LIU Tao,ZHANG Wei-jing,MA Jie,ZHANG Guang-lei

(State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University,Shanghai 200030,China)

1 Introduction

Cable structures are extensively used in mechanical,civil,electrical,ocean,and space engineering due to their capability of transmitting forces,carrying payloads,and conducting signals across large distances.From a cable mechanics perspective,the towed cable can be divided into the two categories:a tensioned(taut)cable and a low tension cable.When a cable is subject to end forces which are much larger than the sum of the forces distributed along its length,its configuration is close to a straight line and it is called a taut cable.Because of the ability of a taut cable to withstand external forces with small changes in its configuration,it is a desirable positioning device both for land and marine applications.

In the case of a cable being towed it-self,the cable becomes low tensioned.The examples of low tension cable include,in military,a towed array sonar system for detecting a moving object or submarine,for civil use,the seismic streamer system which is the main approach in ocean resource survey and the umbilical tethers of remotely operated vehicles ROV,etc.The efficiency of the low tension cables depends on accurate data of cable position,velocity and tension.In order to obtain accurate data,it is necessary to analyze in depth the three-dimensional dynamic behavior of a low tension cable.

Weakly nonlinear taut cables have been investigated extensively and people have achieved many meaningful results[1],in contrast,until recently the attention to low tension cables was started.A low tension problem means that dynamic tension is of the same order as the static tension.Low tension cable problems are particularly complex as linear solution are unobtainable in most cases,due to the lack of a meaningful static configuration.Due to the relatively small restoring force,large amplitude displacement maybe occur,thus giving preeminence to the effect of geometric nonlinearities.The onset of large displacements also prevents developing solutions which are based on linearizing the equations of motion about some static configuration as a meaningful static configuration does not exist.The nonlinear dynamics of taut cables,on the other hand,are only weakly nonlinear.

Meanwhile,the dynamics of low tension problem is more complicated by isolated points of zero tension.For example,near the free end of towed array,energy travels at very small speed because the speed of propagation is proportional to the square-root of the tension.It is essential to introduce the bending stiffness of the cable near zero tension points.The resulting solutions were found to contain boundary layers,demonstrating the importance of the underlying physical mechanism.

Leonard[2]considered the dynamics of a low tension cable by formulating the problem for a slack cable.He investigated the nonlinear response caused by the presence of a moderately large dynamic tension.Dowling[3]and Triantafyllou[4]showed that it is essential to introduce the bending stiffness of the cable near zero tension points for eliminating the ill-posed problem due to zero tension.Howell[5],Burgess[6]and Grosenbaugh et al[7]investigated in detail the dynamic modeling of low-tension cable including bending stiffness,and the application the umbilical tethers.Wang et al[8]investigated the towed-tension cable by lumped mass method,and analyzed the effect of bending stiffness.Sun et al[9]studied the oceanic cable/body system with a localized low-tension region adopting a hybrid model and a solution scheme by direct integration.Park et al[10]presented a numerical and experimental investigation into the dynamic behaviour of a towed low tension cable,measured the tension and shear forces at the top end.Numerical results were compared with experimental data.Good agreements were only achieved by employing enhanced drag coefficients.Patel et al[11],Vaz et al[12-13]numerically solved the two,three dimensional transient motion of a marine cable during installation in the case of a cable laying vessel arbitrarily changing speed and direction while paying out cable with eabed slack.Grosenbaugh[14]described a numerical solution for the transient motion of marine cables being towed from a cable ship which was changing speed,and changing course from a straight-tow trajectory to one involving steady circular turning at a constant radius respectively.Huo et al[15]addressed numerical analysis of the dynamic transient behaviors of undersea cables under various environments.

In general,the governing equations for low-tension cable dynamics are coupled and highly nonlinear.Analytical solutions are unavailable and numerical means have to be used.The finite difference method and finite element method are two widely used numerical methods to solve such nonlinear dynamics problems.The finite difference method approximates the governing differential equations by some difference equations,whereas the finite element method discretizes the continuum(cable)into a finite number of elements.In practice,each method has its merits and limitations.The main advantage of the finite element method is that the procedure can be implemented in general purpose analysis programs for a wide range of applications.In contrast,the finite difference method is more suitable for some specific situations where numerical instability may pose convergence problem to the finite element method.

A popular finite difference scheme is the second-order accurate box method,in which the governing equations are discretized on the half-grid point in both space and time,while it has advantages of easy implementation and two order accuracy.This method was first employed for the solution of tow cable dynamics by Ablow and Schechter[16],then it was adopted by Park et al[10],Huo et al[15],Howell[5],and Burgess[6].But this method has poor stability,and the convergence depends on initial value heavily.Koh et al[17-18]developed the modified box method.Compared with the original box scheme,the modified box scheme has been found to give more stable and more accurate results,and to eliminated spurious high-frequency oscillations.However,the method only had one order accuracy.

Gobat et al[19]proposed the generalized α method which has tow order accuracy and is more stable.Liu et al[20]analyzed the stability and accuracy of the generalized-α method,and applied the method to the three-dimensional low-tension cable dynamics equations.Comparing to the other algorithms commonly used in cable problems,numerical solutions demonstrate that the new scheme not only has second-order accuracy,but takes less computing time and exhibits superior stability properties.In addition,some different finite element methods have been developed by Buckham et al[21],Zhu et al[22],and Montano et al[23]to investigate low-tension cable respectively.

Though numerous studies have dealt with towed cable dynamics,there is still a paucity of publications devoted to the analysis of the transient dynamic behavior about towed low tension cable in detail.In this paper,the transient dynamic behavior of a towed low tension cable is analyzed by numerical and experimental methods.The remainder of the paper is organized as follows.In section 1,coordinate system and governing equations with bending stiffness are formulated in detail.Section 2 presents numerical modelling of towed low-tension cable and application of the generalized α method to solve the dynamic equations.In section 3,experiment was carried out for 3 m cable in towing tank in Shanghai Jiao Tong University.Numerical results are compared with experimental results.Excellent agreement is reported.The possible reasons for these results are further explained.Finally conclusions are drawn in section 4.

2 Governing equations

In this paper,the set of governing equations for the transient behaviour of low-tension cables comprises of compatibility equations as a result of geometric continuity and equilibrium equations from the conservation of momentum.These equations are formulated in the following sub-sections.

The following practical assumptions are made in the formulation:

(1)The towed cable is continuous and flexible.

(2)Plane sections of cable remain plane after deformation.

(3)The cable material is homogeneous,isotropic,and linearly elastic.

(4)Cable strain is small though the displacement and rotation may be large.To improve numerical stability as mentioned earlier,the effect of flexural stiffness is taken into consideration in the formulation though the bending moment may be small.

Firstly the reference systems need to be established.In the analysis of the cable′s dynamic response,two coordinate systems of a fixed one in space and a local one on a cable element are used.In this study,the fixed coordinate system is labeled as X,Y,Z and the local coordinate system is denoted as t,n and b,where t,n and b are the tangent,normal and binormal unit vectors respectively in the local system.The tangential direction is defined as tangent to the cable axis,pointing in the direction of increasing s.The normal direction is perpendicular to t and binormal direction is defined as b such that the system of vectors t,n and b is orthogonal and right-handed.The local system t,n and b changes with time and space as it is attached to a cable element.

Fig.1 Sketch of ship model towing tank in Shanghai Jiao Tong University and coordinate system on the tank vehicle

Fig.2 Euler rotation sequence

The coordinate system is given in Fig.1.The transformation between the fixed coordinate X,Y,Z and the local coordinate t,n and b is accomplished through a set of rotations known as Euler angles.The following rotation sequence has been chosen in this work:first rotate the local axes by the angle φ about the Z-axis,then rotate the local axes on angle θ about the new Y-axis.In this experiment,the cable is just towed and thus the rotation about the new Z-axis is not considered.The rotation sequence is shown in Fig.2.These rotations can be expressed in matrix form as follows:

The governing equations of towed low-tension cable have been investigated by numerous authors,Howell[5],Burgess[6],and Park[10]et al.The motion equations for a cable element in local coordinate system are as follows:

The motion equations can be casted into Matrix format:

where,

M,K,F(see appendix I).

In the above equations,the subscripts,t,n and b denote the directions in the local coordinate system.T is the cable’s effective tension,Snis the normal shear stress,Sbis the binormal shear stress.Vt,Vnand Vbare the cable tangent,normal,binormal velocity at the local coordinate respectively.Ωnand Ωbare the cable normal,binormal curvature.m is cable mass per unit length,w0is the cable weight per unit length in water,R is the fluid drag force,mais the added mass per unit length,E is the Young’s modulus,A is the cross-sectional area of cable,I is moment of inertia of area and u is the flow velocity.ρωrepresents the density of water.d denotes the diameter of cable.Ct,Cnand Cbdenote the cable tangent,normal,binormal hydrodynamic coefficients respectively.

To complete the mathematical formulation we must consider boundary conditions.One end of the cable(s=0)is considered as a free boundary while the other end(s=l)is fixed at the towed carriage.At the free boundary,the tension,moment,and shear forces are all zero.At the fixed end,the three velocities are prescribed and the Euler angels are equal to defined angels.Mathmatically,the boundary conditions are expressed as follows:

where u（t）,v（t）and w（t）are towed carriage velocities at the X,Y,Z coordinate system respectively.φ0and θ0are the mounted angel at the fixed end.

3 Numerical investigation

Generalized α method retains the box method′s spatial discretization,but employs the generalized α method for temporal discretization.This method is employed for solving the three dimensional cable equations in this paper,and the Newton-Raphson iteration scheme is adopted for solving the non-linear problem.

Unlike traditional box method solutions in which the governing equations are discretized in both space and time in one step,generalized α method is chosen to discretize the equations in two distinct steps.The spatial discretization is used the same as that in the box method.Because the discretization is applied on the half-grid points,the method is second order accuracy.

If there are n nodal points along the cable,then we can derive a set of n-1 matrix equations(one equation per half grid point)and Eq.(19)can be rewritten as:

where the overdot signifies differentiation with respect to time and the subscript j defines the spatial node number.The matricesandand the vectorare modified forms of Eq.(19)(see appendex II),containing constants and variables associated with nodes j and j-1.The matrices and vector have dimensions N×2N and N×1 respectively,where N is the number of dependent variables at each node.

Compared to universal box method,generalized α method has many advantages,such as strong stability,fast convergence,and eliminating Crank-Nicholson noise[19].Applying temporal weighted averaging of the velocity,displacement and force vectors to Eq.(22)leads to a semi-discrete equation of the form

where,

In the above equations,the superscript i defines the temporal node number,Δt is the time step size,three temporal weighted averaging parameters αm,αkand γ construct the generalized α method.The coefficient matrices of Eq.(19)vary in time due to high nonlinearity of towed low-tension cable,and the stability becomes conditional with time.For avoiding this problem,Eq.(25),Eq.(26)are adopted to calculate average coefficient matrices using the same weights that are used in averaging the velocity and displacement vectors.

At the fixed end,the tension and shear force have large deflection and sudden variation,namely,they have large gradient.Thus,it is necessary to employ denser grid.If the whole cable is meshed according to this section cable,the number of grid will become very large,and computing time will be very long.Thus a non-uniform grid is employed.In the final version of the program Sjtu-cable,a non-uniform grid which linearly increases from top to bottom is employed,and choose αm=0.333,αk=0.167 and γ=0.167.

4 Experiment

The cable used in the experiment is made of silicon pipe filled with water and counterweight blocks.The properties of the cable are specifically described in Tab.1.

4.1 Test set-up

Experiments were carried out in a 3 m deep towing tank that was designed and constructed for ship model research in Shanghai Jiao Tong University.The top end of the model was fixed to the connected rod and was towed in any prescribed velocity by control signals from the console of tow carriage.The data acquisition system of towed carriage can acquire the velocity signal form encoder mounted on the motor of towed carriage.An acceleration sensor was mounted on the connected rod and used for the effect of towed carriage vibration.The clock synchronization between towed carriage data acquisition system and stress measurement data acquisition system was implemented through wireless network.The strain gauges were attached on the surface of polyethylene rod at the fixed end to the cable in a symmetrical manner.In order to observe the effect of vortex shedding,some strain gauges were mounted on the one meter distance from the top end.

Tab.1 The parameters of low-tension cable

Fig.3 Side view of towing carriage tank in Shanghai Jiao Tong University and experimental apparatus

The side view of the tow carriage,tank and experimental apparatus is already shown in Fig.3.In addtion,an experimental scene is shown in Fig.4.

Fig.4 Cable towing experiment

The towed carriage started from static state,accelerates to steady state,and retained 50 seconds.The velocities of steady state were 0.55 m/s,1.01 m/s and 1.5 m/s,respectively.Each of them was repeated three times for the experimental repeatability.In the data processing equipment,a standard software program is created to process the data sample.The sampling rate was 256 times per second for each channel and a low-pass filter was designed.

4.2 Experimental results

4.2.1 Mean tension for constant towing speed

The Fig.5 reveals the tension and shear force at three towing velocity and their FFT spectrums.When the velocity arrives at steady value,as Fig.5 shows,the tension and shear force still have some fluctuations.These fluctuations may be caused by vortex induced vibration.Comparing to the spectrums for three towing velocities,ones can find that the energy distributes at wider frequency range and has larger value at corresponding frequency when the velocity was 0.55 m/s.It means that large normal and tangential drag coefficients should be adopted at 0.55 m/s.

It can be seen from the tension and shear force time history in Fig.5 that the mean tension becomes smaller and the shear force becomes lager when the towing speed goes faster.The reason is that when the velocity becomes faster,the cable angel becomes larger,namely,the cable floats up,the component of cable weight becomes small.But larger hydrodynamic drag is divided into the tangential directions,and it causes larger shear force.

Fig.5 Experimental results of tension and shear force time history and their FFT spectrum at the top end of towed cable for different steady towing velocity

It can be found from spectrum Fig.5(b)that when the velocity is 1.01 m/s,there is special obvious mono-frequency vortex shedding effect.Fig.5(c)illustrates that the response of the cable contains more frequency components at 1.5 m/s.And it means that when the velocity increases,the fluid field around low-tension becomes complex,mono-frequency vortex does not shed.

Tab.2 Hydrodynamic coefficient for different speeds

4.2.2 Comparison between numerical and experimental resuts

a)Mean tension and shear force

The mean hydrodynamic drag plays a critical role in the small-diameter lightweight cable.It is very important to select accurate hydrodynamic coefficients in a towing cable analysis.The normal and tangential coefficients are known to be dependent on Reynolds number and a towing angle in the towed cable case.As the towing carriage accelerates to a prescribed speed,the hanging cable becomes gradually slanted until it reaches a steady state position.The inclined angle of the cable depends on the normal and tangential velocities.Unlike an ordinary approach,in which a normal coefficient is chosen by using the curve of normal drag coefficient versus Reynolds number of an inclined cable and a tangential coefficient is taken as 1%of the normal coefficient.The normal and tangential drag coefficients must be properly adjusted based on the velocity in this paper.According to the suggestion in Park et al[10],the coefficients used to simulation for constant towing speeds are given in Tab.2.

Fig.6 describes the results of tension and shear force at different towing velocity.As illustrated in Fig.6,the tensions increase as velocity increase,in contrast,the shear forces decrease.Fig.6 also shows that numerical and experimental results have good agreement for tension,including some discrepancy.

Fig.6 Mean tension and shear force of numerical and experimental results for different towing speeds

b)Transient tension and shear force

Fig.7 represents the transient tension and shear force of numerical and experimental results for different towing speeds and towed velocity variation.As experimental results illustrated in Fig.8,ones can observe that,whatever tension or shear force,there is an overshot domain for towing velocity at 1.01 m/s and 1.5 m/s.However,this phenomenon does not happen at 0.55 m/s.In numerical simulation,the input velocities are acquired from the data acquisition system of towed carriage.We adopt two kinds of calculation method:constant hydrodynamic coefficient method and variable hydrodynamic coefficient method.

Fig.7 Comparison of numerical and experimental results at transient situation for different towing speeds

When the cable is towed from static state to steady state,the inclined angel of towed cable varied continuously.Thus,the tangent and normal hydrodynamic coefficients are not a constant value.The variable tangent and normal hydrodynamic coefficients are adopted when the transient dynamics of the towed cable are simulated.The coefficient is assumed as the linear function of velocity.Meanwhile,for contrast,we give results which adopt the constant coefficient for steady towing at three towing velocity(Tab.2).

By looking at Fig.7,it can be seen that variable coefficient method is closer to experimental result than constant coefficient method,and shear force has better result than tension.The simulations of overshoot domain in shear force have excellent agreement with experimental results,but those in tension have relatively large deviations.The reasons for these discrepancies need to be studied further in future.

5 Conclusions

In this paper,a numerical and experimental investigation for transient dynamic behavior of towed low-tension cable is considered.The generalized α method is employed to solve the transient dynamic behavior of towed low-tension cable.The tension and shear force at the top end were measured for different towed speeds and the velocity of towed carriage was recorded.The tension and shear force fluctuations and their FFT spectrum under steady towed velocity demonstrate the effect of vortex shedding.During trainsient process,whatever tension or shear force,there is an overshot domain for towing velocity at 1.01 m/s and 1.5 m/s.However,this phenomenon does not happen at 0.55 m/s.In transient behavior simulation,two kinds of calculation method are adopted.Although variable coefficient method has better agreement with experimental value,there is still certain discrepancy.Thus it needs to pay more attention to the hydrodynamic coefficient in towed cable transient dynamics.

Acknowledgement

The authors gratefully acknowledge the support of The Ship Model Towing Tank Laboratory in Shanghai Jiao Tong University.

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AppendixⅠ

AppendixⅡ