Applied Ocean Research 15 (1993) 235-242

Technical Note

A numerical method for predicting snap loading

of marine cables
Shan Huang & Dracos Vassalos
Marine Technology Centre, University of Strathclyde, Glasgow, G40LZ, UK,

(Received 14 June 1993 revised and accepted 4 August 1993)

A numerical approach for predicting the snap loading of marine cables operating
in alternating taut-slack conditions is presented. The modelling is based upon the
lumped-mass-and-spring method with modifications to take into account the bi-
linear axial stiffnessof the cable. The resulting governing equations are integrated
in the time domain using the modified Euler method. Numerical examples are
given and comparisons of the numerical results and the experimental results
available are made which demonstrate the validity of the present method.

1 INTRODUCTION a more complex multi-degree-of-freedom model for

predicting the snap loading qualitatively should it
Cables are widely used for offshore and subsea occur using the Newmark-Beta method for the
engineering. In applications such as mooring a buoy numerical integration in the time domain. Milgram
or tethering a subsea unit, marine cables can become et al. 2 and Shin 3 considered two-dimensional cases.
slack if the tension temporarily falls to a level which is Using various assumptions such as a uniform distri-
comparable to the distributed drag force along the bution of dynamic tension along the cable and a high
cable, and thus operates in alternating taut-slack ratio of static tension to cable weight, they simplified the
conditions under periodic environmental excitations. governing equations and solved them by the spectral
In the slack state, the motion of the cable is dominated method with the Newmark's method for the time
by its own inertia force and the fluid drag force. integration.
Depending upon the rate at which the cable becomes The present study focuses upon a general three-
taut, the transition from the slack to the taut state may dimensional modelling and an alternative numerical
cause high tension in the cable which can have integration scheme. The so-called lumped-mass-and-
detrimental effects and may even cause cable breakage. spring model, which discretises the cable into a series of
It is therefore of paramount importance for the safety concentrated masses connected by massless springs, is
and the proper design of marine cable systems to predict employed and modified here. The resulting second-order
cable tensions, taking into account the snap loading ordinary differential equations are solved using the
caused in the transition between slack and taut cable modified Euler method. 4 The approach adopted here is
conditions. capable of tackling the pertinent nonlinearities of
In spite of the practical significance of snap loading of marine cable dynamics including damping, geometric
marine cables, only a few theoretical investigations have nonlinearity, and the bi-linear axial stiffness of cable
been conducted addressing the problem. Focusing operating in alternating taut-slack conditions.
exclusively on a one-dimensional case, Niedzwecki and The lumped-mass-and-spring model has been used
Thampi I used a single-degree-of-freedom model for the widely by many authors predominantly for two-
preliminary assessment of snap loading occurrence, and dimensional dynamic analysis of taut cables. 5-7 The
simplicity and flexibility of this model allows many
Applied Ocean Research 0141-1187/93/$06.00 modifications which enable the treatment of bi-linear
© 1993 Elsevier Science Publishers Ltd. axial stiffness of the taut-slack cables. For numerical
235
236 S. Huang, D. Vassalos

Z
solutions of second-order systems in the time domain,
the Newmark family of methods is perhaps most widely
used for its accuracy and stability. For the highly
nonlinear marine cable dynamics, however, the appli-
cation of the Newmark's methods is not convenient as
additional computational effort is needed for iteration.
The application is further complicated by the response-
dependent nature of the forces such as tension and fluid
drag, especially for cables operating in taut-slack Fig. 1. Coordinate system and discretisation
conditions. In comparison, the modified Euler method
is considerably simpler than the Newmark's methods. massless elastic segments. All forces along the cable are
The explicit nature of the scheme makes it easily assumed to be concentrated at the mass points which are
adaptable to nonlinear dynamic analysis. numbered by i running from 0 at one end to N at the
other. By invoking Newton's law of motion for the i-th
node on the line, we have the following equation of
motion in the Cartesian coordinate system:

1
2 MATHEMATICAL MODELLING

2.1 Governing equations A21 A22 A23 f;i = Fyi (1)

A31 A32 A33 zi Fzi
The problem is formulated in a general manner allowing
for the following: where the coefficients A~, A12,..., A33 contain the mass
and added mass of the i-th cable segment. The added
1. Three-dimensional motion. mass is assumed to be independent of the component of
2. Large displacements. No linearization is made motion parallel to the segment line. Thus only when an
based on small amplitude motion assumption. element of the cable has transverse acceleration, does it
3. Inclusion of forces due to the weight of the cable, possess added mass, whilst if it is accelerated long-
buoyancy, drag and virtual inertia of the fluid. itudinally no hydrodynamic reaction due to inertia
4. Non-uniform cables. The approach has the occurs, xi, Yi and z i are coordinates of the i-th node. The
capacity to include any subsystems, such as forces Fxi, Fyi, Fzi include tension forces in the two
hanging weights on the cable. segments, drag force, gravitational force and buoyant
5. Bi-linear axial stiffness of the cable operating in force. Any other external forces, if they are present, can
alternating taut-slack conditions. be included in this term. The tension forces are
determined by the elastic properties of the cable
Figure 1 shows how a cable is replaced by a segments and their deformation. If the cable is taut,
discretised model consisting of many point masses and tensions are determined by Hooke's law; if slack they are

400

EXPERII4ENT (FYLLIN6 G WOLD t979)

380 -
• PRESENT IIETHO0
320 -

280

z~ 240

bl
t- 180
HAXZHUH

120
B - m
/ _, -
80
NZNINIJI4 ~-4~--~'--1~--"---~
4O

0 I I I I I I I
-.I .t .3 .5 .7 .9 1.1 t.3 .5
FREGUENCY (Hz)

Fig. 2. Comparison between the numerical and experimental results (excitation amplitude 0.025 m)
 A numerical method for predicting snap loading of marine cables 237
400

380 EXPBIINENT ( D ' L L I N 6 E N(X.D 1979)

-, PRESENT 14ETHO0
320

280

240

200

i- t00
NAXINUN

t20
m []
m
80
NZNZNUN
40

0 I I I I I I I
t .1 .3 .5 .7 .9 t.t 1.3 t.5
FREQUENCY (Hz)

Fig. 3. Comparison between the numerical and experimental results (excitation amplitude 0.05 m)
set to zero. The hydrodynamic drag on the i-th node is known motions is considered. In this case,
expressed as one-half of the drag acting on each of the
two segments on either side of that node. The drag on Xo(t) = 0
each segment is resolved into two components, namely, yo( t) = 0 (2)
the normal and tangential, each a function of the
relative velocity in that direction. Zo( t) = 0

and
2.2 Boundary and initial conditions XN(t ) : XtN

Boundary conditions must be given at both ends of the yu(t) = y~ (3)

cable. Different types of boundary conditions occur zN(t) : z~
depending on the physical conditions at the ends. In the
following numerical examples, a horizontally placed where X~v,Y~v and z~v are given beforehand.
sagged cable with one end fixed and the other subject to To complete the formulation, a set of initial

EXPERZNENT (FYI.LZNG G NOt.IX t079)

ai PFIESENT HETHO0

280

240

8O m -- II
NZNZHUN
4O

0 I I ! I I I I
--.t .1 .~ .5 ,7 .9 1,4 1.3 .5
w Y (~)
Fig. 4. Comparison between the numerical and experimental results (excitation amplitude 0"075m)
238 S. Huang, D. Vassalos

m
400
EXPERINENT (FYLLIN6 & WOLD 19791
380 " PRESENT NETHOD

320

280

240

2OO

uJ
p-
t60

t20 B

80

40

0 I I I I I IIr''nk I
- I .I .3 .5 .7 .9 1.1 '1.3 4.5
FREGUENCY (Hz)

Fig. 5. Comparison between the numerical and experimental results (excitation amplitude 0-1 m)

conditions must be specified for each node on the line. At must be sufficiently small to describe the response. A
This includes both the initial positions and initial recommended value for At is one eightieth of the small
velocities, as required by the second order ordinary period in the discretised cable system.
differential equations.

4 R E S U L T S AND D I S C U S S I O N

3 NUMERICAL INTEGRATION Model tests on cable snap tension were performed in the
Ship Research Institute of Norway 8 which consisted of
For the numerical integration of the mixed initial- restraining one end of a horizontally placed sagged cable
boundary-value problem defined by Eq. (1) and its and exciting the other end sinusoidally in the horizontal
associated boundary and initial conditions, the modified direction with known amplitude and frequency. The test
Euler method is used here. The numerical scheme can be results are used here to verify the validity and accuracy
expressed as follows:

A131[/] of the present numerical approach.

The principal parameters of the cable and test are:

ii
J~i
zi n
= A21 A22
LA31 "/132 A33Jn
A23[
~i n
(4) diameter:
length:
span:
Young's modulus:
0"01m
10.9774m
10.792 m
l0 ll N/m 2
Yi At (5) mass distribution: 0-61kg/m
Yi = Yi d-
range of amplitude: 0.025 m to 0"1 m

[xiI [xi]
-7i n + l -7i n range of frequency: 0 Hz to 1"2 Hz
Figures 2 to 5 show the maximum and minimum
Yi = Yi + Yi At (6) values of tension in the cable under excitations of
different amplitudes and frequencies for both the
Zi n+l Zi n Zi n+l numerical results and the experimental results. The
where At is the time step, and subscripts n and n + 1 agreement is good. It is. easy to see that when the
indicate their operant taking its value at time t = n A t or excitations are not severe, i.e., the excitation frequency is
t = (n + 1)At respectively. The procedure defined by low or the amplitude is small or both, the maximum and
equations (4)-(6) is carried out for all the nodes where minimum tensions are symmetrical with a constant
new displacements, velocities and accelerations are to be mean value which is equal to the static tension. When
evaluated. the excitation becomes severe, however, this symmetry
As the scheme is conditionally stable, At must be can no longer hold. As the minimum tension approaches
selected to satisfy the stability condition. 4 In addition, zero, that is the cable starts to operate in alternating
 A numerical method for predicting snap loading of marine cables 239

-.4

-.6

-.8

-t .0

- t .2 I I I I
240 242 244 246 248 2110
TZHE (SECONDS}
(a)

&

-t

-2 | I | I
240 242 244 248 248 250
TZI~ (SECONDS)
(b)

t0

i -2

-4

"6

-8

-10 I I I I
240 242 244 248 248 26O
TZ~ (s[co~sl
(e)

Fig. 6. Time record of the vertical responses of the mid-point of the cable. (a) Displacement; (b) velocity; (c) acceleration.
(Excitation amplitude 0.05 m, frequency 0-75 Hz.)
240 S. Huang, D. Vassalos

-.4

-.8

-.8

-t .0

-1.2 | | I I
190 t92 t94 t98 tM 200
TZNE (SECONDS)
(a)

3.0

2.5

2.0

1.5

t.0

.5

0.0

-.5

-1.0 ! l I I
190 tg2 194 t96 t98 200
TI~ (SECONO)
(b)

20

15

,y // /
t0

5
iI i

i
/
0
#
n
/ /
n
-5 ! /
-t0
/ /

-t5

-20 i i I I
190 111~ 104 tM 4M
TZPlE ( ~ )

Fig. 7. Time record of the vertical responses of the mid-point of the cable. (a) Displacement; (b) velocity; (c) acceleration.
(Excitation amplitude 0.075 m, frequency I Hz.)
 A numerical method for predicting snap loading of marine cables 241

-.4

-.O

1
X

-.8

- I .0

I I I I
-1.2t90
t92 194 196 t98 2OO
TZNE (SL=COmS)
Ca)

3.0

2.5

2.0

1.5

1.0

0.0

- t .0
190 192 194 196 t96 200
TTHE (SECOND)
(b)

i -:~
jjj
ij l ljl
-t5

-25

-30 I I I I
t92 ~94 t96 198 200
TIlE (SI~C0ND)
{c)

Fig. 8. Time record of the vertical responses of the mid-point of the cable. (a) Displacement; (b) velocity; (c) acceleration.
(Excitation amplitude 0.1 m, frequency 1 Hz.)
242 S. Huang, D. Vassalos

taut-slack conditions, the maximum tension increases 5 CONCLUSIONS

significantly.
It is interesting to examine the kinematic responses The lumped-mass-and-spring model has been employed
of the cable under small, average and severe excitations. for marine cable dynamics and extended to incorporate
To this end, Fig. 6, 7 and 8 illustrate the vertical the tension-related nonlinearity which is important in
displacements, velocities and accelerations of the mid- predicting snap loading. The modified Euler method,
point of the cable under three different situations. In which has certain advantages over the more popular
these figures the positive direction corresponds to Newmark's methods, is suited for analysing nonlinear
upward motion, and negative direction to downward. marine cable dynamics. A good agreement has been
In Fig. 6 where the cable is under small excitation achieved between available experimental results and the
(amplitude 0'05 m and frequency 0-75 Hz), the responses numerical results obtained by using the present method.
are almost sinusoidal with small distortion. In Fig. 7, the
cable is under far more severe excitation (amplitude
0.075m and frequency 1Hz). The responses become REFERENCES
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