Paul A Palo.

The 1980 CEL mooring dynamics seminar online

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the entire structure and its dynamic equations are obtained by summing the
individual element contributions under the obvious assumption that the elements
are joined at the nodes.

Element forms of higher order that the 1-D simplex can be used. The most
readily defined forms are based on polynomials, and the next logical element
form is one which uses a parabola for the geometry and the displacement fields.
Such an element uses one more node along the line between the two end nodes,
and the functional form is obtained using Lagrangian interpolation on the
three nodes. The process could be extended to cubics and higher orders, but
there appears to be no justification for doing so. Even though the higher
order fields are more capable of modeling complicated geometries and responses
with fewer elements , the number of nodes required does not change much while the
complexity of the equations and the cost of their calculation increase

The very regular and orderly form of the stiffness method makes it very
attractive for coding on a digital computer. The method also makes it very


easy to introduce discrete bodies and special constraints. A very attractive
feature is that the coding is insensitive to the geometric complexity. To be
sure, more nodes and elements means more computation cost, but the method is
insensitive to the degree of interconnection, multiplicity of materials and
the irregularity of the geometry and boundary conditions.

A very strong feature of the stiffness method is the ability to develop
the governing discrete equations directly for a variety of solution forms.
Since it works with the governing equations from mechanics, it is as easy to
get the incremental equation form as it is to get the total response equations.
Frequency domain or time domain forms may be chosen. This is of particular
value in large displacement solutions where dynamic effects occur relative to
some static preloaded state. This allows the static and dynamic analyses to be
done using consistent models. This is of very specific value in mooring
analys is .

Two very real problems with the use of the finite element method deserve
mention. First is the fact that this approach (like most discrete methods)
tends to produce large order simultaneous coupled equations, and solution
of these equations can be expensive. Often the novice code user will tend
to "shot gun" the problem with many nodes, many design perturbations and many
debug runs. Cost may not be a dominant factor if there is no other way and the
answer must be had; however, one tends to vault from very crude models to
excessively complex models with little thought about what is in between. Once
the ability to analyze is given, one may tend to over- analyze or expect far
too much from the analysis. The second problem is related to the first. When
a very complicated problem is solved on the computer, the input generation is


a major task and the output is voluminous. One tends to be overwhelmed by it
all and jumps to the very convenient conclusion that since a successful run
is finally obtained and it was done using an "all powerful black box" using a
computer, then the answers obtained must be correct. These are two aspects of
the "black box syndrome" and they tend to reduce the amount of intelligence
put into the formulation of an analysis .... Without some careful control, they
may lead to a very costly pile of garbage!

Typical Solution Forms

The most common approach to analyzing mooring dynamics begins with a
static analysis to establish a stable initial or reference state of the cable
system. The dominant nonlinearities are present in the static equations and
all effects mentioned earlier must be accoionted for except the dynamics.
Because of the strong geometric nonlinearity present in most deep water moors,
this may be a very difficult step in the analysis. Often the static reference
state is not well defined by a simple connection of the unstretched elements.
Usually the unpreloaded system represents a mechanism which is unable to
support loads in one or more directions and/or it violates boundary conditions
and compatibility constraints. Load resistance (stiffness) is only developed
as the elements rotate and stretch.

Once the static reference state is obtained, various dynamic solution
options are available. Some of them are:

1. Linearize equations and solve for small displacement perturbations
about the static reference in the time domain.

[M]{Aq} + [C]{a4} + [K]{Aq} = {Af (t) } (3)


2. Linearize equations as above, transform to frequency domain for
quasi-linear solution where equations are frequency dependent.

{Af} = Re ({F} e^'^^)

{Ail = Re ({Q} e^'^'') (4)

(-a)^[M] + io) [C] + [K]){Q} = {F}

3. Direct numerical integration of nonlinear time domain equations.

[M] {q} = {f} - {g} (5)

where [M] is the position dependent virtual mass matrix.

{f} represents the time and position dependent external
forces (dragloads, point loads, etc.)

{g} represents the internal loads from the elements,

reflecting the material and geometric nonlinearities and
material damping.

The finite element method allows direct calculation of any of these terms
for the mooring lines given the material properties (EA, mass, etc.), the
nodal positions and the unstretched element lengths. The effects of lumped
bodies; such as, buoys, platforms and ships, can be readily included if they
are described in functional or tabular form appropriate to the solution form.
Special rigid link multi-point constraints are used to tie the bodies into the
mooring model.

Some Demonstration Solutions

Single Degree of Freedom System With Geometric Nonlinearity
Figure 1 shows a single degree of freedom system composed of two linear
springs attached to a single mass point. It also shows the nonlinearity


of its static response. When the mass point is released from a deformed state,
it will oscillate about the reference state. Without damping, the oscillation
will continue indefinitely with a period which is dependent on the magnitude
of the initial displacement. The natural period for this system is 0.2639
seconds for an initial displacement of 20 cm. If the mass point is forced at
some other frequency with a force magnitude equal to that required to produce
the initial static defection, some interesting things occur. Two examples are
shown in Figures 2 and 3. Excitation below the natural frequency induces a
response similar to the linear case where the impressed frequency is dominant
and the response amplitude approximates the static response. The little
ripple in the response is at the natural frequency and would be expected to
die out in the presence of appropriate damping. Excitation above the natural
frequency introduces a new phenomenon. As before, there are two frequencies
present: one at the imposed frequency and the other at a varying frequency.
The variable frequency response appears as a damped transient because of the
numerical damping that was included in the integrator. The varying frequency
is a direct result of the geometric nonlinearity which causes the natural
frequency to be a function of the amplitude of the response. An important
aspect of this behavior is that the decay of the transient is long compared
to the period of the excitation.

Figure 4 shows the results of an attempt to force the system at the
apparent natural frequency. In this solution there is no damping in the model
nor is there any intentionally in the numerical integrator. Although the plot
is a crude one which attempts only to show the peaks and valleys, it shows
behavior not found in the linear problem. The response is not unbounded and is
quite complicated in form. There is not a single amplitude, and for the most


part the frequency of the response is higher than the frequency of the input.
It does appear that two amplitudes are in the response: one at about 35 cm
and the other at about 15 cm. In addition, it appears that the oscillation
pattern repeats on a period of about ten times the apparent natural period
(coincidence?) .

Taken together, these three figures are indicative of the typical response
of a nonlinear oscillator as represented in Figure 5. They also show some of
the difficulties in using numerical calculations of transient responses to
correlate with theoretical steady state responses.

Static Excursion of a Moored Ship

A diagram of the DD692 Destroyer in a four point moor is shown in
Figure 6. The lines are essentially catenaries in the quiescent state, and
substantial lengths of line lie on the bottom. Figure 7 shows the combined
effects of a 2 kt. surface current and a 30 kt. wind versus the heading relative
to the ship. Figure 8 shows two calculations of the excursion the e.g. of the
ship takes as the heading of the wind and current is varied through 180
relative to the original quiescent position of the ship. The effect of
neglecting the bottom interaction with the lines is clearly shown. The moor
appears much stiffer without the bottom interaction. The differences in
stiffness as well as the change in ship position could have significant influence
on dynamic response calculations. See Reference 1 for more details.

Frequency Domain Dynamic Response Calculations for Moored Ships

Following the approach represented by Equations (4) for some basic mooring

configurations offers some insights. Reference 2 gives more detail and presents

the figures which will be commented on briefly here.


The paper notes three important results :

First - Contrary to the assumption made in many mooring analyses, the
mooring has significant influence on the dynamics of the moored vessel.
This can be seen in the moored and free frequency response curves (Figures
3-6, 9, 10, OTC) . For example, there is a significant reduction in peak
response and a shift in the frequency where this occurs. This means it is not
appropriate to analyze mooring legs by simply imposing the free ship motions
at the upper end of the line.

Second - Contrary to the assumption made by some to get from 6 to 3 degrees
of freedom, the use of mooring buoys and hawsers does not effectively isolate
the moor from the heave/pitch/roll motions of the vessel. See OTC Table 4
and the discussion. This is due to the geometric stiffening effect of the
hawser preload.

Third - At some frequencies, the mooring legs act in a nearly linear
fashion while at others the dynamic behavior is decidedly nonlinear (see
Figures 13 and 14, OTC). This particular phenomenon is quite difficult to
predict and probably involves multi-frequency responses and resonances along
with other large displacement effects. This calls into question the validity
of the entire frequency domain solution procedure. The OTC paper suggests
there is general qualitative agreement with the results obtained with the
frequency domain solution and established mooring design procedures (DM-26) ,
but the calculation of specific responses may be erroneous or difficult to



This brief discussion and the examples presented suggest that proper
modeling of the deep sea mooring problem (and perhaps shallow water problems
as well) requires very careful consideration of the nonlinear dynamics of the
mooring lines. At present, this means that time domain models are preferred
above frequency domain models. This further means there is a need to develop
appropriate time domain models of the ships, platforms, buoys and other bodies
used and to develop the appropriate description of the environment. It may
also mean that there is a need to develop new solution techniques. Although
not dealt with in this discussion, low frequency effects such as wave induced
drift forces and swells acting in combination with wind and waves may require
that large displacement responses of the mooring be considered even in the
situations where the frequency domain solution may be an adequate model of the
first order wave responses.


1. Webster, R. L. , "Finite Element Analysis of Deep Sea Moors and Cable
Systems," Preprint 3033, ASCE Fall Convention and Exhibit, San Francisco,
California, Oct. 17-21, 1977.

2. Webster, R. L. , McCreight, W. R. , "Analysis of Deep Sea Moor and Cable
Structures," Paper OTC 3623 Presented at the 11th Annual Offshore
Technology Conference, Houston, Texas, 30 April- 3 May 1979.



Figure 1. Stretched String with Point Load




Zi. ^^




ir> to
CD c^

II 1




















> -





■- —











- ^




-C M




(iu3> j.N:rw:iovi<isi(i








f/^o) - _ir*3yi3o\f~i<i'=,x<i





WEIGHT = 35 lb/ft
EA = 3x10^ lb

= 25000 lb

WATER DEPTH = 1000 ft

Figure 6. Finite element model of moored ship.









20 - 10 -


5 -


-10 -




-5 -




120 140 160 180


CURRENT = 2 knots
WIND = 30 knots

Figure 7. Combined ship loads versus heading.




C. J. Garrison

3088 Hacienda Drive
Pebble Beach CA 93953


The mathematical formulation and solution of the boundary-
value problem for the hydrodynamics associated with the motion
of a moored vessel in a seaway is rather complex due primarily
to the nonlinear free surface boundary condition. Difficulty with
the free surface boundary condition has impeded progress on the
exact solution for wave/body interaction problems and little pro-
gress has been made. Thus, the more fruitful approach has been to
pursue linearized solutions as an approximation. The linearized
problem is also difficult but computer solutions can be obtained
for bodies of practical interest. Moreover, linearization admits
the concept of superposition of motions and waves, with which
comes the powerful concept of wave excitation spectra and the
motion response spectra. Although some rather broad assumptions
are made in order to linearize the boundary value problem, linear
solutions have been found to give physically realistic results for
cases of practical interest.

In addition to the dynamic response of a moored vessel to
wave motion at the frequency of the waves, a second-order effect
referred to as slowly-varying drift motion also occurs when the
vessel is subject to random waves. This is a phenomena which has
received a great deal of attention in recent years and is an area
of ongoing research.


The theory of the motion of a floating vessel is based on the
following assumptions:

(a) Inviscid fluid and irrotational flow.

( b) Small amplitude waves and resulting small amplitude
response .

(c) Wave motion and response motion r epresentable by a super-
position of regular sinusoids.

The notion of superposition of both the incident waves and
the response of the vessel allows one to view the motion of a .
moored vessel in waves as: (a) the wave interaction with the
vessel held fixed and ( b) the motion of the vessel oscillating in
each of its six degrees of freedom separately in otherwise still
water. From consideration of (a) , the wave excitation forces and
moments are determined, and from ( b) the reaction forces and
moments resulting from the motion of the vessel are determined.
The latter are characterized by use of added mass and damping

A numerical procedure based on distributed three-dimensional
sources has been presented by Garrison (1974) and Faltinsen and
Michelsen (1974) for three-dimensional bodies of arbitrary shape.


Bai and Yeung (1974) also have developed a numerical procedure
based on the Green's function method (or boundary integral method
as it is sometimes called) which utilizes simple sources dis-
tributed over the surface of the vessel as well as the free sur-
face, the bottom and an enclosing vertical cylindrical surface- far
from the vessel. A third numerical method for solution of the
three-dimensional free surface problems is referred to as the
hybrid-element method. This procedure, which has been applied by
Berkhoff (1972), Chen and Mei (1974), Bai and Yeung (1974),
Chenot (1975), Yue , Chen and Mei (1977) and Bettess and Zienkie-
wicz (1977), is based on the finite element method and uses a
"super-element" at the outer boundary of the discretized region to
infinity. Of the available methods indicated above, the distri-
buted source procedures of Garrison, and of Faltinsen and Michel-
son is believed to be the most versatile and simplest in appli-
cation, and has been most v/idely used in practice.

2.1 Strip Theory

The solution of the three-diiaensional boundary-value problem
for bodies of arbitrary shape requires computer runs, considerable
CPU time, and until recent years numerical methods for solving
three-dimensional problems were not available. Thus, it has been
common practice to use a strip-theory analysis for elongated
(shiplike) bodies in which the hydrodynamic coefficients are
determined by subdividing the body into a number of slices or
segments and assuming that each segment acts as a two-dimensional
body and that segments do not reflect mutual interaction effects.
The hydrodynamic coefficients for the complete body are obtained
by summing up the coefficients associated with each segment.

Clearly, strip theory represents a valid approximation to a
truly three-dimensional hydrodynamic analysis provided the vessel
is highly elongated. Of course, one would expect the strip theory
approximation to break down as the length to beam ratio decreased
and it would be of practical value to know what value of the
length to beam ratio might represent a limit on the strip theory
approximation. An absolute limit for all vessels does not exist
since it is presumably dependent, if only mildly, on the hull
shape in addition to the overall proportions, but it appears that
few studies comparing three-dimensional theory with strip theory
h.ave been made. In fact, the only such comparison known to this
writer was made by Migliore and Palo (1979) for rectangular
barge configurations. For the series of cases considered, the
results indicated that the strip theory analysis tended to break-
down when compared to the three-dimensional theory for length to
beam ratios of less than 8. Thus, it would appear that for
barges, most cases of practical interest would require the appli-
cation of three-dimensional theory for predicting hydrodynamic co-

2.2 Comparison with Experiment

Experimental results for hydrodynamic coefficients for three-
dimensional bodies are very limited but results of a few studies
have been reported. Faltinsen and Michelson (1974) have presente-
ed experimental results for a model of a simple barge 90 meters
by 90 meters by 40 meters draft. In general, although the scatt-
ter in the experimental data is large in some cases, the agree-


ment with calculations based on linear theory is good as indicat-
ed in Figures 1 and 2.

However, the measured heave damping is substantially greater
than the predictions of linear theory. In cases such as this,
where a rather large difference occurs between experiment and
calculated results, the cause can generally be traced to viscous
effects. In the present instance, the bottom surface of the
barge acts as the wave generating surface in heave but since it
is rather deeply submerged its wave-making ability is diminished.
In this connection it may be noted that the damping coefficient
in heave is about one-fifth that of surge. Since the wave-making
damping in heave is very small the importance of viscosity is rel-
latively large and this presumably accounts for the experiment-
al values being considerably above the values based on the linear,
inviscid theory.

Faltinsen and Michelson (1974) present no pitch data but
since motion of the barge in pitch typically produces a very
small radiated wave, the damping coefficient predicted by linear
theory is normally very small, except in the case of very shallow-
draft bodies where the wave-making surfaces are very near the
free surfaces. Thus, a similar situation to the above may be ex-
pected. In view of the small radiation damping in pitch, theory
generally predicts a very large resonance peak which is not ob-
served in reality. However, it is well-known that damping is
only important near resonance and, therefore, the motion response
is generally in error on this account only near resonance.

Pinkster and van Oortmersen (1977) have also presented ex-
perimental results and comparisons with linear theory for excitation
loads and response motions of a barge of 150m length, 50m breadth
and 10m draft. In general, the linear theory agreed very well
with the experimental results. The only significant discrepancy
was the rather large resonance peak in roll which is to be expect-
ed in view of the above comments regarding roll damping.


There are two rather well-known and commonly applied methods
for treating the motion of a vessel in a seaway. These are refer-
red to as frequency-domain and time-domain analyses.

The frequency domain analysis is based strictly on the
assumption that all forces acting on the floating body are linear
functions of displacement, velocity or acceleration, and as a re-
sult the response is directly proportional to the amplitude of
the incident wave. For a given frequency, the equations of mot-
tion for the floating body would appear as follows:

(pn^i-Mi^('r))j:j(.±) tN^/^)^jW -^ K^jJTjW^ FT Co-) (1)

in which fy^ii denotes the mass matrix of the body, McjCO') den-
notes the added mass matrix, N^^^ denotes the damping matrix,
Ki'i denotes the restoration force matrix due to buoyancy and ela-
astic forces, and F^ denotes the wave excitation force.

To examine the difficulty in application of Eq.(l) to random
waves it is enough to consider two frequencies, cr, and OV • It
is generally assumed that the response associated with the two





1 r-








^— $ P ^














8.0 10.0 12.0 14.0 16.0 18.0 20.0

PERIOD, sec.




ao 10.0 12.0 14.0 16.0 18.0 20.0
PERIOD, sec.

Figure 1 Heave Added Mass and Damping Coefficient for a
90ra X 90m x 40m Barge.





2.5 -


0.5 J

T 1 1 T



■ '

8.0 10.0 12.0 14.0 16.0

PERIOD, sec.

18.0 2ao



I r



-I L.

8.0 lao 12.0 14.0 16.0 lao 20.0

PERIOD, sec.

Figure 2 Surge Added Mass and Damping Coefficients for a
90m X 90m x 40m Barge.


wave components, CT, and O^ , is represented by the sum of the re-
sponse due to O", alone, which we laay call Xj" f and the re-
sponse due to g-j^ alone, X';'*' • However, as pointed out by
Wehausen (1971) it is only in the special case that Mij (-07) = /Mij(0"i)
and Ntj (-^1 ) ■= Mii(0"i.) that this could in fact be the case since
it is only then that ^xj'** V- ^j'^ ) could represent a solution
to an equation of the form of (1) . In spite of this, such equa-
tions have been used with some success to describe the motion of
a vessel in random waves (see, e.g. Fuchs and Mac Camy (1953),
Fuchs (1954), St. Denis and Pierson (1953) in which the values of
/V ■ ■ and A7<-j were taken as constant at some average value. It
appears, however, that it is currently common practice to utilize
the superposition discussed above regardless of the difficulty
associated with frequency dependent coefficients of mass and damp-
ing. Wehausen (1971) has discussed a further method of treating
the linearized motion of floating bodies in random seas when the
added mass and damping coefficients are frequency dependent as,

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Online LibraryPaul A PaloThe 1980 CEL mooring dynamics seminar → online text (page 3 of 11)