Seakeeping design of a high-speed autonomous semi-submersible vehicle

Seakeeping Design of a High Speed Autonomous Semi-Submersible Vehicle P.A. Wilson and D.A. Hudson ]

ABSTRACT

Increasingly in the @eld of ocean surveying, both scientl~c and commercial!, autonomous underwater vehicles (AUVS) are being employed as a means to reduce the problems associated with operatingfmm su<ace ships, such as ship motions, engine noise, hull vibration and high operating costs. Whilst an AUV addresses some of these problems they are expensive to design and build, mechanically complicated and have a limited speed.

This paper looks at the design of an alternative survey vehicle, a small, fast semi-submersible capable of au-tonomous operations and able to achieve acceptably low motion characten”stics. Mth such a novel vehicle it is important to asse.w, early in the design process, whether ornot the intended design can fulfill the requirements oflow

motions in a relatively severe sea-state.

To this end an extensive parametric stui$ was undertaken using a three-dimensionalpulsating source distribution method to evaluate the vessel responses, This studv investigated changes in body iength, diametec depth below the surface and position of the centre of gravity, as well as othec more detailed, changes of vehicle contguration. The small vehicle size coupled with a requirement to transit to andfmm a survqv area leads to a high Froude number for the vessel and changes in forward speed thus formed a furthec important, investigation.

Comparison of vessel Response Amplitude Operators (RAOS) and lLMS motion values in realistic operating sea-states con@-tns the 10w motion characteristics of the vehicle and allow an optimum configuration for a particular sea-state to be selected. Comparison of theoretical data with experimental measurements confirms, within limits, the suitability of the adopted mathematical model. The shortcomings of the present analysis are discussed, with particular emphasis on prediction of roll motion and the effects of control systems.

1 INTRODUCTION

Recent years have seen an increased requirement for effective surveying of ocean and coastal waters, both from the scientific community and commercial opera-tors. These requirements can range from sea-bed sur-veys in areas of underwater exploration or operations, to measurement of wave and current conditions. increas-ingly, in order to meet the demands autonomous under-water vehicles (AUVS) are being employed as a means to reduce the problems associated with operating from surface ships, such as ship motions, engine noise, hull vi-bration and high operating costs, Additionally AUVS can have wider applications due to their ability to perform missions submerged. Unfortunately, whilst an AW ad-dresses some of these problems they have become expen-sive to design and build, mechanically complicated and have litnhed speed and endurance.

This paper looks at the design of an alternative sur-vey vehicle, a small autonomous semi-submersible. Such

IFl~id.S~CtUreIntemction Research Group, School of

Enghreer-ing Scienees, University of Southampton, Highfield, Southampton, S017 IBJ, UK.

a design can achieve the desire for low motion character-istics, at least as good as those for a survey vessel, whilst allowing the use of ‘off the shelf components’ to min-imise manufacturing and operational costs. Additionally such a vessel can reduce ship hire time, further reducing operational costs. It is possible that the semi-submersible concept could be realised in different sizes, suited either to operation from a base port or handled from a ‘mother’ ship capable of operating more than one vehicle at the same time. Speed, endurance and range can be chosen to satisfi the particular survey mission envisaged. The small vehicle size, and requirement to transit to and from a survey area, leads to a high Froude number for the ves-sel.

When considering such a novel vehicle design it is

important to confirm, as early in the design process as

possible, whether or not the intended concept can fulfill the requirement of low motions in a relatively severe sea-state. With low motions being vital for the success of the concept, and many parameters in the design not fixed, it is desirable to consider alternative vessel shapes and choose one giving minimum motions. To undertake this exercise experimentally would be prohibitive in terms of

both time and cost. This study must therefore be under-taken using theoretical methods.

When looking to calculate the motions of the semi-submersible survey vehicle it is clear that to employ a

two-dimensional method, i.e. a strip theo~, however

computationally desirable this may be, is fruitless since the body shape cannot be adequately represented by such

a theory. Equally, application of a speed dependent

three-dimensional solution, whether in the time or fre-quency domain, would be too time-consuming to enable investigation of sufficient vehicle parameters. The com-promise adopted in this study has been to use a three-dimensionrd method solving the zero forward speed free-stuface boundary condition, accounting for forward speed in a limited manner similar to strip theory [1]. Such methods are adequate for conventional ships, find common application in the offshore industry and have been applied successfully to SWATHS [2], catamarans [3] and trimarans [4].

Clauss and Birk [5] discuss a technique to minimise downtime of offshore structures based on optimizing an objective fi.mction subject to constraints and employ a three-dimensional method for the hydrodynamic calcula-tions, Such an approach may well be feasible for the pro-posed semi-submersible but for such a novel hull shape, at such an early stage of the investigation, a simpler but more readily controlled parametric study was undertaken. This allowed detailed assessment of the Response Am-plitude Operators (RAOS) in regular waves to help un-derstand the effect of changes in vehicle configuration and speed on its dynamic response.

Calculation of vessel RMS response in realistic sea states was used as a means to compare different configu-rations.

The parametric study investigated changes in body length (L), body diameter (D), submergence of the main body (S), vertical position of the centre of gravity (VCG), waterplane area, pitch and roll moments of inertia and

forward speed. It was impossible to test every single

combination of the above. However, over 200 variations were tested and this gives a good indication of the ef-fects of changing the main parameters on the motions of the vehicle. A selection of the results is presented here. More detail may be found in Wilson and Hudson [6].

For important configurations, an ITTC sea spectrum was used to calculate RMS heave and pitch responses in different realistic sea-states. A selection of RMS values is presented, confirming the low motion characteristics of the vehicle. Comparisons with limited experimental data illustrate the suitability of the theoretical approach adopted. Limitations of the current theoretical analysis are discussed.

2 Mathematical Model

2.1 Equations of Motion

The motions of a rigid vessel undergoing small perturba-tions, in regular sinusoidal waves about an equilibrium position can be represented by the coupled linear equa-tions of motion

k==l

= _{F/+ FjD, j=l,2,...,6 (2.1)}

where,

&fjk= element of mass or ine~ia ~hix, we = frequency of encounter,

Ajk = added mass in the jfh mode due to unit motion

in the ldh direction,

~~k = damping coefficient in the jih mode due to unit motion in the 1# direction,

Cjk = hydrostatic restoring COeffiCltUIt,

qk = complex motion amplitude,

F: = complex amplitude of the incident wave exciting

force (Froude-Krylov),

FjP =complex amplitude of the difl?raction wave exciting force.

The terms in this equation can be evaluated by a number of methods, such as two-dimensional strip the-ory and three-dimensional potential flow analysis; each assuming the fluid homogeneous, inviscid and the fluid motion irrotational [7, 8, 9]. The fluid motion can now be represented by a velocity potential satis~ing the Laplace equation throughout the fluid domain. Unfortunately cal-culating the total velocity potential in its most general form is difficult and, for practical use, some

simplifica-tion is necessary. Thus, the total potential can be

ex-pressed as a linear summation of components:

~(z, W,z, ~) = (–UZ + ~. (z, y, z)) + #~eiW’t (2.2)

where,

U is the forward spee~

#@is the perturbation potential due to steady translation, 4T is the unsteady perturbation potential which can be decomposed to give:

with,

#1 as the incident wave potential,

4D as the diffraction potential and

~j denoting the radiation potential due to unit motion

In equation (2.2) the first two terms represent the problem of the ship advancing at steady forward speed in calm water. These can be determined separately from the unsteady potentials. Using equation (2.3) and appro-priate boundary conditions, solutions to equations (2.1) are obtained.

2.2 Methods of Evaluation

Evaluation of the radiation and diffraction potentials, which are used to calculate the added mass and damping coefficients and the distraction component of the wave excitation force respectively, can be carried out in sev-eral different ways [10]. For this investigation a three-dimensional method was used.

The three-dimensional analysis is a

boundary-element metho~ whereby the problem of modelling the whole fluid domain can be reduced to that of modelling the boundaries of the fluid, in this case by application of Green’s 2nd theorem. By suitable choice of the singu-larity to be used, the problem can be fbrther reduced to modelling the body surface only.

Thus, in the three-dimensional approach adopted, the wetted surface of the hull is discretised by four cor-nered panels each of which contains a singularity at its centre. In this method the singularity is a pulsating source [2, 8, 9]. Such a singularity satisfies the Laplace equa-tion throughout the fluid domain, a radiaequa-tion condiequa-tion at intinity and the linearised free surface condition with zero forward speed. This distribution neglects interaction effects between the steady and unsteady components of the velocity potential. This arises from the assumption, when formulating the boundary condition on the body

stiace, that the perturbation of the steady flow due to

the presence of the hull is negligible. A full description of the boundary value problem can be found in Bishop et al. [2].

3 Vehicle Idealisation

The theoretical analysis requires the wetted surface of the hull to be adequately described for numerical purposes. In this context the vehicle is represented as a number of flat quadrilateral panels, ideally each having an aspect ratio of unity. In order to allow the dimensions of the ve-hicle to be changed readily, the process of creating this wetted surface idealisation was automated as far as pos-sible. For this study the forward and aft ends of the ve-hicle were treated as ellipsoids and the centre as a cylin-der. The keel was modelled as tapering towards its base in the fore and aft direction. Whilst on a practical de-sign the keel may have a bulb on the base, at this stage no bulb was included as the size and shape are undeter-mined. The hydrodynamic influence of such a bulb is

judged to be small compared to the rest of the body. Sim-ilarly, lacking a detailed waterline shape, the strut was simply represented as tapering at its fore and aft ends. If the panel length is sufficiently small a parallel mid-body is inserted in the strut. At this stage no appendages (other than the keel) were included in the model. Care was taken throughout the modelling process to use pan-els with aspect ratios close to unity. New configurations were generated as variants of a basis vehicle; changes in body length, diameter etc. being input and a new wetted surface idealisation generated automatically.

Figure 1 illustrates the wetted surface idealisation adopted for the basis vehicle configuration. This ideal-isation has 344 panels, of which 240 are on the main body. This was felt to give an adequate description of the hull without using an excessive number of panels. A more refined idealisation having 624 panels (464 on the main body) was investigated to determine the effect of improving the modelling of the hull. This confirmed the suitability of the initial idealisation and did not in-troduce significant variations to the results subsequently discussed.

Figure 1: Wetted Surface Idealisation (344 panels)

4 Analytical Predictions

The characteristics of the basis vehicle (and all subse-quent variants) were evaluated in regular head and beam waves and are presented in figures 2 to 7. Wave period was varied from 3.0 seconds to 31 seconds, an adequate

plotted as a Response Amplitude Operator, that is; heave amplitude (m)/wave amplitude a (m). Pitch and Roll are plotted as motion amplitude (rad)/equivalent wave slope ka (rad). A speed of 12 knots was taken as the basis vehicle speed. Variations on the basis configuration are discussed below.

4.1 Geometrical Variations

Three different lengths of the main body were consid-ered L=2.OW L=5.Om (the basis) and L= 10.Om. For each of the lengths considered, three different diame-ters were also investigated. The same L/D ratios were tested for each length. Whilst the length was varied the strut length (and breadth) were also altered to maintain

the strut lengthhehicle length ratio as a constant. Keel

length, breadth and depth remained unaltered. The re-sults of the variation in length are shown in figure 2.

From figure 2 it is clear that the longer the vehicle is made, the higher the heave and pitch natural periods. The effect of increasing length on the roll response is the same as for heave and pitch. This is the case for all diameters investigated.

Figure 3 illustrates the effects of the changes in di-ameter for the basis vehicle length. For this length the strut length and breadth and the keel depth were not al-tered. It is clear from figure 3 that the larger the diameter of the vehicle, the longer its natural period in heave and pitch. This is a result of increasing the displacement of the vessel, whilst the waterplane area and VCG remain unchanged. For roll motion, the effect of changing the body diameter appears to be minimal.

The effects of changing the vertical centre of gravity of the vessel were considered for the basis length/diameter ratio. This investigation, considered in more detail in Wilson and Hudson [6], demonstrated that the higher the VCG, the longer the heave, pitch and roll natural periods. The effect being greatest for roll. The basis position of the VCG was –S – 0.25m below the free surface, where S is the submergence of the main body below the surface.

For each of the configurations of length and diame-ter, the submergence of the main vehicle was changed.

Three submergence of the main body, of 0.2L, 0.4.L

and 0.6L were investigated. The effect of the change

in submergence of the main body is shown in figure 4. Whilst the main body submergence has varied, the depth of the keel has remained unaltered (relative to the body). Changes in submergence appear to produce only mini-mal changes in the natural periods of heave and pitch. The magnitude of the resonant response does, however, increase slightly as the main body approaches the sur-face. This effect is more marked in heave. For roll mo-tion, the effect of increasing the submergence of the main

body is to increase the natural period of the vehicle, For the basis vehicle configuration, the influence of

changing the waterplane area was studied. In order to

do this three different strut lengths were considered. As the strut length was altered the breadth of the strut was changed to retain a constant strut length/breadth ratio. These variations are presented in figure 5. As one may intuitively expect, the smaller the strut size, the higher the natural periods in heave and pitch of the vehicle. The largest change occurs between a strut length of 0.75m and 1.Om. The effect of the strut size on roll is minimal, probably due to the small absolute change in the breadth of the strut.

4.2 Forward Speed

Figure 6 illustrates the effect of varying the speed of ad-vance of the vessel. Vehicle speeds of 8 knots, 12 knots and 16 knots were considered. If the basis strut length of 1.Om is used to define the Froude number then these

speeds correspond toaFroudenumberof1.31, 1.97 and

2,64 respectively. If the overall vehicle length of 5.Om is used then the Froude number corresponding to each of the speeds is reduced to 0.59, 0.88 and 1.18

respec-tively. Given the submerged nature of the main bodyj

it is not clear which definition of Froude number is the most appropriate and the absolute speed is therefore used in discussion. However, it is clear that the vehicle has a high speed for its size. As one might expect the natural ffequency of the vehicle remains unchanged with

chang-ing forward speed. However, the increase in forward

speed results in this natural frequency being encountered in longer wavelengths.

A second effect of forward speed is to increase the response of the vehicle at the resonant frequency, al-though this is not clear from figure 6. The forward speed effect means, however, that an increase in speed will re-sult in less wave energy at this natural frequency.

4.3 Changes in Inertia

For the basis vehicle configuration, changes in the roll moment of inertia are illustrated in figure 7. Three values

of roll radius of gyration, k44,were investigated, ranging

from 0.5S to 1.0S. An increase in the moment of inertia will increase the natural period of the vehicle in roll. The effect of increasing pitch inertia would intuitively be ex-pected to be the same for the pitch motion as roil inertia is for the roll motion. To a limbed extent this is the case, but the effect is small when the vessel has forward speed. At zero forward speed the effect is much more marked. With forward speed, at the low frequency of encounter at which the natural frequency is excited, the pitch added inertia is large and dominates the total pitch inertia. With

zero forward speed the pitch added inertia is much less and the influence of changing moment of inertia conse-quently greater.

5 Responses in a Seaway

For all of the variations discussed in section 4 the Root Mean Square (RMS) of the responses of the semi-submersible have been calculated for an ITTC sea spec-trum. The spectrum is defined using two parameters; sig-nificant wave height (hi/3) and modal wave period (Z’). In this case responses were calculated for a significant wave height of 1.Om, to allow easy scaling to any re-quired sea-state, and for three modaI periods of 8.0 sees, 10.0 sees and 12.0 sees, which are broadly representa-tive of the periods one may expect in conditions ranging from shallow coastal waters to the open ocean. Results were evaluated for head seas only; as the roll response prediction is likely to be the least reliable due to the po-tential flow assumptions adopted in section 2. Results for a selection of important configurations are presented in Table 1.

Responses to an irregular sea state defined by this spectrum are generally low, particularly considering the size of the vehicle. For the basis vehicle length of 5.Om, a submergence of 40’?40of the body length and a strut length of 0.75m gives the best (i.e. lowest) response for the three wave periods investigated. This is true for both heave and pitch responses. For this, and the strut length of 1.Om,the response gets worse as the wave period is in-creased. This is a consequence of the modal wave period approaching the vehicle natural period, For practical pur-poses a strut length of 0.75m on a vehicle length of 5.Om may be a little too small, making the vehicle too sensitive to changes in load and difficult to construct and handle.

As the speed of the vehicle is increased the responses decrease for the two lowest wave periods, despite the resonant response increasing. This is a consequence of the wave period required to excite the natural frequency

increasing as speed is increased. As this wave period

increases the energy contained in the spectrum at that

period decreases. This is especially the case when the

modal period of the wave spectrum is fhrther from the natural period of the vehicle, hence the large decrease in the vehicle response as speed increases when the modal wave period is 8.0 sees. When the modal wave period is 12.0 sees the wave encounter effect is more complex as the natural period of the vehicle is close to the modal wave period of the spectrum.

Decreasing the diameter of the vehicle is beneficial for the longest wave period, as the natural frequency of the vehicle is increased, moving the resonant response fiwther from the modal wave period.

For the shorter vehicle length of 2.Om, the responses

are increased compared to the basis length. This is true for all diameters apart from the smallest combined with

a spectrum of modal wave period 12.0 sees. For the

longest vehicle, of 10.Om, responses are very low when the largest diameter is also considered.

In all cases, since the response of the vehicle is sharply tuned to a particular wave period the proximity of this natural wave period to the modal wave period of the sea spectrum determines the severity of the response in that seaway. It would appear from the investigation that a length of 5,0m, submergence of 40% of the length, diameter of 1.Om and a strut length as small as practi-cally possible will give the best vehicle performance un-less the extra costs incurred by a 10.Omversion could be justified. If this was the case, then a 10.Om vehicle with a diameter of 2,0m would give the best vehicle response by a significant amount.

6 Comparison with Experiments

In order to provide additional confidence in the ca-pabilities of the mathematical model adopted, particu-larly in predicting the motions of such a novel vehi-cle, experiments were carried out on a scale model. A 0.75m (1:6.667) scale model of the basis configuration of the semi-submersible was constructed and tested at the Southampton Institute towing tank. This tank has a length of 60.Om and a width of 3.5m, it is equipped with a wave-maker capable of generating both regular and ir-regular waves. Cahn water resistance tests were under-taken in addition to seakeeping experiments. Seakeeping experiments were performed at zero forward speed as no means of attaching it to the tow fitting was possible al-lowing the model to be free to pitch. Experiments were performed at zero forward speed in regular head waves and the theoretical method applied to a model scale rep-resentation of the vehicle.

Figure 8 illustrates the comparison between the the-oretical and experimental data. For the heave response the agreement is very close with the exception of the low-est wave frequency tlow-ested. This frequency corresponds to a wavelength of 18.74m and, in a 60.Om tank, the quality of the wave and the measurement cannot be guaranteed. No experiments could be carried out at a lower frequency due to limitations with the wave-maker. The predicted pitch response shows the same trends as the experimen-tal data, although it appears as though the predicted res-onant frequency is too high. Once again the validity of the measurement at the lowest frequency must be open to question.

Open water tests on a larger self-propelled model are currently being undertaken and further comparisons with theoretical predictions will be of interest; but it is be-lieved that the theoretical models adopted are accurate

enough for the purposes of an initial design study. More advanced theoretical analyses may be appropriate for ac-curate prediction of vessel responses later in the design process.

7 Conclusions

A theoretical study into the motions of a novel survey vessel of semi-submersible form has been undertaken. A three-dimensional potential flow analysis was used for this study. Such methods provide a reliable and efficient means to investigate the seakeeping behaviour of novel vessel types from a desigw as well as a research, view-point.

A large parametric study was conducted to assess the effect of varying key vehicle parameters such as vessel length, diameter, submergence, waterplane area, centre of gravity and moments of inertia, as well as operational parameters such as forward speed.

From this study important vehicle parameters can be determined and the characteristics of any configuration investigated in a realistic sea-state. This was done using vessel RMS responses in representative sea-states.

The investigation confirms the low motion charac-teristics of the proposed semi-submersible concept,

Although the analysis adopted proves adequate for an iritial investigation, detailed comparisons of vessel re-sponse with corresponding experimental data at a range of forward speeds would be needed to better assess the accuracy of such theoretical methods. It is likely that the potential flow method adopted will require some means

to account for the effects of viscous damping. With

a semi-submersible vehicle the predicted resonant re-sponse is sharply tuned around the natural ffequency. This is a consequence of the small amount of wave (po-tential flow) damping present in the system. In reality the wave damping will be augmented by viscous flow effects. This will tend to reduce the peak responses pre-dicted by a potential flow method, particularly the peak roll response. A practical application of this concept will require fins to be placed on the vehicle, at the very least to provide directional stability and control. For a direct comparison to be made between theoretical predictions and experimental measurements, the theoretical method will need a means of accounting for the effects of such control fins on the motions of the vehicle.

8 Acknowledgements

Financial support for this work was provided by the Nat-ural Environment Research Council (NERC) through a DTI LINIVSEASENSE programme.

The authors would like to thank the members of the SASS consortium for their co- operation in the

prepara-tion of this paper, in particular Mr. Stephen Phillips of Seaspeed Technology Ltd. U.K. and Mr. Hugh Young of Hugh Young and Associates, U.K.

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Table 1: RMS Responses of Semi-Submersible variants

Basic Configuration L=5.Om, VCG=-2.25m, strut length (s1) = 1.Om.

L=5.Om, VCG=-2.25nL S/L=O.4

L=5.Om, VCG=-2.25m, strut length (s1) = 1.Om.

L=5.Om, VCG=2.25m, strut length (s1) = 1.Om.

L=2.Om, WL=O.4, strut length (s1) = 0.4m.

L=1O.OW S/L=O.4, strut length (s1) = 2.Om.

Variant SIL=O.4 s/L=o,2 SIL=O.6 sl=O.75m s1=1.25m sl=l.50m U=5kts U=l 2 kts U=l 6 kts D=O.40m D=O.70m D=O.16m D=O.28m D=O.40m D=O.80m D=l .40m D=2.Om Motion RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad) RMS Heave (m) RMS Pitch (rad)

T=8 sees T=1O sees T=12 sees

0.169 0,368 0.456 0.022 0.024 0.023 0.195 0.383 0.448 0.021 0.024 0.022 0.187 0.393 0.468 0.023 0.025 0.024 0.086 0.093 0.179 0.012 0.013 0.015 0,335 0.372 0.341 0.032 0.029 0.024 0.454 0.464 0.418 0.050 0.040 0.031 0.357 0.406 0.384 0.044 0.039 0.033 0.169 0.368 0.456 0.022 0.024 0.023 0.094 0.288 0.413 0.017 0.020 0.020 0.468 0.408 0.357 0.057 0.043 0.032 0.401 0.451 0.422 0.039 0.035 0.029 0.388 0.331 0.300 0.109 0.072 0.051 0,462 0.397 0.348 0.075 0.053 0.038 0.451 0.460 0.415 0.058 0.045 0.034 0.416 0.451 0.416 0.034 0.032 0.027 0.183 0.388 0.466 0.016 0.021 0.020 0.080 0.117 0.295 0.008 0.011 0.012

1.04

k?

@ll

EzEl

WavePerl~ -=secs> Wave Period <sacs>

Figure 2(a): Heave RAO for Varying Length (max. D.), Figure 2(c): Roll RAO for Varying Length (max. D.),

VCG=-S-O.25U U=12kts, WL=O.40. VCG=-S-O.25m, U=12kts, S/L=O.40.

N

-e- L-5 ,0-1 h -& L.1O ,D-20rr

0

5

Figure 2(b): Pitch RAO for Varying Length (max. D.), VCG=-S-O.25m, U=12kts, S/L=O.40.

U

Wave Pariod <sees

Figure 3(a): Heave RAO for L=5.Om, VCG=-S-O.25m, U=12kts, S/L=O.40, varying diameter.

10

1

bd

D

5

Figure 3(b): Pitch RAO for L=5.Om, VCG=-S-O.25m, U=12kts, WL=O.40, varying diameter.

lo– 8– ~ ‘-a ~ 4-2–

U

Wave Period <seca>

Figure 3(c): Roll RAO for L=5.Om, VCG=-S-O.25m,

U=12kts, WL=O.40, varying diameter.

4,0-3.5– 3.0-~ 2.5-a a al 2.0 – $ 0 = 1.5 -1,0– I \ )

n

+

S

Wave Pariod wca>

Figure 4(a): Heave RAO for L=5.Om, VCG=-S-O.25nL U=12kts, varying S.

(

r

o

5

Wave Period <seca>

Figure 4(b): Pitch RAO for L=5.OIrL VCG=-S-O.25m, U= 12kts, varying S.

1111

Wave Pericd <sees>

Figure 4(c): Roll RAO for L=5.0m, VCG=-S-0.25m,

U=l 2kts, varying S. 4.0-3.5– 3.0- 2.5-2 E e 2.0– g m = l.5– 1.0 0.5 0.0 i_.slBJ 0

5

Wave Period <sacs>

Figure 5(a): Heave RAO for L=5.Om, VCG=-S-O.25m, U= 12kts, S/L=O.4, varying strut length.

0

5

Figure 5(b): Pitch RAO for L=5.Om, VCG=S-O.25m,

U=12kts, S/L=O.4, varying strut length.

L---J

Encounter Frequency (rad/aec)

Figure 6(a): Heave RAO for L=5.Om, VCG=-2.5~

El

Encounter Frequency (radkec)

Figure 6(b): Pitch RAO for L=5.Om, VCG=-2.5m, S/L=O.4, varying U.

rl

+

Wave Period <ewe>

Figure 7:Roll RAO for L=5.0m, VCG=-S-O.25m,

U=12kts, S/L=O.40, varying k~.

o.2- A

0.0 1 I I I I

0.0 0.2 0.4 0.6 0.8 1.0

Wave Frequency <Hz>

Figure 8(a): Model Scale Heave Wo, S~=0.4,