In the last decade of the 19th century the dreging market saw an change towards bigger projects in a shorter period of time and increasing sailing distances

ON THE MOTIONS OF SUCTION PIPE CONSTRUCTIONS

A DYNAMIC ANALYSIS

ir. O.W.J. ten Heggeler¹, ir. P.M. Vercruijsse¹, dr. ir. S.A. Miedema²

Abstract: With the introduction of jumbo trailing suction hopper dredgers and the development of deep sea dredging equipment came suction pipe constructions and gantries of ever increasing dimensions.

The implementation of submerged dredge pumps and the application of longer and greater number of pipe sections of large diameter resulted in suction pipe constructions which distinguish themselves from traditional constructions by their bigger mass and deviating mass distribution.

Given the fact that these characteristics have changed significantly and are features that determine the dynamic motion behaviour of a suction pipe construction, one enters a new field concerning the dynamic behaviour of these constructions.

This paper describes a mathematical model, developed at IHC Hollands R&D institute MTI Holland in co-operation with Delft University of Technology.

With this mathematical model one can perform calculations in the frequency domain to determine the eigenvalues and eigenvectors of an arbitrary suction pipe construction during the hoisting or lowering operation above the water line. These eigenvalues and eigenvectors determine the dynamic motion behaviour of the suction pipe construction.

By implementing the motions of the vessel into the model it was furthermore made possible to determine the motion behaviour of suction pipe constructions while working in offshore conditions. The probability of the draghead and gimbal joint colliding with the hopper dredgers hull in a certain seastate can be calculated as well as the occurring collision speeds. This knowledge can be used to evaluate performance and operating limits of dredging equipment working in offshore conditions. With the mathematical model a tool is developed to analyse the dynamic behaviour of existing suction pipe constructions or suction pipes which are still in an early stage of the design phase.

Keywords: trailing suction hopper dredger, suction pipe, dynamics, hoisting/lowering operation, operation limits, design tool

¹ IHC Holland NV

MTI Holland BV

Smitweg 6, 2961 AW Kinderdijk

The Netherlands

Tel.: +31-(0)78-6910342

Fax: +31-(0)78-6910331

E-mail: o.tenheggeler@partsservices.ihcholland.com

E-mail: p.m.vercruijsse@mtiholland.com

² Delft University of Technology

Faculty of Mechanical Engineering and Marine Technology

Section of Dredging Technology

Mekelweg 2, 2628 CD Delft

The Netherlands

Tel.: +31-(0)15-2786529

Fax: +31-(0)15-2781397

E-mail: s.a.miedema@wbmt.tudelft.nl

NOMENCLATURE

Symbols

A₀ orientation angle [rad]

b length trunnion elbow [m]

C_i coefficient [-]

d cable length [m]

g gravitational acceleration [m/s²]

H wave height [m]

H_s significant wave height [m]

k_ij stiffness matrix coefficient [Nm]

K stiffness matrix [Nm]

length [m]

L length pipe section [m]

m mass [kg]

m_ij mass matrix coefficient [kgm²]

M mass matrix [kgm²]

P probability [-]

s distance between hoisting point and hinge [m]

S_z spectral density [m²s]

R length [m]

t time [s]

T natural or eigenperiod [s]

T kinetic energy [Nm]

T wave period [Nm]

T_p peak frequency [Nm]

v velocity [m/s]

V potential energy [Nm]

V sailing speed [kn]

x surge [m]

y sway [m]

y_i displacement in y direction [m]

y_a displacement amplitude in y direction [m]

z heave [m]

e_f_z roll phase angle [rad]

f roll angle [rad]

f angle in vertical plane [rad]

f_a roll amplitude [rad]

g phase angle [-]

wave direction []

q pitch angle [rad]

q angle in vertical plane [rad]

w eigenfrequentie [rad/s]

w radial frequency [rad/s]

w_e wave frequency [rad/s]

w_n wave frequency [rad/s]

y yaw angle [rad]

y angle in horizontal plane [rad]

y_A swing amplitude [rad]

z wave elevation [m]

z_a wave amplitude [m]

z_an wave amplitude [m]

Abbreviations

cog centre of gravity

RAO Response Amplitude Operator

1. INTRODUCTION

Over the years dredging industry witnessed an enormous increase of scale. Especially at the closing of the century marine engineering projects have been realised of unimaginable scale. Land reclamation projects in South East Asia particularly in Hong Kong, and recently Singapore are good examples of this development, where hundreds of millions of cubic meters of dredged material were involved.

This increasing demand for enormous amounts of sand for major infrastructual projects have resulted in a rapid expansion in size of the hopper dredger. Longer sailing distances justified the huge investment costs of larger trailing suction hopper dredgers which resulted in the introduction of the Jumbo trailing suction hopper dredger some ten years ago.

The growth in hopper capacity still continues today. In the last decade, the maximum hopper capacity more than doubled, well exceeding the 20.000 m² nowadays (see table 1). In the mean time dredging depths increased as well, as more material was needed and one ran out of easy accessible places (see table 1). Dredging depths now reach more than 100m as demonstrated in recent projects in Taiwan and near Newfoundland, Canada

Table 1 Overview of the 13 largest trailing suction hopper dredgers

	Name	Capacity [m³]	Depth [m]¹⁾	2)	Owner	Built
1	Rotterdam	21.500	40/60	n³⁾	Ballast Nedam Dredging, the Netherlands	2001
2	HAM 318	23.700	55/70	y	HAM Dredging, the Netherlands	2001
3	Vasco da Gama	33.000	45/131	n³⁾	Jan de Nul, Belgium	2000
4	Nile River	17.000	40/60	n	DEME, Belgium	2000
5	Queen of Penta Ocean	20.000	35/60	y	Penta Ocean, Japan	1999
6	Volvox Terranova	20.000	40/105	y	AZC van Oord, the Netherlands	1999
7	Queen of the Netherlands	23.425	35/120	y	Boskalis, the Netherlands	1998
8	Fairway	23.425	35/120	n	Boskalis, the Netherlands	1997
9	Amsterdam	18.000	50/75	y	Ballast Nedam Dredging, the Netherlands	1996
10	Gerardus Mercator	18.000	30/102	y	DEME, Belgium	1996
11	Pearl River	17.000	40/60	n	DEME, Belgium	1994
12	JFJ de Nul	11.750	45/75	y	Jan de Nul, Belgium	1992
13	Vlaanderen XVIII	11.300	30	n	DEME, Belgium	1970

¹⁾ Standard/maximum

²⁾ Submerged dredgepump

³⁾ Optional

Not only the dredging vessels became bigger, the dredging equipment onboard the hopperdredgers changed accordingly. Suction pipe diameters increased, suction pipes became longer, hydraulics were installed and the total amount of available power grew. Deviating constructions as four-armpieces were introduced as well as submerged dredgepumps and extra jet water to assist the excavation process. As the size and total weight of the construction grew, consequently gantries had to become higher and stronger and pulley blocks had to be used to support the suction pipes. Given the fact that all these changed features determine the dynamic motion behaviour of suction pipe constructions, one enters a new field concerning the dynamic behaviour of these constructions.

One of the situations of interest concerning the dynamic behaviour of the suction pipe is the hoisting/lowering operation above the waterline. With an empty vessel and winch speeds of around 10 m/min this situation normally lasts for about 1 or 2 minutes. During this situation the suction pipe is free to swing under the influence of the ships movements, as damping is hardly present in the pipes construction. Problems may arise, limiting the operation of the equipment. This was reason for IHC Hollands R&D institute MTI Holland, in co-operation with Delft University of Technology, to examine the dynamic behaviour of a suction pipe during lift/lowering operation. This resulted in a mathematical model of the suction pipe describing the dynamics of the constrution (ten Heggeler 2000).

2. EQUATION OF MOTION

As a basis for the mathematical modelling of trailing suction pipe installations use was made of the theory for a single pendulum (see in figure 1) as the construction of a suction pipe shows many similarities with a common pendulum. A closer look at a suction pipe (consisting of two pipe sections) shows two interconnected pendulae which are attached to the hopper dredgers hull.

For a pendulum, as for every mechanism, holds that the total amount of energy within the system is constant. In other words: the sum of the kinetic energy T and potential energy V is constant:

constant or (1)

(2)

Figure 1 Single pendulum

In mechanics the latter equation is referred to as the equation of motion, which is often printed in the following format:

(3)

In which M is the mass matrix and K is the spring or stiffness coefficient matrix.

From equation 3 one finds one of the best known expressions in this world, i.e.; the expression for the natural frequency w and natural period T of a pendulum:

(4)

and (5)

The number of natural, or eigenfrequencies, for a mechanism equals the number of degrees of freedom. The pendulum shown in figure 1 for instance has one degree of freedom, i.e.; q, therefore one will find only one eigenperiod given in equation 5.

Figure 2 gives a sketch of a typical suction pipe. At first glance the trailing suction pipe installation shown has four degrees of freedom, i.e. f₁, f₂, y₁ and y₂. However as y₁ is a function of f₁ and y₂ is a function of f₁ and f₂, the system has not four but just two degrees of freedom. For the mathematical modelling of the suction pipe installation one can therefore use the equation of motion for a double pendulum, see figure 3.

The basic theory for the determination of the two natural frequencies and periods for this double pendulum is similar to the theory for the single pendulum and well documented in handbooks for engineers (Reynen 1993). For this reason the derivation of the eigenfrequencies and periods of the double pendulum are not printed here in this article.

Figure 2 Trailing suction pipe installation

Figure 3 Double pendulum

The equations of motion for a typical suction pipe construction in matrix form become:

(6)

The two eigenvalues of this homogeneous equation represent the eigenfrequencies of the system. The eigenfrequencies and eigenperiods can be derived as follows:

and (7)

With each eigenfrequency comes a accompanying eigenvector which represents the swingform which belongs to this eigenfrequency. The eigenfrequencies and so-called eigen-swingforms are characteristic for a certain suction pipe construction.

3. EIGENFREQUENCIES AND EIGEN-SWINGFORMS

Figure 4 shows the two eigenperiods of a typical suction pipe printed against the free cable length. The dotted black line represents the eigenperiod of a single pendulum, which is a function of the cable length d₁ (see eq.5). It shows that the two eigenperiods of the typical suction pipe vary slightly around the eigenperiod of the single pendulum and are proportional to it.

Figure 4 Eigenperiods and eigen-swingforms of a typical suction pipe construction

Figure 4 shows a top view of the accompanying eigen-swingforms of the suction pipe. These eigen-swingforms are symmetrical around the pipes equilibrium position (equilibrium position = dotted line) and constant over the whole interval of cable lengths. For both eigen-swingforms the angles y₁ and y₂ are out phases (y₁/y₂ < 0). Unlike the single pendulum the eigenfrequencies of a suction pipe are, just as the double pendulum, dependent on its mass and mass distribution.

Both eigenfrequencies and eigen-swingforms calculated with the mathematical model have been verified with the use of a multi body dynamics programme. In practice the calculated swingforms have been recognised.

4. EXCITATION

The suction pipe is excited by the motions of the dredging vessel. The several gantries and the suction inlet support the suction pipe construction and transfer the ships motions to the suction pipe. The ship itself is excited by periodic loads due to ocean surface waves. The ships motions can be defined by three translations of the centre of gravity (cog), surge x, sway y and heave z and three rotations around cog, roll f pitch q and yaw y. Using superposition one can calculate the motions in any point on the ship (Journe 1999).

Figure 5 Single pendulum on a rolling ship

Figure 5 illustrates a single pendulum supported by a gantry onboard a rolling ship. The roll response of a ship to a regular harmonic wave is given by:

(8)

with the harmonic wave elevation z given by:

(9)

Assuming small angles of rotation, the equation of motion of the pendulum onboard a rolling ship is:

(10)

The left side of this equation is equal to equation 4. On the right side of the equal sign appears a force term dependent on the motion of the top of the gantry caused by the rolling of the ship (Glansdorp and Remery 1965). As all six degrees of freedom of a ship are involved in the motion of a suction pipe. Assuming small angles of rotation one finds an equation of motion for a typical suction pipe that has the following format:

(11)

Different models were derived depending on the on the hinging connection between the pipe sections. A four-armpiece instead of a gimbal joint results in different matrix coefficients. Also the orientation of the four-armpiece is an important role.

Coefficients C₁ till C₄ depend on the position of the gantries and suction inlet with respect to the vessels centre of gravity (cog). These constants also depend on the ships motions in the six degrees of freedom due to a regular wave. The particular solution of equation 11 is know as the steady state solution and is of most interest. The particular solution (see equation 12) can be found using the Laplace transformation.

(12)

The behaviour of the suction pipe is linear. This means that at each wave frequency the ratio between the pipes motion amplitudes and the wave amplitudes and also the phase shifts between the motions and the waves are constant. As a consequence of the linear theory, the resulting motions of the pipe in irregular waves can be obtained by adding together results from regular waves of different amplitudes, frequencies and possible propagation directions of the waves.

5. APPLICATIONS OF THE MODEL

With the solution of equation 11 the movement of any point on the suction pipe due to a regular wave or a summation of regular waves can be calculated. Now this is possible one is interested in determining the limits to the hoisting/lowering operation of a suction pipe in terms of seastate and orientation of a hopper dredger in a wave field. To do so one should be able to calculate the chance of a collision between a swinging suction pipe and the hopper dredger and the accompanying collision speeds. Furthermore one is interested in the influence of the various design parameters on the swinging behaviour of a suction pipe and of course the possibilities to affect this behaviour.

5.1 Collision chance

Frequency characteristics

In order to calculate any collision chance the so-called frequency characteristics of a suction pipe should be determined. These characteristics exist of the transfer functions (also called response amplitude operators, RAOs) and the phase shifts. The transfer functions are the ratio between the suction pipe motion amplitudes and the wave amplitude and follow just as the phase shifts from the particular solution.

The largest motion amplitudes occur at the end of the pipe sections, at the gimbal joint and the draghead. As these points are the first to collide into a hopper dredgers side they are of most interest. Figure 6 gives transfer functions and phase shifts of the gimbal joint and the draghead of a typical suction pipe.

Figure 6 The frequency characteristics of a typical suction pipe

These frequency characteristics apply only to a certain load case and speed of the hopper dredger and to a certain cable length d₁ and wave direction m. For wave frequencies equal to the eigenfrequency of the suction pipe, resonance occurs. In that case the amplitude ratios approach infinity as can be seen in figure 6. In practice of course the amplitudes of the suction pipe are limited. For wave frequencies slightly lower or higher than the eigenfrequencies the amplitude ratios decline rapidly.

Besides the two peaks at the eigenfrequencies of the pipe, a third peak (w = 0,81 rad/s) can be found in the transfer function. This one is especially distinct in a beam wave situation (m = 90 or 270). This peak is caused by the roll eigenfrequency of the vessel. As the cable length of the suction pipe is not constant the resonance occurs at lower frequencies when the cable length increases. In the situation that the eigenfrequencies of the suction pipe are equal to the roll eigenfrequency of the vessel, one wide peak appears in the transfer function. In this situation the cable lengths is well above 10 m. Normally the suction pipe reached the waterline by then so the motions of the pipe are damped by the water.

The motions of the of the suction pipe are, among others, dependent on the incoming wave direction m. Extreme motions of a pipe happen when the hopper dredger is in beam waves. In theory when a ship is in following or head waves no motion transition occurs to suction pipe. However, in practice, due to wave spreading there will always be some motion transition to the pipe.

Energy spectra

To determine the dynamic behaviour of a suction pipe in an irregular sea it is necessary to describe the condition of the sea. This is done with the use of wave energy spectra. A wave energy spectrum gives the spectral density S_z as a function of the wave frequency w_e. This spectral density is in fact the average energy per unit area of waves of which the frequencies lie between w_n and w_n+ (Journe 1999) (see figure 7).

Figure 7 Definition of spectral density

In equation form the spectral density is equal to:

(13)

These spectra can be determined from measured wave elevation records.

In literature several mathematical formulations for wave spectra are found. Most of them are based on a certain significant wave height H and its period T. An example of these so-called standard wave spectra are the JONSWAP spectra (Joint North Sea Wave Project) for the North Sea. At the top of figure 9 a JONSWAP spectrum is given for windscale Beaufort 4.

With the transfer functions between wave motion and the pipes response motion one can transform any wave energy spectrum to a corresponding motion energy spectrum:

(14)

Figure 9 shows the construction of the motion energy spectra of the draghead and gimbal joint of a typical suction pipe for cable length d₁ = 7,5 (left) and 15 m (right) in Beaufort 4 and wave direction of = 60

Chance calculation

As the instantaneous wave elevation has a Gaussian distribution relationship with statistics can be found from computing the area under the motion spectrum m_0y. Expressed in terms of m_0y the probability P that a motion amplitude of the suction pipe y_a exceeds a certain threshold value, a, is given by:

(14)

Figure 8 Collision chance versus cable length

The chance that a draghead or gimbal joint collides with the hull of the hopper dredger is exactly a half times the chance P of the draghead or gimbal joint exceeding the distance between the draghead respectively gimbal joint and the hull of the hopper dredger. Figure 8 gives the collision chance of the draghead and the gimbal joint of a typical suction pipe with the hopper dredger in a beam wave situation, Beaufort 4. It shows that the collision chance of the draghead has a maximum for a cable length of 8,5 m and the gimbal joint for a cable length of around 11,5 m. For these cable lengths one of the eigenfrequencies coincides with the peak frequency of the wave energy spectrum. However these maxima are dependent on several factors it is well within the interval of cable lengths for which the suction pipe is still above the waterline and free to swing.




Figure 9 Construction of the motion energy spectra of a typical suction pipe for a cable lengths f d₁ = 7,5 and 15 m and incoming wave direction of = 60

5.2 Collision speed

Besides collision chances the occurring collision speeds are important as well. They are necessary to determine the possible consequences of a collision or for example to determine the characteristic of a hydraulic buffer to prevent a suction pipe from damaging a hopper dredgers hull or gantry (see figure 10). Collision speeds can be calculated by differentiation of the particular solution found in equation 12.

Figure 10 Buffer at the base of a gantry

Figure 11 Distance between the upper and lower pipe and there supporting gantries

Figure 10 illustrates a suction pipe onboard a hopper dredger. Imagine that at t = 0 sec the suction pipe is lifted from its saddles and starts swinging due to the movement of the hopper dredger caused by a wave with a frequency of w = 1,05 rad/s and a wave height H = 2 m. Figure 11 shows the distances, y_s1(t) and y_s2(t), between the upper and lower pipe and their supporting gantries. It shows that the suction pipe initially moves away from the gantries. After t = 3,5 sec however the upper pipe first hits a gantry. The collision speed v_{y s1}(3,5) equals 0,98 m/s in this specific situation. Of course the worst case scenario will be normative for the choice of a buffer.

5.3 Design parameters

The eigenfrequencies and eigen-swingforms are features that are specific for a particular suction pipe. These features depend on various design parameters and can be affected by changes in these design parameters. As the model is analytical it can be used for studying the influence of the various design parameters on the dynamic behaviour of the suction pipe. The location of the gantries, suction inlet or hoisting points can be varied easily as well as length proportions and mass properties of the suction pipe.

Figure 13 and figure 14 show the influence of the location of the hoisting point on the eigenfrequencies and eigen-swingforms. The ratio s₂/L₂, in which s₂ is the distance between the gimbal joint and the hoisting point and L₂ the length of the lower pipe (see figure 12), is varied between 0,5 and 1,0.

Figure 12 Location of the hoisting point on the lower pipe

This example shows that theres a significant spreading in possible eigenfrequencies. The eigenperiod T₂ for s₂/L₂ = 0,5 (T₂ var3) is nearly 5/3 times the T₂ for s₂/L₂= 1,0 (T₂ var1).

This holds also for the both eigen-swingforms. Swingform 2 in figure 14 for example, changes even from a out phases situation to a situation in which the angles y₁ and y₂ are in phases (y₁/y₂ > 0).

Figure 13 Eigenperiod versus cable length for various values of s₂/L₂

Figure 14 Eigen-swingforms for various values s₂/L₂

Extensive analysis showed that changes in mass and mass distribution have just little influence on the eigenperiods and eigen-swingforms. This clearly indicates the relation between the suction pipe and a single pendulum. In the contrary to changes in mass properties, the location of the hoisting points have, as shown in figure 13 and 14, a major influence on the eigenperiods and eigen-swingforms of a suction pipe. The ratio between the length of the upper pipe and the length of the lower pipe, or in other words the location of the gimbal joint, has no influence on the dynamic properties at all. The difference in height between separate gantries have a special effect on the course of the eigenperiods as the two eigenperiods approach one another and cause rapid changes in swingforms. Another consequence is that the course of the eigenperiods are no longer proportional to the eigenperiod of a single pendulum.

6. CONCLUSIONS

As shown above, with the mathematical model one can easily perform calculations in the frequency domain to determine the eigenvalues and eigenvectors of an arbitrary suction pipe construction. Collision chances can be calculated as well as collision speeds in any seastate and wave direction. This provides a tool for analysing the dynamic behaviour of both existing suction pipe constructions as suction pipes that are still in the design phase.

From the results of he parameter study can be concluded that a scaling of the size of the suction pipe it self has no or little influence on the eigenperiods and eigen-swingforrms. Longer and heavier suction pipes constructions have nearly the same dynamic properties as smaller and less heavy suction pipes, as long as the ratios s₁/L₁ and s₂/L₂ (see figure 12) remain the same. The use of a submerged dredgepump in upper or lower pipe or extreme heavy dragheads have no significant influence either. However as a consequence of larger and heavier suction pipe constructions gantries become higher. Together with the increase in hopper size this means that the distance to be bridged between waterline and the top of the gantry increased considerably. In case of a constant winch speed the suction pipe is excited for a longer period of time before the motions of the suction pipe are damped by the water. Another consequence of the increased distance between the top of the gantry and the waterline is that the interval of eigenfrequencies changed. The maximum cable length grew which results in lower eigenfrequencies of the system. In general these lower eigenfrequencies correspond better with the peak frequencies of wave energy spectra of higher Beaufort wind scales, so resonance behaviour is more likely to happen. A higher Beaufort windscale means more wave energy and therefore more motion transition to the suction pipe resulting in a higher risks at collisions between suction pipe and the hoppers side.

One of the possibilities to affect the dynamic behaviour of the suction pipe construction, as mentioned above, is to shift the hoisting points. One can influence the eigenperiods as well as eigen-swingforms. Eigenperiods determine the frequencies at which resonance occurs and the eigen-swingforms qualify whether the draghead or gimbal joint has a higher chance on a collision with the hopper dredgers hull. Besides the hoisting points a different construction of the hinging connection between the separate pipe sections can be used to change the dynamic behaviour as well. Other considerations are gantry types that reduce the maximum distance between gantry top and the waterline or one can simple increase the available winch power shortening the available time to build up a significant swing motion. A more complex solution may be to introduce damping in the suction pipes construction but so far theres no practical implementation known to do so.

Besides all these technical considerations of course one should always seek to lower the pipe in a head or following sea situation as much as possible as this reduces the risk at a serious collision significantly. However due to wave spreading this will not completely take away the risk at a collision. Thereby theres always the possibility of a failing winch and this may not jeopardise the safety of the hopper dredger in any situation.

REFERENCES

Glansdorp, C.C., and G.F.M Remery (1965). Botsingsnelheid van een hangende sloep tegen de huid van een slingerend schip. Schip en Werf, Vol. 32, No.1.

Heggeler ten, O.W.J. (2000). Het slingergedrag van zuigbuisconstructies (deel I). Report 2000.BT.5325, Delft, Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, Section of Dredging Technology.

Heggeler ten, O.W.J. (2000). Het slingergedrag van zuigbuisconstructies (deel II). Report 2000.BT.5326, Delft, Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, Section of Dredging Technology.

Journe, J.M.J, W.W. Massie (1999). Offshore Hydromechanics. Delft, Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology.

Reynen, J.W. (1993) Mechanica, College dictaat Mechanica II (deel B2). Delft, Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, Section Technical Mechanics.