Analysis of non-linear vessel motions: experiments and predictions

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ANALYSIS OF NON-LINEAR VESSEL MOTIONS:

EXPERIMENTS AND PREDICTIONS

JOSHUA BOYD

Postgraduate Research Student, Australian Maritime Engineering CRC Ltd (Perth Node), Curtin University of Technology, Perth, Western Australia

KIM KLAKA

Assoc-Director, Australian Maritime Engineering CRC Ltd (Perth Node)

GILES THOMAS

Research Associate, Australian Maritime Engineering CRC Ltd (Perth Node)

Nomenclature

Introduction

Considerable advances have been made over the past four decades in the field of theoretical predictions of vessel motions in a seaway. Strip theory is now the primary tool available to naval architects for predicting the motions and seakeeping performance of a vessel operating in waves. There is particular interest in Australia in the seakeeping characteristics of high speed craft and craft required to operate in large sea states. Many vessels currently in production are designed to travel at Froude numbers approaching or exceeding unity (equivalent to 38 knots for a 40 metre vessel or 54 knots for an 80 metre vessel). The relatively small size of these vessels and the operators' desire to maintain speed even in rough seas may mean that numerousassumptions of linear behaviour implicit in conventional strip theory are invalid. This may result in significant inaccuracies in modelled seakeeping behaviour. One of the basic assumptions of conventional strip theory is that the hydrodynamic coefficients are calculated assuming a flat waterline over the length of the vessel, regardless of its speed or the prevailing sea state.

a33 = Sectional added mass coefficient F3, F5 = Wave exciting heave force and pitch

A33 = Total added mass coefficient moment

A35 = Heave/Pitch added mass cross-coupling Fn = Froude number, Fn =

U/JZ

coefficient _g = Acceleration due to gravity

A53 = Pitch/Heave added inertia cross-coupling _{= Pitch moment of inertia}

coefficient = wave number, k0 = 2,t/X

b33 = Sectional damping coefficient _L _{= Vessel length}

B33 B35

= Total damping coefficient

= Heave/Pitch damping cross-coupling coefficient M U x = Vessel mass = Vessel speed

= Longitudinal distance from LCB B53

C33 C35

= Pitch/Heave damping cross-coupling coefficient

= Sectional restorihg force coefficient = Total stiffness coefficient

z

z,

13' 15

= Total local waterline offset

= Local waterline offset due to incident wave = Phase angle of wave

(zero = equilibrium at LCB) = Heave and pitch displacement = Heave/Pitch stiffness cross-coupling

coefficient

C53 = Pitch/Heave stiffness cross-coupling

coefficient Ci), W

= Wave length

= Wave frequency, encounter frequency C55 = Total pitch stiffness coefficient = Wave amplitude

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This paper presents a study into the wave height dependence of the motions of a high speed displacement craft and a hull with significant flare and overhang. A series of seakeeping

towing tank experiments on these craft has been performed with the

objective of determining the wave height dependence of their vertical motions. A pseudo non-linear strip theory is presented which models effects not treated by linear strip theory, including the Kelvin (ship generated) wave pattern, emergence of the bow or stern due to motions, the profile of the incident wave and the amplitude and phasing of the resultant motions. It is shown that consideration of these effects may have a significant influence on predicted motions, particularly for large wave heights.

Strip Theory

The AM1ECRC implementation of strip theory described in this paper follows that given by Salvesen, Tuck and Faltinsen (1970). This is generally considered to offer the most consistent description of ship motions and it follows on from earlier work, most notably Korvin-Kroukovsky and Jacobs (1957) and Gerritsma and Beukelman (1967). To predict

the motions of a vessel,

it is divided into a series of 2-dimensional transverse strips (typically 21 strips are used). Regular waves are assumed and the heave and pitch motions are determined by solving the coupled two degree of freedom (heave and pitch) equations of the forced spring, mass and damper system representing the ship, where the exciting force is produced by the incident regular waves.

The equations of motion for heave and pitch are given in (1) and (2) below, together with their forcing functions, equations (3) and (4). A further four equations of motion exist for sway, roll, yaw and surge. However, heave and pitch are usually considered uncoupled from sway, roll, yaw and surge, and so can be considered independently. Heave and pitch displacements are represented by the terms ri3 and i respectively. The mass coefficient for the vessel includes the added mass (A33), or the equivalent virtual mass of water which is accelerated with the vessel due to its vertical motion in the water. Viscous forces on the hull are neglected, with damping (B33) considered solely due to the production of surface waves. The spring term (C33) represents the buoyancy of the submerged portion of the hull.

Added moment of inertia (A55) and equivalent pitch damping (B55) and stiffness terms (C55) are present in the moment equation (2). The remaining terms on the left hand side of (1) and (2), with '35' and '53' subscripts represent the cross-coupling between heave and

pitch.

The full coefficients of equations (I) and (2) are derived from the sectional added mass, damping and stifThess coefficients, a33, b33 and c33. They will not be restated here, and readers are urged to consult the original Salvesen et al paper where they are given in full. (M ± A33 ) + B333 + C33i13 + A35i3 + B353 + C3513 = F3e°'

(15 + A55 )ij3 + B55i5 + C5515 + A53i5 + B53i5 + C53r = F5e'°'

The force and moment equations are similarly given as:

= aJ

e"{c33 ci0(0,a33 _ih33)]d

10)

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U

=

J e'e

c33 - a33 -ih33)

-

ib33)

10) A o) a. 11)33 ) ' , 10)

The vessel is modelled as a linear system, but with frequency dependant coefficients. In the above equations, the 'A' superscript represents the added mass and damping of theaftmost station, which is generally zero except for vessels with transom sterns. The sectional stiffness coefficient, c33, is determined directly from the hydrostatics of the submerged portion of the hull. Calculation of sectional added mass, a33, and damping, b33, coefficients

is more complex: they are derived from the potential flow solution for a cylinder oscillating in a free surface (Ursell, 1949). The AMECRC program determines the conformal mapping coefficients required to map the shape of the hull onto a cylinder, for which the potential flow solution is known. This enables a unique set of hydrodynamic coefficients to be determined for that hull for a given speed and wave frequency. In their simplest form, the hull may be described by low order conformal mapping coefficients known as Lewis sections or Lewis coefficients. The Lewis coefficients for each section are simple analytical functions of the sectional underwater area, beam and draft. As with higher order conformal coefficients, they are independent of the shape of the hull above the waterline.

Non-Linear Effects

Non-linear motion effects may manifest themselves as a dependence of the non-dimensional heave and pitch transfer functions on the wave height and wave slope respectively, so that the dimensional response is not directly proportional to wave height, or by a heave or pitch response at frequencies other than the wave encounter frequency, If the response is linear then the non-dimensionalised heave and pitch transfer functions are independent of wave height for a given frequency and speed regardless of the wave height.

Implicit in conventional linear strip theory is the assumption Lewis mapping operations and hydrostatic calculations of stiffness terms are performed on the calm water submerged

portion of the hull. That is, a flat waterline is assumed over the length of the vessel

regardless of its forward speed or incoming waves. Equations (1) and (2) are solved using the sectional added mass, damping and stiffness coefficients determined from the calm water Lewis coefficients, and therefore only the hull shape below the calm waterline can affect the motions of the vessel. Hull forms with overhangs above the waterline or which are not wall-sided at the waterline, in particular, are not fully modelled.

The true instantaneous waterline of a vessel operating in waves will be strongly dependant on both its forward speed, the incoming wave and the amplitude and phasing of the resultant heave and pitch motions. In the regular waves which we are considering, the true waterline will oscillate about a mean position which will in general not be coincident with the calm water line, nor will it be flat over the length of the vessel. At various phases of the motion, the vessels overhangs may become immersed or the local waterline may rise well above the calm waterline for that section, submerging much of the hull in the process.

d

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In this pseudo non-linear implementation of strip theory, the assumption of a calm waterline is abandoned. The Lewis section coefficients, and hence the hydrodynamic coefficients

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representing added mass, damping and stiffness may be calculated at any waterline for each section along the hull. The location of the appropriate sectional waterline is determined by the operating conditions of the vessel and the degree of non-linearity the user wishes to consider. As the instantaneous waterline is considered, the hull above the calm waterline can, as it does in reality, significantly affect the motions of the vessel.

At the first stage, only the Kelvin wave pattern may be considered. Kelvin waves are produced by changes in the pressure distribution around the hull due to its forward motion, and are present regardless of whether it is operating in calm water or a wave field. The Kelvin wave pattern may be modelled in this pseudo non-linear theory by applying a fixed vertical waterline offset to each section along the hull ofthe vessel. The time dependant Kelvin wave pattern of a vessel which is not operating in calm water is very difficult to determine experimentally or theoretically, and the effects of a vessel's changing Kelvin wave pattern are not treated in this theory. Therefore the Kelvin wave offset applied to the calm waterline is a constant value applied over all phases of the vessel's motion cycle, and it is equivalent to the calm water Kelvin wave offset.

The incoming wave may also be considered. For a vessel is operating in regular waves, the elevation of the incident wave at any time and position isgiven by:

C=caS(0et4'4,w)

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To model the incoming wave, its sinusoidal profile is superimposed onto the hull, and the new waterline is determined. Since the motions are computed in the frequency domain, the time dependant part of this equation is ignored, and the new waterline with respect to the waterline of the vessel in calm water is written as:

z(x) =

a

sin(kx -)

(6)

The changed waterline due to the incident wave will generate a trimming moment and change in underwater volume of the vessel. The trim and sinkage required to balance the vessel on this new waterline is calculated by the program from hydrostatics. The sectional waterline is adjusted by this amount so that there are no vertical hydrostatic forces or moments acting on the hull. This is not a true physical representation of the vessel in a wave field, since dynamic effects (in particular the inertia of the vessel) may be important, however it is assumed to be an adequate first approximation.

If both the Kelvin wave and the incoming wave are modelled,

then the waterline is

determined from a linear superposition of the changes due to each of these two wave effects. The Lewis coefficients are recomputed for the new underwater shape, and the added mass, damping and stiffness coefficients are determined for each 2-dimensional transverse strip using the same method employed by linear theory. The hydrodynamic coefficients following this step are representative only of the vessel

waterline for a

particular phase angle, of incident wave, where a phase angle of zero corresponds a zero waterline offset at the centre of buoyancy. To account for this, the phase angle of the incident wave is stepped with respect to the vessel, the balancing repeated and new hydrodynamic coefficients computed for this phase angle. Stepping of the incident waveis repeated a number of times and mean hydrodynamic coefficients are computed from the

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array of sectional hydrodynamic coefficients obtained for each wave phase, Heave and pitch motions are calculated from equations (I) - (4) only after the mean hydrodynamic coefficients have been determined, since solving for the motions at each wave position has

no physical meaning if the equations are solved in the frequency domain.

The motions of the vessel may also be included in calculation of the dynamic local waterline. In this case, the amplitudes and phases of the resultant motions derived from the mean hydrodynamic coefficients arecomputed and the changed local waterline determined. Using the linear superposition principle, the new local waterline is determined as in equation (7) below, where z%(x) represents the offset due to the incident wave computed in (6), but with appropriate trim and sinkage to rebalancethe vessel applied.

z(x) = z(x)' + 113 sin + rl5x 1c (7)

Lewis section and hydrodynamic coefficients are computed for this new waterline. As before, the phase angle of the wave and motions is stepped to obtain mean coefficients which are then substituted into the motion equations for the new heave and pitch motions to be determined. In an iterative procedure, the Lewis coefficients and the hydrodynamic coefficients are recomputed and the motion equations resolved until convergence is achieved, however in this case the program does not re-balance the vessel before the hydrodynamic coefficients are recalculated.

Waterline changes due to wave radiation and wave diffraction are not directly treated by this code. The wave radiation forces, arising from the production of free surface waves due to the vertical motion of the vessel, are determined from the calculated sectional damping terms. The damping terms reflect the change in waterline due to the incident wave and resultant motions which are themselves dependant on wave radiation and diffraction forces, however, an offset is not directly applied to the local waterline due to the diffracted or radiated waves produced by the vessel.

This method represents a frequency domain equivalent to the time domain pseudo non linear theory given by Kuhn and Schlageter (1992). The use of averaged hydrodynamic coefficients and forces in the standard linear motions calculations mean that this is a pseudo non-linear prediction method. Computation times are increased by approximately a factor of 20 over a pure linear code. However, the very low CPU demands of strip theory in general mean that a complete transfer function curve for a given vessel is still obtainable in only a few minutes of CPU time. This is very much faster than would be possible with a fully 3-dimensional code or even a time domain strip theory simulation incorporating similar non-linear effects.

Towing Tank Experiments

A series of towing tank experiments on two types of ship models were conducted at the Australian Maritime College towing tank, Launceston, Tasmania in January and October 1994. Two AMIECRC Systematic Series Displacement hulls and anAMIECRC series yacht hull were tested, with the yacht hull tested at two different drafts. Results of the second systematic series hull have not yet been analyses and are not included in this paper. The AMIECRC high speed systematic series of displacement hulls is derived from the MARIN

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systematic series (Blok and Beukelman, 1984), developed in cooperation with the Royal Netherlands Navy, the Royal Australian Navy, and the Canadian Navy. Details of the models tested are given in table 1 and 2 below. Full lines plans for both hulls are given in figures 8 and 9. Body plans of the parent model are given in Blok and Beukelman, 1984. The AJvIECRC series yacht hulls follows asimilar plan to the Delft yacht hulls, described in Gerritsma, Keuning and Versluis, 1993.

Model Dimensions: Syst Series

Length, L 1600mm Beam, B 200mm Draft,T 80mm Displacement 10.123kg Gyradius 400mm L/B 8 B/T 2.5 CB 0.4 LCB 86.6mm aft of midships

Table 1 Dimensions of systematic series model

Table 2. Dimensions of yacht model

The purpose of these tank tests was to investigate the extent of the non-linear response of a vessel to waves and to provide data for the validation of the non-linear predictions of the AMIECRC strip theory code.

Experimental transfer functions for heave and pitch are plotted against non-dimensional wave amplitude in figures 1-5c & d and against wave frequency in figures 1-5a & b. In both sets of plots, non-linearities are indicated by a vertical spread of experimental data points.

Regular wave response for a yacht with overhang

The yacht hull was tested at two drafts in order for the effects of bow and stern overhangs on the motions to be determined. The overall length of this vessel was 2.25 metres. In the case of the deep draft, thewaterline length was 2.22 metres, with the waterline extending fully over the length of the vessel from the transom aft to 0.03 metres aft of the forward perpendicular and with the transom itself immersed to a depth of 7mm. For the shallow draft there were significant bow and stern overhangs, with the rear waterline ending 0.3 metres forward of the transom and the forward waterline approximately 0. 15 metres aft of the forward perpendicular, giving a waterline length of 1.8 metres. Additionally the hull had significant flare when ballasted to the shallow draft. All tests were at a forward speed of

Model Dimensions: Yacht

Length, L I 800mm12220mm Beam, B 495.8rnm/569.6mm Draft, T 64.6mm/I 19.2mm Displacement 22.54kg/63.66kg Gyradius 48 7mm L/B 3.63 / 3.90 B/T 7.67 / 4.78

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1.52ms-', corresponding to 6.6 knots full scale (Fri = 0.362 for shallow draft, Fn = 0.325 for deep draft). The waterline position for the yacht is shown in side view in figure 8b

In the case of the shallow draft, non-linearities appear to be significant only for heave at the 0.815Hz and 0.9 1Hz wave frequencies tested, which correspond to the peak in the average vertical acceleration curves (not shown). At other frequencies tested the heave response is

linear (within experimental uncertainty) and even at the peak vertical

acceleration frequencies the pitch response was linear, even as wave heights were increased to quite large values . This was an unexpected result: it was envisaged that the emergence and

submersion of the bow and stern overhangs would give rise to significant non-linear behaviour. In particular, immersion of the stern would cause a rearward shift in the LCB and LCF positions and also generate a large pitching moment. Video of the tests indicated that large wave heights did induce motions which caused large changes in waterline ending

positions throughout a motion cycle, however these tests. indicated that

it did not

significantly affect the non-dimensional response of the vessel:

The response of the yacht when ballasted to the deeper draft indicates a greater degree of non-linearity in the motions than the shallow drafted yacht, particularly for pitch. Overall however, the response is still generally linear, with the maximum change in heave and pitch response under 10 percent over the range of wave heights tested. The transom of the yacht remained immersed during all motion tests, regardless of wave height or wave frequency used. Conversely, during some phases of the motion, the bow of the vessel was clear of the water for close to 10 percent of the vessel's waterline length when tested at the frequency of peak vertical acceleration, 0.75Hz. As with the shallow draft tests, the degree of non-linearity was an unexpected result. It was thought that the flare at the waterline would cause rapid changes in hydrodynamic coefficients, and therefore motions, throughout a cycle of oscillation. It is possible that this is due to a number of separate effects cancelling out and producing a linear response.

The greater degree of non-linearity exhibited by the yacht in the deep draft configuration was due in part to the higher wave heights tested, a maximum of 85mm compared with 65mm for the shallow draft. During testing at the shallow draft, freeboard constraints were encountered and wave heights were limited to avoid the possibility of deck wetness. For the deep draft tests, a spray guard was added to the bow of the model, enabling larger wave heights to be used.

Regular wave response for a high speed systematic series

Seakeeping tests were carried out on the systematic series displacement hull at model speeds of 1. l3ms-'/Fn0.285, 2.26ms'/Fn0.57 and 3.39ms4/Fn0.855, corresponding to 15.5 knots, 31 knots and 46.6 knots full scale respectively. Wave heights tested were in the range 16mm - 64mm model scale, corresponding waves up to 3.2 metres full scale. Greater wave heights could not be tested due to bow submergence on the model. Despite the smaller range of wave heights tested (compared to the yacht), the systematic series hull exhibited a greater degree of non-linearity in its response. It also showed a larger degree of spread in the data, particularly at the slowest speed tested (figure 3).

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Significant non-linearities were present in both heave and pitch response. Interestingly, the pitch and heave dependence on wave height did not necessarily follow the same trend, in some cases resulting in a reduction in non-dimensional pitch while heave increased or vice versa. At the O.81-Iz resonant peak at the mid speed tested, 2.26ms', there is a modest increase in heave response with increasing wave height, but a significant reduction in pitch response. However at this speed, there is no pitch resonant peak and the pitch response simply falls away from its maximum value of unity at low frequency.

No resonant peak exists for pitch at any of the speeds tested. In general the heave response shows a modest increase with increasing wave height at and below the frequency of the heave resonant motion peak. Above the resonant peak, the heave response is linear or shows a modest decrease with increasing wave height.

Comparison Between Theory and Experiment

Comparisons are given between experiment, linear strip theory and non-linear strip theory. Due to time constraints, Kelvin wave effects have not yet been included in strip theory

predictions. Yacht

The effects of modelling non-linearities in the seakeeping code are most significant at the frequencies around resonance, with the greatest changes in the predicted responses evident at these frequencies. The difference between linear and non-linear predictions is most significant for pitch, in particular for the shallow draft condition. In this case the program predicts cyclic immersion of the stern overhang with a corresponding pitching moment being generated. The programs predictions for the deep draft reflect the much smaller change in immersion depth of the transom throughout a motion cycle: in this case the predicted heave and pitch responses show approximately the same magnitude of change, with this change due largely to the flare of the hull around the calm waterline. The change in hydrodynamic coefficients as a ftinction of wave height for the yacht is shown in figure 7,

giving the sectional breadth coefficient (proportional to the sectional

stiffness, c33,

coefficient). At the smallest wave height shown, the rear two sections have a zero breadth coefficient, indicating they so not become immersed at this wave height. Recall that the calm waterline ending for the yacht at the shallow draft is at the third last section, so this Situation is expected. At larger wave heights, the stern of the vessel becomes immersed, resulting in non-zero hydrodynamic coefficients.

Figures la,b and 2cz,b show predicted responses for the linear and non-linear code and experimental results plotted against wave frequency. The predicted responses generated from the non-linear program are closer to the experimental data than the linear predictions for both drafts for heave and pitch. It is encouraging to note that the non-linear predictions result in a reduction in response compared with linear theory for the over-predictedshallow draft case and an increase in predicted response for the under-predicted deep draft case. That is, the program does not simply predict either an increase or decrease in response regardless of hull configuration.

The response of the yacht in deep draft configuration is relatively linear. This is reflected by the linear predictions, which remain roughly constant when plotted against

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non-dimensional amplitude (figure 6b). The non-linear program appears to be an inaccurate predictor of changes in response for the shallow draft case (figure 6a). Trends in response are not well predicted with theoretical and experimental responses diverging at a number of the frequencies studied. In some cases, the program predicts a significant non-linear response when the true response is linear. The Kelvin wave pattern was not considered for the yacht. It would be desirable to include this in the predictions as this may improve the predictive capabilities of the non-linear code. It is anticipated that the Kelvin wave will affect the motions of the yacht more than the systematic series displacement hull due to the much lower length to beam ratios applicable to the yacht.

High Speed Systematic Series

As with the predictions for the yacht hull, the effect of modelling non-linearities is most pronounced for pitch around resonance. Trim and sinkage of the vessel, determined from calm water tank tests, were included in linear and non-linear predictions for the systematic series, however the Kelvin wave pattern was not included. At the two slower speeds tested, 1.l3ms-' and 2.26ms-', use of the non-linear code yields improvements in the predicted responses, compared to the experimental data. At 3.39ms'(Fn = 0.855), while the heave predictions are improved, the pitch response of the vessel is significantly over-predicted, with a transfer ftmnction peaking at 1.5. This is compared with an experimental responseof

1. 1. The frequency of the peak at this speed is also over-predicted, possibly indicating that the stiffness to added mass ratio is too high.

Strip theory is generally considered applicable up to Froude numbers of approximately 0.5, after which 3-dimensional, viscous or other effects may become important and lead to inaccuracies in predictions. However accurate results have been obtained up to Froude numbers of 1.14 (Blok and Beukelman, 1984; Faltinsen, 1993) for the parent MARIN systematic series displacement hull form. These results have been reproduced by the AMECRC strip theory program, with the same close agreement between experiment and theory. It is interesting to note there is discrepancy between linear theory predicted and measured results for the systematic series model used in the AMECRC tests at a Froude number of 0.855, considering this systematic series hull was a derivative of the same parent hull. By comparison to the parent hull (Blok and Beukelman, 1984), this systematic series hull has markedly inferior seakeeping properties, particularly for heave. It follows that even within one family of hull forms, there is a considerable variation in the limits of applicability of strip theory as a function of forward speed.

The non-linear predictions for the three speeds tested show a general agreement with experimental data when plotted against non-dimensional wave amplitude (figures 6c,d,e). In a number of these plots, the predicted pitch response shows a sudden step to a higher or lower value. This is an ai-tefact of the finite number of sections used to describe the vessel. As the wave height is increased by a small amount, a bow or stern station either becomes submerged or emerges with a corresponding large change in pitching moment and pitch response. This problem may be minimised by using a sufficiently fine section spacing. A sudden change in response should be treated as a more gradual response over a wider wave height range.

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Conclusion

Towing tank experiments were performed on two hull forms for which non-linear motions effects were expected to be significant. Tests in multiple wave heights were carried Out Ofl

a yacht hull with large

overhangs and flare, and on a systematic series high speed displacement hull. It was found that the motions of the yacht, despite its extreme hull shape, were relatively linear except close to the resonance condition. The overall response, particularly pitch, for the displacement vessel was relatively small, indicating its good seakeeping characteristics (although poorer than the parent hull of the series). Its motion indicated a significant non-linear response, particularly around the heave resonant peak. The degree of non-linearity was such that, when the sea-kindliness of such a vessel is under

consideration, some stipulation must be made as to the sea state th vessel would be expected to operate in.

A pseudo-nonlinear strip theory has been developed, which goes some way towards modelling extreme hull shapes such as yachts and or other vessels operating in heavy seas. In contrast to conventional linear theory, this allows hydrodynamic coefficients which reflect the true instantaneous waterline position of the vessel to be calculated. This is a more physically realistic model of the system since the hydrodynamic coefficients used are representative of the true submerged portion of the hull, rather than just the calm water submerged portion. In theory, the shape of the hull above the calm waterline can, unlike conventional theory, have a significant effect on the seakeeping characteristics of the vessel. Similarly, the presence of bow or stern overhangs can be included and their effects on the motions modelled. This frequency domain approach to non-linearities is significantly faster in computer time than either a time domain strip theory or a three dimensional simulation. For the hull tested, the nonlinear strip theory can often correctly predict trends in the motions: in particular it is capable of determining how the heave and pitch response will change with increasing wave amplitude.

The pseudo non-linear program has been shown to improve the seakeeç'ing predictions of the yacht and the systematic series hull, with predicted responses moving significantly

closer to experimental data. The program was also used to predict the trends in response of the two models with increasing wave height. The program was not successful in predicting the response trends for the yacht in shallow draft configuration,

where a large stern

overhang was present, however general agreement was found with the yacht in deep draft configuration and with the systematic series hull.

Future Work

This pseudo non-linear strip theory code is still currently under development. Planned future developments include:

- modification of non-linear code to prevent jumps in predicted response due to small

changes in wave height

- inclusion of Kelvin wave effects in predictions for the yacht

- further validation using towing tank data from the second systematic series displacement hull

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References

Blok, JJ; Beukelman, W, "The High-Speed Displacement Ship Systematic Series Hull Forms Seakeeping Characteristics", Trans. Society of Naval Architects and Marine Engineers, Vol 92,

1984, pp 125-I50

Faltinsen, OM, "On Seakeeping of Conventional and High Speed Vessels", Journal of Ship Research, Vol 37, No 2, June 1993, pp87-101

Gemtsma, J Beukelman, W, "Analysis of a Modified Strip Theoty for the Calculation of Ship Motions and Wave Bending Moments", International Shipbuilding Progress, Vol 14, 1967,

pp3 19-337

Gerritsma, J., Keuning, LA., Verstuis, A., "Sailing Yacht Performance in Calm Water and Waves", 11th Chesapeake Sailing Yacht Symposium, Society of Naval Architects and Marine Engineers, 1992, pp233-245

Korvin-Kroukovsky, BV; Jacobs, WR, "Pitching and Heaving Motions of a Ship in Regular Waves", Trans. Society of Naval Architects and Marine Engineers, Vol 65, 1957, pp590-632

Kuhn, JC; Schiageter, EC, "The Effects of Flare and Overhangs on the Motions of a Yacht in Head Seas", 11th Chesapeake Sailing Yacht Symposium, Society of Naval Architects and Marine

Engineers, 1992, pp261-276

Salvesen, N; Tuck, EO; Faltinsen, 0, "Ship Motions and Sea Loads", Trans. Society of Naval

Architects and Marine Engineers, Vol 78, 1970, pp250-287

Ursell, F, "On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid", Quarterly

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1.4 1.3 1.2 1 C 0 0.9

Fig Ia: Yacht Shallow Draft HEAVE Transfer Function for Various Wave Heights

Experiment and Theory

0 linear theory 0 non-linear 20mm -I non-linear 40mm o non-linear 60mm A non-linear 80mm <25mm 25mm - 35mm 0 35mm - 45mm X 45mm - 55mm :;l( >55mm I I I 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Wave Frequency (Hz) > 0.6

I

0.5

0.4

0.3

0.2

-0.1 I 0 I I I

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Fig Ib: Yacht Shaflow Draft PITCH Transfer Function for Various Wave Heights

I linear theory 0 non-linear 20mm non-linear 40mm O non-linear 60mm A- non-linear 80mm . <25mm 25mm - 35mm 0 35mm - 45mm X 45mm - 55mm *( >55mm 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Wave Frequency (Hz) C 0 C U- I-C I-1.4 1.3 1.2 1.1 1 0.9 0.7 0.6 0.5 0.4 0.3

0.2

-0.1 0

-0

-x I

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1.4 1.3 1.2 1.1 1 0.9

Fig Ic: Yacht Shallow Draft HEAVE LINEARITY Experimental Transfer Function for Various Frequencies

0 0 0 0 0 0 0.505Hz D 0.605Hz 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Wave Number * Wave Amplitude A A 0.7Hz 0 0.815Hz A _0.91Hz 0.4 A A A 0.3 0.2 0.1 I 0 I I

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Fig Id: Yacht Shallow Draft PITCH LINEARITY Experimental Transfer Function for Various Frequencies

C 0 C.) C U-C 4-1.4 1.3 1.2 1.1 1 0.9 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

.

0 A A 0

.

00

A 0 0 A 0 A 0.505Hz 0 0.605Hz 0.7Hz 0 0.815Hz A _0.91Hz 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

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C

(0

1

I-1

Fig 2a: Yacht Deep Draft HEAVE

Transfer Function for Various Wave Heights Experiment and Theory

0 A o x linear theory -- non-linear 20mm

.

- non-linear 40mm o non-linear 60mm A non-linear 80mm A 25mm - 35mm 35mm-45mm o 45mm - 55mm X 55mm - 65mm 65mm - 75mm 75mm - 85mm >85mm F I I I F I 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Wave Frequency (Hz) (0 > 0.6 (0 C,

I

0.5 0.4 0.3 0.2 0.1 0 I I 0.9 C 0

(17)

Fig 2b: Yacht Deep Draft PITCH

S linear theory o non-linear 20mm non-linear 40mm O non-linear 60mm A non-Linear 80mm 25mm - 35mm 35mm - 45mm 0 45mm - 55mm X 55mm - 65mm 65mm - 75mm Z 75mm - 85mm - >85mm 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Wave Frequency (Hz) 1.4 1.3 0

1.2

(18)

-0.5 0.4 0.3 0.2 0.1 0 I 0 0.01 002 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Wave Number * Wave Amplitude

Fig 2c: Yacht Deep Draft HEAVE LINEARITY

Experimental Transfer Function for Various Frequencies

C 0 C.' C U- I-1.4 1.3 1.2 1.1 0.9 0.8

-0 0 U 0

.

0 0 0 0 0 0 0 0 0

I

0 U 0.6Hz 0 0.655Hz 0.7Hz 0 0.75Hz A _0.81Hz 0.9Hz U, C NO.6 A A A A A

(19)

Fig 2d: Yacht Deep Draft PITCH LINEARITY

Experimental Transfer Function for Various Frequencies

0 0.01 0.02 003 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

1.4 1.3 1.2 1.1 1 0.9 C 0

tO.8

C 0.7 C I-0.6 0.5 0.4 0.3 0.2 0.1 0 U 0 0

.

0 U 0 A A 0

.

0 0

I

0

.

A

.

0 0 0 A 1 I A 0 A

I

A A A 0.6Hz 0 0.655Hz 0.7Hz 0 0.75Hz A _0.81Hz A 0.9Hz 0

(20)

linear theory 20mm nonlinear 40mm nonlinear 60mm nonlinear 55-65mm 48-55mm 42.5 - 48mm 35 - 42.5mm 25-35mm <25mm I

Fig 3a: Systematic Series V1.13m/s HEAVE

S A 0 X 0 SW 0 X AX 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 Wave Frequency (Hz) 1.1 1.05 1 0.95 0.9 0.85 0.8

0.75

0.7

-.2 0.65 ° 0 = A . 0.5 > _o 0.45 X

0.4

-:*

0.35

-0.3 0.25 0.2 0.15 0.1 0.05 0 I I

(21)

1.1 1 05 1 0.95 0.9 0.85

0.8 -

linear theory

0.75 -

0 20mm nonlinear

0.7 -

40mm nonlinear 0.65 0 60mm nonlinear 0.6 A 55-65mm

U-055 -

48-55mm C 42.5 - 48mm I-

0.5

-.2 35 - 42.5mm 0.45 X 25-35mm 0.4 * _<25mm

0.35

-0.3 0.25 0.2

0.15

-0.1 0.05 0 0

Fig 3b: Systematic Series V=1.13m/s PITCH

A

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

(22)

J

0.5

0.4

0

Fig 3c: Systematic Series V1.13m/s HEAVE LINEARITY Experi mental Transfer Function for Various Frequencies

1.2Hz

0 1.407Hz

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Wave Number * Wave Amplitude 0.3 0.2 0.1 A A A A

.

0 A

.

A 0 ₀ 0 I I I I 1.1 1 0.9 0.8 0.7 0.6

U-.

0

.

0

.

0 0

.

0.65Hz 0 0.7Hz 0.8Hz 0 0.9Hz 1Hz A _1.11Hz 0 A 0

₀₀

0 A 0 A

(23)

Fig 3d: Systematic Series V=1.13m/s PITCH LINEARITY

Experimental Transfer Function for Various Frequcencies

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 012

C 0 C-, C LI C, C C5 C-, 0 1.1 1 0.9 0.8 0.7 0.5

0.4

0.3

-0.2

0.1

-0

-.

0 0 U o0 A A 0 A U A A 0 0 0 0 S U 0

.

A 0 0

.

A A S A 0 S 0.65Hz 0 0.7Hz 0.8Hz 0 0.9Hz A _1Hz A 1.11Hz 1.2Hz 0 1.407Hz 0

(24)

0.9 C 0 Z 0.8 C LJ C I-G) 0.6 C, = 1

0.5

0.4

-0.3

0.2

-0.1

Fig 4a: Systematic Series V2.26m/s HEAVE Transfer Function for Various Wave Heights

0 A 0 + 0 S K 0 0.3 0.4 0.5 0.6 0.7 0.8 Wave Frequency (Hz) i near o 20mm nonlinear - 40mm nonlinear O 60mm nonlinear A _>65mm 55-65mm 48-55mm 0 42.5 - 48mm X _35-42.5mm ( 25-35mm + _<25mm 0.9 1 1.1 1.2

(25)

1.4 1.3 1.2 1.1 1 0.9 C 0 0.8 C 0.7 I-o 0.6 0.5

Fig 4b: 93-13 Systematic Series V=2.26m/s PITCH

0.4

0.3

-0.2 0.1 0 0.3 0.4 0.5 0.6 0.7 0.8 Wave Frequency (Hz) linear 20mm nonlinear 40mm nonlinear O _{60mm nonlinear} A _>65mm 55-65mm 48-55mm o 42.5 - 48mm >< 35 - 42.5mm 25-35mm ± <25mm 0.9 1 1.2

(26)

Fig 4c: Systematic Series V=2.26m/s HEAVELINEARITY

o I I I F

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

1.4 1.3

-0 o 0 0 0 4) 0 1.2

.

1.1 -

0 0

.

0 0 1 0 0.6Hz 0.9 _{0 0.7Hz} A 0 0.75Hz 0.8 _A A A 0 0.8Hz U-I- A A _0.9Hz U) C '5 I- A A 1.0Hz I-a, a,

x

0.6 0.5 0.4 A 0.3 _A A A _A 0.2 0.1

(27)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

A A A 0.5 A A 0.4 0.3 A A A A A 0.2 0.1 0 1.1 1 0.9 C 0 0.8 C Li 0.7 o 0.6 a.

'

o 0 0 0 0 0 00 0

00

0.6Hz 0 0.7Hz 0.75Hz 0 0.8Hz A _0.9Hz 0.917Hz

(28)

C 0 a

Cl

I-'I-. U)

-C 1 I-w 0.8 > C.' 0.7

Fig 5a: Systematic Series V=3.39m/s HEAVE Transfer Function for Various Wave Heights

x S x A in ear 0 20mm nonlinear - 40mm nonlinear O 60mm nonlinear A 55-65mm 48-55mm 42.5 - 48mm 0 35-42.5mm X _25-35mm l( _<25mm > 1.2 1.1 0.9 1 0.7 0.8 Wave Frequency (Hz) 0.4 0.3 0.5 0.6 0.6 0.5 0.4 0.3 0.2 0.1 0

(29)

1.8 1.7 1.6 1.5 1.4 1.3 1.2 0.6 0.5 0.4 0.3 0.2 0.1 0

Fig 5b: Systematic Series V3.39m/s PITCH Transfer Function for Various Wave Heights

I in ear o 20mm nonlinear 40mm nonlinear 60mm nonlinear A _55-65mm 48-55mm 42.5 - 48mm o _35-42.5mm X _25-35mm <25mm 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Wave Frequency (Hz)

(30)

Fig 5c: Systematic Series V=3.39m/s HEAVE LINEARITY

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Wave Number Wave Amplitude

0 0 U- I-U) C CU C) > CU C) 0.8 0.7 0.6

0.5

0.4

-0.3

O.2-0.1 0 0 0 0 A 0 A 0 A 0.7Hz 0 0.8Hz A _0.9Hz 1.8 0 1.7 0 1.6 U 1.5 U U 1.4

.

1.3 1.2 0.624Hz 1.1 ₀ _0.65Hz

(31)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

0 0 C U- I- 3-1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 0.9 .8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

.

0 0 0

I

0 A

.

0 A 0 A 0.624Hz 0 0.65Hz 0.7Hz 0 0.8Hz A _0.9Hz I 0 I

(32)

I-0 0.9 0.8 I-I-. 0.6 0.5 0.4 0.3 0.2 1 0

Fig 6a: Yacht Shallow Draft

Motion Linearity at 0.815Hz Wave Frequency Experiment and Theory

0 F/theory - H/theory PIexp o _H7exp 0 0

0

---0 0.01 0.02 003 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

0.1

0

I I I I F

(33)

1 0.9 0.8 C.) C LL I-C) U) C 1 I.- 0.6 0.5 0.4 0.3 0.2 0.1 0

Fig Gb: Yacht Deep Draft

-

0

----c--

0 0

.

P'/theory - H'/theory P/exp 0 H'/exp 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

(34)

1.1 1.05 0.95 0.9

0.85

-0.8 0.75

!05

0.45 ₀

0.4

-0

0.35

0.3

-0.25

0.2

0.15

0.1

0.05

-0

Fig 6c: Systematic Series W1.13m/s Motion Linearity at 1.0Hz Wave Frequency

P/theory - H/theory P/exp 0 _H/exp I I I 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Wave Number * Wave Amplitude 0.7 0.65 C 0 0.6 C

.

0 0 0

(35)

1.4

1.3

1.2

-1.1 0.9 0 0 0

Fig 6d: Systematic Series V2.26m/s Motion Linearity at 0.9Hz Wave Frequency

t.

0.5

-0.4 0.3

0.2

-0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

P/theory - H'/theory P/exp o _H'/exp 0 OO 0 0.8 0 C, 2

---I

:.:

-0.09 0.1 0.11 0.12

(36)

1.6 1.5 1.4 1.3 1.2 1.1 1 C o 0.9 C.) C LL. "-en C 0.7 0.6

0.5

0.4

0.3

-0.2

0.1

-0

Fig 6e: Systematic

_{Series V3.39m/s}

P/theory - H/theory P/0.aHzlexp 0 _H/0.8Hz/exp 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 _0.08 _0.09 _0.1 0.11 0.12

Wave Number * Wave_Amplitude 0

0

.

(37)

0.6

0.5

0.4

0.2

0.1

Fig 7: Yacht SECTIONAL BREADTH COEFFICIENT

Shallow and Deep Draft 0.7Hz Wave Frequency Linear and Non-Linear Theory

linear/shallow O 20mm/shallow 40mm/shallow 60mm/shallow linear/deep 20mm/deep 40mm/deep O _60mm/deep 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Station

(38)

U 0 .0 1

-.2. e.o

04

WI. WI.

-BODY SECTXONS

Figure 8a Lines plan of yacht hull (normalised dimensions)

Figure 9 Lines plan of systematic series hull (normalised dimensions)

5 B 0.5 1

BODY SECTIONS

0 0.1 I

0 I. n,I.. In

I .0 2.1

N