TESTS WITH A SIX DEGREES OF FREEDOM FORCED
OSCILLATOR TO INVESTIGATE THE NONLINEAR EFFECTS IN
SHIP MOTIONS
Pepijn de Jong
Jan Alexander Keuning
Ship Hydromechanics Laboratory Delft University of Technology Mekelweg 2, 2628 CD, Delft, The Netherlands
ABSTRACT
In the last decade a growing interest can be observed in research aimed at the survivability of ships. Ship survivability is associated with larger amplitude motions and combined motions, such as capsizing (heave-roll) and broaching (sway-yaw-roll). Since large amplitude motions are inherently nonlinear, assessment of the seakeeping behaviour by the commonly used linear approach can be problematic. Complex effects, such as the viscosity dominated roll damping and the coupling effects in the horizontal plane are disregarded. This may result in limited accuracy of these methods.
The current research project is aimed at quantifying the importance of the nonlinearities in the hydrodynamic reaction forces due to large amplitudes, high(er) forward speed and nonlinear coupling effects between simultaneously performed motions by means of forced oscillation tests. A segmented model of the Royal Netherlands Navy M-frigate was mounted to a newly developed 6-DOF forced oscillator in the towing tank of the Delft Ship Hydromechanics Department. Each of the seven sections was fitted with a 6-component force transducer. The new oscillator renders the possibility of performing both large amplitude and arbitrary motions in six degrees of freedom. In this paper the results of this research will be presented, analysed and the possible importance of nonlinear effects quantified. For now the scope will be limited to combined sway-yaw motions. Significant nonlinearities, mostly related with unsteady lift and viscous forces, will be quantified and qualified.
1. INTRODUCTION
In the assessment of the seakeeping behaviour of ships, calculations based on the linear approach are commonplace. Methods according to the linear approach are based on linearization of the seakeeping problem, making extensive use of the superposition principle.
In the first place, irregular seaways are decomposed in individual wave components by spectral analysis. Spectral analysis is a method developed by Barber and Ursell [1] and introduced by St.Denis and Pierson [2] in the shipbuilding community. The motions at different wave frequencies are evaluated separately with the subsequent superposition of their resulting motions.
Secondly, the contributions of the forces from different sources are evaluated individually. This leads to the so-called load superposition, the separation of the forces acting on a ship travelling in waves in two parts:
1. The reaction forces generated by the motion of the body itself; i.e. the hydromechanic reaction forces.
2. The forces generated on the body by the incident waves; i.e. the wave exciting forces.
This separation allows considering the problem of ship motions in a seaway in two different parts: 1. The oscillation of the body in still water.
2. The waves coming in at the body fixed in its reference position.
Finally, the reaction forces generated by motions in the six degrees of freedom are superposed to yield the total reaction force on the body, disregarding (nonlinear) interactions between motions in different directions. This means that the pressure distributions caused by the individual motions are superposed to yield the total hydromechanic reaction force on the body, allowing for linear coupling between the motions (a forced caused by one motion in the direction of another), but disregarding any nonlinear interaction.
Superposition requires that only small perturbations from the steady state are to be considered. This
restriction means that the input has to be limited to waves with small wave steepness (the ratio of wave height and wave length) and wave amplitude (with respect to the draft of the floating body under consideration), while the output in the form of the resulting motion amplitudes and velocities has to be equally small.
As an indication St. Denis [3] gives amplitudes of heave of less than one-fifth of the draft, amplitudes of roll of less than 15 degrees and amplitudes of pitch less than 5 degrees. The linear approach is only applicable for moderate forward speeds; when the forward speed of the ship increases beyond moderate, the motions of the ship become of a strong nonlinear nature. This is shown by Keuning [4] for heave and pitch motions of fast mono hulls in head waves. The linear methods to evaluate the seakeeping behaviour (in this paper referred to as the linear approach) are commonly based on linear potential theory. They can be in the form of linear striptheory or three-dimensional diffraction codes. In order to include forward speed in linear potential theory, two extra flows are introduced besides wave excitation, diffraction and radiation. These flows are a uniform incoming flow with a velocity equal to the forward speed of the ship and the disturbance of this flow by the moving ship. As this last disturbance is of a highly nonlinear nature (it is affected by the instantaneous position of the ship) it is neglected in the linear approach.
One important issue in the seakeeping calculations is how to deal with simultaneously performed motions (simultaneous swaying, yawing, rolling and so on). In the linear approach motions are simply linearly superposed, as outlined above. That is, the pressure distributions caused by motions in different directions are simply added up to yield the total pressure distribution, allowing for linear coupling but ignoring nonlinear interaction effects between motions. At forward speed, both the neglect of the nonlinear interactions between the motions and the neglect of nonlinear interactions of the ship motions with the forward speed can be expected to cause large differences between practice and calculation.
Nonlinear methods allow for motions to be realistic imposed, accounting for interactions caused by motions in other directions and viscous forces (lift, viscous drag). Nevertheless, the implementation depends on the accuracy of the description. With a lack of analytical descriptions, empirical descriptions are often used, that are not necessarily accurate. The research presented in this paper aims at identify-ing, quantifying and where possible explaining nonlinear behaviour in ship motions occurring due to motion superposition. Tests were performed in the towing tank facility of the Ship Hydromechanics
Laboratory of the Delft University of Technology in order to analyse the seakeeping behaviour model oscillation. A recently acquired hydraulic oscillator enabled the forced oscillation of the model in the tank in six degrees of freedom simultaneously. This renders the possibility to superpose harmonic motions at different frequencies, at different phase angles and in different directions. This allows for superimposing not only coupled motions, but also uncoupled motions can be superimposed, as for instance heave and roll.
First, single mode oscillations were performed. The results of these tests were analysed with respect to nonlinearities due to motion amplitude and forward speed, but their main purpose was to serve as a reference for the multiple mode oscillations.
Next, multiple mode oscillations were performed for superposition in heave-pitch, heave-roll and sway-yaw. Within the scope of this paper the focus will be on the results of combined sway-yaw oscillations. A more elaborated description and analysis of the results of the other forms of motion superposition is given by De Jong [5].
2. METHODS
2.1. Experimental installation
A model of the M-class frigate of the Royal Netherlands Navy was used for the oscillation tests. All appendages as bilge keels, rudders and propellers were removed, except for a small skeg. The scale of the model was 1:40. Figure 1 shows a rendering of the model and table 1 provides the main particulars.
Description Value
Length over all 122 m
Beam over all 14.4 m
Draft 6.2 m
Speed 29 kts
Tab. 1: Main particulars for the M-class frigate of the Royal Netherlands Navy
Fig. 1: Rendering of the model
The model was divided in seven separate sections, each fitted with a 6-component force-torque transducer. A rigid `backbone’ of steel attached to the motion platform connected the seven sections. The division in sections is shown in Figure 2, along with a
representation of the full-scale M-frigate. As can be seen in figure 2 the foremost and aftmost sections were approximately half of the length of the sections in the midship. This was done for the reason that the gradient of the forces along the length of the model was anticipated to be the largest at the bow and stern.
Fig. 2: Side view full-scale ship and model in sections
The six-degrees-of-freedom forced oscillator of the Ship Hydrodynamics Laboratory of the Delft University of Technology was used to oscillate the model.
The main part of this oscillator consists of a 6DOF ‘Stewart platform’ or ‘Hexapod’ depicted in figure 3, commonly used in applications as aircraft simulators. For the oscillation of ship models in a model basin, the structure is mounted topside down, with the base frame attached to the towing carriage and the model fitted to the motion platform.
Fig. 3: Hexapod (Stewart platform)
The oscillator is capable of carrying out any motion within its envelope, whether it is a pulse motion, a steady translation, a harmonic motion or a combination of the former. The oscillator is software-controlled and in these tests the oscillator was instructed to carry out harmonic oscillations.
The motions performed by the model were obtained by using the instantaneous cylinder lengths to derive the actual orientation of the model.
The force-torque transducers that were used are able to register forces and moments in all 6 degrees of
freedom. The force and moments of all sections along with the motions obtained from the cylinder lengths were digitally stored on a data-acquisition computer. As a backup, an optical position and orientation measurement system was included on the towing carriage.
2.2. Evaluation of the measured data
In figure 4, the reference frames used are depicted. Two right-handed orthogonal reference frames can be defined:
1. A steady translating reference frame OXYZ with the X-axis remains parallel to the forward speed vector V of the ship, while the Z-axis remains vertical. This system is travelling with the forward speed V of the model. The origin O is the vertical projection of the mean position of the centre of gravity on the water surface.
2. A body-fixed reference frame Rxyz with its origin R, the rotational centre of the ship, defined as the vertical projection of the centre of gravity on the water plane area in rest. The frame Rxyz not only travels with the forward speed of the ship, but also follows the motions of the ship. In rest, systems Rxyz and OXYZ coincide. The local ship motions are defined as the orientation of the body-fixed system Rxyz relative to the steady translating system OXYZ. The translations in the X, Y and Z-directions (respectively surge, sway and
heave) are given by vector rOR, while rotations are
described by the angles , and , respectively for rolling, pitching and yawing.
Fig. 4: Reference frames
Consequently, a sway motion always remains a horizontal displacement perpendicular to the forward speed vector and a heave motion always is a vertical displacement. When the body is oriented at a roll, pitch or yaw angle, the translation directions do not change. This in accordance with the linear approach, as this approach neglects the presence of other
motions than the one being considered at that moment.
All signals were sampled with a rate of 1000 Hz and subsequently were digitally filtered using a low-pass filter with a cut-off frequency of 5 Hz to remove unwanted noise, while retaining the data of interest as much as possible.
As the forces were measured with body-fixed transducers, they needed to be transformed to axis system OXYZ. This transformation was carried out by using Euler angles. In addition, the influence of the model mass was removed from the measured forces and moments were transformed to yield the moment around the centre of the rotations R.
The amplitudes of both motions and forces along with the phase angles of the measured forces with respect to the performed motion were calculated by a harmonic analysis of the signals.
The harmonic analysis was carried out by a Fourier transformation at one frequency. The co and quad terms of the force were derived from the signal:
n s quad a n s co as
t
s
F
n
F
s
t
s
F
n
F
1 , 1 ,cos
2
sin
2
(1)Now, the amplitude and the phase angle with respect
to
t
0
could be calculated by:
2
arctan
, , 2 , 2 ,
quad a co a F quad a co a aF
F
F
F
F
(2)This harmonic analysis can only yield accurate results for a complete number of oscillation cycles, as otherwise rest terms would disturb the obtained amplitude. In the case of multiple mode oscillations, this means that the length of the signal analysed should be equal to an integer number of cycles for each of the two oscillation modes. When an integer number of cycles did not correspond with an exact sample, small corrections were implemented.
Finally, the added mass and damping distributions could be calculated from the hydromechanic reaction forces and the motions. All of these were expressed in terms of a frequency, an amplitude and a phase angle. In the general case of a multiple mode oscillation test of a motion in direction x combined with a motion in direction y, the added mass and damping per section were calculated as follows:
y xx x F xa xx y xx xx x F a xa xx
b
F
b
a
m
x
F
a
x x
cos
sin
2 (3) Where: axx and bxx: the added mass and damping of
motion x.
Fxa: the force amplitude, xa: the motion
amplitude.
Fx: the phase between the force and the motion.
to the oscillatory frequency.
mxx: the mass in direction x.
axx-y and bxx-y: the contribution of motion y to the
added mass and damping in direction x (coupling
coefficients, only when the oscillation
frequencies of both motions are equal).
3. RESULTS
The results will be represented by non-dimensional added mass and damping values per unit length. The factors used to make the measured added mass and damping non-dimensional values, are listed in table 2 for sway and yaw. The density of the water is denoted
by , the gravitational acceleration by g, the volume
displacement by , the oscillatory frequency by
and the yaw radius of gyration by rzz2.
Description Divided by
Added mass sway
g
[kg]Damping sway
g
[kg/s]Added mass yaw 2
zz
r
g
[kgm2] Damping yaw
2
zzr
g
[kgm2/s]Tab. 2: Determining dimensionless added mass and damping
Two different types of motion superposition can be distinguished:
1. Superposition of motions both performed at the same oscillation frequency.
2. Superposition of motions each performed at a different oscillation frequency.
The difference between both types lies in the fact that for a motion superposition of the first type linear coupling has to be taken into account. That is, the force generated by the one motion in the direction of the other has a contribution in the added mass and damping of the other motion. This contribution was
Added mass combined sway vs single sway fsway=fyaw -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional added m ass
Sway-Yaw f2 Eps = 90 deg Sway-Yaw f2 Eps = 180 deg Sway-Yaw f2 Eps = 270 deg Sway single f2
Fig. 5: Sway added mass distributions for superposition of type 1
Damping combined sway vs single sway fsway=fyaw
-0.5 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional damping
Sway-Yaw f2 Eps = 90 deg Sway-Yaw f2 Eps = 180 deg Sway-Yaw f2 Eps = 270 deg Sway single f2
Fig. 6: Sway damping distributions for superposition of type 1
Added mass combined sway vs single sway fsway~=fyaw
-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional added m ass Sway-Yaw f2f1 Sway-Yaw f2f3 Sway single f2
Fig. 7: Sway added mass distributions for superposition of type 2
Damping combined sway vs single sway fsway~=fyaw
-0.5 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional damping Sway-Yaw f2f1 Sway-Yaw f2f3 Sway single
Fig. 8: Sway damping distributions for superposition of type 2
Added mass combined yaw vs single yaw fyaw=fsway
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional added m ass
Yaw-Sway f2 Eps = 90 deg Yaw-Sway f2 Eps = 180 deg Yaw-Sway f2 Eps = 270 deg Yaw single f2
Fig. 9: Yaw added mass distributions for superposition of type 1
Damping combined yaw vs single yaw fyaw=fsway
-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] No ndimensiona l damping
Yaw-Sway f2 Eps = 90 deg Yaw-Sway f2 Eps = 180 deg Yaw-Sway f2 Eps = 270 deg Yaw single f2
Fig. 10: Yaw damping distributions for superposition of type 1
Added mass combined yaw vs single yaw fyaw~=fsway
0.0 1.0 2.0 3.0 4.0 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] No ndimensiona l add ed mass Yaw-Sway f2f1 Yaw-Sway f2f3 Yaw single f2
Fig. 11: Yaw added mass distributions for superposition of type 2
Damping combined yaw vs single yaw fyaw~=fsway
-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0 10 20 30 40 50 60 70 80 90 100 Xstation [%] Nondime nsional damping Yaw-Sway f2f1 Yaw-Sway f2f3 Yaw single f2
Fig. 12: Yaw damping distributions for superposition of type 2
Sway-yaw oscillations have been performed at combinations of three non-dimensional oscillation frequencies listed in table 3.
Component
g
L
f1 0.491
f2 0.653
f3 0.980
Tab. 3: Non-dimensional oscillation frequencies
Figures 5 and 6 show the added mass and damping distributions for sway and 9 and 11 for yaw for combined sway-yaw oscillations performed at oscillation frequency f2 with three different phase angles between the sway motion and the yaw motion The phase angles are 90, 180 and 270 degrees, respectively denoted by , and .
Figures 7 and 8 show the added mass and damping distributions for sway and 11 and 12 for yaw for combined sway-yaw oscillations performed at different oscillation frequencies. The combinations depicted here are f2-f1 and f2-f3, respectively denoted with and .
The dashed lines with square-shaped markers give the added mass and damping distributions determined with single mode oscillations. These lines serve as the reference case. Any deviation of the multiple mode (solid) lines with respect to the single mode (dashed) lines indicates a nonlinearity that occurs due to the presence of the other motion.
When looking at the results of motion superposition of the first type (both oscillation modes have the same oscillation frequency) large differences with the single mode oscillations are visible, especially in the added mass and damping of sway. These differences are mostly located at the bow (the right-hand side of each graph) and section 2 shows a small deviation, especially in the sway damping.
In particular, when integrated over the length, the deviations of the total added mass and damping of multiple mode oscillations with the single mode added mass and damping are very large. For example, a measure of difference of the multiple mode added
mass amultiple with the single mode added mass asingle is
defined as:
100
%
single single multiplea
a
a
(4)Figure 13 shows that this percentage can be up to 200% for the total damping for sway and up to 100% for the sway added mass. In contrast, figure 14 indicates that the yaw added mass and damping are much less affected by the presence of the sway
motion at the same oscillation frequency. However, the relative difference for both added mass and damping is still up to 30%.
Motion superposition of the second type (super-position of modes performed at different oscillation frequencies) yields far smaller differences than motion superposition of the first type. The differences of the solid lines of the multiple mode results with the dashed lines of the single mode results are far less (figures 7, 8, 11 and 12) than for superposition of the first type. This is even clearer shown by figure 15 presenting the relative difference of the total sway added mass and damping for motion superposition of the second type.
0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Added mass Sway-Yaw = 90 deg Sway-Yaw = 180 deg Sway-Yaw = 270 deg 0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Damping Sway-Yaw = 90 deg Sway-Yaw = 180 deg Sway-Yaw = 270 deg
Fig. 13 Relative differences total added mass and damping for sway in combined sway-yaw motions –
type 1 0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Added mass Yaw-Sway = 90 deg Yaw-Sway = 180 deg Yaw-Sway = 270 deg 0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Damping Yaw-Sway = 90 deg Yaw-Sway = 180 deg Yaw-Sway = 270 deg
Fig. 14 Relative differences total added mass and damping for yaw in combined sway-yaw motions –
type 1
The yaw relative difference for the second type of motion superposition is similar to the sway added mass relative difference shown in figure 15. The sway added mass, the yaw added mass and yaw damping seem to be hardly influenced by each other’s presence. However, differences for the total values of the sway damping are large: up to 150% for the lowest oscillation frequency.
Moreover, regarding the influence of motion superposition of the second type on the sway added mass interesting results are found. In the total value no influence of the presence of the yaw motion is visible: the relative difference remains smaller than 10%. Still, locally the combined sway added mass shows large differences compared to the single sway added mass. The distribution over the length is influenced, not its total value.
0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Added mass y a = 0.06*B ya = 0.12*B ya = 0.18*B 0 0.5 1 1.5 2 -200 -150 -100 -50 0 50 100 150 200 (L/g) Sin g le m o d e s [ % ] Damping y a = 0.06*B ya = 0.12*B ya = 0.18*B
Fig. 15 Relative differences total added mass and damping for sway in combined sway-yaw motions –
type 2
Other results obtained in these series of forced oscillation tests include multiple mode superposition of heave with pitch and heave with yaw. In addition, extensive single mode oscillations have been performed in sway, heave, roll, pitch and yaw at medium and high forward speed.
4. DISCUSSION
4.1. Summary of observations
In the preceding section the results of the research have been presented for sway and yaw motions along with a quantification of the differences. Based on these results two observations can be made:
1. When oscillating at equal frequencies, large differences with the added mass and damping of single motions occur, especially in the sway added mass and damping.
2. When oscillating at different frequencies
interactions are small, except for the sway damping. The sway added mass shows locally
large differences, while these differences
disappear when integrated over the length.. In the next subsections each of these observations will be discussed with respect to the underlying mechanisms.
4.2. Motion superposition of the first type
Large differences were found in the added mass and damping, when superposing motions with equal oscillation frequencies, even while linear coupling is taken into account. The differences were found both
locally on each section as in the total values. The deviations were very much dependent on the chosen phase angle between the superposed motions.
The deviations signify that besides linear coupling between sway and yaw, which is of significant importance, there are additional, nonlinear interaction effects present. These nonlinear interactions can be related to the disturbance of the flow generated around the body by its forward speed. As pointed out in the introduction this disturbance is neglected in the linear approach.
For instance as the body is proceeding through the water with a constant yaw angle it functions as a low aspect ratio wing with an angle of attack relative to its inflow. This will cause lift and drag forces, both of potential nature and of viscous nature (sometimes referred to as cross-flow drag). Also when the body has a sway velocity when moving forward at is basically a low aspect ratio wing with a angle of attack, generating lift and drag forces.
Both mechanisms have a slightly different nature. The lift force generated by yaw is primarily due to the motion amplitude (the yaw angle causing an angle of attack), while the lift force generated by sway is caused by an angle of attack due to a sideways velocity together with the forward speed. The lift force caused by yaw could be looked upon as a form of restoring; a force acting on the body due to a static motion amplitude. However, this force is pushing the model further from its reference position.
When oscillating in sway or yaw these lift and drag forces become unsteady forces, strongly varying with the instantaneous orientation of the body, causing large nonlinear variations in the hydromechanic reaction forces. These variations are not anticipated by the linear approach and will show up as amplitude dependencies of the added mass and damping. For superposed motions the unsteady forces not only result in large amplitude nonlinearities, but also cause large nonlinear interactions between both motions. That the generation of lift and viscous effects (flow separation, eddy-forming, cross-flow drag) are the most probable sources of nonlinearity is confirmed by the differences that mostly show up at the foremost sections and at the section 2, which has a skeg. The sharp V-shaped bow sections and the skeg will cause flow separation and the subsequent creation of large eddies (during the tests large eddies were visible at the bow when swaying and yawing).
The phase angle between both motions is of great importance in determining whether interactions show up in the added mass or in the damping of the motions. This effect could be amplified by the fact that the phase angle between the motions is chosen from the sequence 0-90-180-270 degrees, each time a
variation of 90 degrees. The phase angle between added mass and damping, being in the in-phase and out-of-phase components of the hydrodynamic reaction forces, is also 90 degrees.
Variation of the phase angle between sway and yaw motions can result in completely different motions. A phase angle of 90 degrees produces a motion similar to pure yawing. When moving in pure yaw the ship has relative to the flow almost no angle of attack (or drift) so that almost no nonlinear unsteady lifting and cross-flow drag forces will be generated. In contrast, phase angles of 180 and 270 degrees result in motions with large angles of attack and drifting relative to the surrounding water. This would produce large nonlinear force contributions due to unsteady lifting and cross-flow drag. This view is confirmed by the fact that the deviations are largest for phase angles of 180 and 270 degrees and smallest for 90 degrees.
4.3. Motion superposition of the second type
Almost no deviations were found in the yaw added mass and when superposing motions that each was performed at its own oscillation frequency. These tests revealed that interactions caused by `leaking’ between different frequencies are much smaller than interactions showing up between motions performed at the same frequency. This supports the idea of the wave superposition, in which each frequency component is evaluated separately, without taking into account interactions.
However, the sway damping still shows large deviations with respect to the single mode results, while the distribution over the length of the sway added mass also is influenced by the presence of the yaw motion. The explanation of these deviations must have its origin in the disturbance of the flow around the body by the yaw motion, even at different oscillation frequencies of both motions. Apparently, the yawing motion generates disturbances of such nature that even when both modes are performed at different frequencies nonlinear interaction occurs. The instationairy angle of attack with respect to the flow will play an important part in these disturbances. During the testing of sway-yaw large eddies were formed at the bow sections, indicating indeed large disturbances of the flow of a viscous nature.
4.4. Recommendations and future research
Based on the research presented in this paper the following can be recommended:
The inclusion of unsteady lift and viscous force
contributions when evaluating the seakeeping behaviour in the horizontal plane at larger motion amplitudes.
The results could be elaborated by including
calculated wave excitation forces and expressing the nonlinearity on the level of the predicted motions. This would enable prediction of the
implications of the nonlinearities occurring in the hydromechanic reaction force on the total seakeeping behaviour.
The definition of yaw restoring terms: a static
yaw angle produces a static yaw moment when the body is moving with a steady forward speed, as noted in section 4.2. This could lead to the definition of a yaw `restoring term’: the resulting moment divided by the amplitude of the static displacement.
For a more complete picture of the influence of
the phase angle on nonlinear interactions testing could be performed at more phase angles, outside the series 0-90-180-270 degrees.
More research into the unsteady lift and
cross-flow drag contributions caused by sway and yaw motions.
REFERENCES
[1] N.F. Barber and F. Ursell. The generation and propagation of ocean waves and swell. Philosophical
Transactions of the Royal Society of London, 240:p.
527-560, 1948.
[2] M. St.Denis and W.J. Pierson. On the motions of ships in confused seas. Transactions Society of Naval
Architectures and Marine Engineers, 61:0. 280-357,
1953.
[3] M. St.Denis. Some observations on the techniques for predicting the oscillations of freely-floating hulls in a seaway. In Offshore Technology Conference, 1974
[4] J.A. Keuning. The nonlinear behaviour of fast
monohulls in head waves. PhD dissertation, Delft
University of Technology, Ship Hydrodynamics Laboratory, 1994
[5] P. de Jong. Investigation of nonlinearities in the superposition of ship motions by 6DOF forced oscillations. Report 1424, Ship Hydrodynamics Laboratory Delft University of Technology, 2005.
