1 Introduction
The behaviour of dredges in off- and near shore conditions is important
in relation to the workability of the dredge. To be able to understand this
behaviour and the processes involved, at the end of the 70's the Laboratories
of Soil Movement and Ship Hydromechanics of the Delft University of Technology
and Delft Hydraulics initiated joint research into this subject.
The objective of this research was the development of a computer program
with which, the behaviour of seagoing cutter suction dredges could be simulated
and thus predicted. In the early 80’s the computer program DREDMO was
operational on a mainframe. There were several publications in 1983 (Journée,
Miedema and Keuning 1983 [4], Keuning and Journée 1983 [7] and de Koning,
Miedema and Zwartbol 1983 [8]) published about this subject. The program,
written in the language ALGOL, required several pre- and post-processing
programs and was only accessible to specialists. In the mid-80’s Dutch dredging
companies expressed the desire to be able to operate DREDMO them selves. The
program was translated to FORTRAN and made operational by Miedema, Koster and
Hurdle 1986 [9] for use on MS-DOS computers. The Dutch contractors joined in
the CSB research group, purchased DREDMO from Delft Hydraulics in 1986.
Since the pre- and post-processing still had to take
place on a main frame and because of the complexity of the programs involved,
the use of DREDMO was rather user-unfriendly. Reason for a number of scientists
of the Delft University of Technology to develop pre- and post-processing
programs and a user-interface for the on personal computers.
These programs became operational in 1990 and are now
used by Delft Hydraulics and Delft University. The mathematical modelling and
structure of the DREDMO package will be discussed in the following paragraphs.
2 Equations of Motion of a Floating Object
Generally, the equations of motion for the six degrees of freedom of a
floating object, influenced by external loads, are written in the frequency
domain. In these equations of motions, the hydrodynamic mass and damping coefficients
and the external loads depend on the circular frequency of oscillation 
. As a result of the formulation in the frequency domain, any
system influencing the behaviour of the vessel should have a linear relation with
the displacement, the velocity and the acceleration of the body. Figure 1 shows
the six degrees of freedom of a floating object. The rotation of the ladder
around the ladder bearings gives the 7th degree of freedom, which
will be considered separately from the body motions.
Figure 1: Degrees of Freedom of a
Floating Object
The equations of motion for the six degrees of freedom
form a system of six coupled differential equations according to:
(1)
However, in many cases there are several complications,
which negate this linear assumption, such as the non-linear viscous damping,
non-linear hydrostatic restoring spring terms or non-linear external forces or
moments such as cutting forces. Dredging vessels, especially cutter suction
dredges and wheel dredges, are in contact with the bottom of the sea by means
of the excavating element, mounted on the ladder and by the anchoring system.
The excavation process causes strong non-linear effects in the equations of
motion and also couples the longitudinal and lateral degrees of freedom of the
vessel. To include these non-linear effects in the vessel behaviour at zero
forward speed, it is necessary to formulate the equations of motions in the
time domain, which relates instantaneous values of forces, moments and motions.
Memory functions have to be used to represent the frequency-dependent
hydrodynamic mass and damping terms.
Referring to the basic work on this subject by Cummins
equations of motions are called the “Cummins Equations”, coupled non-linear
integral-differential equations:
(2)
In this system of coupled equations of motions the
following terms can be distinguished:
1.
The inertial forces and moments caused by accelerations.
2.
The potential damping forces and moments caused by
velocities (the convolution integral).
3.
The restoring forces and moments as a result of
displacements.
4.
Wave forces and moments as a function of time acting on
the vessel.
5.
External forces and moments as a function of the
accelerations, velocities and displacements of the vessel and of time.
3 The DREDMO User-Interface
From the previous paragraph it will be clear that
solving equation (2) requires much knowledge related to ship hydrodynamics, soil
mechanics, mathematics, etc. The objective of the MS-DOS version of the DREDMO
package was, however, to have a user-friendly software package. To meet this
requirement the DREDMO user-interface was developed. The user-interface has
been developed on the basis of the philosophy that the user should not be
concerned too much with the theoretical backgrounds of the calculations. The
user should also not be concerned with the way the subsequent programs
communicate with each other. The only thing the user should be concerned with
is the input of the geometry of the cutter suction dredge and operational input
like wave spectrum, haulage velocity and type of soil to be dredged. A first
step in reaching these requirements was to modify the calculation programs in
such a way that the programs do not communicate with the user by means of the
keyboard. Only the user-interface communicates with the user.
The other programs run in the background or obtain
essential information from the user-interface. The user-interface also takes
care of a correct communication between the calculation programs (see Figure
2). The result of this all is that the user only has to deal with the
user-interface.
A calculation should be carried out with the following
steps:
1.
Enter the hull form in the user-interface.
2.
Preview the hull form with the SEAHULL program.
3.
Enter data with respect to the calculation method of the
hydrodyna-mic coefficients in the user-interface.
4.
Calculate the hydrodynamic coeffi-cients with the
SEAWAY-D program. The hydrodynamic coefficients are stored in a so-called
Hydrobase.
5.
Define a wave spectrum in the user-interface.
6.
Calculate wave force time series with the WAFOR program.
7.
Enter the dredge layout and the conditions of operation
of the cutter suction dredge in the user-interface.
8.
Solve the Cummins equations of the cutter suction dredge
with the DREDMO program.
9.
Enter data with respect to the layout of the graphical
output in the user-interface.
10.
Create graphical, statistical and spectral output with
the PLOSIM program.
In the following paragraphs the programs used will be
discussed. The different steps of the calculation will be illustrated with
graphs in the frequency and time domain.
3.1 The SEAWAY-D Program
When carrying out time domain calculations with a
program like DREDMO, first the potential mass coefficients for an infinite
frequency have to be calculated. Then the potential damping coefficients for a
range of frequencies have to be calculated, followed by the determination of
the retardation functions. These calculations have to be carried out by an
external program.
Figure
3: Added Mass of Roll In the Frequency Domain
For this a new pre-processing program, named SEAWAY-D, has
been written, which calculates the hydrodynamic mass coefficients and the
retardation functions at zero forward speed. The program has been derived from
the frequency domain ship motions personal computer program SEAWAY, Journée
1990 [6], which has recently been made suitable for twin-hull ships and
semi-submersibles too.
The use of the pre-processing program SEAWAY-D makes no
high demands on the ship hydromechanical knowledge of the user of the DREDMO
program.
The ship hull form and some parameters concerning the
calculation method are input to the program. The program first calculates the
hydrodynamic coefficients in the frequency domain.
Figure 3 shows the added mass of the roll motion as a
function of the frequency.
Figure
4: Damping of Roll in the Frequency Domain
Figure 4 shows the potential damping of the roll motion
as a function of the frequency.
Figure
5: Wave Force Transfer Function in Roll (Amplitude Operator)
The wave loads (wave force transfer function) of the roll
motion as a function of the frequency are illustrated in Figure 5. The linear
restoring spring coefficients follow from the underwater geometry of the ship.
It may be noted that this approach leads to linear left-hand sides of the time
domain equations (2).
To calculate the frequency-depending hydrodynamic mass and damping
coefficients of a ship, two- or three-dimensional potential theories can be
used. Here use has been made of the relatively simple two-dimensional or strip theory
method to calculate the sectional sway, heave and roll coefficients. For the
determination of the two-dimensional coefficients of ship-like cross-sections
that are not fully submerged, the cross sections are conformably mapped to the
unit circle by a two- or three-parameter Lewis transformation or by a 
-parameter Close Fit conformal mapping technique.
The advantage of conformal mapping is that the velocity
potential of the fluid around an arbitrarily shape of a cross section in a
complex plane can be derived from the more convenient circular section in
another complex plane. In this manner hydrodynamic problems can be solved
directly with the coefficients of the mapping function only. The theory for the
calculation of the two-dimensional hydrodynamic potential coefficients is given
by Ursell 1949 [18] and Tasai 1959-1961 [16, 17]. All algorithms, necessary to
derive these coefficients, are described in detail by Journée 1990 [5].
Another very suitable method is the Frank Close Fit
method (Frank, 1967 [2]), especially advised for fully submerged
cross-sections. This method determines the velocity potential of a
two-dimensional cross section by an integral-equation method, utilising the
Green’s function, which represents a pulsating source below the free surface.
To suppress the so-called “irregular frequencies” in the operational frequency
range, not fully submerged cross sections have to be closed at the free surface
with some additional points. This results into a shift of these irregular
frequencies towards a higher frequency region. A separate method determines the
two-dimensional potential surge coefficients.
Then, according to the strip theory, the total
hydrodynamic potential coefficients of the ship for surge, sway, heave and roll
can be found easily by integrating the sectional values over the ship length.
The pitch and yaw coefficients follow from an integration of the moments caused
by the sectional surge, sway and heave coefficients over the ship length.
Studies, carried out in the past on this subject, have shown that this approach
leads to a fairly well prediction of the hydrodynamic potential coefficients.
When comparing the linear frequency domain equations (1)
with the time domain equations (2), the time domain coefficients can easily be
found from the frequency domain coefficients.
It is found that the hydrodynamic mass coefficients in
the time domain equations are defined by:
(3)
and the retardation functions by:
(4)
Figure 6 shows the retardation function of the roll
motion as a function of time. The calculated data are stored in the required
format in a file named “hydrobase” and DREDMO and WAFOR can read this file.
Figure 6:
Retardation Function of Roll in Time Domain
Verifications of time domain calculations with results
of frequency domain calculations have been carried out for the linear case at
zero forward speed. Wave loads, calculated by the frequency domain program,
have been input in the time domain program. In spite of errors caused by
numerical integration, truncations and differences expected by using two
different techniques to solve the differential equations of motions, the two
approaches showed a remarkably good agreement. The differences between the
amplitudes calculated in two manners, of the harmonic surge, sway, heave, roll,
pitch and yaw motions are within 1.0 percent. The differences between the
calculated phase lags belonging to these motions with respect to the exciting
wave loads are within 1.0 degrees.
3.2 The WAFOR Program
As described above, the wave force transfer functions
are computed in the frequency domain with the program SFAWAY-D. Based on these
transfer functions, the user can compose time-series of the wave excitation
forces on the barge by means of the program WAFOR.
In the computational process to obtain the time series of the wave
forces, the following steps can be distinguished:
·
Determination of the wave
spectrum.
·
Determination of the wave force
spectra (6 components).
·
Determination of the wave force
time series.
The wave conditions at the dredging location are defined
by a wave spectrum. WAFOR is able to generate a Pierson-Moskowitz spectrum, a
JONSWAP spectrum or regular waves. It is also possible to use a file containing
a user-defined spectrum.
The Pierson-Moskowitz spectrum is derived for a fully
developed sea and originally has the wind speed as the only free parameter
(Pierson 1964 [15]). For engineering purposes however, it is more convenient to
have the significant wave-height and the peak-period as free parameters. WAFOR
applies this two-parameter Pierson-Moskowitz spectrum.
The JONSWAP spectrum has been derived from a large
volume of data in a major international project (Hasselman 1973 [3]). This
spectrum uses three free parameters, the significant wave height, the peak
period and the peak enhancement factor 
. The peak enhancement factor defines the shape of the
spectrum. For 
the JONSWAP spectrum
is equal to the Pierson-Moskowitz spectrum. Factors larger than 1.0 will
enhance the shape of the peak and characterise the stage of development of a
sea.
Figure 7: JONSWAP Wave Spectrum
Figure 7 shows the wave energy density of a JONSWAP
spectrum with a significant wave height, 
m, a period of the
peak of spectral density, 
s, and a peak
enhancement factor, 
. The peak enhancement factor used is applicable for a sea in
development.
If a user-defined spectrum originating for instance from
measurements in the operational field is to be used, the user has to create a
file containing a tabulated wave spectrum. WAFOR will read this file to
generate a wave spectrum.
If the wave spectrum has been calculated, the spectra of the wave forces
have to be determined. Since the natural frequencies of response of cutter
suction dredges are usually close to the peak of the wave spectrum, only
first-order wave forces are considered.
The wave forces acting on the body of the dredge consist of six
components:
·
, wave force acting in the
longitudinal (surge) direction.
·
, wave force acting in the
transversal (sway) direction.
·
, wave force acting in the
vertical (heave) direction.
·
, wave moment about the
longitudinal (roll) axis.
·
, wave moment about the
transversal (pitch) axis.
·
, wave moment about the vertical
(yaw) axis.
The energy density spectra of these six components of
the wave force can be obtained by multiplying the energy densities of the waves
by the square of the amplitude operator of the wave force transfer functions at
the required frequency:
(5)
An example of the amplitude operator is given in Figure 8.
Figure 9 shows the resulting energy density spectrum of the wave-forces
when the amplitude operator of Figure 8 is applied to the wave spectrum of
figure 7.
Figure 8: Amplitude Operator of Wave
Force for Sway
Figure 9: Energy Density
Spectrum of Sway Wave Forces
The final step in the computation of the wave forces concerns the
determination of the wave force time series. The fundamental obtain time series
from a energy density spectrum is:
(6)
The phase angle 
is determined on the
basis of a random phase shift for the wave frequency components and the phase
operator of the wave-force transfer functions.
Special care should be taken to avoid the time-series
repeating themselves within the required computation interval. This repetition
can be avoided in the following ways:
-
The use of non-equidistant frequencies in equation (6).
This method has the advantage that a relatively small number of frequencies may
be used. The determination of the time series however has to be carried out in
the time domain and is relatively inefficient (time consuming). Moreover, the
set of frequencies should be dense enough to cover the frequency band of
interest.
Figure 10 shows a typical
example of this method using a frequency step too large. In this figure, the
amplitude spectrum of a typical ship movement is illustrated for a case where
the energy of the waves is concentrated in a small number of frequencies (in
total 20 frequencies). These frequencies can be deduced from the amplitude
spectrum of the ship movements.
-
Using a large number of
equidistant frequencies (
frequencies with a constant step 
) in equation (6). The time series will repeat themselves
with a period equal to 
, so the frequency step can be adjusted with respect to the
required time interval of the time-domain computations. Since the frequency step
is constant, a Fast
Fourier Transform can be used to solve equation (6).
WAFOR offers the user a dense representation of the spectrum that is efficiently used to create wave force time-series, which are statistically correct, with a minimum on input from the user-interface. The wave force time-series are stored in a file named the “wavebase”.
Figure 10: Response
on a Small Number of Non-Equidistant Frequencies
3.3 The
DREDMO Program
The DREDMO program solves the Cummins equations. DREDMO reads the mass,
added mass, restoring spring coefficients and retardation functions from the
“hydrobase”. The wave force time series are read from the “wavebase”. The
dredge geometry and the operational parameters are read from a file produced by
the user-interface.
This file contains information with respect to:
-
the current,
-
the dynamics and kinematics of
the ladder,
-
the geometry of the cutter head,
-
the geometry of the face,
-
the spud or X-mas anchoring
system,
-
the swing wires and hoisting wire,
-
the type of soil to be dredged,
-
the operational parameters and
-
the calculation.
The cutting forces are implemented for the cutting of
water saturated sand according to Miedema 1987-1989 [12, 13, 14]. The Cummins
equations are solved in the time domain by an implicit Newton-Raphson method
for the prediction and correction of the acceleration vector. Integrating the
accelerations and velocities can derive velocities and displacements. For
numerical stability the “teta” integration method is used. Figure 11 shows a
flow chart of the solution method. At 
the dredge is
considered to be in a static equilibrium. For 
the dredge is excited
by the wave-forces (depending on time only). For each time-step the iteration
process, as illustrated in Figure 11, is repeated until the predicted
acceleration vector 
and the calculated
acceleration vector 
match within the convergence
criterion. Then the next time step is executed. This is repeated until the
behaviour of the dredge is calculated for the required time interval.
Figure 11: Flow Chart of Solution Method in DREDMO
In DREDMO the ladder is not considered as the 7th
degree of freedom, but as an external influence on the body of the dredge. To
achieve this the non-linear equilibrium equation of the ladder is solved by an
implicit Newton-Raphson method every iteration step of the main program. The
reason for this is, that different ladder constructions can be used while the
main program remains unchanged.
The output of DREDMO consists of a number of files
containing time-series of:
-
the motions of the dredge,
-
the motions of the ladder,
-
the motions of the cutter head,
-
the forces on the cutter head and the cutter torque,
-
the loads on the spud keepers or the forces in the X-mas
tree wires,
-
the loads on the ladder bearings,
-
the forces in the swing wires and hoisting wire,
-
the swing velocity at the cutter position,
-
the number of revolutions of the cutter head and
-
the production.
Figure 12 illustrates the DREDMO output for the motions
of a dredge during 30 seconds simulation.
Figure
12: Roll Motions Obtained by DREDMO
3.4 The PLOSIM Program
As mentioned in the previous paragraph, the results of
the DREDMO program are stored in a number of files. To evaluate the results of
the calculations the results have to be made visible for interpretation. For
this purpose the PLOSIM program is added to the DREDMO package. All graphs in
this paper have been created with PLOSIM. Except for creating graphs, PLOSIM is
also able to smooth signals, apply a low pass filter on a signal, carry out
linear and non-linear curve fitting and perform spectral analysis (amplitude
spectrum) on signals with a varying time step. Figure 10 illustrates the
spectral analysis.
PLOSIM can process files with up to 100 channels and up
to 16000 samples per channel. Graphs can be produced with a maximum of 12 view
ports and 6 channels maximum per view port on A4 or A3 format. A spread sheet
function is added, which permits operation with deduced variables. The size of
the graphs can be adjusted as required as is being used in this paper.
Figure 13 illustrates the motions of a dredge, anchored
on a spud pole, as a function of time, during 100 seconds simulation. Figure 14
illustrates the amplitude spectra of the motions. As can be seen, the heave,
roll, pitch and surge motions have most of the energy concentrated in a small
area around their natural frequencies following from the hydrostatic restoring
spring stiffness and the anchoring system. The behaviour of the sway and the
yaw motions depend on the anchoring system and the cutting process and has more
natural frequencies.
4 Conclusions
The separation of the communication of a user with
application programs by means of a user-interface has proven to allow
non-specialists to operate the DREDMO package and make them use it. This should
however also be valid for other software packages.
The DREDMO package is now a self-contained software
package, so no external programs have to be used.
The different programs, the DREDMO package consists of,
can also be used for other applications.
5 List of Symbols
Frequency dependent
added mass matrix in kg, kgm or kgm2
Acceleration
vector in m/s2 or rad/m2
Amplitude
operator of wave force transfer function in N/m or Nm/m
Frequency
dependent potential damping matrix in Ns/m or Nms
, 
Spring coefficient
matrix in N/m or Nm/rad
Frequency
independent added mass matrix in kg or kgm2
Frequency
in Hz or rad/s
External
loads in N or Nm
Significant
wave height in m
Retardation
functions in N/m or Nm
Mass matrix
in kg, kgm or kgm2
Spectral
density of wave forces in N2s
Spectral
density of wave energy in m2s
Time in s
Peak period
in s
Velocity
vector in m/s or rad/s
Wave loads
in N or Nm
Displacement
vector in m or rad
Phase
shift in rad
Peak
enhancement factor
Time in s
Infinite
6 Bibliography
[ 1] W.E. Cummins, The Impulse Response Function and Ship Motions, Symposium on Ship
Theory, Institüt für Schiffbau der Universität Hamburg, Germany, 25-27 January
1962.
[ 2] W. Frank, Oscillation of Cylinders in or below the Free Surface of a Fluid,
Naval Ship Research and Development Center, Washington, U.S.A., Report 2375,
1967.
[ 3] K. Hasselman et.al., Measurements of Wind Decay during Joint North Sea Wave Project
(JONSWAP), Erganzungsheft zur Deutschen Hydrographischen Zeitschrift, Reihe
A (8), Nr. 12, 1973.
[ 4] J.M.J. Journée, S.A. Miedema and J.A.
Keuning, DREDMO, A Com-puter Program for
the Calculation of the Behaviour of Seagoing Cutter Suction Dredges, Delft
University of Technology & Delft
Hydraulics, 1983.
[ 5] J.M.J. Journée, Theory and Algo-rithms of Two-Dimensional Hydro-dynamic Potential
Coefficients, Delft University of Technology, Ship Hydromechanics Laboratory, Delft, The Netherlands,
Report 884, November 1990.
[ 6] J.M.J. Journée, SEAWAY-DELFT, User Manual and Theoretical Background of Release 3.00,
Delft University of Technology, Ship
Hydromechanics Laboratory, Delft, The Netherlands, Report 849, January 1990.
[ 7] P.J. Keuning and J.M.J. Journée,
Calculation Method for the Behaviour of a Cutter Suction Dredge Operating in
Irregular Waves, Proceedings 10th WODCON, Singapore 1983.
[ 8] J. de Koning, S.A. Miedema and A.
Zwartbol, Soil-Cutterhead Inter-action under Wave Conditions, Proceedings 10th
WODCON, Singapore, 1983.
[ 9] S.A. Miedema, A.W.J. Koster and D.
Hurdle, DREDMO-V3, MS-DOS Version of the
DREDMO Program (FORTRAN), Delft Hydraulics, 1986.
[10] S.A. Miedema, DREDMO-V4, User Interface for the DREDMO Package, Delft University
of Technology, The Netherlands, 1990.
[11] S.A. Miedema, PLOSIM-V4.0011, Graphical
Plotting Program, Delft, Holland, 1987-Now.
[12] S.A. Miedema, The Calculation of the Cutting Forces when Cutting Water Saturated
Sand, Basic Theory and Applications for 3-D Blade Movements with Periodical-ly
Varying Velocities for in Dred-ging Usual Excavating Elements (in Dutch).
Doctors Thesis, Delft, 1987, The Netherlands.
[13] S.A. Miedema, On the Cutting Forces in Saturated Sand of a Seagoing Cutter Suction
Dredge, Proceedings 12th WODCON, Orlando, Florida, USA, April
1989.
[14] S.A. Miedema, On the Cutting Forces in Saturated Sand of a Seagoing Cutter Suction
Dredge, Terra et Aqua 41, December 1989, Elseviers Scientific Publishers.
[15] W.J. Pierson and L. Moskowitz, A Proposed Spectral Form for Fully Developed
Seas Based on the Similarity Theory of S.A. Kitaigorodskii, Journal of
Geophysics, Res., Vol. 69, No. 24, pp 5181 - 5190, 1964.
[16] F. Tasai, On the Damping Force and Added Mass of Ships Heaving and Pitching,
Research Institute for Applied Mechanics, Kyushu University, Japan, Vol. VII,
No 26, 1959.
[17] F. Tasai, Hydrodynamic Force and Moment Produced by Swaying and Rolling
Oscillation of Cylinders on the Free Surface, Research Institute for
Applied Mechanics, Kyushu University, Japan, Vol. IX, No 35, 1961.
[18] F. Ursell, On the Rolling Motion of Cylinders in the Surface of a Fluid,
Quarterly Journal of Mechanics and Applied Mathematics, Vol. II, 1949.
