dynamic model. One is a global coordinate system ˄,,,E [K ]˅fixed at the ocean surface ship with origin at E ,]axis pointing vertically down to the water, and the K [, axes being in two mutually perpendicular horizontal directions. The other is a local coordinate system (o ,x , y , z ) fixed on the vehicle with origin at o ,x axis pointing to the nose of the vehicle, z axis pointing to the belly of the vehicle, and the y axis completing the right-hand system with the other two axes. The transformation of forces and motions from global to the local coordinate system can be fulfilled by the transformation matrix [T B ].»111213212223313233[]=B T T T T T T T T T T ªº«» ««» ¬¼(1)where11=cos cos T T \,12=cos sin T T \ˈ13=sin T T ˈ21=sin sin cos cos sin T M T \M \ ˈ22=sin sin sin cos cos T M T \M \ ˈ23=sin cos T M T ,31=cos sin cos sin sin T M T \M \ ˈ32=cos sin sin sin cos T M T \M \ ˈ33=cos cos T M Tand ,,\T \ are the Euler angles. The orientation of the vehicle in global coordinates can be specified bythe vector Eo r fromE to o .2.2 Inertia force on ROVSince all vehicles have a longitudinal symmetric plane xoz (), the added mass coefficients A =0G y ij with (I +j ) odd are all zeros. The mass matrix [M ]consisting of all the coefficients of inertia and added mass terms is defined as11223342445153556266000symmetric[]=000000m m m M m m m m m m m ªº«»«»«»«»«»«»«»«»¬¼(2)where1111=+m m A ˈˈ2222=+m m A 3333=+m m A ˈˈ4444=+xx m I A 5555=+yy m I A ˈˈ6666=+zz m I A 5115=+G m mz A ˈˈ4224=+G m mz A 6226=+G m mx A ˈ5335=+G m mz A212The remaining added inertial terms are^`^`T12345661=I I I I I I I F F F F F F F u where221[(+)+I G ]+G F m qw vr x q r z pr 632=1(jj j qArA V ¦)Rj +G )Rj ]+G (3a)2()I G F m ru wp z qr x pq(3b)613=1(jj j qApA V ¦223[(+)+I G F m pv uq z q r x pr621=1()jj Rj j pArA V ¦ 3(c)4()+[()I zz yy G F I I qr m z pw ru ]++) 22()+yx xx yx r q J J qp J pr66532=1(++jj Ry j Rz j R j qArA V A V A V ¦j (3d)5I F =()+[()xx zz G G I I rp m z qw rv x <+G 22()]+()zx xy qw rv p r J J rq 6461=1+(++xy jj Rz j j J qp rApA V A ¦ (3e)3)Rx j Rj V A V 6()+[I yy xx F I I pq m x <22()]+()xy yz ru pw p q J J pr +6542=1+(++xx jj Rx j j J rq qArA V A ¦1)Ry j Rj V A V (3f)where V Rj is the velocity of the vehicle relative to the fluid, and u,v,w,p,q,r are the motion parameters of ROV in the local coordinate system. 2.3 Weight, buoyancy and thruster forcesThe gravitational force and buoyant force are defined in the global coordinate system, and therefore, they must be transformed to the local coordinate systems:+=()G B B m g U <F F ^`Tsin cos sin cos cos TT MT M (4)(+)=(G G B B G B B g mz z )U u u <R F R F ^ (5)`T Tcos sin sin 0T MTThese forces can be combined into a dynamic vector {F CB }.The resultant force and moment of N thrusters can be expressed as=1=i Nr i ¦F F (6)=1=1=+i i N Nr T T i i u ¦¦M M R F (7)The magnitudes of thrust and moment owing to the i th thruster are2=i i T T i 4i F K n D U < (8) 2=i i T Q i 5i M K n D U < (9)where D i is the diameter of the i th thruster, n i is the rotate speed of the thruster shaft, and K Ti , K Qi are the coefficients of the thrust and the torque, respectively, expressed as functions of the advanced coefficient J=V n /(nD ). These two forces are then combined into a dynamic vector {F T }.2.4 Non -inertia hydrodynamic forcesAll the non-inertia hydrodynamic forces can be expressed as:^`,,223,,=2x y z F RR X Y ZF V F C U(10a)213^`,,523,,=22x y z F R R N K M N F V C R UU<^`T,,,,p q r N N N p p q q r r C < (10b)where(,)Fx C D E ,(,)Fy C E J ,(,)Fz C D J , , , Np C Nq C Nr C are six hydrodynamic coefficients of ROV and1=tan ()RzRxV V D (11a) 1=tan ()Ry RxV V E (11b)1=tan (RzRyV V J (11c) Hydrodynamic coefficients owing to the speed of transport motion such as drag coefficient, moment coefficient etc. can be obtained by model tests. The hydrodynamic coefficients owing to the accelerations and angular accelerations of the vehicle can be obtained by means of Planar Motion Mechanism. All these forces are then combined into a dynamics vector ̗F F ̙.2.5 Umbilical cable forceThe umbilical cable force is one of the most important nonlinear disturbance force on ROV when the current is strong and the cable length is long enough. The movement of a flexible cable moving in fluid can be described by the following lumped parameter equation [4]:212[]d ()=++d i i i i i i t tO r W D T T (12) where []i O is the cable mass per unit length, is the position vector of the i th lumped mass with respect to the fixed inertial reference, and is the gravitational force of i th lumped mass.i r i W The drag force lumped onto the i th node can be expressed as follows:11=(+)(+4i i i i i ti Rti Rti l d l d C SD U U )ni Rni Rni C U U where i ,C t C ni are the tangential and normal drag coefficients of the cable in the i th link, respectively,and Rti U ,Rni Uare the tangential and normal components of the relative speed at the i th node.d =[()]d iRti c i i t<r U U W W (13a) d =d iRni Rti tr U U 1) (13b) The unit tangential vector [4] at the i th node is defined as21=(=i i r r W (14a)1+1+11=(=2i i i i i i N ,1)r r r r W (14b)1=(=i N N i N ) r r W (14c)The tension force exerted in the i th link can be calculated by the position of the nodes+12+1=(14i ii i i i id E l S)ir r T r r (15) The cable force on ROV can be obtained by the transformation matrix [T B ] as>@^T=c B N c c c T X Y Z {F T ` (16a) ^`T=c c cz ccx c cz ccx c R Y R Z R X R Y u R F (16b)where ^`T=c cxcycz R R R R is the position vector oftethered point at the body coordinate system of ROV. Based on the above analysis, the motion equation of TROV can be written as^`^`^`^`^`6161616161[]+=++I GB T c M QF F F F u u u u u (17)T ()=()+[]N bo B c t t T r r R (18a)214T d ()d =[][+d N B c t T t d c tu r RU R :(18b)where^`^`61=,,,,,Qu v w p q r u ` 3.2 Co ˈˈ^`T=,,u v w U ^T=,,p q r : (19)3. Numerical results3.1 The numerical integration methodA set of coupled differential equations of the multi-body system can be solved simultaneously with dynamic equilibrium at each time step. A static equilibrium position of system can be used to provide the initia1 condition.All nonlinearities (material, geometric, explicit loads, and hydrodynamic loads) are treated in a consistent way. A computer program has been developed to integrate these governing equations by the Runge-Kutta method. The time step size for the numerical integration can be changed automatically. To save the CPU time, the Euler integration method can be used to surface ship and underwater ROV.Fig.1 Simulated configuration for anchor -last deployment testAn ROV is connected to the cage by a cable of 300 m length. There is a clump weight in the middle of the cable. The cable diameter is taken as 32 mm and the weight in water is 2.25 N/m. The Young’s modulus of the cable is taken as 2u 10Fig.2 Comparison of the present theory with the testmparison with the experimentAn anchor-last deployment experiment is chosen for comparison. The tether is fully immersed and allowed to reach an initial static configuration at the beginning (Fig.1). The results with changed drag coefficient are in fairly good agreement with the experiment (Fig.2). The comparisons also indicate that different choice of the drag coefficients can bring the simulated and measured results into better agreement. Some transient behavior is different from the measured values. For example, the oscillation in the measurements is not completely present in the simulation. This discrepancy may be due to the inaccurate drag coefficients or certain non-modeled effects such as bending stiffness of the cable.Fig.3 The configuration of a three-dimensional submersiblecable3.3 Numerical computations of a TROV11 N / m. The surface excitation end is taken to be fixed. The current speed is taken as 0.5 knots. The lower ROV moves in a circular path. Figure 3 shows how current affects theconfiguration obviously.Fig.4 Static equilibrium cable configurationThe motions of the upper cable nodes lag the motion of the lower ROV end clearly. The largest215tension appears when the ROV is just against the current. It is clear that the current plays a significant role to the cable tensions. Figure 4 shows the static equilibrium configuration of the cable at different current speed, which can be used as initial values in the dynamic simulation to save computation time. The cable tension forces are sensitive to the surging when the ship is located in the upstream of ROV and sensitive to heaving motion when the ship is located in the downstream of ROV.In a towing case, the tension will fluctuate in the upper and lower end owing to the pure heaving or / and the surging motions of the surface towing ship. The calculation is done for the severe oceanic conditions with the amplitudes of the ship heaving and surging taken as 5 m and the motion period taken as 6 s. The calculation shows that the surface ship excitation, which has minimum angle between the excitation and the tangent of the cable of upper end, will induce the peak tension of the cable. A further comparison of the model predictions with experimental data is intended for the future.4. Conclusions(1) The lumped-parameter model is still an effective and practical tool to simulate the nonlinear coupling movement of a multi-body system when various components such as ROV cages or other in-line clumps are included in the model.(2) Model test basins can make use of coupled analysis for calibration of models. The efficiency of coupled analysis proves that it can be applied even in early design stages and thereby reduces uncertainties in analyses, and a more optimal design can be achieved.(3) The current has great effect on the cable tension in a tethered underwater vehicle problem. The cable tension are sensitive to the surging and heaving motion of the surface ship when the ship is located in the upstream and downstream of ROV, respectively.(4) The surface ship excitation that has minimum angle between the exciting direction and the tangent of upper end will induce the peak tension value of the cable in a towing problem.(5) There is still considerable study to be done about drag coefficients or certain non-modeled effects such as bending stiffness of the cable. It will be very useful to give a more simple representation of bending force used in the lumped-parameter model. References[1] HOU Guo-xiang, LI Hong-bin and ZHANG Sheng-jun.Behavior of elastic towing cables in shear currents [J].Journa l of Hydrodyna mics, Ser. B, 2005, 17(5):590-595.[2] XIE Nan, GAO Huan-qiu. Two-dimensional timedomain dynamic analysis for the generalbuoy-cable-body system[J]. Journal of Hydrodyna mics, Ser. A, 2000, 15(2) : 202-213 (inChinese).[3] ZHU Ke-qiang. 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