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Small craft motion in reflected long-crested seas Kennedy, Andrew B. 1993

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SMALL CRAFT MOTION IN REFLECTED,LONG-CRESTED SEASbyAndrew Brian KennedyB.Sc.E., Queen's University at Kingston, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust, 1993© Andrew Kennedy, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at The University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may be granted by the Headof my Department or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Civil EngineeringThe University of British Columbia2324 Main MallVancouver, British Columbia, CanadaV6T 1Z4Date:  465z.,1 /7) /Q'Z AbstractThis thesis attempts to provide an analysis of several aspects of moored small craftmotion, concentrating on vessel response to wave action and wave reflection, andbriefly examining the effects of irregular waves.Experiments were performed in the National Research Council multidirectionalwave basin in Ottawa to determine the response of a moored small craft model in long-crested waves. The SELSPOT imaging system was used to measure vessel motions.With several exceptions, model response was found to be highly linear. The mostnotable of these exceptions were the resonance peak in surge, and all yaw motions. Themodes of sway and roll also exhibited nonlinearities but, surprisingly, these were fairlyA numerical program was undertaken to attempt to correlate and extend the resultsof the experiments. Linear diffraction theory as represented by the programsFACGEN3 and WELSAS3 was used to calculate vessel movements for various waveperiods, incident angles and mooring conditions. Various perturbations were made toboat properties to find the sensitivity of response. Response in head seas generallyagreed with experimental results with the notable exception of the surge resonancepeak, which was predicted to be much too high. However, the addition of an extra 5%damping lowered this peak to experimental levels without substantially affecting anyother modes of motion. Response in beam seas was not as well predicted. For bothsway and roll, long wavelength response was predicted to be much too low. This wasthought to be due to poor representation of the boat hull by Green's functions andneglect of viscous effects. Calculations performed with a widened hull showed betteragreement with experiment.Expressions were developed to relate vessel motion in a partially reflected andunreflected wave field and were found to agree well with experiment. The effect ofirregular waves on wave criteria developed for monochromatic waves was alsoconsidered. For both cases, recommendations were made toward extending existingwave criteria in these directions.iiiTable of ContentsPageAbstract^ iiTable of Contents^ ivList of Tables viiList of Figures^ viiiNomenclature xiAcknowledgements^ xiv1.^Introduction 11.1 General^ 11.2 Scope of Project^ 22^Literature Review 32.1 Acceptable Wave Climate in Small Craft Harbours^ 32.2 Determination of Vessel Response for a Given Wave Climate^62.2.1 Physical Models^ 62.2.2 Numerical Models 73. Theoretical Development^ 93.1 Dimensional Analysis 93.2 Vessel Response 113.2.1 Linear Diffraction Theory for Floating Bodies^ 113.2.2 Effects of Random Waves^ 173.2.3 Reflection Effects^ 203.2.3.1 Regular Waves 203.2.3.2 Random Waves 224. Experiments^ 244.1 General 244.2 National Research Council Hydraulics Laboratory^ 244.2.1 Wave Basin^ 244.2.2 Data Acquisition System^ 25ivv4.2.3 Wave Measurement^ 254.3 Model Testing Program 264.3.1 Vessel^ 264.3.2 Mooring System^ 264.3.3 Measurement of Model Motions^ 274.3.4 Estimation of Moments of Inertia 284.3.5 Measurement of the Model Subsurface Profile^ 294.4 Experimental Setup and Procedure^ 304.4.1 Still Water Tests^ 304.4.2 Model Setup 304.4.3 Moored Tests Under Wave Action^ 305.^Results and Discussion^ 325.1 General^ 325.1.1 Experimental 325.1.2 Numerical^ 335.2 Head Seas^ 355.2.1 Surge 355.2.2 Heave 365.2.3 Pitch^ 375.3 Beam Seas 385.3.1 Sway 385.3.2 Roll^ 405.3.3 Yaw 415.4 Quartering Seas 425.4.1 Surge^ 425.4.2 Pitch 435.4.3 Sway 435.4.4 Roll^ 445.5 Reflected Seas 445.5.1 Head Seas 455.5.2 Beam Seas^ 465.6 Analysis^ 475.6.1 Vessel Motions 475.6.2 Wave Reflection^ 515.7 Extension of Results to Full Scale 52vi5.8 Acceptable Wave Climate-Limits of Motion^ 566.^Conclusions and Recommendations^ 61Appendix A: Estimation of Centre of Gravity and Moments of Inertia^65References^ 69Tables 72Figures^ 80List of TablesPage2.1 Provisionally Recommended Criteria for a "Good" Wave Climate in SmallCraft Harbours (from Northwest Hydraulic Consultants Ltd. (1980)) 722.2 Provisional Recommended Criteria for "Good" Wave Climate in SmallCraft Harbours for Resisting Vessel Surge and Sway (from Taylor (1983)) 732.3 Allowable Maximum Significant Wave Height (from Fisheries andOceans (1985)) 742.4 Classification of Harbours (from Fisheries and Oceans (1985)) 742.5 Recommended Allowable Wave Agitation Criteria for STACAC Class 2and Class 3 Fishing Vessels (from Fournier et al (1993)) 753.1 Commonly Used Wave Height Parameters 764.1 Experimental Conditions 775.1 Linearity of Response with Wave Height 785.2 Comparison of Model Mooring Stiffness in Surge with Full Scale ElasticBehaviour of Commonly Used Ropes 79viiList of FiguresPage3.1 Definition Sketch for Vessel Motions 803.2 Definition Sketch for Oblique Wave Reflection 803.3 Variation of Motion Amplitudes with Full, Normal Reflection 813.4 Variation of Motion Amplitudes with Partial Reflection, n=-1 814.1 Sketch of Multidirectional Wave Basin and Model Setup 824.2 Control Room for Multidirectional Wave Basin 834.3 Boat Model with SELSPOT Frame Attached 844.4 Sketch of Mooring Structure 854.5 SELSPOT set up for Measurement of Model Motions 864.6 Determination of Model Centre of Gravity 864.7 Measurement of the Moment of Inertia in Roll 874.8 Measurement of the Moment of Inertia in Yaw 874.9 Measurement of Subsurface Profile Using a Digital Micrometer 884.10 Mooring Setups 1-4 894.11 Model under Wave Action, Setup 4 904.12 Model under Wave Action, Setup 1 904.13 Model under Wave Action, Setup 3 914.14 Model under Wave Action, Setup 4 915.1 Facet Representation of Subsurface Hull Shape 925.2 Surge Response in Head Seas, Mooring Type 1 935.3 Surge Response in Head Seas, Various Moorings 935.4 Surge Response in Head Seas, Various Perturbations 945.5 Surge Response in Head Seas(from Northwest Hydraulic Consultants Ltd. (1980)) 945.6 Heave Response in Head Seas, Mooring Type 1 955.7 Heave Response in Head Seas, Various Moorings 955.8 Heave Response in Head Seas, Various Perturbations 965.9 Heave Response in Head Seas(from Northwest Hydraulic Consultants Ltd. (1980)) 965.10 Heave Response in Beam Seas(from Northwest Hydraulic Consultants Ltd. (1980)) 975.11 Pitch Response in Head Seas, Mooring Type 1 97viii5.12 Pitch Response in Head Seas, Various Moorings 985.13 Pitch Response in Head Seas, Various Perturbations 985.14 Pitch Response in Head Seas(from Northwest Hydraulic Consultants Ltd. (1980)) 995.15 Sway Response in Beam Seas, Mooring Type 1 995.16 Sway Response in Beam Seas, Various Moorings 1005.17 Sway Response in Beam Seas, Various Perturbations 1005.18 Sway Response in Beam Seas(From Northwest Hydraulic Consultants Ltd. (1980)) 1015.19 Roll response in Beam Seas, Mooring Type 1 1015.20 Roll response in Beam Seas, Various Moorings 1025.21 Roll response in Beam Seas, Various Perturbations 1025.22 Roll Response in Beam Seas(From Northwest Hydraulic Consultants Ltd. (1980)) 1035.23 Example of the Nonlinearity of Measured Yaw Motions in Beam Seas 1045.24 Yaw Response in Beam Seas, Various Moorings 1055.25 Yaw Response in Beam Seas, Various Perturbations 1055.26 Surge Response, 0=45°, Mooring Type 1 1065.27 Surge Response, 0=45°, Various Moorings 1065.28 Surge Response, 0=45°, Various Perturbations 1075.29 Pitch Response, 045°, Mooring Type 1 1075.30 Pitch Response, 0=45°, Various Moorings 1085.31 Pitch Response, 0=45°, Various Perturbations 1085.32 Sway Response, 0=45°, Mooring Type 1 1095.33 Sway Response, 0=45°, Various Moorings 1095.34 Sway Response, 0=45°, Various Perturbations 1105.35 Roll Response, 0=45°, Mooring Type 1 1105.36 Roll Response, 0=45°, Various Moorings 1115.37 Roll Response, 045°, Various Perturbations 1115.38 Surge Response in Fully Reflected Head Seas, Mooring Type 1,1/L=1.07 1125.39 Heave Response in Fully Reflected Head Seas, Mooring Type 1, VL=1.07 1125.40 Pitch Response in Fully Reflected Head Seas, Mooring Type 1, 1/L=1.07 1135.41 Sway Response in Fully Reflected Beam Seas, Mooring Type 1, 1/L=0.30 1135.42 Heave Response in Fully Reflected Beam Seas, Mooring Type 1, l/L=0.30 1145.43 Roll Response in Fully Reflected Beam Seas, Mooring Type 1, 1/L.30 1145.44 Nonlinearity of Measured Surge Reponse at Resonance 115ix5.45 Theoretical Response of Model-Scale Mooring in Surge 1155.46 Common Moorage Conditions (from Taylor (1983)) 1165.47 117Example of Small Craft Mooring Configuration5.48 Example of Small Craft Mooring Configuration 1175.49 Example of Small Craft Mooring Configuration 1185.50 Example of Small Craft Mooring Configuration 1185.51 Elastic Behaviour of Nylon Rope (from Taylor (1983)) 1195.52 Definition Sketch for Nonlinear Mooring 1205.53 Schematic Response of a System with Nonlinear Restoring force to HarmonicExcitation 120A.1 Definition Sketch for Centre of Gravity Calculation 121A.2 Definition Sketch for Calculation of Moment of Inertia in Yaw, Plan View 121xNomenclatureA^amplitude of incident wave traina^amplitude of elastic motion, integer exponentaij^added mass coefficient[a] added mass matrixa^angle of mooring lines to boatslack length, integer exponentbij^radiation damping coefficient[b] damping matrixstill water depthdid^distance to masses added for determination of moment of ineritia in yawdi^distance to restoring springs for determination of moment of ineritia in yawbandwidth parameterhydrodynamic force{Fex}^real vector of exciting forcesphase anglevelocity potential arising from incident wave trainvelocity poential arising from diffraction of 00 about motionless rigid body4'0 +4)j^forced velocity potential arising from the sinusoidal rigid body motion ofmode i with unit amplitudegravitational accelerationwave heightHi^incident wave heightxiXliHiR^effective wave heightHr^height exceeded by fraction r of wave heightsHs^significant wave height[h]^hydrostatic stiffness matrixdiagonal moment of inertiaindicial counterindicial countermodel spring stiffness, wave numbermooring stiffnesswaterline vessel length1^positions of masses added for centre of gravity testing, distance fromreflecting wallA,^wavelengthmass, applied momentMij^vessel inertial properties[M]^mass matrixmn^spectral moment11.^dynamic viscosityunit vector normal to body surface pointing inwardfluid pressureangle of wave incidence relative to vesselhorizontal distance from body, reflection coefficientposition vector ( x, y, z)fluid densityPB^local body densitybody surface, waterline areaSB^submerged body surfaceS(co)^frequency spectrum[S]^mooring stiffness matrixs^distance of centre of gravity from centrepoint of pivot linean^standard deviation of position about the meanT^wave periodT( k, 1, R)^transfer function;^peak period of random recordt^timet^record lengthV^volumeVB^submerged body volumeVT^total body volumeW^waterplane areaCO^radial frequencyXi^complex valued exciting force{Xi}^complex vector of exciting forcesxg^x-position of centre of gravityx'^x-position of centre of gravity with pivot point as origin4i^six-degree-of-freedom rigid body positionYb^y-position of centre of buoyancyYg^y-position of centre of gravityY^y-position of centre of gravity with pivot point as originNi^angle of wave incidence relative to reflecting wallAcknowledgementsThe author would like to express his appreciation to his supervisor, Dr. Michael Isaacson, forhis valuable advice throughout this project, which kept this thesis on the right track andprevented long delays. Thanks must also be given to the staff of the Hydraulics Laboratory ofthe National Research Council in Ottawa, particularly to Mehernosh Irani, Geoff Mogridge,Etienne Mansard, Peter Launch, Ed Funke, and Bruce Pratte, for their generous assistance withthe experimental portion of this study.Finally, acknowledgement must be made to the Natural Sciences and Engineering ResearchCouncil and the National Research Council for their generous financial support.xiv1 Introduction1.1 GeneralMoored vessels may be subjected to a number of factors that may cause them distress.Among these are wind, ice, currents, and surface waves. Protection against surface wavesrequires structures such as breakwaters, which are very expensive, and it is the aim of anydesigner to keep this to a necessary minimum. The importance of efficient design was ablypresented by Huntington and Thorn (1989), where the results of a study performed for TorquayMarina in England were given. It is stated "the outcome of the study was a recommendation thatthe breakwater extension should be 40 m rather than 50 m, to achieve the specified reduction inwave height inside the harbour. The cost of the structure was about £20,000 per metre. Areduction of 10 m saves £200,000. The study cost is 3% of the capital cost: the saving to theclient 7%. Of course, if the studies had found that a 10 m extension was necessary the studieswould have been even more cost effective, as the benefit then would be the avoidance of afinancial disaster."As the demand for small craft moorage fills existing sites, more new harbours requiringthis sort of augmented protection will undoubtedly have to be built. The necessary waveprotection is a function of two parameters: the outer wave climate and the desired climate insidethe harbour. This is to say the outer wave climate and the wave protection geometries determinethrough the diffraction problem the inner wave climate which itself must meet limits determinedby vessel response in the harbour. The wave diffraction problem is fairly well understood but,although some attention has more recently been focused on it, the acceptable wave climate hasnot been critically examined as thoroughly and itself consists of two separate problems: theresponse of a moored vessel to wave action, and the limits of acceptable motion of a mooredvessel.11.2 Scope of ProjectThis project examines the problem of moored small craft motion under wave action witha focus on wave agitation criteria, their extension to take into account reflected seas, irregular seastates and the effect of changes in the configuration of the vessel-mooring system. A literaturereview is performed to survey published results. An experimental program is undertaken togather long-crested wave response data, which are compared with numerical results. Thesenumerical calculations are further extended to consider the effect of changes in boatconfiguration and mooring on vessel motions. Reflection effects and the influence of randomwaves are examined. Considering all of the above, wave agitation criteria and acceptable limitsof motion are reviewed and conclusions are made.22 Literature Review2.1 Acceptable Wave Climate in Small Craft HarboursWhen any harbour is being planned, one of the design criteria will be theacceptable wave climate inside that harbour. In the past, the design limit has usually beenchosen on the basis of experience at other locations, usually by specifying the waveheight (implicitly or explicitly causing a certain boat motion) to be less than a certainmaximum value.Lundgren (1972) states "The permissible disturbance is usually specified in termsof a maximum wave height with a recurrence interval of 10 or 100 years. However, a 5-second period wave of maximum height 0.3 m is more injurious than a 7-second wave ofheight 0.5 m, provided that the tendering allows the vertical motion. The 'permissiblewave height' depends also upon the angle between wave direction and boat axis." Hethen continues: "It seems more rational to prescribe certain angles of rotation for therolling and pitching of selected craft. These movements can be determined from testswith model boats where typical mooring and fendering systems are simulated."Dunham and Finn (1972), in their treatise on small craft harbours, say that thenormal acceptable maximum wave height is "about 2 to 4 feet in the entrance channel and1 to 1.5 feet in the berthing areas, depending on the characteristics of the using craft.Generally , if waves can be attenuated to a height of about 1 foot in the berthing areas,their horizontal oscillations will not be troublesome and any longer-period effects will gounnoticed."Bruun (1976) states in a section on fishing harbours: "Short wind waves ofperiods up to 6 seconds should not exceed about 1/2 foot for smaller vessels and 1 footfor larger vessels at any berthing place in a fishing port. For periods above 6 seconds,3wave heights should not exceed about 1/2 foot."Le Mehaute (1977) states that "20 cm would be considered the maximum bymany pleasure boat owners, even though harbour masters may consider that 40 cm isacceptable, if the boats are properly moored-which is often not the case..., harbourdesigners may accept an 80 cm wave as the maximum wave height criteria forcommercial fishing boats." He adds that "moored small craft have a period of resonancein their mooring of 4 to 8 seconds. They are not susceptible to seiche", which has a muchlonger period.In Kamphuis (1979), it is stated that unobstructed "incoming waves larger than0.5 m will create enough disturbance within the marina, especially if the perimeter is veryreflecting, that small craft, even when moored properly will sustain considerabledamage." He goes on to say "In our basic research we have come across very littleliterature concerning marina design specifically. Most work refers to agitation inharbours and marina design is often considered to be `mini-harbour' design." He thenasks "how can we reduce the short, steep, 'unimportant' waves to below a threshold ofabout 0.3 m at which time lines and fittings begin to break? The solutions are not simple,neither are they cheap."A desired wave height of less than 0.3m (1 ft) is also mentioned in passing inthree separate papers presented by engineers involved in marina development at aconference on marina design and operation (Huntington and Thom (1989), Boc et al(1989), and Bentley (1989).At the same conference Cox (1989) cited as more comprehensive a standarddeveloped by Northwest Hydraulic Consultants Limited (1980). Here, a large studywhich included physical and numerical modeling was undertaken to determine theacceptable wave climate in small craft harbours. As part of its conclusions, it gave a table4reproduced as Table 2.1 and which lists wave limits inside a marina for a variety of waveperiods and angles of incidence.An extension of this study was performed by Taylor (1983) which concentratedon motion in surge and sway. More model testing and numerical analysis was performed,with a comprehensive treatment of moorings, critically examining the effects of mooringconfiguration, line stiffness, and amount of slack. Wave height and frequency effectswere also considered. All tests were performed using regular, long-crested wavesincident for head and beam seas. Taylor also gave acceptable wave limits for surge andsway, which may be found in Table 2.2.Canadian Fisheries and Oceans (1985), as part of their Guidelines for HarbourAccommodation, gave allowable wave agitation criteria for recreational and fishing boatsbased on a simplified version of the Northwest Hydraulic Consultants (1980) results.These criteria, and their allowable frequencies of exceedence, are presented in Tables 2.3and 2.4.Recently, Fournier et al (1993) published wave agitation criteria applicable forSTACAC class 2 (35 to 45 ft) and STACAC class 3 (45 to 60 ft) fishing vessels. Thesewere based on a field study of non-directional wave measurements in several AtlanticCanada harbours which were correlated with observations from harbourmasters on theacceptability of the wave field. Results are presented in Table 2.5.Rosen and Eliezer (1984) gave motion limits for craft of up to approximately 800DWT as beingMaximum Linear Acceleration:Maximum Angular Acceleration:Maximum Peak to Peak Roll:Maximum Force in a Mooring:Maximum Force in a Fender:0.4 m/s22.0 deg/s26.0 degrees20% of breaking load60% of ultimate load5but did not give any specific wave conditions which might bring about these motions, andinstead presented a method in which the response might be determined through modeltests, a summary of which is given in the next section.In Jensen et al (1990), acceptable wave limits for small craft were given a smallpart in a study more concerned with larger vessels. For open boats of 5-12 m, safemooring conditions at berth were given as Hs 5 0.20 m, where Hs is the significant waveheight. For other boats of length 5-12 m, Hs 5 0.30 m. For small fishing vessels oflength 15-30 m, Hs 5 0.30 m, and for coasters of less than 2000 DWT, Hs 5 0.45 m.2.2 Determination of Vessel Response for a Given Wave Climate2.2.1 Physical ModelsThe response of a prototype vessel to wave action can be determined by scale-model tests in a wave basin. The procedure for this is very well known (e.g. Streeter andWylie (1981), pp. 175-6), with variables scaled on the basis of a constant Froude number.It is very important to note that it will be impossible to scale viscosity due to the fact thatwater is likely to be used in both the prototype and the model. Once the models andsurroundings have been constructed, the boats are put in a scaled wave field and theresults (forces, accelerations, displacements) are factored up to prototype size. The maindistinguishing characteristics of any individual model test will be the details: how thevessel was moored, what type of vessel was modeled, the details of the instrumentation,and the nature of the incident wave field.It is common to use Linear Variable Displacement Transducers or potentiometersin various configurations to measure model motions. This method was used byNorthwest Hydraulic Consultants Ltd. (1980), Taylor, (1983), Rosen and Kit (1984) andMansard and Pratte (1982), in which capacitance-type probes were also used to measurethe vertical motions. A disadvantage of this method is that forces exerted between themodel and the ground may not be negligible and are generally not accounted for in the6analysis. Forces in mooring lines in the above studies were measured with strain gaugesor load cells, or were back calculated from mooring line displacements and a previouslymeasured load-extension curve.A special case of a model testing program is the measurement of motion underwave action of full-scale craft. The major advantage of this method is that there will beno scale effects, and the measurement of vessel movements will be unlikely to affect thatmotion. A major disadvantage is that there is no control over wave conditions and theremay be a long wait for significant data. Measurements of this kind were taken in Jensenet al (1990) as part of a study to determine criteria for ship movements in harbours.For cases when measurements are not made at full scale, it is generally difficult toperform tests of a harbour model with a ship model inside due to the constraints of space.Harbour models are generally built at a scale of 1:60 to 1:125, while ship models mustgenerally be built at a much larger model scale in order to minimize the scale effects ofviscosity. In Rosen and Kit (1984), where wave action in a fishing harbour was studied, aharbour model and a moored vessel model were built at different scales. First the vesselresponse to wave action was determined through a testing program of the moored model.Next, the wave climate inside the harbour was obtained and the results were applied tothe response spectrum gained from the vessel motion tests.2.2.2 Numerical ModelsThe numerical calculation of the movements of a floating body has been wellestablished, at least for the case of case of small amplitude waves and motions. Mostmethods are based on potential flow in which diffraction and radiation potentials areestablished and are used to obtain the exciting forces and harmonic motions of the body.Diffraction potentials refer to disturbances to an incident wave train, proportional to waveheight and due to the presence of a fixed body, while radiation potentials are disturbances7proportional to and resulting from body movements in its six degrees of freedom. Due tothe small amplitude assumption, these may be superposed with magnitudes and phasesobtained from the equations of motion.An exception to this is found in Raichlen (1966), where a semi-empirical methodwas given for calculating boat motion in standing waves. Here, the added mass wasestimated through comparison of periods of oscillation in air and water. A linearised dragcoefficient was used to represent viscous effects, and driving forces were found fromthe pressure acting on the ends of the vessel. A closed-form solution was thus obtaineddependent on frequency and position in the standing wave field, and this generally agreedwith experimental data, except at resonance.In two extensive papers John (1949, 1950) established all of the basic conditionsfor solution of the velocity potential and equations of motion, but presented no results.These assume an inviscid fluid and irrotational flow, with long-crested, small amplitudewaves incident on a rigid floating body with zero average forward speed. Haskind (1953)presented a method which Newman (1962) used to calculate forces on a fixed body bythe use of the radiation potential. Kim (1965, 1966) used this method to giveapproximate solutions for added masses, radiation damping coefficients, body response,and phase lags for semi-submerged, freely floating elliptical cylinders and ellipsoids.Many other authors have presented added masses and damping coefficients for variousgeometric configurations.Wehausen (1971) gives a very complete overview of linear theory and gives somecomparisons of numerical and experimental results. Furthermore, he presents a summaryof various methods used in the solution of velocity potentials. Another good overview ofsmall amplitude theory is given by Newman (1977), and Sarpkaya and Isaacson (1981)present floating body theory and give more recent results in a manner well suited toengineering readers.83 Theoretical DevelopmentFigure 3.1 provides a definition sketch for the motion of a rigid floating vessel, whichhere is represented by three translational and three rotational degrees of freedom. TheCartesian coordinate system Oxyz is defined such that the origin 0 is located at the still waterlevel. In the first problem to be considered, the vessel is excited by regular, small amplitude,long-crested waves incident at an angle 0 relative to the x-axis. The vessel will thus undergosinusoidal motions in its six degrees of freedom with amplitudes 4i, where i is an indicialmarker varying from 1 to 6 representing, in increasing order, surge, sway, heave, roll, pitch,and yaw. The vessel is restrained by a linearly elastic mooring system that may becharacterised by a stiffness matrix with components kij, where kij gives the force in mode icaused by a unit displacement in mode j. The inertial properties of the vessel may berepresented by a mass matrix with components Mij, with Mij giving the body force in mode jresulting from a unit acceleration applied to mode i.3.1 Dimensional AnalysisIn order to gain a better understanding of the problem of small craft motion underwave action it is useful to perform a dimensional analysis. This is a well understoodprocedure which may be used to relate results at different scales and is described in manytexts (see for example Sarpkaya and Isaacson (1981) ch. 9).The motions of a moored vessel in regular waves may be represented as a function ofvarious parameters:( 41, 42,43,4,45,46 ) = H, X, d, g, p. L, geometry, kij, 0, Mij, )^[3.1]whereis the incident wave height910A,^is the incident wavelengthd^is the still water depthg^is gravitational accelerationp is fluid densityL^is the waterline length of the moored vesselgeometry^accounts for hull shapeil^is the dynamic viscosity of the fluidChoosing L, p, and g as the basis for carrying out a dimensional analysis, Eq. [3.1]can be expressed in terms of dimensionless parameters as41 42 43H A, d^kii^m..^11244 , 45 , 46 ) = G( r, r, T, geometry, —L—, 0, ---11- - ) [3.2]pgLa pLb ' p2gL3^'witha^= a(i)+a(j)+2b^= a(i)+a(j)+3a(i)^=0, 0, 0, 1, 1, 1 for i=1, .., 6.In the special case where response is linearly dependent on wave height, Eq. [3.2] maybe simplified toi 41 42 43 441- 451- 461- \^X a^kii 0^)L4ii. 112 ,k IT, IT, Fr T_T- , 11- , Tr j 1= G ( r, E., geometry„ „ — [3.3]pgLa pLb p2e)This gives the ratio of vessel response to wave height as a function of dimensionlesswave and vessel characteristics, which may be easily calculated for model testing, and theresults of which may be applied at different scales. Assuming that gravity is constant andfluid of the same density is used throughout all comparisons, the only free scaling factorbetween any two similar situations will be a length scale from which all other quantities ofinterest are derived. It should be emphasized that it will be very difficult to scale viscositydue to the fact that the same fluid will likely be considered for both prototype and model.3.2 Vessel Response3.2.1 Linear Diffraction Theory for Floating BodiesThe behaviour of a floating body in regular waves is a subject which has received agreat deal of attention. While it is not yet possible to calculate the full non-linear response forall conditions, a complete linearised theory based on the assumption of small wave height andbody response has been developed, and is applicable to a wide range of situations. A goodpresentation of this is found in Newman (1977), whose derivation is generally followed.This theory takes all of the conditions of section 3.0, with the additional assumptionsof an inviscid fluid and irrotational flow. It thus follows that all fluid forces on the body willbe normal to its surface and fluid motions may be represented through the use of a velocitypotential which may be expressed in the form6(x, y, z) = Re {(E 0;(x, y, z) + A 0A(x, y, z))ei"^[3.4]j=1whereis the amplitude of motion of mode j4)i^is the forced velocity potential arising from the sinusoidal rigid body motion of mode jwith unit amplitudeA^is the amplitude of the incident wave trainOA^is the velocity potential arising from the incident wave train and its interaction withthe motionless structure=Re( c } designates the real portion of complex number c11OA may be further separated into two components: 00, the velocity potentialcorresponding to a linear wave train, and 07, the diffracted stationary body potential. 4)7 isdefined as the velocity potential which satisfies the followingd07 = --a00 on the submerged body surface Ssan^an[3.5]which states that the fluid velocity on the body surface corresponding to OA must have nonormal component. There are similar boundary conditions for the forced velocity potentials.These areaciPi iconi, j 1, 2, 3= ico(r x n); - 3, j = 4, 5,6whereis the position vector ( x, y, z)is the unit vector normal to the body surface pointing inwardand all conditions are imposed only on the body surface SB.In addition, all potentials must satisfy the Laplace equationV2, =^j = 0, 1,^7at all points within the fluid and must satisfy the bottom boundary conditionacik = 0, y = -d12aY[3.6][3.7][3.8]or, for infinite depthOi -4 0, y-->^. [3.91At the free surface, the linearised boundary condition to be satisfied isg^aY^ony=0, j=0,1, ...,7^[3.10]The final boundary condition to be imposed states that all disturbances created by thebody must, in the far field, radiate away from the body, and may be written asocit-Y2e ^as R -->^j =1, 2, ..., 7^[3.11]where R = (x2 + z2)The velocity potentials 01, 02, ..., 07 may now be solved for, and the resulting pressureat any point in the fluid is given by the linearised Bernoulli equation asa0P = —P(—dt gY[3.12]6= —pRe{(EjØj + A(00 + 07)}coei0}— pgyBy integrating over the wetted body surface, the hydrodynamic force is found to be6p Re[E ia4; et( n(M)=—Pgif(rx n)YSi^rxn ^(PidS— pRe[iwA ei" .11( n )(0o + 07)dS]x nSa r[3.13]Of the three integrals, the first represents the hydrostatic force; the second gives added massesand damping coefficients from the real and imaginary portions of the integral respectively,and the third gives the exciting force on the body proportional to the incident wave amplitude.For a floating body with non-zero waterplane area, hydrostatic restoring forces areproportional to displacements in roll, pitch, and heave; no other modes of motion havingrestoring forces. Gravitational force on the body may also provide restoring moments for roll13and pitch movements, and these will be combined here with hydrostatic forces. A bodysymmetric about the x-y plane, such as a boat, will have a simpler hydrostatic stiffnessmatrix, and this configuration will henceforth be assumed. With the origin taken along thecentreline, the matrix will have as its only non-zero componentsc33 = pgS^ [3.14]c pgSn + M(yb— yg)C55 = PgS11+ M(Y b — yit)cm= c53=—pgS1wheres = ff cfrdzwsi = ffxdrdzws„ = f x2 dxdzwS22= 5z dxdz.wyb = --111 ill ychdydz.P v.[3.15]whereW is the waterplane areaVB is the submerged volume of the vesselyb is the vertical coordinate of the centre of flotation,yg is the vertical coordinate of the vessel centre of gravityM is the object mass.The added masses and damping coefficients together are associated with the fluidforce due to the sinusoidal velocity of the six modes of motion such that the added mass forceis in phase with acceleration and the damping force is in phase with velocity. Both the addedmasses and damping coefficients are frequency-dependent. Once the velocity potentials for14r a0iCO2 —^= —pn—dn 15; dSSb[3.16]all modes of motion have been determined, the values of the added masses and dampingcoefficients may be found to be15whereau^is the added massis the damping coefficientand both matrices are symmetric. Thus, the component of body force in any mode due toadded mass and damping may be represented in the form6= ->(a4 + b4).J.1[3.17]The exciting forces are given by the last integral in Eq. [3.13]. As they are linearisedthey are independent of body response and thus, the exciting forces on a fixed or free objectare identical. They may be more easily written asF. = Re{A^i = 1, ..., 6^[3.18]where Xi is the complex amplitude of the exciting force of mode i for an incident wave trainof unit amplitude such that= —p (itn) + Ø7).^ [3.19]S.Once the hydrodynamic forces have been established, it is possible to combine themwith the inertial and mooring forces to solve for the amplitudes of motion 41, .., 46. Theequations of motion may be expressed in matrix form as([a]+[M])fil +[b]fil + ([h]+[5]){x} = {F.}^[3.20]or, in complex form as(— (02 gai+ [M])+ ico[b]+ ([11]+[.5])){x,} = {X}^[3.21]where[a] is the added mass matrix[b] is the damping coefficient matrix[h]^is the hydrostatic stiffness matrix[S]^is the mooring stiffness matrix[M] is the mass matrix.{x} is the real vector of vessel motions{70 is the complex vector of vessel motions{F.} is the real vector of exciting forces{X} is the complex vector of exciting forcesThe mass matrix can be very complicated. However, if the body is symmetric aboutx, as is the case for ships, and is also symmetric fore and aft, which may not be strictly truebut is usually a reasonable approximation, the cross products of inertia will be zero. Themass matrix is then much simpler and contains as elements_Ai 0 0 0 Myg 0 -0 M 0 —Myg 0 Mxg0 0 M 0 —Mxg 0[M] = 0 —Myt 0 I. 0 0 [3.22]Mys 0 —Mxg 0 /y 00 Mxg 0 0 0 L _wherexg is the longitudinal location of the centre of gravity, and Ix, 1y, Iz, are the mass moments ofinertia of the object which are defined as1617h= pB[{x} {x}- x2PV=^pB[{x}. {x}-^ [3.23]L. pk{x}. {x} - z2k1Vwhere VT is the total vessel volume and Pb is the local density of the body. Once all of theinputs for Eq. [3.21] have been determined, the complex amplitudes of motion i, .., 46 maybe determined through the use of any complex matrix inversion technique.3.2.2 Effects of Random WavesWhile it is helpful to be able to predict vessel response in regular, long-crested waves,naturally occurring conditions seldom approach this. Therefore, an extension of these resultsto irregular waves would be very useful. Irregular waves may be either short-crested or long-crested, with short-crested waves corresponding to a range of wave directions, while long-crested waves are unidirectional. While naturally occurring waves are always short-crested tosome degree, an assumption of unidirectionality is generally a good approximation, and ismuch easier to treat mathematically.A signal which varies randomly in time, which may be taken to correspond tounidirectional random waves or to the corresponding vessel motions, may be characterised bya frequency spectrum and a groupiness factor. However, for floating body applications thegroupiness factor has little use except for drift motions, which are unimportant here, and willnot be used. The frequency spectrum is a widely used method of describing the distributionof energy of a signal through a frequency range, and is discussed well in Newman (1977),Sarpkaya and Isaacson (1981), and many other references. Its derivation will not beperformed here but one useful property in this context will be given. Assuming a zero mean,the standard deviation of the signal is given asan=(.1 S(co)duo0[3.24]18wherean^is the standard deviation of positionco^is the angular frequencyS(o) is the frequency spectrumIf a wave spectrum is narrow-banded, i.e. its energy is concentrated within a smallfrequency range and the water surface elevation follows a Gaussian, or normal, distribution,then wave heights will follow a Rayleigh distribution. Due to linearity, this means that vesselmotions will also have the same distribution. The assumption of narrow-bandedness is notstrictly true in general, but the results it gives are in good enough agreement with reality suchthat its use can usually be justified (Sarpkaya and Isaacson (1981), Newman (1977)). TheRayleigh distribution depends only on the standard deviation of the water surface elevation,chi, to arrive at a probability distribution for wave height, namelyp(H)— 2^ eXp(-1/2/8^ [3.25]The probability of exceedence for wave heights then becomesODP(H' > H)= p(H)dil^ [3.26]= eXp(-112 180-),and the average of the highest fraction r of wave heights isf Hp(H)dHThr^H, P(Ht > H r)•[3.27]Table 3.1 provides a listing of some commonly used values of Hr.The Rayleigh distribution may also be used to estimate the height of the largest waveof a particular record. The expected maximum height may be estimated asL1'2 'mafax = 1121n17-1^ [3.28]whereis the length of the record; is the peak period of the (narrow-banded) spectrum.This equation is only an approximation, and becomes more valid as T increases.For spectra with a significant bandwidth, Cartwright and Longuet-Higgins (1956) givea modification to Eq. [3.28]. First, a spectral width parameter must be introduced aswhere2 MOM4 '"2 E —mornam„ = coRS(co)dco[3.29][3.30]is the nth moment of the spectrum. The spectral width parameter varies from 0 for a narrow-banded spectrum to 1 for white noise. The expected maximum value of a long record lengthmay now be estimated asEH" )=11214 1/1--E1an^Tp[3.31]19It must be noted that this is not the maximum trough to crest height H, as in Eq. [3.28], but isinstead the maximum expected distance of the function 1 from the mean.3.2.3 Reflection Effects3.2.3.1 Regular WavesReflection plays an important part in determining the wave climate and, hence, vesselmotion inside a small craft harbour. For normal reflection in head or beam seas, the ratio ofvessel motions in reflected seas to those in unreflected seas depends on the reflectioncoefficient R, the wavelength X, and the distance from the reflection source /.Consider a wave train with free surface elevation r propagating x-ward, with vesseland wave coordinates having identical origin. In the absence of reflection, the free surfacewill be of the form?I= A cos(kx — cot)^ [3.32]with corresponding vessel motions= cos(—wt —8)^ [3.33]where vi are the six degree of freedom positions of the vessel and Eli are phase differencesbetween vessel motions and wave peaks.Now consider that this wave train is reflected normally at x=/ with reflectioncoefficient R. The free surface elevation of this reflected wave train will be?I= RAcos(kx —2k1+ cot).^[3.34]If the vessel is symmetric both fore and aft and port and starboard (a not unreasonableapproximation), this will produce vessel motionsvi =^cos(2k1— wt — Si)^[3.35]where n accounts for symmetric effects in the various modes and is given as201, i=3,6ni =1-1, i=1,2,4,5[3.36]21Vessel motions in reflected seas will now be the sum of the motions caused byincident and reflected waves, i.e.vi = 4, cos(—cot — Si) + niRi cos(21c1— or —The^Si). amplitude of this motion, tilt, will beill = i 111+R2+2RCOS(2k1), n. =1.L = 4i All + R2 - 2R cos(2k/), n. = —1. [3.38b]The response obtained will be strongly dependent on wavelength, distance from the reflectingwall, and reflection coefficient. Figures 3.3 and 3.4 show the variation of Eq. [3.38] withdistance from the reflecting wall. For fully reflected waves, i.e. R=1, Eq. [3.38] may beexpressed in simpler form asL = 24 ilcos(kI)I, ni = 1. [3.39a]L = 2ilsin(01, ni = —1. [3.39h]The previous equations [3.38]-[3.39] are valid for any angle of wave incidencerelative to the vessel as long as reflection is normal. However, if the vessel is not completelysymmetric fore and aft, as was earlier assumed, these equations will give poor results forsurge and pitch in the area of beam seas. Since surge and pitch are of little importance inbeam seas, this is not a great limitation.Expressions may also be derived for the case of oblique reflection with the addedlimitation that the vessel must either be parallel or perpendicular to the reflecting wall. Forvessels parallel to the reflecting wall, the original assumption of symmetry fore and aft may[3.37][3.38a]be relaxed but this is not true if the vessel is perpendicular to the wall. Figure 3.2 shows adefinition sketch of vessel setup. In this, waves are incident from a direction v on a vessel adistance 1 from a reflecting wall. The behaviour of a body in this reflected wave field may bedescribed by the equationsandil? =^R2 + 2R cos(2k/ cos yr)= i-N11+ R2 - 2R cos(2k/ cos yr)[3.40a][3.40b]For vessels parallel to the reflecting wall, Eq. [3.40a] is valid for the modes of surge, heave,and pitch, while Eq. [3.40b] applies for sway, roll and yaw. For vessels perpendicular to thereflecting wall, Eq. [3.40a] applies for sway, heave, and roll, while Eq. [3.40b] may be usedfor the modes of surge, pitch, and yaw.In its interaction with incident waves, the vessel will cause disturbances to theincident wave train which will also be reflected and thus affect vessel motion. It must benoted that all of the expressions in this section assume that these diffracted and forcedpotentials are small in comparison with the incident waves and their reflections will causenegligible exciting forces upon the vessel.3.2.3.2 Random WavesRandom waves may be treated as a superposition of linear waves. Using techniquesdescribed in sections 3.2.1 and 3.2 2, the spectral response of each mode of motion Si(f) maybe determined for a given incident wave spectrum. With the previous assumptions, behaviourin a regular wave field is described by Eq. [3.38] and Eq. [3.40]. Spectral response in anormally-reflected random wave field SIR is then described by very similar equations, namelySiR(f)= S1(f)T(kl,R)2^ [3.41]22where T(kl,R) is the transfer function between motions in incident and reflected seas and isgiven asT(k/,R) = + R2 + 2R cos(2k/), n =1.^[3.42a]T(k/,R)= V1+ R2 — 2R cos(2k/), n = —1.^[3.42h]for normal reflection, andT (kl,R, vf) = +R2 + 2R cos(2k/ cos ty)T(kl,R,O= Ji + R2 — 2R cos(2k1 cos v)for oblique reflection and to be used in the same manner as Eq. [3.40].[3.43a][3.43b]If the calculated spectral response may reasonably be classified as being narrow-banded, the results of section 3.2.2 may be applied, but if the reflecting wall is relatively faraway from the location considered and reflection is strong, the response spectrum maycontain several peaks with very low response in between. In this case, most of the results ofsection 3.2.2 are not valid, but the standard deviation of position, an, may be obtained fromEq. [3.24]. As the distance between the reflecting wall and floating object increases, thephase behaviour with respect to frequency becomes increasingly more sensitive until, in thelimit, there is no real correlation. In this case, the vessel motions correspond to those of anobject floating in a sea consisting of two uncorrelated wave trains advancing in oppositedirections. The resulting response spectrum iss,, (f) = (f)(1 + R2)^ [3.44]and the standard deviation of position in any degree of freedom will beaiR =^+ R2 .^ [3.45]234 Program of Experiments4.1 GeneralA program of experiments was undertaken at the multidirectional wave basin at theNational Research Council Hydraulics Laboratory in Ottawa. All experiments were performedto determine the motion of a 0.76 m model of a fin-keel sailboat under wave action, using bothregular and random waves, with variables of wave period, wave height, angle of wave incidence,wave reflection coefficient, and model mooring condition. Model motions were measured usingthe SELSPOT imaging system, while wave heights were measured using capacitance-type waveprobes.4.2 National Research Council Hydraulics Laboratory4.2.1 Wave BasinThe multi-directional wave basin is a state-of-the-art installation for scientific andengineering research and testing of structures in the ocean environment. The facility, shownschematically in Figure 4.1 with the model in position, consists of a 50 m long, 30 m wide, and 3m deep inground tank with a covered centre pit. The basin is surrounded on three sides byvertical wave absorbers which have low reflection characteristics for any depth of water. For allwave conditions the average reflection coefficient is less than 10%, and for the most commonlyused wave heights and periods it is less than 5%. Waves are generated by a segmented wavemaker consisting of sixty 0.5 m wide by 2 m high wave boards. The boards may be raised orlowered depending on the water depth used and all wave generators may be operated in piston,flapper, or combination mode. The multi-directional wave basin has comprehensive wavegeneration programs available for control of the wave machines, providing simulation of naturalsea states as defined by spectra or time series. Wave grouping can be specified separately and is24modeled with simulation of bounded long waves without introducing spurious free waves in thebasin for most conditions (National Research Council, 1989).4.2.2 Data Acquisition SystemThe data acquisition system captures analog signals given by instrumentation such aswave probes, converts them to digital information and transmits them to a computer system. Inthe present case the analog to digital (A/D) converter is a Neff Series 100 conversion unit. Thedigital output is then transmitted to a VAX computer system and stored. Figure 4.2 shows aview of the control room which houses the computer and A/D converter. Control of the systemis through the use of the GEDAP software package developed by NRC. For these experiments,all sampling was performed at a frequency of 20 Hz, which gave good resolution but did notprovide unnecessarily large output files.4.2.3 Wave MeasurementWave elevations were measured using capacitance-type wave probes, sampled at afrequency of 20 Hz. Due to the constraints of the SELSPOT system, it was impossible to placethe probes close to the model. They were thus placed between the model and the wave generatorin the directions of wave propagation. Reflection analyses were performed using a five probearray. Calibration of wave probes was performed using spacing elements so as to obtainmeasurements for immersions of 0 m, 0.05 m, and -0.05 m. A best fit line was then computedusing linear regression. The calibration was repeated if any of the measured points lay furtherthan 1% of the range of measurement off the line. For some of the tests, reflection from the wallcaused the probes to give meaningless data. For these tests, the incident wave heights and periodswere taken to be the same as those measured using the same drive signals while the reflectingwall was not in place. This undoubtedly introduced some additional error into the measurements.However, in this case, this was the only possible method. Wave heights parallel to the wavecrest differed significantly across the basin so it was not possible to take measurements in a clear25section of water and correlate them to the incident wave in the area of the model. The only otherpossible option would have been to perform a reflection analysis for each wave coming off thereflecting wall. This would likely have introduced more error than the method used andexperiments were performed under time restrictions which made reflection analysis of eachexperiment impractical.4.3 Model Testing ProgramExperiments were carried out to determine the motion of a moored model of a fm-keelsailboat under long-crested wave action of both regular and random waves.4.3.1 VesselThe boat used was a modified version of a model originally intended for radio-controlledoperation and is shown in Figure 4.3. The keel and rudder were both shortened to give a betterrepresentation of a full-scale boat, and ballast was added front and back in an attempt to increasethe radii of gyration in pitch and yaw. The model was 0.76 m long, 0.229 m in breadth, with awaterline length of 0.692 m. It had a draft of 0.143 m and was tested in a water depth of 0.5 m.The mass was measured to be 1.61 kg, while the centre of gravity was 0.037 m below thewaterline. The radii of gyration were estimated to be 0.144 m, 0.071 m, and 0.103 m in pitch,roll, and yaw respectively.4.3.2 Mooring SystemThe boat model was restrained by a four point mooring system consisting of linearsprings and fine steel wire attached to the bow and stern, spreading out horizontally at a nominalangle of 35 degrees as indicated in Figure 4.4. To improve the linearity of response, all mooringlines were pretensioned 0.05 m. The springs used were highly linear and almost identical,having a stiffness constant of 16.2 N/m. This was the standard condition and was designatedmooring type 1. The mooring structure itself was made in one piece, having a bottom made of26thin plate and mooring posts of steel rod. It measured 1.35 m in length by 0.36 m in width. Thewhole structure was braced and weighted to make it rigid in comparison with the mooring lines.For several tests, springs of different stiffness were used, having a spring constant of 8.1 N/m;that is, half of the previous value. This mooring was designated mooring type 2.4.3.3 Measurement of Model MotionsAll model motions were measured using SELSPOT, an optical spotting system whichprovides an accurate means of measuring the six-degree-of-freedom motions of a rigid body witha minimum of connecting cables which could affect that motion. It consists of two cameras, alight-emitting-diode (LED) control unit which handles up to eight LED's, an administration unit,a computer, and various cables. The two cameras point at the targets: up to eight LED's attachedto the model and which flash on and off at times specified by the administration unit andcommunicated to the LED control unit through cables. This allows each camera to obtain a two-dimensional position for each target. This information is communicated to the computer and isconverted to provide the location of each target in three dimensions, and then the six-degree-of-freedom motions, through the use of software routines which access previously createdcalibration files.For the purposes of these experiments, a rigid frame was built on the model to houseeight LED's which were placed at known locations. The LED's were connected to the LEDcontroller by thin wires. The effect of these on the model motions was not measured but likelysmall for low-to-medium amplitude boat motions. The cameras were placed at a distance ofabout 2 m from the model, each at about 30 degrees to the side as shown in Figure 4.5. Thisgave the best compromise between a wide angle, which would provide the best three dimensionalresolution, and a narrow angle, which gives the highest LED signal strength.One drawback of the SELSPOT system is that any reflections off the water surface oradjacent surfaces caused by the LED's which were registered by a camera would distort the27readings, giving greater error. To minimize unwanted LED reflections off the water surface thecameras were placed as near to the water as was practical and aimed at an upward angle. When awall was placed behind the model to create wave reflection, parts of it were painted black for thesame reason.In general, readings returned by SELSPOT were of very high quality, with root-mean-square error readings for the translational degrees of freedom almost always less than 0.1 cm.On occasion, an erroneous reading would be taken due to the presence of some external lightsource. However, this was relatively infrequent and the resulting spike was easily disposed of.4.3.4 Estimation of Moments of InertiaTo give meaning to any of the model tests, it was necessary to determine the dynamicproperties of the model. These properties include the model mass, moments of inertia about theprincipal axes, and the location of the centre of gravity. Cross products of inertia could not bemeasured and, as they were likely small for reasons given in section 3.2.1, were assumed to bezero. As was previously mentioned in section 4.2, the model was attached to the LED controlunit by thin wires. This made the measurement of the model's dynamic properties a necessarilyimprecise operation. To achieve best results, all measurements were made with the wires held inthe same position as when the boat was moored. The mass was measured on a Mettler PC 24scale, accurate to ± 0.5 g, and found to be 1.610 kg. This estimate is likely reliable to ±10 g,again due to the attached wires.In order to determine the location of the centre of gravity and the moments of inertia, itwas necessary to have a swing frame on which the model could freely rotate. For this purpose, arigid frame was built as part of the model, consisting of hollow aluminum rods runninglengthwise and crosswise to the boat. The ends of these rods projected clear of the hull, and hadhardened screws driven through them, point down. As shown in Figures 4.6 and 4.7, thesescrews could rotate freely on a hard base, giving a system that could move in either pitch or roll.28To find the location of the centre of gravity, the model was set up so that it could rotate inpitch as shown in Figure 4.6. Static angles of tilt were measured by SELSPOT with knownmoments applied to the boat, and thus the location of the centre of gravity was determined. Themoments of inertia in pitch and roll were determined in a similar manner. First, as depicted inFigure 4.7, the model was set up so that it could freely rotate about one axis. Next, smallpendulum motions were induced about this axis. The natural period of oscillation was found bytiming a number of cycles with a stopwatch. From a knowledge of this and the location of thecentre of gravity, the moment of inertia about that axis could now be calculated. However, themoment of inertia in yaw could not be measured in the same manner, due to the fact that it wasimpossible to attach knife edges to allow rotation in this mode. Therefore, as shown in Figure4.8, the boat was hung from a string about the centre of gravity and was attached to the mooringsystem. To have the mooring line forces act through the centre of gravity, attachments weremade to both ends of the model to allow the mooring lines to attach at the appropriate location.The model was then made to oscillate in yaw and the natural period of motion was determinedwith a stopwatch. From this, the overall moment of inertia was found, and the contribution fromthe added attachments was subtracted to get the moment of inertia in yaw. Details of allcalculations may be found in Appendix A.4.3.5 Measurement of the Model Subsurface ProfileAs part of the input into the numerical model, it was necessary to determine thesubsurface profile of the boat. First, the location of the waterline was found and marked. Next,the model was set up out of water and lines inscribed on it at intervals of 2.00 cm below thewaterline using a Mitutoyo digital height gauge as shown in Figure 4.3. Figure 4.9 shows how,to get a complete representation of the hull, cross sectional widths were then taken along itslength using a Mitutoyo digital micrometer.294.4 Experimental Setup and Procedure4.4.1 Still Water TestsTests were performed to determine the natural period of oscillation in several modes.The natural period in roll was found by giving an initial rotation to the unmoored model and thentiming the subsequent cycles. This gave a natural period of 0.73 s. Free oscillations in heaveand pitch were too damped to get any meaningful data. The natural period in surge of themoored model was determined by timing free oscillations after an initial displacement and wasfound to be 1.27 s. Again, oscillations in sway were too damped to get meaningful results. Therestoring moments in pitch and roll were found by applying known moments and recording thedisplacements.4.4.2 Model SetupFigure 4.10 depicts the four setups used for tests under wave action. In the first setup, themodel was moored with the bow pointing at an angle of 22.5 degrees from the perpendicular tothe wave generators. In the second setup, the bow was pointed at an angle of 112.5 degrees fromthe generator. The third setup was the same as the first but a reflecting wall was added behindthe boat. For the last setup the boat was moored at a 67.5 degree angle to the perpendicular andthe reflecting wall was placed on the side of the boat. The reflecting wall itself was a 2.5 m longrigid structure with a vertical steel plate as the reflecting surface.4.4.3 Moored Tests Under Wave ActionA variety of tests were performed for comparison with calculated motions and resultsfrom other studies. Regular wave tests with a variety of incident angles and wave periods wereperformed with no reflection for comparison with linear floating body theory presented in section3.2.1. To give an idea of the linearity of response, several tests were repeated with more than30one wave height at each period. The expressions of section 3.2.3 were examined through wavetests in setups 3 and 4, which, as previously stated, used a wall to create a reflected wave field.To provide a comparison with regular wave tests, and to give an uninterrupted record of responseover the range of frequencies considered, random wave tests were also performed in all setups.A summary of all tests performed can be found in Table 4.1. Photographs of boat motion underwave action may be found in Figures 4.11-4.14.315 Results and Discussion5.1 GeneralIn this section, the results of the experimental program and numerical simulations arepresented and compared, both to each other and to published data. Their applicability isdiscussed and results are extended to full scale.All motion amplitudes are plotted as dimensionless response amplitude operators(RAO's), with the abscissa being in all cases the ratio of wavelength to vessel waterlinelength. The ordinate used depends on the mode of motion considered. Translational modes(surge, sway, heave) are plotted as the corresponding motion amplitude to incident waveamplitude; i.e. 4f=24/H, i = 1, 2, 3. Rotational modes (pitch, roll, yaw) are plotted as theratio of corresponding motion amplitude in radians multiplied by an appropriate length scaleto incident wave amplitude. For pitch, the length scale is the vessel waterline length, soi=5. For roll and yaw, the appropriate length scale is the vessel waterlinebreadth B; i.e. 4i1=21B/H, i=4, 6.It is important to note that, for simplicity, the term 'wavelength' is sometimes usedduring the presentation of results instead of the more correct term 'dimensionlesswavelength'. When presenting or discussing motion results, both the terms 'wavelength'and 'dimensionless wavelength' refer to the abscissa of the plot considered as defined in theprevious paragraph. At all other times, 'wavelength' should be understood to refer to itsnormal dimensional form.5.1.1 ExperimentalResults are presented for the set of experiments described in Chapter 4. Responseamplitude operators for mooring types 1 and 2 are plotted separately, but the results ofrandom and regular wave tests for mooring type 1 are always plotted on the same graph.32Comparisons are made with the results of similar experiments performed by NorthwestHydraulic Consultants (1980). These were performed using regular waves on a model thatwas similar, although somewhat wider and with greater moments of inertia. However, themain difference between the two experiments was associated with the moorings. Thepresent study used moorings with highly linear response, while the Northwest HydraulicConsultants experiments were performed with nonlinear moorings of a configuration morelikely to be used in a small craft harbour.5.1.2 NumericalAs a tool to better understand the problem of small craft motion, a numerical analysiswas undertaken. Runs corresponding to the experimental program were performed and werefurther extended to consider different mooring types, dynamic properties, and hull shapes.Response was computed using linear diffraction theory as set forth in section 3.2.1 and asrepresented by the programs FACGEN3 and WELSAS3.FACGEN3 is a facet generation program which computes characteristics of facets ofthe submerged surface of the vessel, such as size, shape, orientation, and location, requiringas input the coordinates of the facets' corners. Figure 5.1 shows a representation of the boatas discretized. WELSAS3 uses the results of FACGEN3 together with specified incidentwave climate and mooring conditions to calculate the added masses, damping coefficients,and exciting forces.Although WELSAS3 may also be used to calculate steady-state response, this featurewas not used, since in many cases the only change between several successive runs was dueto non-hydrodynamic differences such as the mooring condition. Due to the nature ofWELSAS3, previously known added masses, exciting forces, and damping coefficientswould have been recalculated had a full run been performed for each small change. Thiswould have required excess computational effort, so the response was instead computed33using a small program which solved the complex steady-state dynamic problem for eachcondition.It must again be noted that results were obtained only for small amplitude regularwaves which, since drift forces do not play a significant role in small craft motion,represents a fair approximation to reality. Thus, irregular waves can be depicted by thesuperposition of regular waves, so that regular wave results can be directly applied toirregular wave problems.Because of the ease of changing conditions using numerical as opposed toexperimental methods, extensions were made to the experimental results for all test setups todetermine some trends for different boat and mooring parameters. The first extension madewas to consider the freely floating response as an extension of the mooring condition. Dueto its lack of a mast, the model used was low in moments of inertia in roll and pitch and hada low centre of gravity when compared to an actual sailboat. To estimate the effect of this,computational runs were performed using greater moments of inertia and a higher centre ofgravity in the boat mass matrix. Another change considered was to increase the boat hullwidth by 50 percent and the appropriate mass matrix entries by the proportionate amount inorder to give some indication of the importance of hull shape on vessel motion. The fmalperturbation was to add extra terms to the damping coefficient matrix along the diagonal toprovide an extra 5 percent of critical damping to approximate viscous effects (see Cloughand Penzien (1975)). This calculation used for its mass term the absolute value of the sumof the mass and added mass of the appropriate entry, and, for stiffness, the sum of thehydrostatic stiffness and that of mooring type 1. It must be noted that this damping, beingproportional to vessel speed, is not of the form which might physically be expected to bemore proportional to the square of the distance between vessel and fluid speeds. However,the latter type of damping could not be easily introduced into a small amplitude, frequencydomain type of solution. Therefore, it was decided that since some form of additional34damping was desirable, the previously described type would be used, notwithstanding itsnoted deficiencies.5.2 Head Seas5.2.1 SurgeFigures 5.2-5.4 give experimental measurements and numerical predictions for surgebehaviour in head seas. Figure 5.2 gives results for mooring type 1. The dominant featurefor this condition is a strong resonance peak located at a dimensionless wavelength of about3. This peak gives a maximum RAO of 12 from experimental results and is much higher inthe numerical prediction. Away from the peak, however, agreement between predicted andmeasured motions is very good and response amplitude operators are less than 2. Linearityof response is also very good in this region, but nonlinear behaviour may be seen at theresonance peak, as results are not colinear when plotted dimensionlessly.Figure 5.3 gives the effect of mooring changes on surge response. Limitedexperimental data shows a strong resonance peak for mooring type 2 at a dimensionlesswavelength of between 5 and 6. This same peak is also predicted numerically, although thepeak value is somewhat less than the measured maximum. However, because thecomputational grid used is fairly coarse, the location of the peak may not be at one of thepoints considered. For short wavelengths, surge response for mooring type 2 is significantlyless than for mooring type 1, but this is completely reversed at long wavelengths.Calculated freely floating response shows very small motions at short wavelengths,increasing steadily to the limit considered. At this dimensionless wavelength limit of about7.5, freely floating response and mooring type 2 have similar amplitudes of motion.Figure 5.4 shows the effects of various perturbations on the standard computationalsolution. The case in which moments of inertia and centre of gravity were changed had nosignificant effect on surge motions. Widening the boat hull moved the location of the35resonance peak to a lower frequency, as was expected, since boat mass increased andmooring restoring force remained constant. For both of these conditions, as with thestandard case, peak response was extremely high. The final perturbation involved addingextra damping amounting to 5% of critical. This had no effect on response except at theresonance peak, the size of which was greatly reduced. In fact, with this perturbation, surgeresults were in good agreement with experiment over the entire range.The above results may be compared with experimental values from a similar studyperformed by Northwest Hydraulic Consultants Limited (1980) presented in Figure 5.5.This shows results similar in form to the present study, but much smaller in magnitude. Thisis due to the mooring condition, which is of utmost importance for surge motion. Aspreviously stated, the Northwest Hydraulic Consultants study used highly nonlinearmoorings of a type commonly found in small craft harbours but, in the present case, themoorings were intentionally designed to be as linear as possible.5.2.2 HeaveFigures 5.6-5.8 present results for heave motion in head seas. Figure 5.6 givesresponse for mooring type 1. Experimental results show heave RAO increasing withdimensionless wavelength to a maximum value of about 1. Random wave results show verygood linearity, while measurements from regular wave tests demonstrate some scatter.Calculated response shows an identical trend, with heave RAO increasing monotonically toan asymptotic value of 1.Figure 5.7 examines the effect of mooring change on heave response. From limiteddata, there is no significant difference in experimental values between the two mooringtypes. The same observation holds true for computed values, where mooring types 1,2 andfreely floating response give almost identical heave results.36Figure 5.8 explores the effect of the various perturbations on heave response.Widening the boat hull causes a drop in response at very short wavelengths, but this is of noreal importance. No other change in boat characteristics causes anything but trivialdifferences from the standard solution.Figure 5.9 gives experimental results for heave motion in head seas from NorthwestHydraulic Consultants (1980). It is very similar to all of the other results, rising to anasymptotic value of 1 at large dimensionless wavelengths. A comparison may be made withFigure 5.10, which gives heave results in beam seas from the same Northwest HydraulicConsultants study. These are of similar form to the others, having an asymptotic value of 1at high wavelengths, but possessing a small peak at a dimensionless wavelength of about 2.5.2.3 PitchFigures 5.11-5.13 give pitch response in unreflected head seas. Figure 5.11 givesresults for mooring type 1. These show a response amplitude operator decreasing steadilywith increasing wavelength, from a response amplitude operator of about 2.7 at adimensionless wavelength of 1.4 to a lower value of 0.7 at a dimensionless wavelength of 8.Linearity of response is generally very good. Computed results in pitch start at a smallerdimensionless wavelength and show an early peak which then decreases steadily.Agreement with experiment is good at long wavelengths, but the size of the computed shortwavelength peak is about one third smaller than measured values. An interesting feature ofcomputed pitch response is the demonstration of coupling between pitch and surge at thelocation of surge resonance. This is evidenced by a sharp dip and then a return to the normaltrend as soon as the surge peak has been passed. This coupling is obvious because of thevery high response of surge at resonance. This causes an otherwise imperceptible linkagebetween surge and pitch (as seen in Figure 5.12 with the freely floating condition, where nosurge resonance exists) to become very evident.37Figure 5.12 lists pitch response for the various moorings considered. From limiteddata, no significant difference is apparent between experimental values for mooring types 1and 2. Computed results for mooring type 2 are very similar to those for mooring type 1 butdo not exhibit the large dip at the location of surge resonance for mooring type 1. However,a small anomaly at the location of surge resonance for mooring type 2 is again visible. Incontrast, the computed freely floating response is very similar to those for the other twomooring types but shows no such anomalies, since no resonance exists in surge.Figure 5.13 gives the effects of various computational changes on pitch response.The only significant change caused by increasing the moment of inertia was to cause thesharp dip at the location of surge resonance to become a sharp peak. Runs with the widenedhull were also very similar to the standard solution, with a peak apparent at the location ofresonance. As stated in section 5.2.1, adding an extra 5% damping caused a large reductionin the size of the surge resonance peak. Because of this, the anomalies in pitch at thelocation of surge resonance are not visible in this perturbation.Figure 5.14 gives pitch response in head seas from the Northwest HydraulicConsultants (1980) study. These results are similar in form to all of the others, having anpeak at short dimensionless wavelengths and then decreasing steadily, but the peakmagnitude is somewhat greater, reaching a RAO of nearly 4. This could be due to adifference in models used, but is more likely because in the present study, the peak had notyet been reached at the allowable high frequency limit for the wave generator.5.3 Beam Seas5.3.1 SwayFigures 5.15-5.17 give sway response in beam seas. Figure 5.15 presents the resultsfor mooring type 1. Experimental data shows a continual increase in the sway RAO withincreasing dimensionless wavelength. As would be expected, behaviour is more linear at38short than long wavelengths. Numerical results also show increasing response withwavelength, but predict magnitudes significantly smaller than those measured. Predictedand observed responses are comparable at short wavelengths but, at the longest wavelengthsconsidered, computed results are less than half experimental values.Figure 5.16 shows sway response for different moorings. Experimental values formooring type 2 are similar in magnitude to those for mooring type 1. Calculated results formooring types 1 and 2 are also comparable, although mooring type 2 has slightly higherresponse at large wavelengths. Freely floating response is also predicted to be similar to thetwo previous, but slightly lower in magnitude.Figure 5.17 presents the effects on sway response resulting from changes to boatcharacteristics. Increasing moments of inertia and raising the centre of gravity producesonly small changes from the standard form, and the addition of the previously describedextra 5% viscous damping is even less significant, resulting in almost imperceptibledifferences. In direct contrast, however, computational runs performed with a widened hullpredict much greater response for long wavelengths and, perhaps coincidentally, veryclosely resemble experimental results.Figure 5.18 shows sway motion in beam seas from the Northwest HydraulicConsultants (1980) study. Response here is much lower than was found in the present study,with an measured RAO maximum of about 1.7 found at a dimensionless wavelength ofabout 4. For short wavelengths, results are comparable to the previuously-describedresponse found in Figures 5.15 and 5.16. However, for long wavelengths, the NorthwestHydraulic Consultants results decrease while motions in the present study continue to grow.As with surge, this was undoubtedly due to the mooring conditions which, in the NorthwestHydraulic Consultants setup, were very restrictive of large displacement sway motions andmore nonlinear than in the present study.395.3.2 RollRoll response in beam seas is presented in Figures 5.19-5.21. Figure 5.19 gives rollresponse in beam seas for mooring type 1. Experimental results show maximum response atthe lowest wavelengths considered, steadily declining to nearly constant values at highwavelengths. Linearity of response is fair, with nonlinear effects becoming visible at eachend of the range. Computed results show a peak at short wavelengths which, as withexperimental results, declines to a nearly constant value at high wavelengths. However,when compared with experiment, computed values at high wavelengths are much lower.Figure 5.20 gives results for different mooring types. From limited data,experimental results for mooring type 2 appear to be slightly less than for mooring type 1.However, this is contradicted by numerical results, which predict higher roll response formooring type 2 than mooring type 1 at long dimensionless wavelengths. Also predicted is amuch higher short wavelength resonance peak. These discrepancies are part of a generaltrend in beam seas and will be discussed at length in section 5.6.1. Calculated results for thefreely floating condition continue this trend, with an even greater resonance peak, and highwavelength response in the general range of experimental values.Figure 5.21 presents the effects of the various perturbations on roll response.Increasing moments of inertia and raising the centre of gravity causes a shift in the locationof the resonance peak to a lower frequency, as is expected. This perturbation also appears tosomewhat lower peak response, but the computational mesh is not fme enough to say thiswith certainty. Runs performed with a widened hull showed a large increase in peakresponse, and also greater high wavelength motions. The addition of an extra 5% dampingproduced minor peak reduction, and otherwise showed little difference from the standardcomputational solution.40Figure 5.22 gives roll response obtained by Northwest Hydraulic Consultants (1980).Magnitudes are very comparable with Figure 5.19, although the peak location is at a higherfrequency in the present study due to the lesser moments of inertia and lower centre ofgravity in the present study.5.3.3 YawYaw results for beam seas are presented somewhat differently from those in the othermodes of motion. Measured response was nonlinear to such a degree that dimensionlessplots would have no meaning. Figure 5.23 shows an example of this nonlinearity. Here, itis seen that the monochromatic excitation of Fig. 5.23(a) produces yaw motions, found inFig. 5.23(b), at two frequencies: the excitation frequency and triple the excitation frequency.Furthermore, the motions at both frequencies are of comparable magnitude. In some cases,three or four higher harmonics were exhibited for monochromatic excitation. Response atseveral frequencies for monochromatic excitatrion is one evidence of nonlinearity. For thisreason, experimental results were not examined closely, and numerical simulation was theonly means used to analyze trends.However, all of the following results must be interpreted very carefully. In beamseas, slight differences in hull shape fore and aft are the driving forces for yaw motions,making it difficult to give generalizations. Additionally, mooring restoring forces in yaw inthis case are different from those expected in a full scale setup. This, along with thedemonstrated nonlinearity of experimental results, suggests that calculated response shouldnot be used except as a general guideline for trends.Figure 5.24 shows computed yaw response in beam seas for different mooring types.For all mooring types, a peak is visible at the location of the roll peak. For mooring type 1,response quickly drops and remains at a low value for long wavelengths. Mooring type 2exhibits the same behaviour, but the decline from the peak is not as sharp and long41wavelength response is considerably higher. Freely floating response shows a small dropfrom the short wavelength peak, with response that increases with wavelength to the limitconsidered.Figure 5.25 gives calculated yaw response for the various perturbations considered.The solution for increased moments of inertia shows a narrower short wavelength peak, butotherwise is little different from the standard results. The addition of an extra 5% dampingproduces minor peak reduction but otherwise is insignificant. The perturbation with thewidened hull, however, shows a significant increase in peak height and very low response athigher wavelengths.5.4 Quartering Seas5.4.1 SurgeSurge results in quartering seas are shown in Figures 5.26-5.28. Figure 5.26 givesresponse for mooring type 1. Regular wave tests show results similar in form to head seasbut magnitudes are smaller, as would be expected. Computational results are in goodagreement everywhere except at the resonance peak, where, once again, predicted responseis extremely high. Figure 5.27 presents computed response for various mooring types.Mooring type 2 again is predicted to have a peak at a dimensionless wavelength of about5.5, and all magnitudes are reduced over head seas. Freely floating results are also ofsimilar form to head seas, with reduced response. Figure 5.28 shows computational resultsfor various perturbations. Increasing moments of inertia and raising the centre of gravity hasnegligible effects on surge motion in this case. The addition of 5% extra damping has littleeffect on non-resonant surge motions, but reduces peak height to a level comparable toexperimental results.425.4.2 PitchPitch results are given in Figures 5.29-5.31. Figure 5.29 shows response for mooringtype 1. Except for one point at the location of surge resonance, measured magnitudes aregenerally smaller than in head seas, but not by a large amount. Calculated results showtrends similar to head seas, with magnitudes approximately one quarter less at highwavelengths but with a similar peak value; the peak being shifted slightly toward lowerwavelengths. The anomaly at the location of surge resonance is still visible. Figure 5.30gives calculated response for various mooring types. The results for both mooring type 2and the freely floating response are very similar to the standard solution, but have no largejumps at resonance. Calculated peak values become slightly greater as mooring restraintsbecome softer. Figure 5.31 shows computed results with various changes to boat properties.Increasing moments of inertia and raising the centre of gravity had no effects other than toslightly increase peak response and make the anomaly at the location of surge resonance aspike instead of a dip. The addition of 5% extra damping caused no changes over thestandard solution other than to eliminate this anomaly.5.4.3 SwaySway response in quartering seas is given in Figures 5.32-5.34. Figure 5.32 presentsresults for mooring type 1. As with beam seas, experimental results show responseincreasing with wavelength, but magnitudes are decreased over beam seas by about a third.Again, computational results also show this trend, but predict much lower response at highwavelengths. Figure 5.33 shows numerical results for various mooring changes. As withbeam seas, neither mooring type 2 nor the freely floating response differs significantly fromthe standard solution. Figure 5.34 gives sway response for various perturbations to boatproperties. Increasing of moments of inertia changes response slightly at short wavelengths,43but not significantly. The addition of extra damping makes almost no difference to thestandard solution.5.4.4 RollRoll response is presented in Figures 5.35-5.37. Figure 5.35 gives results formooring type 1. Experimental results show response declining with increasing wavelengthfrom a short wavelength peak; with a smaller secondary peak at a dimensionless wavelengthof 7. Compared to beam seas, magnitudes of response are about one third smaller.Numerical values show an early peak, with low response at high wavelengths. Peak heightis greatly reduced over computed values for beam seas, but this could be due to thecoarseness of the computational mesh. As with beam seas, numerical results are much lessthan measured values at high wavelengths. Figure 5.36 gives calculated response forvarious moorings. Mooring type 2 shows a much larger peak than the standard mooringcondition and also shows greater response at high wavelengths. The freely floating responsecontinues this trend, with even greater peak and high wavelength response. Figure 5.37shows computed response with the usual perturbations. Increasing the moments of inertiaand raising the centre of gravity causes both a reduction in peak height and an increase inlong wavelength response over the standard solution. The addition of an extra 5% dampingmakes only minor differences.5.5 Reflected SeasTrials were performed to determine vessel response in reflected seas. Experimentswere performed for mooring type 1 in head seas, with the ratio of the distance of reflectingwall from origin of boat coordinates to vessel waterline length as 1.07, and in beam seas,with the same ratio being 0.30. Numerical estimates for the same conditions were also madeusing the results of section 3.2.3 and the 5% additional damping perturbation was used forall trials, as it seemed to give the best overall results. For computational purposes, full44reflection was assumed. Due to the previously mentioned peculiarities of a segmented wavegenerator, experimental measurements of reflection ranged from 80% to 120%, so it seemslikely that reflection was close to 100%. It must also be noted that all of the results in thissection would change with distance from the reflecting wall as noted in section 3.2.3 so caremust be taken in their interpretation.5.5.1 Head SeasVessel response in fully reflected head seas is given in Figures 5.38-5.40. Figure5.38 shows surge response. Experimental results show much the same form as unreflectedseas, but have a much higher resonance peak, reaching a response amplitude operator ofover 20 due to reflective amplification. Nonlinear response is visible at the peak asevidenced by the non-colinearity of experimental results, but away from resonance, linearityis good. Computational results agree very well with experiment, even at resonance, wherethey plot in the range of experimental values.Figure 5.39 gives heave response in fully reflected head seas. Experimental resultsshow a peak at short dimensionless wavelengths, quickly dropping as reflective effectscreate destructive interference in heave, then rising again to the limit considered, with amaximum response amplitude operator of about 2. From random wave results, linearity isvery good, but tests from regular waves show some scatter. Numerical results show thesame trends, but tend to predict smaller response, most noticeably at high wavelengths.Figure 5.40 shows pitch response. Experimental results give two peaks withmaximum RAO of over 4, followed by a steady decline at high wavelengths. Again,random results are very colinear, while regular wave tests show scatter. Computationalresponse follows measured values fairly closely, but predicts smaller peak values, acarryover from numerical results for unreflected seas, which also produced lower resultsthan experiment at short wavelengths.455.5.2 Beam SeasFigures 5.41-5.43 give results for fully reflected beam seas. Figure 5.41 showsresponse in sway. Experimental results show an increase in response amplitude operatorwith wavelength, possibly leveling off at the highest wavelengths considered. In any case,high wavelength response is much less than in unreflected seas. Random results show verygood linearity at short wavelengths, but start diverging as wavelength increases. As usual,results of regular wave tests show some scatter. Numerical results show a small peak atshort wavelengths, followed by a slow decline. As with unreflected seas, numericalresponse is much smaller than experiment for long wavelengths, but short wavelengthagreement is somewhat better.Figure 5.42 gives heave response in fully reflected beam seas. Measured valuesshow response amplitude operator steadily increasing with wavelength to a maximum valueof near 2.5. Again, random results appear to be quite linear, while monochromatic datashow some scatter. Computational results show very similar form to experimental data,although, for long wavelengths, experiment generally gives higher response than predicted.Figure 5.43 gives roll response with full reflection. Measured values show themaximum response amplitude operator at the shortest wavelengths considered, declining toa fairly constant value at high wavelengths. At short wavelengths, response in reflected seasis greater than without reflection, while the opposite is true for long waves. Linearity forboth random results and regular waves is quite good over the range plotted. Numericalresponse follows the same form, and agreement at short wavelengths is fair. However, aswith unreflected seas, roll response is predicted to be much lower than measured values forlong wavelengths.465.6 Analysis5.6.1 Vessel MotionsAn overall examination of the experimental results shows vessel response amplitudessimilar to those obtained in other studies, with the exception of surge and sway, which weremuch greater in the present case. This is definitely due to the mooring condition used, andwill be discussed thoroughly in section 5.7. A high degree of linearity was found inmeasured response, with the notable exception of the surge resonance peak, the results forwhich are shown in Figure 5.44, and yaw motions. This linearity includes the modes ofsway and roll, which is somewhat unexpected, since turbulent viscous forces on the keel andrudder would be proportional to the square of the wave height and would not lead tocolinearity when results are plotted dimensionlessly. However, regular wave results doshow some of what might be expected for this type of behaviour, and consequently showgreater scatter.In general, regular wave tests did not give as good results as irregular waves, likelybecause of the characteristics of the wave generator. Due to the finite width of the waveboards, there are frequency-dependent spatial variations in wave height throughout thebasin. Also, due to time constraints, all waves were measured between the model and thewave generator along the theoretical direction of wave propagation, and not at the locationof the model itself. Furthermore, the processing of irregular wave results causes somesmearing across frequency bands that tends to damp out large variations in wave energy overa small frequency. This lessens the effects of spatial variability, but with monochromaticwaves there was no such averaging effect, so the wave climate where measured and at themodel did not correlate as well.47Due to the intrinsic nature of a segmented wave generator, if all of the wavegeneration boards do not perfectly follow their assigned stroke, or if there exist unevenlysized gaps between the wave boards, spatial variations in wave height will also occur. Thisis to say that wave heights will differ along the crest of a wave, and also along the path of awave if a point on the crest is followed. Since wave board stroke lengths were small, anydifference in displacement would be relatively more important. In addition, if the depth ofwater in the basin were different from what was assumed when the wave board drive signalswere calculated, the angle of wave propagation would not have been correct. As the wavebasin slowly leaked water over the course of a day, and the floor varied in elevation in theorder of several percent of water depth, this undoubtedly occurred to some degree. All ofthis shows that the measured wave climate and that which actually occurred at the modelwere not necessarily the same. This was demonstrated by Shaver (1989), who performed aseries of regular wave tests in the multi-directional basin, and also in this study duringinformal measurement of waves at different points in the basin, where wave heights werefound to vary by as much as a third.The resonance peak in surge was also the location of discrepancies in the numericalsolutions. As clearly seen in Figure 5.2, linear diffraction theory tends to predict anunrealistically high resonance peak for the standard mooring condition. This is likely due toa number of factors ignored; the first and most obvious of which is viscosity, which causesforces which move in phase with and opposite in direction to vessel motion relative to thewater. As models become smaller, the ratio of viscous forces to inertial forces becomesgreater and viscosity tends to damp out resonance. The other major failing of linear theoryin this case is large amplitude motions. One of the assumptions made in the development ofthe theory states that the body motions are small, and thus all conditions imposed on thebody may instead be imposed at its equilibrium position. The large resonance peak in surgeviolates this condition, so it is not surprising that differences should exist. Anotherconsideration is the nonlinearity of the moorings for high displacements, which also tends to48diminish resonance effects. Figure 5.45 shows the theoretical surge resistance of mooringtype 1, and a clear change in slope may be seen at the point where line pretension ends.Vessel motions in fully reflected seas generally showed good agreement betweenmeasured and calculated motions. Although discrepancies are visible, they are generallyattributable to discrepancies in unreflected seas being carried over. However, onediscrepancy of note concerns heave response in head seas. Here, the locations of thecomputed and measured minimums differ somewhat, and measured long wavelengthresponse is significantly greater. Part of the former is likely due to the coarseness of thecomputational mesh but the main reason for both discrepancies is probably uncertain waterdepths.As previously noted, the basin leaked water over the course of a day. This meantthat the basin was usually filled to slightly over the desired depth and the water level wouldlower with time. If a test were run with the water level higher than assumed, two effectswould be evident. The first is that wave heights would be slightly larger than anticipated,with the second being that wavelengths would be slightly longer than assumed. Since, intests with the reflecting wall, wave heights could not be accurately measured, they weretaken to be the same as measured using the same drive signal when the reflecting wall wasnot in place. If water levels with the reflecting wall in place were indeed higher, calculateddimensionless response would be greater than in reality. Transfer functions for reflection,found in Eqs. [3.38] and [3.40], which are highly dependent on wavelength, would also beaffected. This would explain both the higher experimental values and the discrepancy inlocation of the minimum. However, since both surge and pitch results from the same test,found in Figures 5.38 and 5 40, generally show good agreement, it seems unlikely that basindepths can have differed by a large amount.Overall, the comparison of computed and measured response can be divided into twocategories: results in surge, heave and pitch, and those in sway and roll. Results in the49former category are generally quite good, especially once the additional 5% damping isadded. Here, the difference between computed and measured response is usually less than10%, especially when compared with results of random tests away from the resonancepeaks. However, the same cannot be said for sway and roll results. Agreement withexperiment at short wavelengths is reasonable, but for longer wavelengths, experimentalvalues are much higher. There is one previously noted exception to this. When the width ofthe boat was increased by 50% as one of the perturbations, computed sway response agreedvery well with experimental results for the unwidened boat, and roll response wassignificantly larger for higher wavelengths.The most likely explanation for this seems to be that widening the boat made for abetter representation using Green's functions. It had been noted that with the standarddiscretisation, some source strengths were very high, while with the widened hull thesevalues were much lower. It seems possible that, since some facets were so close together, afiner grid for representation of the subsurface profile, especially on the keel and rudder,would have given better results.However, the other, and more obvious, explanation for the lack of good sway androll results is viscous effects on the keel and rudder. Forces caused by flow separation werenot taken into account in the numerical model and undoubtedly were of importance but, asstated previously, experimental results were quite linear, which would not be expected ifturbulent viscous effects were important. Still, if Figures 5.15 and 5.19 are closelyexamined, it will be seen that the random results occupy a smaller portion of the graph thanis normal, as the ends of each range tended to diverge and were not plotted.The effects of the perturbations were varied. Heave was very robust in that it did notappreciably change for any difference in vessel properties. Surge was greatly affected byboth mooring changes and the widening of the boat, as the natural frequency of the systemwas changed. The only other perturbation with any significant effect on surge was the50additional 5% damping, which acted to reduce the resonance peak to a reasonable level.Pitch was not greatly affected by anything except as it affected the anomaly at the locationof surge resonance. Mooring had uncertain effects on sway, possibly due the coarseness ofhull discretisation, but the widening of the hull may have remedied this in that it gave valuesmore in line with experiment. Softer mooring lines appeared to generally give higher rollresponse, as did widening the boat hull. An increase in moments of inertia and height of thecentre of gravity shifted the location of the roll resonance peak to a lower frequency. Adecrease in mooring stiffness seemed to increase yaw response at high wavelengths, butlittle else may reliably be concluded for this mode of motion.5.6.2 Wave ReflectionThe expressions of section 3.2.3 are valid for normal reflection for any angle of waveincidence on a vessel, or for oblique wave reflection with the vessel either parallel orperpendicular to the reflection source. All expressions assume symmetry fore and aft withthe exception of oblique reflection with the vessel parallel to the reflecting wall, whichrelaxes this condition. This assumption of symmetry is a good approximation to reality andshould not induce undue error if the differences fore and aft are not large.Considering Eqs. [3.38] and [3.40], it is readily obvious that both are highly sensitiveto both wavelength and distance from the reflecting wall. For example, an eight secondwave in 3 m of water has a wavelength of 42 m. If a boat were to be thus moored at adistance of 42 m from a reflecting wall which was experiencing full normal reflection, it isseen from Eq. [3.38] that its roll motions from incident waves with eight second periodwould be very small. However, if a 7 second wave were instead incident, roll magnitudeswould be 1.6 times greater than the unreflected case. Furthermore, if the water depth were 5m with the original 8 second wave incident, roll motions would be nearly double theunreflected values due the change in wavelength.51This poses a dilemma. Varying water depths due to tides and seasonal variationsmake it so that a wave of constant period and incident angle may, at one location of thebasin, at times have its effects amplified and at other times find them diminished. A wave ofslightly different period may produce completely opposite effects. Lastly, the spatialvariation of reflection effects is very high.There are several possible solutions to this. The first assumes the mean amplitude ofeffective wave height is given by Eq. [3.45] as 'N/ 1+R2 times the standard deviation ofsurface elevation. There will be areas where wave action is higher or lower than normal, butthis gives a good general idea of the wave severity. The other method is to assume the worstcase scenario and consider the effective wave height to be (1+R) times that incident. Thiswill overestimate wave strength in most areas but will never underpredict it. Where each ofthese would be applied depends upon the situation, and necessitates a knowledge of wavelimit criteria. It will therefore be discussed in section 5.7, in which criteria are examinedand improvements suggested.5.7 Extension of Results to Full ScaleBefore they may be used, all results must first be converted to prototype scale. InFigures 5.2-5.43, all responses are plotted dimensionlessly and may easily be applied at anyscale desired. However, plots such as these imply linearity with response, so one piece ofinformation is left out, namely wave height. To convert general information from one scaleto another with experiments of this type, a scaling factor, here called 2c, is first chosen,usually for length. All time scales are then scaled by a factor of Viand mass scales by afactor of 2c3, as is seen from the dimensional analysis of section 3.1. It must again be notedthat due to the fact that water is utilized for both model and prototype, viscous effects are notscaled in this method, a limitation which may or may not become important. It is of somesignificance for roll, sway, and yaw motions, and would likely affect the magnitudes to52some degree, but not the general trends. Surge, pitch and heave responses are not likely tobe as greatly influenced by viscous scale effects.For an example of scaling, the results of model tests applied to a prototype scaletwelve times greater, (x = 12), would then be representative for a boat with a waterlinelength of 12x0.692 m = 8.3 m and mass of 123x1.61 kg = 2780 kg. The location of theresonance peaks in surge, found at periods of about 1.2 s and 1.9 s for mooring types 1 and 2respectively, would be changed to 4-12x1.2 s = 4.2 s and 412x1.9 s = 6.7 s, in the rightrange for small craft (Le Mehaute (1977)).These surge resonance periods are highly dependent on mooring configuration,which in the model case consisted of a pretensioned four-point system, shown in Figure 4.4,and designed to have highly linear response. While real boat-mooring systems will havesimilar scaled periods of resonance in surge, they are unlikely to use this configuration.Figures 5.46-5.50 show examples of common mooring configurations, which are morelikely to use an asymmetric shape with slack-elastic behaviour. These have highly nonlinearresponse arising from both the geometry and slackness of the mooring lines, and from thenonlinear stress-strain behaviour of nylon rope at large displacements, which is pictured inFigure 5.51.Because full scale mooring systems usually have slack-elastic behaviour, in order tohave a natural period in surge in the same scaled range as the model, the full scale mooringsetup must be proportionally much stiffer in the elastic range, as may be seen in Table 5.2.Stiffnesses at small displacements for commonly used sizes of nylon rope are up to fourtimes greater than the scaled stiffness of mooring type 1. This obviously provides adifferent response, especially in surge and sway, where the natural period of resonance itselfis dependent on the amplitude of motion.53The effect of this slackness may easily be shown by considering the system in Figure5.52, where a one dimensional system of mass M and stiffness Ice experiences periodicmotion with a magnitude of Ae and a slack length of b. The amplitude of the elastic motionis therefore (Ae-b)/2 or a. If there were no slack or damping, the system would experiencefree vibration of the formx = acos(w,t)^[5.1]with corresponding velocityi =—acoesin(coet)^[5.2]whereArg:eW = m. [5.3]is the circular natural frequency of the system.The maximum velocity acoe would be reached at the static equilibrium positionwhere restoring force is zero. Now imagine the undamped system in free vibration withslack b and an initial elastic displacement of a. The maximum velocity will still be awe andthe time to cross slack b would be tb = b/awe. The total time to complete one period wouldbeTo =24 +27rIco,=2-NIMIke[2bAA, — b)+ 7r]which may be referred to as the effective amplitude dependent natural period. The effectiveamplitude dependent stiffness may be then estimated as\2R. ke = ke (20(21, — b)+ ir) .54[5.4][5.5]Knowing this, the motion of a single degree of freedom system may be estimated insinusoidal form by substituting the effective amplitude dependent stiffness into the SDOFharmonic equations of motion and performing trial and eiror convergence with the standardsolutions (see Clough and Penzien (1975) ch. 4). This approximation is of course moreaccurate as the slack length becomes less. The results can be very interesting in thatsometimes there may be multiple solutions as shown in Figure 5.53. Both may be valid, butthe one that occurs will depend on initial conditions.For example, consider an system (undamped for simplicity) with a mass of 1 actedupon by a sinusoidal force with unit amplitude and frequency. The system has a nonlinearrestoring force with a total slack length b of 2.2, followed by a linearly elastic spring ofstiffness 2. This system may experience two different steady state responses. The firstoccurs with the object oscillating back and forth in the slack portion of the system with anamplitude of 1, not reaching the elastic part of the curve. The second response showsresonant behaviour, with the body venturing well into the elastic portion of the curve andexperiencing an amplitude of motion of 3.8. Alternatively, the system may experiencechaotic behaviour, with the system jumping back and forth between the two levels inresponse to linear excitation.These possible multiple amplitudes of motion along with high nonlinearity ofresponse with wave height (mainly in surge and sway) makes it more difficult to apply theresults of random waves and reflection, although the latter is more easily handled asdescribed in the previous section.It is a more difficult problem to characterize the effects of random waves. Due tononlinearity of response, some vessel motions will not follow a Rayleigh distribution.However, incident wave height probability distribution is not affected by any of this andmay still be characterized by the results of section 3.2.2 if applicable. This leads to apossible solution to the problem in the form of a characteristic wave height which may be55applied to the various modes of motion. These characteristic heights would cause acharacteristic movement which could be applied to motion limits. In all cases, if more thanone solution for motion amplitude exists, the result with greatest magnitude should beconsidered.For modes such as heave and pitch, where excessive motion may cause discomfortand anxiety, the significant wave height 1/,, or Ho, seems to be the best height to use tochoose a characteristic motion caused by a sea state. For roll, motions may be either anuisance, as with heave and pitch, or a hazard to boat integrity, as may happen if sailboatmasts become tangled. For the former case, it seems best that the significant wave heightagain be used, but with the latter, a higher standard seems more appropriate. A bettermeasure for this would be Hu , the height exceeded by 10% of the waves in a train, which isequal to about 1.1 times the significant wave height. This is not a huge difference, but ismaterial. This statistic also seems appropriate for motions in surge and sway during largestorms, which are of the most concern when they cause lines to come loose or physicaldamage to the boat. Surge and sway are still of concern on a day to day basis, asembarkation and debarkation may be made difficult, but here they should be put back intothe nuisance category, as no real damage is likely to occur, and thus the significant waveheight again would be best used to define a characteristic vessel motion.5.8 Acceptable Wave Climate-Limits of MotionAs shown in chapter 2, for many years a rule of thumb by marina designers andoperators has been that wave heights inside a marina should be less than about 0.3 m (1 foot)to ensure safe berthing conditions. Although easy to use, this criterion needed improvement.Accordingly, new wave limit criteria were introduced by Northwest Hydraulic Consultants(1980) to take account of wave direction, period, frequency of occurrence and the quality ofmoorage desired. An extension to this was given by Taylor (1983), who considered surgeand sway motions exclusively and attempted to include reflection effects.56The limits of motion presented by Northwest Hydraulic Consultants are semi-empirical, relying on vessel motion tests and user experience supplemented by limitednumerical analysis to determine response in a variety of conditions. This was applied toacceptable motion limits, which were chosen on the basis of serviceability and integrity ofthe vessel. An identical approach was used by Taylor, who used most of the sameequipment but a somewhat more sophisticated numerical analysis.Both sets of experimental results follow in form the results of the presentexperimental and numerical analyses, with mainly minor differences. The majordiscrepancies occur in surge and sway, due to the large influence of moorings. Since full-scale vessels would be more likely to use mooring methods found in the previous studies,and numerical perturbations performed do not offer great enough differences to justify theadoption of entirely new standards, possible extensions will instead be suggested based onthe results of the present investigations.Both studies were generally well conducted and complete, but are not withoutshortcomings and possible improvements. Both were conducted in a basin that experiencedconsiderable reflection at some periods which was not properly accounted for, althoughsome attempt was made by Taylor. The fifty year wave limits by Northwest HydraulicConsultants do not have as firm a basis as might be wished due to assumed linearity in aregion where nonlinear effects were highly pronounced, and reflection was properlyincluded in neither set of wave criteria. Reflection was mostly ignored by NorthwestHydraulic Consultants, while Taylor included it by giving a reduction factor linearlycorresponding to the reflection coefficient, as shown in Table 2.2. This factor is valid onlyat the node or antinode of a reflected train, depending on the mode of motion considered,and is similar to one of the options considered in section 5.6.2.In order to develop a method in which reflection effects may be incorporated intowave limit criteria, it is necessary to have a knowledge of the forms these limits take. Both57the criteria developed by Northwest Hydraulic Consultants and by Taylor have guidelinesfor three return periods: one week, one year, and fifty years. The purposes of the one weekand one year limits are to prevent excessive motion and boat damage on a regular basis,while the purpose of the fifty year criteria is to prevent catastrophic damage. As such, itwould be best to use Eq. [3.45} as discussed in section 5.6.2 to apply to one week and oneyear conditions. These return periods will deal with shorter wavelengths than the fifty yearevent, so there will tend to be less correlation between incident and reflected wave phaseand, hence, greater spatial variation in effective wave height. As well, the one week and oneyear events are not of the sort for which one localized area of larger than normal effectivewave heights would be of utmost importance.However, this is completely different for the fifty year criteria. Here, larger thanaverage effective wave heights could cause great damage to moored small craft, which iswhat the criteria try to prevent. As well, with longer wavelengths arising from the fifty yearstorm, increases and decreases in local wave action will become more defined. Therefore,the worst case scenario of section 5.6.2, which increases effective wave height linearly withthe reflection coefficient and is equivalent to Taylor's factor, should now be applied.There is one situation to which neither of these should be applied: that of vesselsmoored near a reflecting wall with long wavelengths incident. For the most commonsituation of boats moored parallel to the wall, roll and sway will be greatly reduced, whilepitch, surge, and heave will increase. Here, the results of section 3.2.3 may be applied withthe confidence that small changes in water depth or wave period will not substantially affectthem. Vessels moored perpendicular to the wall will experience greater variability, as theyare likely in this case to be further out and their sensitivity to wavelength is thus increased,but section 3.2.3 still applies. In both situations, the effective wave height should beincreased by 0.1 or 0.2 over that calculated to account for frequency spreading effects, andshould never be less than 0.5 or greater than (1+R).58Another area of concern deals with the effects of random waves. Both of the above-mentioned studies dealt solely with sinusoidal wave trains, the results from which must becarefully considered before applying them to irregular waves. As discussed in the previoussection, the best way to do this seems to be by choosing an equivalent sinusoidal waveheight causing a characteristic movement, which may then be applied to the limits ofmotion. Looking at the Northwest Hydraulic Consultant criteria in Table 2.1, for the onceper week limits, the operative concern is the nuisance factor and thus, instead of reading Hto be less than some limit, it should be instead Hs. For the one year return period, nodamage is to be tolerated, but still Hs seems to be the proper irregular wave statistic to applyto the wave criteria. The fifty year case is something different. Its aim is to prevent seriousdamage and, thus, applying Hs to the wave criteria would allow many waves considerablylarger than intended. It seems better to apply some sort of reduction factor and let WIapply to the wave criteria, where Hs is about 10% less, bringing motions more in accordancewith intentions.Table 2.2 gives Taylor's criteria with respect to irregular waves, which are somewhatmore difficult. These consider only surge and sway, and hence are not as broad as thoseprevious. They factor in wavelength directly, giving a gradual change in acceptable waveheight over the ranges considered. However, for head seas, the acceptable wave height forone week return period is only one third less than that for fifty years. Intuitively, this seemswrong. Indeed, by using the quality of moorage factors given, the fifty year standards at onemarina could be lower than the one week limits at another. As with Northwest HydraulicConsultants, limits of motion in surge from which the criteria are derived are given as ±0.5,1, and 2 feet for 1 week, 1 year, and 50 year return periods respectively. However, listedwave standards applied to these limits do not match up with the experimental data given.These show a response amplitude operator which decreases with increasing height, leadingto a situation in which the 50 year limits should be more than four times larger than the oneweek limits. This is not the case and, unfortunately, it is stated that the basis for these limits59came from numerical simulations, the results of which are not given. There is therefore nochoice but to disregard all criteria for head seas. The criteria given for beam seas seem morereasonable, but in light of their source must also be viewed with caution. If they are to beused, it is recommended that Hi be replaced by Hs for 1 week and 1 year return periods, andby H01 for the 50 year return period, as in the Northwest Hydraulic Consultants limits. It isalso further recommended that the reflection factor as shown be included through themethod introduced several paragraphs earlier.Both of the criteria reviewed are only for head and beam seas, with no explicitcriteria for oblique angles. In the Northwest Hydraulic Consultants report, it is suggestedthat appropriate motions will decrease with the cosine of the incident angle from head orbeam seas. This has some theoretical basis, and is in general accordance with resultspresented earlier. However, to account for nonlinear response, a small modification shouldbe made such that the effective incident wave height instead of the motions will decrease foruse in the wave limits criteria, a small but relevant point.A final area of concern is the applicability of results to more general vessels, whichmay differ widely. Perturbations about the standard condition performed during numericalanalysis show small changes in peak heights and locations, but the general form remainsconstant, and it is not felt that there is any need to modify wave height criteria based onthese. In Northwest Hydraulic Consultants (1980), several different hull shapes were alsoanalyzed, including examples of both planing and non-planing powerboats, with similarresults.606 Conclusions and RecommendationsThe investigations of the previous chapters lead to a number of conclusions. Theserelate to assessments of experimental techniques, their use, and the results obtained.Numerical analysis of vessel motion is scrutinized, suggesting conditions leading to optimalperformance. Recommendations are made regarding the treatment of irregular, reflected,and oblique wave trains with respect to wave criteria, which themselves are examined.An important conclusion to be made from the experimental program is that, with theexception of conditions close to resonance, motions vary linearly with wave height, withresults of random wave tests especially giving good results. This linearity applies itselffairly well even to sway and roll motions, where viscous forces might be expected toincrease with the square of the wave height. However, yaw never exhibited linearity and,under sinusoidal excitation, featured well defined responses at several harmonics.The segmented wave generator provided great ease of operation with respect tochanging incident wave angles and minimizing reflective effects. However, due to spatialvariability in the wave field, it led to significant difficulties in measuring wave heights. Inthe future, if such a basin is to be used for similar experiments, it would be preferable toperform all wave measurements before the model is installed, at the exact spot where it is tobe placed.The SELSPOT system proved to be highly acceptable for the measurement of modelmotions with very little interference. It provided very accurate six-degree-of-freedomtracking, with an adequate field of view, and allowed a good range of model motion. Forexperiments of this type, its use is strongly recommended.The numerical model using linear floating body theory through WELSAS3 andFACGEN3 generally proved to be useful in predicting model motions. However, peak61responses in surge were highly overpredicted unless extra damping amounting to 5% ofcritical (based on diagonal mass, added mass, mooring and hydrostatic stiffnesses) wasadded. This additional damping may be considered an approximation to viscous forcesalthough, as stated earlier, these forces would, in general, be of a different form. However,this damping is recommended for future studies in the frequency domain, as it has littleeffect on any motions except for surge near the resonance peak, which it reduces toreasonable levels. Discretization of the keel was likely too coarse in the numerical model,leading to some very high source strengths, which probably contribute to discrepancies withrespect to sway and roll measurements. In any future use of these applications, special careshould be taken to ensure that numerical resolution is adequate. Effects of perturbationsabout the standard shape were generally small, except for the effects of the moorings onsurge and sway. This suggests that good mooring practice is the best way to control thesemotions. Two other significant changes are the previously mentioned additional viscousdamping, and the effect of widening the boat, which brought model motions more in linewith measurements at higher wavelengths in beam seas.The effects of random waves may best be accounted for by choosing a characteristicwave height to be applied to the criteria developed on the basis of regular waves. For oneweek and one year criteria, this should be the significant wave height, Hs, whereas for thefifty year criteria, all heights should be replaced with H01.Reflection effects should be included in wave criteria by a factor which, whenmultiplied by the incident wave height, returns the effective wave height. For one week andone year criteria, this factor should be 111+R2, whereas for the fifty year criteria, this shouldbe (1+R). However, for the case of a boat moored closely to a reflection source in longwaves (1 5 L16) the effective wave height factor should be calculated as follows:62For normal reflection with any angle of wave incidence on the boatHiR=H(-41 +R2-2cos(2k1)+0.2)for all modes but heave and yaw, which are not likely to be limiting factors, with Hi beingthe incident wave height, and HiR the effective wave height.For oblique reflection with the boat parallel to the reflecting wallHiR=Hbl 1+R2+2cos(2k1 cos(y))-1-0.2) for head seasHiR=HiNl+R2-2cos(2k1 cos(y))+0.2) for beam seasFor oblique reflection with the boat perpendicular to the reflecting wallHiR=HiN 1+R2-2cos(2k1 cos(y))+0.2) for head seasHiR=HiN1+R2+2cos(2k1 cos(y))-f0.2) for beam seasAll factors carry the limitations HiR 0.5 Hi and HiR (1+R)H1.As suggested in passing by Northwest Hydraulic Consultants (1980), oblique anglesof wave incidence should best be treated by decreasing the effective wave height as with thecosine of the wave angle relative to either head or beam seas, depending which isconsidered. For example, if a wave is incident at an angle of 200 off the bow of a boat, theeffective wave height with respect to head seas will be cos(20°) times the incident height.The effective wave height with respect to beam seas would then be cos(90°-20°) times theincident height.Overall, the Northwest Hydraulic Consultants criteria (Table 2.1) are reasonablygood, although they may be improved through the above-mentioned extensions. However,due to assumed linearity in a region where nonlinearities are highly pronounced, the basisfor the fifty year criteria for head seas is not as strong as for the one week and one yearreturn periods. Due to a serious error in Taylor's (1983) limits for head seas, they should not63be used, and his criteria for beam seas should be viewed with caution. If used, they shouldalso be modified in accordance with the above extensions.64Appendix A: Estimation of Centre of Gravityand Moments of InertiaIn order to determine the dynamic properties of the model, tests were conducted as describedin section 4.3.4. In the following section, calculations are given for the estimation of the locationof the centre of gravity, moments of inertia in roll and pitch, and moment of inertia in yaw of thetested model.A.1 Location of Centre of GravityAs described in section 4.3.4, the model was built with an integral rigid aluminum frameextending out of the boat which allowed it to rock freely in either roll or pitch whenappropriately placed on a hard surface, as shown in Figure 4.7. Under an applied externalmoment it would displace to a new equilibrium position dependent on the mass of the model,location of the centre of gravity and magnitude and sense of the moment, shown schematically inFigure A.1. The moment balance equation may be writtenMl cos(e) + m(x'cos(e)+ y'sin(0)) = M212 cos(0).^[A.1]whereM is the model massM1, M2 are the masses added to apply moment12 are the x-positions of masses ml, m2 with respect to the pivot point0 is the angle made with the horizontalx' is the x-position of the centre of gravity with the pivot point as originy' is the y-position of the centre of gravity with the pivot point as origin65All quantities may be measured except for x' and y'. However, two tests with different values ofMI and M2 will yield different values for 0, and give simultaneous equations in x' and y', whichare easily solved.Example: ii = 41.9 cm 12 = 38.9 cm M = 1610 gTest 1:^= 0 g M2 = 0 g 0=16.096°Test 2: M1 = 33 g M2 = 0 g 0=23.706°Substituting into [A.1]1610(x'cos(16.096°)+ y'sin(16.096°))= 033(41.9)cos(23.706°)+1610(x'cos(23.706°)— rsin(23.706°))= 0^[A.2]These solve to y' =-5 .70 cm x' = 1.65 cm.A.2 Moments of Inertia in Pitch and RollMoments of inertia in pitch and roll were determined in identical manner from the results ofcompound pendulum tests, using the rocking surfaces of the boat frame as pivots (Fig. 4.3). Forsmall displacements, the natural frequency of motion of a compound pendulum like this is givenasW2 = MgS n MS2 +[A.3]where s =11.e2 + Y2 and I is the moment of inertia of the mode considered. Knowing all othervariables, the moment of inertia may now be calculated.Example: Natural Period=1.287 ss=0.0587 mM=1.610 kg66Substituting into [A.3]( 2 X \2 1.610(9.81)(0.0587) ‘1.287) — 1.610(0.0587)2+! .^ [A.4]This solves to I = 0.0034 m2kg.A.3 Moment of Inertia in YawAs described in section 4.3.4, the moment of inertia in yaw was found from the natural periodof oscillation of the boat with restoring force provided by springs acting through the centre ofgravity as shown in Figure 4.8, and schematically in Figure A.2. For a small displacement of theattached springsFR = 2ksin2(a)Ay^ [A.5]whereFR is the restoring force at one endk is the spring constanta is the angle at which the springs are attachedAy is the displacement at that end due to a small yaw motionTherefore the total restoring moment will beMR = 2k sin2(a)(42 + ci)Ayaw^ [A.6]orMR = KeAyaw^ [A.7]where1C9= 2k sin2(a)(42 + d )^ [A.8]67and di, d2 are the distances from the origin to the attached springs as shown in Figure A.2.This gives a natural period of2 Ko Wm=I+IA[A.9]where IA is the added moment of inertia in yaw due to the testing equipment. The aboveequation may be solved for I+IA, and IA itself may be estimated as/A = MA14 + MA24•242^ [A.10]with dA 1, dA2 the distances to the centres of gravity of the attached masses. I may now becalculated. It must be noted that this method is strictly true only when the object in question issymmetric fore and aft. However, although the model was not symmetric in this way, there wereonly small differences, and the above process should be a good approximation.Example: k = 16.2 N/mdi =0.413m, d2 = 0.421ma = 36.5°MA1 = MA2 = 0.042 kgdAi = 0.395 m, clA2 = 0.419 mcon = 11.35 rad/sKe = 2(16.2)sin2(36.5°)((0.413)2 + (0.421)2) [A.11]= 3.995 Nm/rad.Substituting into [A.9]= 0.03104 [A.12](01'133995)52^m2kg.From [A.10]IA= 0.42((0.395)2 + (0.419)2) = 0.0139 m2kg. [A.13]Thus I = 0.03104 m2kg - 0.0139 m2kg = 0.0171 m2kg.68ReferencesBentley, L. H., 1989. Design and Evaluation of Marinas, Marinas, Design and Operation:Proceedings of the International Conference on Marinas, Southampton, U.K., Vol. 2, pp.223-239.Boc, S. J., Farrar, P., and Chen, H. S., 1989. Numerical Modeling Investigation of WaveResponse of AGAT Small Boat Harbour, Guam, Marinas, Design and Operation:Proceedings of the International Conference on Marinas, Southampton, U.K., Vol. 2, pp.379-400.Bruun, P., 1976. Port Engineering, Gulf Publishing Co., Houston, Texas, USA.Cartwright, D.E., and Longuet-Higgins, M.S., 1956. The Statistical Distribution of theMaxima of a Random Function, Proceedings of the Royal Society A, Vol. 237, pp. 212-232.Clough, R. and Penzien, J., 1975. Dynamics of Structures, McGraw-Hill Book Company,New York.Cox, J. C., 1989. Breakwater Attenuation Criteria and Specification for Marina Basins,Marinas, Design and Operation: Proceedings of the International Conference on Marinas,Southampton, U.K., Vol. 1, pp. 139-155.Dunham, J.W., and Finn, A.A., 1974. Small Craft Harbours: Design, Construction, andOperation, U.S. Army Corps. of Engineers, Special Report No. 2, Coastal EngineeringResearch Center, Fort Belvoir, Virginia.Fisheries and Oceans, 1985. Guidelines of Harbour Accomodation, Small Craft HarbourDirectorate, Ottawa, Canada.Fournier, Charles P., Mulcahy, Michael W., Chow, K. Ander, and Sayao, Otavio J., 1993. AField Monitoring Program to Determine Wave Agitation Criteria for Fishing Harbours inAtlantic Canada, Proceedings of the 1993 Canadian Coastal Conference, Vancouver,British Columbia, Vol. 1, pp. 27-38.Hasldnd, M. D., 1953. Oscillation of a Ship on a Calm Sea, English Translation, Society ofNaval Architects and Marine Engineers, T & R Bulletin 1 - 12.69Huntington, S. W., and Thorn, M. F. C., 1989. Marina Hydraulics for Optimal Design,Marinas, Design and Operation: Proceedings of the International Conference on Marinas,Southampton, U.K., Vol. 2, pp. 213-222.Isaacson, M, and Mercer, A.G., 1982. The Response of Moored Small Craft to WaveAction, Proceedings, 18th International Conference on Coastal Engineering, Cape Town,South Africa, Vol. 3, pp. 2723-2742.John, F., 1949. On the Motion of Floating Bodies. I, Communications in Pure and AppliedMathematics, 2, pp. 13-57.John, F., 1950 On the Motion of Floating Bodies. IL Communications in Pure and AppliedMathematics, 3, pp. 45-101.Jensen, 0.J., Viggosson, G., Thomsen, J., Bjordal, S., and Lundgren, J., 1990. Criteria forShip Movements in Harbours, Proceedings, 22nd International Conference on CoastalEngineering, Delft, The Netherlands, Vol. 3, pp. 3074-3087.Kamphuis, J.W., 1979. Hydraulic Design of Small Craft Harbours, ConferenceProceedings, Coastal Engineering, Design and Construction, Queen's University, Kingston,Ontario, Canada, pp. 133-152.Kim, W. D., 1965. On the Harmonic Oscillations of a Rigid Body on a Free Surface,Journal of Fluid Mechanics , 21, pp. 182-191.Kim, W. D., 1966. On a Freely Floating Ship in Waves, Journal of Ship Research, 10, pp.182-191.Le Mehaute, B., 1976. Wave Agitation Criteris for Harbours, Proceedings, Ports '77-Fourth Annual Symposium of Waterways, Ports, Coastal and Ocean Division of ASCE, pp.366-372.Lundgren, H., 1972. Coastal Engineering Considerations, Marinas and Small CraftHarbours, Proceedings of Symposium, Southampton University Press, UK, pp. 77-104.National Research Council Canada, 1989. Coastal and Multidirectional Wave Basins,Public Relations Pamphlet.Newman, J.N. 1977. Marine Hydrodynamics, MIT press, Cambridge, Massachusetts, USA.70Northwest Hydraulic Consultants Ltd, 1980. Study to Determine Acceptable Wave Climatein Small Craft Harbours: Final Report for Small Craft Harbours Branch, Fisheries andOceans, Vancouver, B.C., Canada.Raichlen, F., 1966. Wave Induced Oscillations of Small Moored Vessels, Proceedings, 10thConference on Coastal Engineering, Tokyo, Japan, pp. 1249-1273.Rosen, D.V., and Kit, E., 1984. A Simulation Method for Small Craft Harbour Models,Proceedings, 19th International Conference on Coastal Engineering, Houston, Texas, USA,Vol. 3, pp. 2842-2856.Sarpkaya, T., and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures.,Van Nostrand Reinhold, New York.Shaver, M. D., 1989. Regular Wave Conditions in a Directional Wave Basin, MA.Sc.Thesis, Department of Civil Engineering, University of British Columbia, Vancouver,Canada.Streeter, V.L., and Wylie, B.E., 1981. Fluid Mechanics, McGraw-Hill Ryerson Limited,Toronto.Taylor, P. S., 1983. The Behaviour of Small Moored Vessels in Surge and Sway, MA.Sc.Thesis, Department of Civil Engineering, University of British Columbia, Vancouver,Canada.Wehausen, J. V., 1971. The Motion of Floating Bodies, Annual Review of Fluid Mechanics,3, pp. 237-268.71Direction and PeakPeriod of DesignHarbour WaveWave EventExceeded Once inFifty YearsWave EventExceeded Once aYearWave EventExceeded OnceEach WeekHead Seas less than2 secondsThese conditions notlikely to occurduring this eventLess than 1 footwave hightLess than 1 footwave heightHead Seas between2 and 6 secondsLess than 2 footwave heightLess than 1 footwave heightLess than 0.5 footwave heightHead Seas greaterthan 6 secondsLess than 2 footwave height or 4foot horizontal wavemotionLess than 1 footwave height or 2foot horizontal wavemotionLess than 0.5 footwave height or 1.5foot horizontalmotionBeam Seas less than2 secondsThe conditions notlikely to occurduring this eventLess than 1 footwave heightLess than 1 footwave heightBeam Seas between2 and 6 secondsLess than 0.75 footwave heightLess than 0.5 footwave heightLess than 0.25 footwave heightBeam Seas greaterthan 6 secondsLess than 0.75 footwave height or 2foot horizontalmotionLess than 0.5 footwave height or 1foot horizontalmotion^_Less than 0.25 footwave height or 0.75foot horizontalmotion* For "excellent" wave climate multiply by 0.75 and for "moderate" wave climate multiply by 125.Table 2.1 Provisionally Recommended Criteria for a "Good" Wave Climate in SmallCraft Harbours (from Northwest Hydraulic Consultants Ltd. (1980))72..Wave Period-(s)WaveDirection nC Value1 Week Return I 1 Year Return I 50 Year Return0-2 Head -- -- -- --Beam 1.2 300 300 4502-6 Head 1 400 500 600Beam 1.5 100 200 3006+ Head 1 400 500 600Beam 1.7 100 200 ,^300C values represents the combined effect of wave height, wave reflection, and water depth such that:(1+KR) —^in min.tanhn(kd)Insignificant vessel response to incident waves* For "excellent" wave climate multiply by 0.75 and for "moderate" wave climate multiply by 1.25.Table 2.2^Provisional Recommended Criteria for "Good" WaveClimate in Small Craft Harbours for Resisting Vessel Surgeand Sway (from Taylor (1983)).73Location All Recreational BoatsFishing Boats < 15 mFishing Boats > 15 mWithin Harbour Entrance 1.00 m 1.00 mMooring Basin 0.50 m 1.00 mBerthing Area 0.25 m 0.50 mTable 2.3 Allowable Maximum Significant Wave Height(from Fisheries and Oceans (1985))Class of HarbourPercentage of the Timewhen the Wave HeightCriteria may be ExceededVessel Metres*,Fishing HarbourVessel Metres*,RecreationalHarbourA0.17 >800 >800B0.87 300-900 100-900C1.74 <400 <200DNo Limit Indicates Potentialfor DisposalIndicates Potentialfor Disposal* Vessel Metres -^The number of vessels using a harbour in a typicalmonth multiplied by the average length of such vesselsTable 2.4 Classification of Harbours(adapted from Fisheries and Oceans (1985))74Location Threshold Significant WaveHeightFrequency of OccurrenceService/ Offloading Wharf 0.40 m 1.0-2.5%1.0-2.5%Mooring Basin 0.50 mTable 2.5 Recommended Allowable Wave Agitation Criteria forSTACAC Class 2 and Class 3 Fishing Vessels(from Fournier et al (1993))75S mbol Descri 'don Value/7.01 Average height o^ig est 1%of waves 6 .7 anHin Height exceeded by 1% ofwaves 6.1 an1702 Average height of highest 2%of waves 6.2an402 Height exceeded by 2% ofwaves 5.6 angm Average height of highest10% of waves 5.1 anH0.1 Height exceeded by 10% ofwaves 4.3 anH =if$^% Significant wave height(Average height of highestthird of waves)4.0 anH= Average wave height 1friTan14.5 Median wave height 2.35 anHmode Most probable wave height 2.0 anH,-42anN76Table 3.1 Commonly Used Wave Height ParametersSetu s 1 and 2Test Full ScalePeriod (s)Full ScaleHeight (m)Sea State Incident Wave_ Angle (deg)Mooring Type1-6 3-8 0.4 Regular 0 17-8 6 0.15, 0.45 Regular 0 19-10 4 0.15, 0.30 Regular o 111-12 4,6 0.3 Regular 0 213-14 4.5, 6 0.3 Random o 115-20 3-8 0.3 Regular 45 121 6 0.15 Regular 45 122-27 3-8 0.3 Regular 90 128-29 6 0.15, 0.45 Regular 90 130-31 4 0.15, 0.45 Regular 90 132-33 4,6 0.3 Regular 90 234 6 0.3 Random 90 1Setun 3Test Full ScalePeriod (s)Full ScaleHeight (m)Sea State,Incident WaveAngle (deg)Mooring Type35-40 3-8 0.3 Regular 90 141-42 6 0.15, 0.45 Regular 90 143 4 0.15 Regular 90 1^.44-45 4.5, 6^_ 0.3 Random 90 1Setun 4Test Full ScalePeriod (s) _Full ScaleHeight (m)Sea State_Incident WaveAngle (deg) —Mooring Type46-51 3-8 0.3 Regular 0 152-53 6 0.15, 0.45 Regular o 154 4 0.15 Regular 0 155-56 4.5, 6 0.3 Random 0 177Table 4.1 Experimental ConditionsSetup # Period (s) * -J6 Motion Number ofPointsRegression_ Coefficient R21 6 Surge 4 0.9991 6 Heave 4 0.9821 6 Pitch 4 0.9951 4 Surge 4 0.8601 4 Heave 4 0.9991 4 Pitch 4 0.9792 6 Sway 4 0.9702 6 Heave 4 0.9992 6 Roll 4 0.9752 4 Sway 3 0.9492 4 Heave 3 0.99972 4 Roll 3 0.9923 6 Sway 4 0.9643 6 Heave 4 0.9813 6 Roll 4 0.9923 4 Sway 3 0.9493 4 Heave 3 0.99973 4 Roll 3 0.9924 6 Surge 4 0.9884 6 Heave 4 0.9894 6 Pitch 4 0.9924 4 Surge 3 0.9384 4 Heave 3 0.99954 4 Pitch 3 0.99978Table 5.1 Linearity of Response With Wave HeightForce @ 5% Elongation (N) J^Restraint Type2100 Nylon, 20 mm 8 Strand Plait2450 Nylon, 13 mm Double Braided5800i1Nylon, 20 mm Double Braided^11560 Mooring Type 1 **** Stiffness scaled from length factor x=12, 5 m length assumed for elongationTable 5.2 Comparison of Model Mooring Stiffness in Surge with Full ScaleElastic Behaviour of Commonly Used Ropes79Figure 3.1 Definition Sketch for Vessel MotionsFigure 3.2 Definition Sketch for Oblique Wave Reflection80-, -Figure 3.3 Variation of Motion Amplitudes with Full, Normal ReflectionFigure 3.4 Variation of Motion Amplitudes with Partial NormalReflection, n=-18130 m50 mSEGMENTED WAVE MACHINEGATE83Figure 4.2 Control Room for Multidirectional Wave Basin84Figure 4.3 Boat Model with SELSPOT Frame Attached85Figure 4.4 Sketch of Mooring StructureFigure 4.5 SELSPOT set up for Measurement of Model MotionsFigure 4.6 Determination of Model Centre of Gravity86Figure 4.7 Measurement of the Moment of Inertia in RollFigure 4.8 Measurement of the Moment of Inertia in Yaw8788Figure 4.9 Measurement of Subsurface Profile Using a Digital MicrometerReflectingWallWave Generator/"4*--- Incident Wave89Setup 1^Setup 2i Setup 3^Setup 4Figure 4.10 Mooring Setups 1-4Figure 4.11 Model under Wave Action, Setup 490Figure 4.12 Model under Wave Action, Setup 1Figure 4.13 Model under Wave Action, Setup 3Figure 4.14 Model under Wave Action, Setup 49192Figure 5.1 Facet Representation of Subsurface Hull Shape• Measured Regular ResponseCalculated ResponseRandom Response: gn/L=5 1.0Random Response: gn/L=2 8.7•X Measured Response. Mooring 2^ Calculated Response, Normal Mooring Calculated Response, Mooring 2^ Calculated Freely Floating Response15.012.5 X10.0bo^7.55.02.5............................0.00 0^1.0^2.0^3.0^4.0^5.0^6.0^7 .0^8.0Wavelength/WaterlineFigure 5.2 Surge Response in Head Seas, Mooring Type 10 0^1.0^2.0^3.0^4.0^5.0^8.0^7.0^8.0Wavelength! Waterline93Figure 5.3 Surge Response in Head Seas, Various Moorings5.015.0^ Otandard ConditionsI ^ Widened Boat1 ^ Increased InertiaI --- 5X Viscous Damping12.50.02.5...---...--'111111111,..,".....`.............O 0^1.0^2.0^3.0^4.0^6.0^8.0^7.0^6.0Wavelength/WaterlineFigure 5.4 Surge Response in Head Seas, Various Perturbations^ Floating Dock, Waterline/Depth = 1.46• Floating Dock, Waterline/Depth = 2.19A Fixed Dock, Waterline/Depth = 1.46• Fixed Dock, Waterline/Depth = 2.19•1.0-V•0.75- • v0.5-^•^• •A^vI6^60.25- 6• a I ,fO 0^1.0Wavelength/WaterlineFigure 5.5 Surge Response in Head Seas (from Northwest HydraulicConsultants Ltd. (1980))941.5 _1.25 _:a il 10.0 1 1 1 1 1 1 12.0 3.0 4.0 5.0 6.0 7.0 8.0• ••..---------• .....,,.:.'---^-.....,• Measured Regular Respon•e^ Calculated ResponseRandom Response: gT:A=5 1.0^ Random Reiponse: gT:A=2 8.71 I 1 I I 1 1 I1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Wavelength/WaterlineFigure 5.6 Heave response in Head Seas, Mooring Type 11 i1.6-1.26 -X0.5 -0.25-0.00 0^1.0^2.0X Measured Response. Mooring 2^ Calculated Response. Normal Mooring Calculated Response, Mooring 2^ Calculated Freely Floating Response1 1 1 1 1 I3.0 4.0 5.0 6.0 7.0 8.0Wavelength/WaterlineFigure 5.7 Heave Response in Head Seas, Various Moorings951.51.26-_--0.5 -0.26 -0.00 0I I I I 1 13.0 4.0 6.0 6.0 7.0 8.01.6 ^1.25 -1.0 -14.,0aC 0.76_4)M0.5 -0.25 -0.000^ Standard Conditions Widened Boat^ Increased Inertia--- 5% Viscous DampingWavelength/WaterlineFigure 5.8 Heave Response in Head Seas, Various Perturbations^ Floating Dock, Waterline/Depth = 1.46• Floating Dock, Waterline/Depth = 2.19A Fixed Dock, Waterline/Depth = 1.46• Fixed Dock, Waterline/Depth = 2.19•A^o^1•A96••0.5 -0.25 -1.25 -1.0 -0.75 -0.000I^I^I^I^I^I^I2.0 3.0 4.0 6.0 6.0 7.0 8.0Wavelength/WaterlineFigure 5.9 Heave Response in Head Seas (from Northwest HydraulicConsultants Ltd. (1980))3.6 -• Meaoured Regular Reepontie^ Calculated ReeponieRandom Responee: gn/L=51.0^ Random Reeponee: gn/L=28.73.0 -\•2.4 ^1.8 -1.2 _0.6 -1 1 I I I 1 1 I1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.00.31.51.8 V Floating Dock, Waterline/Depth = 1.46A Floating Dock, Waterline/Depth = 2.19A Fixed Dock, Waterline/Depth = 1.46• Fixed Dock, Waterline/Depth = 2.19971.20.90.6-----^I I I I I 1 I I1.0 2.0 3.0 4.0 5.0 8.0 7.0 8.0Wavelength/WaterlineFigure 5.10 Heave Response in Beam Seas (from Northwest HydraulicConsultants Ltd. (1980))Wavelength/WaterlineFigure 5.11 Pitch Response in Head Seas, Mooring Type 10.000^ Standard Conditiond Widened Boat^ Increatled Inertia--- 5% Vircour Damping2.4 -•^.... \.• .••••••••^\%.•••"'^•^\-.......•••••••\. z.....„ ............." s1 I i i 1 1 12.0 3.0 4.0 5.0 6.0 7.0 8.00.0' I/III//I3.6 -3.0 -1.2 -0.6 -3.6 -983.0 -2.4 -X Mearured Rerponae, Mooring 2^ Calculated Reeponee. Normal Mooring Calculated Rerponse. Mooring 2^ Calculated Freely Floating Reeponse1.81.20.6^_-I I I I I 1 1 i1.0 2.0 3.0 4.0 0.0 6.0 7.0 8.0Wavelength! WaterlineFigure 5.12 Pitch Response in Head Seas, Various MooringsWavelength/WaterlineFigure 5.13 Pitch Response in Head Seas, Various Perturbations0.0004.03.22.41.6-_-_I^I^I0.0^ Floating Dock, Waterline/Depth = 1.46A Floating Dock. Waterline/Depth = 2.19A Fixed Dock, Waterline/Depth = 1.40• Fixed Dock, Waterline/Depth = 2.192•:! A2 A• • •XX• A • • • A• • • B• •• VA. A•Aer •A99t••2XV0.8 _0 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0^7.0 8.0Figure 5.14 Pitch Response in Head Seas (from Northwest HydraulicConsultants Ltd. (1980))/////6.26- /• Measured Regular Response^/^ Calculated ReOponise^ //^5.0-^ Random Response: gn/L=51.0^//^ Random Response: gTpz/L=28.7^/^ •=^ 0/N..th /is^3.75-^ •R / .•tn / up.'/ •/ ..•2.5-^ / ..."0.0 I I 1^1^1 i I^I 0 0^1.0^2.0^3.0 4.0 5.0Wavelength/Waterline6.0^7.0 8.0Figure 5.15 Sway Response in Beam Seas, Mooring Type 1^ Standard Conditiond Widened Boat^ Increased Inertia--- 5% Viscous Damping.0/..........................0.00 0^1.0^2.0^3.0^4.0^6.0Wavelength/Waterline6.0 7.0 8.06.09.752.51.267.56.257.56.25-6.0 X Measured Reaponse Mooring 2Calculated Response Normal Mooring-- Calculated Respon•e Mooring 2P./• 3.76 - ^ Calculated Freely Floating ResponseIt2.5 -X••■••■■•••■•••••1.26 -J"."'" ..........................................................................0.0 1 1 I I^I1.0 2.0 3.0 4.0 5.0^6.0^7.0 8.000Wavelength/WaterlineFigure 5.16 Sway Response in Beam Seas, Various Moorings100Figure 5.17 Sway Response in Beam Seas, Various Perturbations• Measured Regular ResponseCalculated ResponseRandom Response: gr:/L=51.0^ Random Reponse: gn/L=28.7• •0.04.8-4.0-3.2-2.4-0.8-1.8101A1.51.2V• •A• A• VVV• AA i • • A• II^•^••I I••• •^ Floating Dock, Waterline/Depth = 1.46el^• • Floating Dock, Waterline/Depth = 2.19•• A Fixed Dock, Waterline/Depth = 1.46• • Fixed Dock, Waterline/Depth = 2.19•0.9-0.8-0.3-0.03.0^4.0^6.0Wavelength/Waterline6.0^7.0^8.000^1.0^2.0Figure 5.18 Sway Response in Beam Seas (from Northwest HydraulicConsultants (1980))0 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0^7.0^8.0Figure 5.19 Roll Response in Beam Seas, Mooring Type 14.84.0X Measured Response Mooring 2^ Calculated Response Normal MooringCalculated Response Mooring 2^ Calculated Freely Floating Re•pon•e3.22.4 -1.6 -0.8-0.0X10200^1.0^2.0^3.0^4.0^5.0^6.0^7.0^8.0Wavelength/WaterlineFigure 5.20 Roll Response in Beam Seas, Various Moorings0.82.4-3.2-4.0^ Standard Conditions Widened Boat^d Inertia--- 5% Viscous Damping............................ .............. 4.8-0.00 0^1.0^2.0^3.0^4.0^6.0Wavelength/Waterline6.0^7.0^8.0Figure 5.21 Roll Response in Beam Seas, Various Perturbations103V^Floating Dock, Waterline/Depth = 1.489.6• Floating Dock, Waterline/Depth = 2.19• A^Fixed Dock, Waterline/Depth = 1.489.0 A • Fixed Dock, Waterline/Depth = 2.19• •2.4 • 2• A •1.8• V• X•1.2 X^A• AOA• 6• •• 2V0.0XS•0.00 0 1.0 2.0 3.0 4.0^5.0^6.0^7.0^8.0Wavelength/WaterlineFigure 5.22 Roll Response in Beam Seas (from Northwest HydraulicConsultants Ltd. (1980))---_0.0030.0024Om,N= 0.0018.,.0.......... 0.0012zn0.00080.00 0^0.3^0.8^0.9^1.2Frequency (Hz)(a) Incident Regular Wave Spectrum2.41.81.20.60.00 0^0.3^0.8^0.9^1.2Frequency (Hz)(b) Resultant Yaw Spectrum1.5 1.8 2.11 A -,....^_^L 11.6 1.8 2.1Figure 5.23 Example of the Nonlinearity of Yaw Motions in Beam Seas1041.0^2.0^3.0^4.0^5.0Wavelength/Waterline00 6.0^7.0^8.07.00 0^1.0^2.0^3.0^4.0^6.0^6.0Wavelength/Waterline1.5 a^ Calculated Response Normal Mooring Calculated Response Mooring 2^ Calculated Freely Floating Response1.26 -0.751.0--0.5 -0.25 -I I i0.0 1 I I IFigure 5.24 Yaw Response in Beam Seas, Various Moorings1.0^ Standard Conditions Widened Boat^ In^ d Inertia- 5% Viscous Damping0.750.25...4-4.00.0Figure 5.25 Yaw Response in Beam Seas, Various Perturbations8.0105••6.0 7.0 8.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/WaterlineFigure 5.26 Surge Response, 0=45°, Mooring Type 18.06.0 7.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/WaterlineFigure 5.27 Surge Response, 0=45°, Various Moorings15.012.55.02.50.0•Calculated Response, Normal Mooring-- Calculated Response, Mooring 2^ Calculated Freely Floating Response•\7 \7^\7 \7 \7^\7 \7^ \77..,10615.012.65.02.5■.............''''''0.0• Measured Regular Response^ Calculated Response15.0^ Otandard Conditions- Increased Inertia- 5X Viscous Damping12.510 .0• 7.65.02.50.01070 0^1.0^2.0^3.0^4.0^6.0^6.0^7.0^8.0Wavelength/WaterlineFigure 5.28 Surge Response, 0=450, Various Perturbations3.6-3.0 •• Measured Regular Response• Calculated Response1.8 ••^•0.60.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0^7.0^8.0Figure 5.29 Pitch Response, 0=45°, Mooring Type 12.4 -6.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline1 I I 11 i 7.0I8.0^ Standard Conditions^-- Increased Inertia 5% Viscous Damping2.4-\.......\, \1.2-0.6-vz \0.0 13.6-9.0-3.63.0--Calculated Response. Normal Mooring- Calculated Response, Mooring 22.4 - Calculated Freely Floating Response... .-....-.\'..1.8 _ N.-\.--:-...1.2 _..,..„...1,....0.8 - . ...............................0.0 1 1 I I^I^i^I I0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00Wavelength/WaterlineFigure 5.30 Pitch Response, 9=45°, Various Moorings108Figure 5.31 Pitch Response, 0=45°, Various Perturbations0.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0 7.0 8.0^ Calculated Response. Normal Mooring Calculated Response, Mooring 2^ Calculated Freely Floating Reponse...................................................0.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0^7.0^6.0•109• Measured Regular Response^ Calculated Response •••Figure 5.32 Sway Response, 0=45°, Mooring Type 1Figure 5.33 Sway Response, 0=45°, Various Moorings6.00.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline5.0^ Standard Conditions- Increased Inertia- 5% Viscous Damping4.03.02.0......■•1.0.........................Figure 5.34 Sway Response, 0=45°, Various Perturbations■••■,........."'''''"........".8.07.0110Wavelength/WaterlineFigure 5.35 Roll Response, 0=45°, Mooring Type 12.52.0--• Measured Regular ResponseCalculated Response1.5 -•1.0 _ •• •0.5 - ••0.01 I I I^t^I0 0 1.0 2.0 3.0 4.0 5.0 6.0•7.0^8.02.5 -^ Standard Conditions- Increased Inertia- 5% Viscous Damping2.0 -1.5 ^1.0 -....... ...... .............0.50.0-/..--"1^i 1^i2.0^3.0^4.0^6.0Wavelength/Waterline6.0^7.0 8.00 02.5 -2.0 _ljI^1Calculated Response. Normal MooringCalculated Response, Mooring 2--I^1 ^ Calculated Freely Floating Response1.5 - I^1I^1I'I1.0 - I1il0.5 _ ................................■- -i •••..... - ......... .. .. . . . .. . . . . . .... . .... ......I - ' - . --0.01 1 I I I I^I^1^10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00Wavelength/WaterlineFigure 5.36 Roll Motion, 0=45°, Various MooringsFigure 5.37 Roll Motion, 0=45°, Various Perturbations1111 I11 1 11 i0.06.0 7.0 8.00 0^1.0^2.0^3.0^4.0^6.0Wavelength/WaterlineFigure 5.39 Heave Response in Fully Reflected Head Seas,Mooring Type 1, 1/L=1.077.06.0 8.024.020.0---• Measured Regular ResponseCalculated Response, 5% DampingRandom Response: gT2/L=51.0PRandom Re•ponse: gn/L=28.78.0-4.0-•0.00 0^1.0^2.0^3.0^4.0^6.0Wavelength/Waterline^Figure 5.38 Surge Response in Fully Reflected Head Seas,Mooring Type 1,1/L=1.07• Measured Regular Response^ Calculated Response, 5X DampingRandom Response: gT2/L=51.0P^ Random Response: gn/L=28.70.4-•...-*"/ ••• ...."'..,'"/•1122.4-2.0---_-_/ ••tI^t^I^I4.0I• Measured Regular Rerpon•eCalculated Response, 5% DampingRandom Response: gn/L=51.0^ Random Response: gn/L=28.7•1 1 I I I t 11.0 2.0 3.0 4.0 6.0 6.0 7.0Wavelength/Waterline3.22.41.60.00.800Figure 5.40 Pitch Response in Fully Reflected Head Seas,Mooring Type 1,1/L=1.072.0^3.0^4.0 6.0^6.0Wavelength/WaterlineFigure 5.41 Sway Response in Fully Reflected Beam Seas,Mooring Type 1,1/L=0.304.03.2--•2.4 _=/hCItto.1.6 -0.8 _0.0 I0 0 1.0Measured Regular ResponseCalculated Response, 5% DampingRandom Response: gni/L=51.0Random Response: gn/L=28.7, •/ .•• y ..--,-------,7-7'.---*w---'t7.0 8.0I11377775.05.01144.0• Measured Regular ReSponseCalculated Response, 5% Damping-- Random Response: gn/L=51.0^ Random Reoponse: gn/L=28.72.03.01.0-•0.00 0^1.0^2.0^3.0^4.0^5.0Wavelength/Waterline6.0^7.0Figure 5.42 Heave Response in Fully Reflected Beam Seas,Mooring Type 1, 1/I.,=0.309.0- •2.5- • •2.0 - • .---*eaP....// • •C 1.6- • • Measured Regular ResponseCalculated Response. 5/1 Damping1.0--- Random Reeponse: gn/L=51.0^ Random Reiponae: g11/1.=28.70.5-0 0^1.0^2.0^3.0^4.0^5.0^6.0Wavelength/WaterlineFigure 5.43 Roll Response in Fully Reflected Beam Seas,Mooring Type 1,1/L=0.307.0^8.00.01150.3 -0.24 -0.18 ^0.12 -0.06 _0.0 I I0 0 0.006 0.012±±±0.018^0.024^0.09Wave Height (m)0.042^0.0460.0967.5.--,Z^5.00.00.0^2.0^4.0^6.0^8.0Extension (cm)12.010.0Figure 5.44 Nonlinearity of Measured Surge Response at ResonanceFigure 5.45 Theoretical Response of Model-Scale Mooring in Surge116Figure 5.46 Common Mooring Conditions (from Taylor (1983))Figure 5.47 Example of Small Craft Mooring Configuration Figure 5A8 Example of Small Craft Mooring Configuration117Figure 5.49 Example of Small Craft Mooring Configuration118Figure 5.50 Example of Small Craft Mooring ConfigurationFigure 5.51 Elastic Behaviour of Nylon Rope(from Taylor (1983))119Iv.Figure 5.52 Definition Sketch for Nonlinear Mooring120dA lRestoring Springs Connector forSpring AttachmentY121li12m2g0^ „A -^.irCentre of Gravity el11^(x', y')MIMMIIM^■•■■•^•••■^OM.. ...■•••^0111.1.migMgFigure A.1 Definition Sketch for Centre of Gravity CalculationFigure A.2 Definition Sketch for Calculation of Moment of Inertia in Yaw,Plan View

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