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`THE SHOCK
`AND VIBRATION
`DIGEST
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`Volume 23, No. 7
`July 1991
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`
`FEATURE ARTICLE
`tutorials and items of special interest
`
`DYNAMICS OF CABLES, TOWING CABLES
`AND MOORING SYSTEMS
`
`M.S. Triantafyllou1
`
`Abstract. This is the third in a series of reviews
`of the mechanics of cables. This review con-
`centrates on the dynamics of towed cables and
`the drag of moorlng systems and covers the
`general literature on cables that has appeared
`since the last review.
`
`The subject of the mechanics of cables and chains
`attracts considerable attention, and the related lit-
`erature is very extensive. The reasons are that
`cablesfind use in positioning or anchoring structures
`of a great variety of shape, in air or in the ocean,
`while their mechanics contain a rich variety of
`interesting or even intriguing properties, which are
`still being discovered.
`If one were to consider also
`the related subjects of fluid—structure interaction and
`hydrodynamically induced Instability, a review would
`be too extensive. As a result the focus here is
`primarily on issues related to the basic mechanisms
`of cable response.
`
`The linear and nonlinear dynamics of cables and
`chains were reviewed by the author in 1984 [1] and
`in 1987 [2]. Since then a number of studies have
`appeared, addressing primarily the behavior of
`cables and mooring lines in water and the nonlinear
`behavior of cables in air.
`
`CABLES AND CHAINS IN WATER
`
`The dynamic behavior of a cable in water is affected
`primarily bythe presence ofthe nonlineardrag force.
`The basic developments have been outlined in the
`second review [2]. The drag force itself is the subject
`of extensive study both for its importance as a force
`resisting any imposed motion and for the complex
`phenomena that cause it. The concentration here
`is on some aspects germane to cable dynamics,
`since the subject of viscous fluid forces is very
`extensive and is covered elsewhere in the literature,
`for example, Sarpkaya's paper [3].
`
`A horizontal hawser used in towing surface vessels
`has similar properties as mooring ines’ in that drag
`Suppresses transverse motions causing the tether
`to stretch substantially. Intact, for higher frequency
`motions the cable employs almost exclusively its
`elastic rather than its catenary stiffness [4,5]. An
`additional phenomenon of particular importance to
`hawsers is snapping. Snapping occurs when the
`dynamic tension exceeds the static tension in
`amplitude, causing the cable to become slack for
`part of the cycle, and then to tighten suddenly,
`possibly causing extreme tensions and failure. The
`principal parameter controlling snapfiing was found
`to be the free-falling velocity of the awser. When
`this velocity is small, the tether cannot achieve a
`sufficiently deep catenary when falling freely, and
`the subsequent tightening is accommodated by
`stretching and hence high tension. On the other
`hand,
`if the free falling velocity is increased for
`example, by making the hawser heavier), the ca Ie
`forms a deep catenary shape when it becomes slack.
`Then as it tightens, the cable absorbs the imposed
`motions through changes in the catenary, resulting
`in small dynamic tensions [6-8]. Milgram, et al. [7]
`explored these properties of towing hawsers and
`developed a methodology for predicting extreme
`forces in towing through a seaway.
`
`The optimization of the dynamics of mooring lines
`with submer ed attached buoys was addressed by
`Mavrakos, 9 al. [9,10].
`It was shown that these
`moored buoys behave as inverted pendula, with the
`buoyanc acting as an equivalent gravityforce. This,
`in turn, a lows, through proper design of the natural
`frequencies, substantial reduction of the dynamic
`tension.
`if proper care is not taken, then buoys can
`cause the dynamic tension to increase substantially
`[11].
`
`Modelin and testing of cables and mooring lines
`was ad ressed by Papazoglou, et al. [8].
`twas
`shown analytically and experimentally that
`the
`principal arameter to model is the elastic stiffness
`of the ca Is, and this can be achieved in practice
`
`1 Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Massa-
`chusetts 02139
`
`
`
`
`
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`

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`only through the insertion of a spring in series with
`the mooring line.
`For hawsers,
`the free falling
`velocity of the cable, as already described, must also
`be scaled properly.
`In a subsequent work, Papa-
`zo lou, et al. [12] consider model testing of a cable
`wit attached buoys.
`It is shown that proper scale
`modeling requires the insertion of a properly
`desi ned spring in each cable span between buoys
`anc or.
`andhn the span between the lower buoy and t e
`Cables towed in water. A large number of publi-
`cations have appeared covering the subject of a
`cable towed in water, usually with a body attached
`at the lower end. Depending on the application, the
`cable may be nearly vertical (as in many applications
`involving remotely operated vehicles positioned
`through a surface vehicle) or nearly horizontal (as in
`the case of towed arrays). The hydrodynamic forces
`are completely different in these two cases.
`In the
`first case, there is a strong resistive in-line force and
`a vortex induced out-of-plane (lift) force, causing
`out-of—plane oscillations.
`In the second case, the
`force is primarily inviscid and acts under certain
`conditionsto destabilizethe motion otthe cable. The
`basic mechanism of destabilization is described in
`a large number of related publications reviewed by
`Paidoussis [13]. A more recent, very elaborate
`analytical-numerical treatment of the dynamics of
`such cables can be found in Dowling’s work [14,15].
`
`Cables towed nearly horizontally are used for a
`variety of applications, such as geophysical explo~
`ration and vehicle detection. Paul and Soler [16]
`considered the two-dimensional dynamics of a
`towed cable-body system. They ignored inertia
`forces and used a finite link approximation, while
`they introduced some clever analytical manipula—
`tions to reduce the computational cost., Sanders [1 7]
`considered the three-dimensional dynamics of a
`towed system usin finite differences, also ignoring
`Inertia forces. De mer, et al. [18] formulated the
`problem of the nonlinear three-dimensional motion
`of a multi-segmented cable subject to fluid and
`weight forces using the finite element method
`(specifically, the method of lines) and allowing for
`cables whose length varies withtime. They reported
`a 10% accuracy when they compared their results
`with ex erimentsinvolvingthede loymentofacable
`[19 .
`blow and Schechter]20 studied the same
`pro lem using the method 0 time differences and
`local cable coordinate formulation (is. projecting
`the equations along the local tangential, normal,
`binormai directions), gaining in speed of execution
`over Sanders [17].
`Further im rovement
`in the
`methodology of Ablow and Sc echter [20] was
`achieved by Milinazzo, el al [21] who employed an
`implicit, second orderfinite difference approximation
`of'the governing equations. Comparison was made
`With the exper mental results 0 Rispin [22]. An
`omisswn In these publicationsthat should be pointed
`
`-
`
`out is in not properly accounting for the hydrostatic
`force. This requires using the concept of effective
`tension as first explained by Goodman and Breslin
`[23]. Erroneous results may be obtained by ignoring
`the hydrostatic force.
`
`Delmer, Stephens, and Tremills [24] studied the
`three-dimensional
`dynamics
`of
`cable—towed
`acoustic arrays using the lumped element method
`and a stiff integration technique. They applied their
`methodology to several examples and compared
`their
`results with experiments,
`including those
`reported by Meggit. et al. [19] and Meggit and Dillon
`[25]. They also comparedtheirresultstoasimulatlon
`of a 180 degree turn maneuver of a towed system.
`Their applications met with varying degrees of
`success. Areasonforsome discrepancies observed
`may lie in the rather primitive modeling of drag
`adopted by the authors.
`
`Chapman [26] studied the effects of ship motion on
`a neutrally buoyant fish. He modeled the cable as
`consisting of straight inextensible rods, freely inter-
`connected at the ends. He discovered the "sheath"
`action of drag, is. the reduction of lateral motion at
`the expense of considerable stretching and the
`considerable effect of cable length on the pendulum
`motion of a suspended fish. Forshort cable lengths,
`the response is that of a damped pendulum, while
`for long lengths,
`the action of drag practically
`decouples bottom and top transverse motions.
`hence causing a length-independent pendulum
`motion of the cable.
`
`Chapman [27] also studied the response of a cable
`to towing ship maneuvers.
`He omitted cable
`stretching and inertia forces on the cable, while he
`considered drag forces only on the cable and not on
`thetowed fish. He distinguished between sharp and
`gradual maneuvers in terms of the ratio of the radius
`of ship turn and the scope of the cable. A distinctly
`different cable response was obtained for sharp
`maneuvers, which caused a rapid descent of the
`cable lower end.
`
`Further, Chapman [28] studied the attenuation of
`ship-induced lateral motions along a towed cable,
`supporting a heavyfish. He employedthe Bath Mk-3
`Sonar Fish to establish the validity of his results.
`In
`an experimental study, Bettles and Chapman [29]
`considered the response of a towed fish to dynamic
`excitation. Delmer and Stephens [30] considered
`the dynamics of a weight towedthrough a long'cable;
`they addressed in particular the response of the
`weight to the cross-tack oscillation of the upper end
`of the cable.
`
`
`
`
`
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`
`
`Schram and Reyle [31] considered the three-
`dlmensional analysis of an inextensible cable and
`towed system, using the method of characteristics,
`which is discussed extensively in the context of the
`application studied. De Laurler [32] studied the
`s ability of towed and tethered bodies whenthe cable
`has general curvature and tension variation. Paul
`and Solar [16], referenced above, considered the
`dynamics of towed cables by ignoring inertia forces
`on the cable and modeling the cable through finite
`interconnected links. They considered also optimal
`strategies for positioning submersibles, and they
`showed moderate advantages in using overshooting
`in the position of the surface vessel to overcome the
`time lag in the response of the submersible.
`lvers
`and Mudie [33] considered the three-dimensional,
`slow (i.e., well below the wave frequency ran e)
`dynamics of towing a very long cable (up to 4 km at
`low speeds (up to 3 knots),
`ignoring inertia and
`tangential drag. The drag coefficient was found b
`minimizing the error of the model predictions wit
`respect to full scale data; its value was estimated to
`be on = 2.7. The authors note the long delays in
`achieving changes in the position of an unpowered
`towed vehicle, which reach,
`in their estimation,
`values of 30 min for a 4 km long cable.
`
`Le Guerch [34] studied the deep towing of fish using
`long cables, 2 to 6 kilometers in length, and at low
`speeds up to 2 knots, using a model based on lvers
`and Mudle’s study. His primary objective was to
`studythe 180 degree turnin maneuverwith a radius
`of turning of the order of 50 meters. He compared
`his
`results with full-scale data, with particular
`attention to the value of the drag coefficient, which
`he reports to be CD = 1.5.
`
`Triantafyllou and Hover [35 and Hover, et al. [36]
`used a simplified version off a governing equations
`fora cable to study the slow and fast dynamics of a
`cable used in towing remotely operated vehicles.
`The simplification consists in assuming moderately
`large deviations from an average configuration and
`quasi-static tangential cable motion. The simu-
`lations were compared with full-scale data, reported
`by Yoerger, et al.
`[37], Triantafyllou, et al.
`[138],
`Grosenbaugh, et al. [39] and Grosenbaugh [40 .
`
`Drag Forces on Mooring and Towing Cables.
`Vessels moored through several mooring lines
`possess a natural fre uency of oscillation. due to the
`low damping availabe in the system. The mass,
`plus added mass, of the vessel and the overall
`stiffness of the mooring system are the equivalent
`mass and spring stiffness, respectively. A variety of
`low-frequency forces, particularly due to the non-
`linear wave forces, cause such vessels to oscillate
`with relatively large amplitudes. The mooring lines
`are subjected to large drag forces as they are orced
`in these large amplitude oscillations, hence provid-
`ing a substantial damping mechanism.
`In the case
`
`of towed cables, the drag coefficient is a principal
`parameter affecting the accuracy of prediction for
`the) ositlon of the towed body and the shape of the
`ca 9.
`
`Despite considerable effort, the value of the drag
`coefficient in a flexible cable still cannot be predicted
`with the required accuracy. There are two basic
`causes for this: first, the well known Karman street
`causes out-of-plane motions, which may cause
`substantial drag increases (up to a factor of 3 or
`more); second, the superposition of wave-induced
`motions and slowly va ing motions results in an
`apparent increase of t e damping of the slowly
`varying motions. The two phenomena together can
`cause an unexpectedly large increase in the
`apparent drag coefficient.
`
`In the case of short cables, low-frequency motions
`are practically absent, hence the principal unknown
`physical mechanism is the appearance of lock-in;
`I.e., vibrations synchronized with the formation of
`vortices. This is a subject of considerable interest,
`but will not be reviewed here since the mechanics
`of the cable itself are of secondary importance.
`
`For long cables, however, both mechanisms men-
`tioned above are present, while the mechanics of
`the cable are of primary importance even for the
`vortex formation problem, since the system is
`practically of infinite extent, and lock-in is continu-
`ous. The significant variation of drag coefficient
`reported in towed cables was noted above (lvers and
`Mudie [33] report 2.7, while Le Guerch [34] reports
`1.5). An apparent explanation is offered by the
`recent work of Yoerger, et al. [37] Triantafyllou, et
`al. [38], Howell [41], Grosenbaugh, et al. [39] and
`Grosenbaugh [40].
`in these papers it Is noted that
`the shear in the tow causes amplitude modulated
`response, which, in turn, lowersthe value of the drag
`coefficient (which also becomes de endent on the
`modulation amplitudes traveling aong the cable
`length). Hence, sharp transient maneuvers will
`result in lowervalues ofdra coefficient, while steady
`towing results in higher va ues.
`
`in the case of mooring lines, Huse [42] and Huse
`and Matsumoto [43,44], Wichers [45], Wichers and
`Huisjsmans [46], Koterayama, et al. [47], and Kot-
`erayama, et al. [48 studied the increase of drag
`coefficient due to t e su erposition of low— and
`high-frequency motions.
`lthough an exact calcu-
`lation is still lacking, the apparent increase in drag
`coefficient of
`the low-frequency motions is well
`documented. An outline ofthe basic mechanism can
`be fl)??? among others, in the paperby Demirbilek,
`eta
`9 .
`
`
`
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`

`

`LINEAR AND NONLINEAR
`MECHANICS OF CABLES
`
`The linear dynamics of cables translating between
`two fixed points were addressed by Perkins and
`Mote [50], using a numerical method, and by Perkins
`[51]. Perkins used a first-order asymptotic expan—
`sion (with respect to the sag to span ratio) and the
`assumption of quasi—static stretching (in essence
`Irvine's equation in a translating frame [52]).
`In both
`studiesthe natural frequencies of atranslating cable
`were studied, while the case of an "inverted cate—
`nary," where the centrifugal force provides a load
`normal to the cable configuration, was addressed
`and in particular the onset of instability when gravity
`overwhelms the centrifugal force as the velocity of
`translation goes to zero.
`
`In an interesting paper, Llu, Huang, and Chen [53]
`showed that lnthe case of achain of vibrating strings,
`coupled dissipatively, the energy always decays
`uniformly if damping is applied only at the ends;
`whereas if damping is distributed at the Internal
`nodes, the energy may decay uniformly or non-
`uniformly, or it may not decay at all.
`
`The well—known quadratic and cubic nonlinearity of
`the planar dynamics of sagging cables and the cubic
`coupling between in-plane and out~of—p|ane motions
`provided further 0 portunity for research. Bene-
`dettinl, Rega, an Vestroni
`[54] considered the
`out-of-plane
`nonlinear
`response
`of
`a
`cable.
`Takahashl and Konishi [55,55] first considered the
`free, nonlinear vibrations of a cable and then
`determined the conditions forobtaining out-of—plane
`response with in-plane harmonic excitation. Bene-
`dettini and Rega studied the nonlinear dynamics of
`an elastic cable under planar excitation [57] and the
`superharmonic planar resonant responses of an
`elastic cable (second and third superharmonic) [58].
`Rega and Benedettinl [59] examined the subhar-
`monic planar resonant responses (one-half and
`one—third subharmonics).
`REFERENCES
`
`1.
`Triantafyllou, M.S.,
`"Linear Dynamics
`of
`83%|: and Chains," Shock Vib. Dig., 15(3). pp 9-17
`
`. Triantafyllou, M.S., "Dynamics of Cables and
`2.
`Chains," Shock Vib. Dig.,1_9 (12), pp 3-5 (1987).
`
`Sarpkaya, T., "Vortex Induced Oscillations," J.
`3.
`Appl. Mech., 45. PP 241-258 (1979).
`
`Bur ess, J.J. and Triantafyllou, M.S., "Time
`5.
`Domain
`imulation of
`the D namics of Ocean
`98 .
`210M739 Lines," Proc. PRADS ’8 ,Trondeim, Nonivay
`6.
`Shin, H., "Nonlinear Cable Dynamics," Ph.D.
`Thesis, M.I.T., Cambridge. MA (1987).
`
`Milgram, J.H., Triantafyllou, M.S., Frlmm, F.,
`7.
`and Anagnostou, G., "Seakeeping and Extreme
`Tensions in Offshore Towing," Trans. Soc. Naval
`Architects and Marine Engrs., 95, pp 35-72 (1988).
`
`Papazoglou, V.I Mavrakos, S., and Trianta—
`8.
`fyllou, M.S., "Non-Llnear Cable Response and
`Model Testing In Water," J. Sound Vib., 151.0 (1), pp
`103-115 (1990).
`
`and
`9. Mavrakos, S.A., Papazoglou, V.J.,
`Triantaf llou, M.S., "An Investigation into the Fea-
`sibilityo Deep Water Anchoring Systems," Proc. 8th
`lntl. Cont. Offshore Mech. Arctic Engrg. (OMAE ’89),
`The Hague, Netherlands, 1, pp 683-689 (1989).
`
`10. Mavrakos, S.A., Neos, L., Papazoglou, V.J.,
`and Triantafyllou, M.S., "Systematic Evaluation of
`the Effect of Submer ed Buoys' Size and Location
`on Deep Water Moor rig Dynamics," Proc. PRADS’
`89, Varna, Bulgaria (1989).
`
`Shin, H. and Triantafyllou, M.S., "Dynamic
`11.
`Analysis of Cable with an Intermediate Submerged
`Buoy for Offshore Applications," Proc. 81h Intl. Conf.
`Offshore Mech. Arctic Engrg. (OMAE), The Hague,
`Netherlands, 1, pp 675-682 (1989).
`
`Papazoglou, V., Mavrakos, S. and Triantafyl-
`12.
`lou, M.S.,
`"A Scaling Procedure for Mooring
`Experiments," Proc.
`uropean Offshore Mech.
`Symp., Trondheim, Norway (1990).
`
`Paidoussis, M.P., "Flow Induced lnstabilities
`13.
`of Cylindrical Structures," Appl. Mech. Rev., Trans.
`ASME. All. (2), pp 163-175 (1987).
`
`14. Dowling, A.P., “The Dynamics of Towed
`Flexible C Iinders.
`Part 1: Neutrall Buoyant
`Elements,"
`. Fluid Mech., 181, pp 507- 32 (1988).
`
`15. Dowling, A.P., "The Dynamics of Towed
`Flexible Cylinders. Part 2: Negatively Buoyant
`Elements," J. Fluid Mech., 191, pp 533-5 1 (1988).
`
`Paul, B. and Solar, A., "Cable Dynamics and
`16.
`Optimum Towing Strategies." Marine Tech. Soc. J.,
`a (2), pp 34-42 (1972).
`
`. Burgess, J.J., "Natural Modes and Impulsive
`4.
`Motions ot a Horizontal Shallow Sag Cable," Ph.D.
`TheSIS, M.I.T., Cambridge, MA (1985).
`
`17. Sanders, J.V., "AThree—Dimensional Dynamic
`Analysis of a Towed System," Ocean Engrg., 9 (5),
`pp 483-499 (1982).
`
`—fl-————n——————~——v~—.A”),
`
`
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`WESTERNGECO Exhibit 2047, pg. 6
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`WESTERNGECO Exhibit 2047, pg. 6
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`

`

`18. Delmer, T.N., Stephens, T.C., and Coe, J.M.,
`"Numerical Simulation of Towed Cables," Ocean
`Engrg., 1Q (2), pp 119-132 (1983).
`
`Schram, J.W. and Fieyle, S.P., "AlThree—
`31.
`Dimensional Dynamic Analysis of aTowed System,"
`J. Hydronautics, 2 (4), pp 213-220 (1968).
`
`19. Meggit, D.J., Palo, PA, and Buck, E.F.,
`"Small—Size Laboratory Experiments on the Large-
`Displacement Dynamics of Cable Systems,"1, Rept.
`No. M-44-78-11,
`Civil
`Engrg.
`Lab., Naval
`aggsgiuction Battalion Ctr., Port Hueneme, CA
`20. Ablow, CM. and Schechter, 8., "Numerical
`Simulation of Undersea Cable Dynamics," Ocean
`Engrg. m (6), pp 443-457 (1983).
`
`21. Miiinazzo, F., Wilkie, M., and Latchman, S.A.,
`"An Efficient Algorithm for Simulating the Dynamics
`of Towed Cable Systems," Ocean Engrg., fl (6), pp
`513-526 (1987).
`
`22. Rispin, P., "Data Package No. 1 for Cable and
`Array Maneuvering," David W. Taylor Naval Ship
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`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2047, pg. 7
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2047, pg. 7
`
`

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`
`
`PGS V. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2047, pg. 8
`
`PGS v. WESTERNGECO (IPR2014-00689)
`WESTERNGECO Exhibit 2047, pg. 8
`
`