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Experimental and numerical analysis of a fishing vessel motions and stability in a longitudinal seaway Allievi, Alejandro 1987

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EXPERIMENTAL AND NUMERICAL ANALYSIS OF A FISHING VESSEL MOTIONS AND STABILITY IN A LONGITUDINAL SEAWAY by ALEJANDRO ALLIEVI DIPL. ING., UNIVERSITY OF BUENOS AIRES, ARGENTINA, 1980 M.ENG., UNIVERSITY OF BUENOS AIRES, ARGENTINA, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard UNIVERSITY OF BRITISH COLUMBIA JANUARY, 1987 ® ALEJANDRO ALLIEVI, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: JANUARY, 1987 ABSTRACT Motions and stability of a typical B.C. fishing vessel were experimentally and numerically investigated in a longitudinal seaway condition. The experimental model was self-propelled, radio-controlled and equipped with an on-board data acquisition system. Pitch, roll, yaw, surge, and heave responses to regular waves of predetermined frequencies and amplitudes generated along a 220-ft model basin were obtained. Different displacement conditions and GM configurations were tested. The numerical model for the dynamic analysis of the fishing vessel motions has been implemented using strip theory. A computer program was developed to study the nonlinear motions of the vessel . The velocity dependent coupling terms, responsible for a major part of the nonlinear behavior, were included. A time dependent component analysis of the roll damping has been performed. Regular linear and nonlinear waves were used. A parametric study of the fishing vessel stability has been carried out by considering its dynamic response in waves of varying characteristics. Unstable behaviour was found to be closely related to waves of length of similar magnitude to the ship length. The effects of wave amplitude and rudder usage were found to be of capital importance in the capsizing process. Experimental and numerical results showed good agreement. Table of Contents A B S T R A C T ii LIST OF TABLES vi LIST OF FIGURES vii NOMENCLATURE x ACKNOWLEDGEMENT xiv 1. OVERVIEW OF PREVIOUS RESEARCH 1 2. INTRODUCTION 4 3. EXPERIMENTAL ARRANGEMENTS 6 3.1 MODEL DESCRIPTION 6 3.2 TESTING SITE 8 4. ELECTRONIC SYSTEM 10 4.1 D A T A COLLECTION SYSTEM 11 4.1.1 THE COMPUTER 11 4.1.2 THE A N A L O G TO DIGITAL CONVERTER (A/D) 12 4.1.3 C A S S E T T E RECORDER 13 4.1.4 THROW SWITCH 13 4.1.5 DISPLAY LIGHTS 14 4.2 RADIO CONTROL SYSTEM 14 4.3 SENSORS : 15 4.3.1 ROLL AND PITCH SENSOR 15 4.3.2 Y A W SENSOR 15 iii 4.3.3 SURGE AND HEAVE SENSORS 16 4.3.4 RUDDER SENSOR 16 4.4 PROPULSION SYSTEM 16 4.5 TRANSFER OF D A T A 17 4.6 D A T A DEMULTIPLEXING 17 5. EXPERIMENTS ". 18 5.1 OBSERVATIONS 18 5.2 TESTING PROCEDURE 21 5.3 EXPERIMENTAL RESULTS 21 6. NUMERICAL SIMULATION 26 6.1 FORMULATION OF THE PROBLEM 26 6.2 SYSTEMS OF COORDINATE A X E S 27 6.3 EULER ANGLES 28 6.4 COORDINATE SYSTEM TRANSFORMATION 29 6.5 ANGULAR VELOCITIES 30 6.6 EQUATIONS OF MOTION 31 6.7 COMPUTATION OF FORCES AND MOMENTS 33 6.8 NUMERICAL INTEGRATION OF THE EQUATIONS OF MOTION 36 7. COMPUTER PROGRAM AND NUMERICAL RESULTS 37 7.1 COMPUTER SOLUTION 37 7.2 TIME STEP AND NUMBER OF HULL ELEMENTS 37 iv 7.3 VALIDITY OF THE NUMERICAL METHOD 38 7.4 ANALYSIS OF INITIAL CONDITIONS 38 7.5 LINEAR W A V E ANALYSIS 40 7.5.1 PARAMETER S 41 7.5.2 LOW CYCLE RESONANCE 42 7.5.3 PURE LOSS OF STABILITY 46 7.6 NONLINEAR WAVE ANALYSIS 47 7.6.1 PARAMETER S 48 7.6.2 LOW CYCLE RESONANCE .48 7.6.3 PURE LOSS OF STABILITY ~ 49 8. CONCLUSIONS 51 REFERENCES 54 APPENDIX A 104 APPENDIX B 106 APPENDIX C 109 APPENDIX D 118 v LIST OF TABLES 1. Ship model characteristics 6 2. Experimental wave characteristics 21 3. Capsizing statistics 22 4. Capsizing statistics 24 5. Prototype characteristics 40 6. Prototype scale wave characteristics 41 vi LIST OF FIGURES 1. Body plan 59 2. Sealing arrangement and rods positioning 60 3. Inclining experiment set-up 61 4a. Righting arm curves for light condition and GM #1 62 4b. Righting arm curves for light condition and GM # 3 63 4c. Righting arm curves for light condition and GM # 4 . 64 4d. Righting arm curves for heavy condition and GM #1 65 4e. Righting arm curves for heavy condition and GM #2 66 4f. Righting arm curves for heavy condition and GM # 3 67 4g. Righting arm curves for heavy condition and GM # 4 68 5a. Testing session 69 5b. Testing session 70 6. Schematic of the electronic system . . 71 7a. Electronic system 72 7b. Electronic system 73 7c. Electronic system 74 8. Longitudinal seaway 75 9. Parameter S as a function of ship length to wave length ratio for GM # 1. Experimental results 76 10. Parameter S as a function of ship length to wave length ratio for GM # 2. Heavy condition. Experimental results 77 11. Low cycle resonance capsizing. GM #1 78 12. Low cycle resonance capsizing with rudder usage. GM # 2 79 13. Wave record for wave frequency 0.8 Hz and amplitude # 4 80 14. Euler angles in the ship system of coordinates 81 15. Coordinate systems and linear translation of CG 81 vii 16. CPU time in seconds to number of time steps ratio as a function of number of hull elements 82 17a. Comparison of numerically and experimentally obtained roll natural periods in still water. Heavy condition . . 83 17b. Comparison of numerically and experimentally obtained roll natural periods in still water. Light condition .84 18. Comparison of numerically and experimentally obtained roll decay curves in still water. Heavy condition 85 19. Motion records for zero-angle roll initial condition. Heavy condit ion,GM # 1, wave length 104 ft, wave height 10 ft 86 20. Motion records, initial roll angle 0.5 deg. Heavy condition, GM # 1, wave length 104 ft, wave height 10 ft 87 21. Motion records, initial roll angle 3 deg. Heavy condition, GM # 1, wave length 104 ft, wave height 10 ft 88 22. Motion records, initial roll angle 5 deg. Heavy condition, GM # 1, wave length 104 ft, wave height 10 ft 89 23. Parameter S as a function of ship length to wave length ratio for GM # 1. Numerical results 90 24. Parameter S as a function of ship length to wave length ratio for GM # 2. Heavy condition. Numerical results 91 25. Low cycle resonance capsizing. GM # 1 92 26. Roll motion record. GM # 2 93 27. Low cycle resonance. GM # 1 94 28. Wave encounter period as a function of wave length and heading angle for a given ship speed 95 29. Wave encounter period as a function of wave length in overtaking waves for a given speed 96 viii 30. Roll record for equal ship and wave speed. Heavy condition. GM # 1 . . . 97 31. Roll record for equal ship and wave speed. Light condition. GM # 1 . . . . 98 32. Free surface elevation for linear and nonlinear waves 99 33. Parameter S as a function of ship length to wave length ratio. Heavy condition. Nonlinear wave. Numerical results 100 34. Low cycle resonance capsizing. GM # 1. Nonlinear wave 101 35. Motion records. GM # 2. Nonlinear wave 102 36. Roll record for equal ship and wave speed. GM # 1. Light condition. Nonlinear wave . . 103 A - 1 . Calibration curves 105 C-1 . Roll damping as a function of Froude number 112 C-2 . Roll damping as a function of roll period 113 C-3 . Roll damping as a function of roll amplitude 114 C-4. Component to total roll damping ratio as a function of Froude number. . 115 C-5 . Component to total roll damping ratio as a function of roll period . . . 116 C-6 . Component to total roll damping ratio as a function of roll amplitude . . 117 D-1. Flow chart for computer program 125 ix NOMENCLATURE a j, b j, c j; i = 1,2,3 ; direction cosines a n : sectional added mass coefficient A : average wave amplitude A(x): cross sectional area Atf : projected lateral area of superstructures and deck houses above main deck A n : added mass in heave AREA : area under the G Z - c u r v e b : A G M J / 2 - G M J b n : sectional hydrodynamic damping coefficient B : beam B n : hydrodynamic damping in heave Bp : hydrodynamic damping in pitch Bg« : hydrodynamic keel roll damping B^ : hydrodynamic eddy roll damping B|_ : hydrodynamic lift roll damping Bp : hydrodynamic friction roll damping Byv : hydrodynamic wave roll damping B r : total hydrodynamic roll damping C^ : block coefficient C n : total heave force C n w : heave force due to wave action C m : midship-section coefficient C u : upper deck area coefficient = deck area/(length x breadth) where length and breadth refer to deck's dimensions x C W p : waterplane coefficient d : water depth D : depth to main deck (molded) ¥ : force vector Fj w ; i = x, y, z : force due to wave action in X , Y and Z directions g : gravitational acceleration G M j : transverse metacentric height GM|_ : longitudinal metacentric height H : wave height H e : effective depth of the ship structure = D + ( A d / L B P ) I : inertia matrix Ijj ; i = x , y , z : virtual moment of inertia about X , Y and Z axes k : wave number = 27T /X kjj ; i = x, y, z : virtual radius of gyration about X, Y and Z axes L : ship length LBP : length between perpendiculars m : virtual ship mass, m = V / g + A n m' : mass of ship particle S : moment vector Mj w ; i = x, y, z : moment due to wave action about X,Y and Z axis Z M X : total moment about X-ax is Z M y : total moment about Y-axis Z M Z : total moment about Z-axis p,p : angular velocity and acceleration about X-axis P :pressure q,q : angular velocity and acceleration about Y-axis xi r,r : angular velocity and acceleration about Z-axis R : average roll amplitude : restoring force coefficient in heave Rp : restoring moment coefficient in pitch S : stability parameter T : ship draft T e : wave encounter period v : translational velocity vector V : s h i p speed x : x-coordiante of ship particle in G - X Y Z system y : y-coordinate of ship particle in G - X Y Z system y(x) : half breadth at cross section x z : z-coordinate of ship particle in G - X Y Z system a : heading angle 0, 4> : sway angle and angular velocity xj/,^ : pitch angle and angular velocity 6, 8 : roll angle and angular velocity p : water density X : wave length C J W : wave frequency in deep water = / (g • k) cj r : natural roll frequency = / ( V G M j / l x x ) c j e : wave encounter frequency = 27r/T e £g : coordinate of center of gravity in 01-^1r)1^1 system rjg : coordinate of center of gravity in O J - ^ T J ^ ! system $g : coordinate of center of gravity in 01-^1v1^l system ? t : coordinate of ship particle in O j - ^ r j ^ ! system xii T?! : coordinate of ship particle in G^-i; j T j ^ j system : coordinate of ship particle in 01-^1r)l^1 system V : displacement xiii ACKNOWLEDGEMENT The author wishes to express his sincere gratitude to Dr. S.M. Calisal for his assistance, advice and encouragement throughout the preparation of this thesis. Mention should also be made of Mr. M. Lefrancois for his valuable work on the electronic system and the software for the data acquisition system. Thanks are also due to Mr. S. Thomson for his help in the preparation of the model. I would also like to thank the staff at the Department of Mechanical Engineering at U.B.C. and at B.C. Research Ocean Engineering Center for the use of their facilities and their assistance. Finally, I wish to thank all those, like I. Blank, F. Namiranian and Gerry Rohling, who helped during actual testing of the model. xiv 1. OVERVIEW OF PREVIOUS RESEARCH The study of the seakeeping qualities of ships has been a major area of research among the naval architecture community in the last 40 years. A complete investigation of this phenomenon would require an analysis of the compounded effect of the steering mechanism and the ship's nonlinear motions in all six coupled degrees of freedom - surge, sway, heave, roll, pitch and yaw - in realistic complex seas of multiple directionality. Such a complex and complete analysis has not yet been made. As a first step, the simpler problems of ship motions have been tackled by several researchers. Kriloff [1.2]1 laid the foundation of what today is known to be the "strip theory" for computing pitching and heaving motions of a ship in regular waves. Kriloff historic achievement remained the state of the art in ship motions for half a century. It was only in 1950 when Weinblum and St. Denis [3] open a new period in seakeeping research. Their work made the first attempt in defining ship motions when advancing on an arbitrary course in regular sinusoidal waves. St. Denis [4] and St. Denis and Pierson [5] further improved these works by extending the problem to an irregular seaway. Korvin Kroukovsky's [6] particular advances in this area were made toward including pitch-heave coupling and an approximate calculation of the interaction between ship and wave that have heretofore been neglected in the "Froude-Kriloff" hypothesis. Experimental and numerical agreement was reasonably good, but large systematic error was found in the phase relationships. Korvin Kroukovsky and Jacobs [7] subsequently corrected this problem. All the above-mentioned works assumed motion amplitudes and wave amplitudes to be small. The above-mentioned works neglected the possibility of rolling in head or following seas. Grim [8] showed that, for certain values of the encounter frequency l u m b e r s in brackets designate References at the end of thesis. 1 2 in a longitudinal seaway, rolling instability could develope. Dispite ship symmetry, due to the fluctuation in metacentric height produced by waves progressing along the length of the ship, unstable roll behaviour was noted. Kerwin [9] solved the roll equation of motion stating that it could be treated as a Mathieu differential equation. This is: d28 — + c; r (1 - b sin 2w et) 0 = 0 dt 2 . Solutions of Mathieu's equation by expansion series, known as Mathieu functions, defined boundary curves of regions of stability and instability depending on the constants u)r and u>e. Unstable roll motion was found to occur when the ratio of the roll natural frequency o>r to the exciting frequency ue took on half integer values n/2 (n = 1, 2, 3, ), - critical values of the frequency ratio. Kerwin concluded that, in a regular seaway, a large number of waves would be required for the roll angle to build up to a large amplitude and therefore the solution for a regular seaway was not of practical use. This conclusion, however, conflicts with some of the experimental and numerical findings of this work and we shall return to this point. Kerwin's and Grim's experimental models were equipped with rotating weights possessing eccentrically located loads which caused the stability fluctuation effect. Unstable roll motion was found to be set up at frequencies predicted by the Mathieu instability chart. Paul I ing and Rosenberg [10] performed experiments in still water on a ship model of known dynamical characteristics which was rigidly clamped in all degrees of freedom except roll and heave. Limits of stability of the rolling motion were determined by varying the heave frequency over a range of values close to the critical. These experiments were performed for a variety of amplitudes. Unstable rolling was again found to correlate with the Mathieu instability chart. Pauli ing [11] studied theoretically and experimentally the change in transverse stability of a ship 3 due to the altered geometry of the hull when immersed in waves. It was shown that significant variations in the righting arm curve were caused by the changes in the immersed geometry as succesive crests and troughs progressed along the length of the ship. Research into ship motions has been mostly concerned with linearized techniques which in turn assume small amplitude motions. A practical implementation of this procedure was remarkably achieved by the new strip theory introduced by Salvensen, Tuck and Faltinsen [12]. This work predicted the six degree of freedom motions in the frequency domain of a ship advancing at constant speed with arbitrary heading in regular waves. This type of analysis, however, is not appropriate for the accurate computation of the time-dependent variation of the roll restoring moment resulting from the wave profile progressing along the length of the vessel. Oakley, Paulling and Wood [13] carried out an analytical and experimental study of ship motions and capsizing in extreme seas. Extensive tests were conducted in the wind generated seas of San Francisco bay. They also developed a simulation to investigate motions and capsizing. However, this numerical model neglected the time dependece of the highly nonlinear roll damping coefficients. A complete simulation of extreme ship motions in a seaway has, at the time of writing this thesis, not been reported. Attempts toward this goal will require extensive computer programs. Techniques to model the hydrodynamic forces and the steering system in an accurate form will demand the greatest efforts. 2. INTRODUCTION The problem of ship motions and ship stability is of fundamental concern to naval architects during the design process. Although conventional techniques and regulations have been reasonably successful in assuring a satisfactory level of stability for large vessel types, the number of ship losses, especially the medium and small sized ones, require new methods of analysis. Model experiments conducted in a controlled environment offer the possibility of gaining an understanding of the capsizing mechanism. A wave basin generally limits the production of a realistic short-crested sea because of its limited dimensions. In contrast, when equipped with a computer controlled wavemaker, it provides the advantage of controlled test conditions so that the response of the model to waves of predetermined amplitudes and frequencies can be assessed and the repeatability of the experiments assured. The experiments described in this work analyze conditions under which a fishing vessel might undergo severe motions that may eventually endanger its safety and may cause capsizing in a longitudinal seaway. Although experiments provide a realistic insight of the significant factors involved in a given physical phenomenon, they are, in general, expensive and time consuming. Therefore, an alternative method should be developed in order to predictjn a faster and relatively inexpensive form, the characteristic factors of that particular phenomenon. This becomes of capital importance in the optimization process followed in the design spiral of a ship. In this work, a time domain numerical simulation of the fishing vessel highly nonlinear motions has been developed as an alternative solution. Regular linear and nonlinear Stokes second order wave theories were used. The computer program was implemented using strip theory and the Froude-Kriloff hypothesis. The numerical technique showed to be a very useful tool in predicting the capsizing characteristics 4 5 of the fishing vessel. During the entire development of the numerical simulation, it has been attempted to avoid using experimental data as input to the computer program. Instead, efforts have been directed toward estimating, in an analytical form, the numerous and complex parameters involved in nonlinear ship dynamics. In so doing, the author has introduced a number of assumptions to simplify the inherent complexity of the problem. These assumptions, however, have been based on accepted facts in ship dynamics. Intact stability of ships has, in general, been defined in terms of static conditions. It is the aim of the parametric study presented in this work, to provide a design tool to determine an approximate minimun value of the dynamic stability required by a fishing vessel in regular waves. 3. EXPERIMENTAL ARRANGEMENTS 3.1 MODEL DESCRIPTION The model used during the experimental part of this work was of a typical B.C. fishing vessel (seiner) with a single chine and a bottom keel. The model was made of wood with a scale factor of 13 and built without bulwarks in order to avoid water-on-deck conditions. Figure 1 and Table 1 respectively present the body plan and the characteristics of the model in both light and heavy conditions. TABLE 1 : SHIP MODEL CHARACTERISTICS Light Heavy Length overall (LOA) 5.923 ft. Length btn. perpendiculars (LBP) 5.381ft. Beam (B) 1.769 ft. Depth (D) 1.154 ft. Draft (T) 0.730 ft. 0.808 ft. Displacement (V) 220.6 lb. 254.5 lb. Block Coefficient (Cb) 0.500 0.531 Midship-Section Coefficient (Cm) 0.756 0.775 Waterplane Coefficient (Cwp) 0.850 0.862 KM 1.006 ft. 0.986 ft. GM # 1 (1.11% B) 0.236 in. 0.236 in. G M # 2 (2.22%B) 0.472 in. 0.472 in. GM # 3 (5.00% B) 1.063 in. 1.063 in. G M # 4 (9.27% B) 1.969 in. 1.969 in. The model was sealed at the weather deck by plexiglass sheets that permitted visualization of the instrumentation installed on board. Figure 2 shows a photograph of the model from which the sealing arrangement can be realized. The vertical steel rods were conveniently placed so as to allow positioning of lead weights. These weights were used to obtain different load conditions and the metacentric height. 6 7 The GM configurations were defined as a percentage of the beam value of the model in order to cover a range of initial stability values. The magnitude of the metacentric height was set by raising or lowering the lead weights and checked by inclining experiments. Figure 3 shows the inclining experiment set-up. It should be noted that GM # 1 and GM # 2 were lower than normally required. However, they are practically possible if one considers effects such as unintentional free surfaces in fish tanks,misplacing of rigging and nets and ice formation on decks. The GZ curves of the prototype are shown in Figures 4a to 4g for both light and heavy load conditions. The areas under the still water curves for GM # 3 (GM = 1.151 ft.), in both full and light load conditions, up to the angle of downflooding (about 30 degrees) are less than 12 f t -deg. The U.S. Coast Guard Navigation and Vessel Inspection Circular [18] suggests the following requirements for fishing vessels: - an initial metacentric height of not less than 1.15 ft. - a minimun area of 16.9 ft-deg under the righting arm curve up to the lesser of 40 degrees or the angle of downflooding. - a maximum righting arm occuring at no less than 25 degrees. Although the second of these requirements was not achieved, the model showed a high stability for GM # 3 throughout all testing. The prototype's downflooding angle was about 30 degrees The model was watertight, therefore, it had no real downflooding angle and the second requirement might be considered applicable to the ship scale only. However, since the experimental conditions did not allow for wind effects or possible operational weight shifts, a 30/35-degree downflooding angle could well be in the right range and the second requirement would still be violated. 8 3.2 TESTING SITE The experiments were conducted in the towing tank at the Ocean Engineering Center at B.C. Research. The main dimensions of the tank are 220 ft in length, 12 ft in width and 8 ft in depth. Figures 5a and 5b show the experimental set up during a testing session. Situated at one end of the tank is a hinged hydraulically operated panel for the creation of waves. The excursion of the hydraulic actuator is electronically controlled via an electro-hydraulic proportioning valve. The entire wavemaker assembly is ultimately controlled by a software driven digital-to-analog board resident in an IBM PC micro computer. Software can create sinusoidal waves of specified frequencies and amplitudes. The wave system was arranged so that the wave length gradually approached a length about equal to the model length. Table 2 lists the average wave amplitudes used during the experimental part of this work. It can be seen that at higher wave frequencies there is a loss of wave amplitude. It was noted that as the wave frequency was increased the hydraulic actuator was unable to maintain the same excursion length as in the lower wave frequencies. As a consequence, the maximum wave amplitudes that could be obtained were lower. Furthermore, these higher frequency waves with a decreased maximum possible amplitude and a lower energy content tended to deteriorate towards the end of the tank. A s a result, it was not possible to test for the condition of equal wave and ship speed. This condition, however, was included in the numerical part of this work. Wave data was obtained by using a resistance wire gauge, as developed by Compton [29]. The gauge consists of two parallel wires separated by a small distance. The wires are connected to a Wheatstone bridge. While the probe is out of the water, the resistance between wires is infinite. When the probe is submerged,the water acts as a conductor, and the resistance in the circuit depends on the length of 9 the wires outside the water. As the wave passes by the probe, the Wheatstone bridge re-adjusts its electrical currents. This change is sensed by a Vishay strain gauge and its output corresponds to the wave amplitude. A computer program to calibrate the wave probe is available at B.C. Research. 4. ELECTRONIC SYSTEM The electronic system is shown in schematic form in Figure 6,Lefrancois [17]. Photographs are also presented in Figures 7a to 7c. The electronic system is composed of the following core components; 1 - A data collection system consisting of the following elements; - a computer IBM PCjr. - an analog to digital converter. - a cassette recorder. - a throw switch. - a set of display lights. 2 - A radio control system formed by: - a hand-held transmitting unit. - a receiving unit. 3 - A group of sensors comprising the following: - an autopilot unit coupled with a magnetic compass. - a vertical axis gyroscope. - two servo-accelerometers. - a precision potentiometer. 4 - A propulsion system composed of; - an electric motor. - a speed controller. The following is an explanation of the above-mentioned components and sub-components. 10 11 4.1 D A T A COLLECTION SYSTEM 4.1.1 THE COMPUTER The computer used was an IBM PCjr because its cordless keyboard, price level and physical dimensions were considered most suitable for installation in the model. It featured 128 K-byte memory, one disk drive and full compatibility with the IBM PC. The cordless keyboard permits communication with the computer with no direct connection, thus allowing operation even through the plexiglass deck. The keyboard emits infra-red light signals which are received by the optical eye mounted in the front part of the computer. The keyboard must be in front of the computer for transmission of commands to occur. In order to ease communication in all directions, the optical eye was removed from its original position and relocated in a more visible location pointing upwards. Modifications to the power supply were also introduced. The computer's original power supply of 17 VAC as output by its transformer power pack was altered to function with a 12 VDC battery. Battery connections were clearly labelled since a reversal in polarity would have damaged the computer. Diodes to protect the circuit could not be used because of their inherent voltage drop that produced computer errors. The computer, as the most important element of the data acquisition system, performed the following functions: - Informing the A / D converter of the computer's state, i.e. data being taken,written to the cassette or ready for data acquisition. - Evaluating the A / D converter responses, i.e. change in polarity. - Storing collected data. 12 - Writing data to the cassette. - Providing power to the receiving unit of the radio control system. 4.1.2 THE A N A L O G TO DIGITAL CONVERTER (A/D) The sensors used in this work output analog voltage signals proportional to their corresponding state. These signals are continuous and small state changes derive in small changes in the output signal. Due to its binary nature, the computer is unable to record small input changes. Information to the computer must, therefore, be provided in a discrete-jump format that is stored as an integer value. The device utilized to convert continuous scalar voltages to integer form was an analog to digital converter. The A / D converter used was a Remote Measurement System's ADC-1 with the following features: - 16 analog input channels. - 6 controlled outputs. - 4 digital inputs that could be used to monitor or count on/off signals. - one instrumentation amplifier providing micro-volt resolution. When purchased, the unit was set to convert between -4095 and 4095 which corresponds to input levels of -0.4095 V and 0.4095 V and a sampling rate of 14 samp/sec. The converter was then modified to convert 70 samples/sec. In order to avoid signal coupling and subsequent malfunctioning, operation was constrained to the interval ±0 .4095 Volts and the amplifier was not connected. The A / D converter was powered by the computer power supply. 13 4.1.3 CASSETTE RECORDER As the computer's random access memory (RAM) was limited, the collected data were stored on a cassette tape at the end of each run. The disk drive resident in the IBM PCjr was not used since it drew more power than the cassette deck. In addition, the high acceleration levels of the model were considered dangerous for disk recording. A cassette deck was not affected by high acceleration levels and was powered by its own batteries. A Panasonic RQ-2736 battery operated cassette recorder and an IBM PCjr cassette adapter cable were used for data storage. It was observed that a 30-minute cassette held up to 150,000 bytes per side. It was also found that the ability to read back the data on to a diskette was very sensitive to the volume setting, the best being in the low volume region (1 to 3). 4.1.4 THROW SWITCH The data acquisition system required an input to activate the sampling routine. The optical eye had a very limited field of view when the boat was in motion and away from the keyboard. For this reason inputting of commands from the keyboard was awkward and unreliable. The radio control provided a much more convenient input device. A radio-controlled switch was installed on the model and proved to be a much more reliable input device to initiate data collection. The switch was connected to the A / D converter and, when thrown, changed the polarity of the ADC-1 .The computer monitored this change and started sampling data. The throw switch was activated by one of the radio control servo motors which in turn was commanded by one of the switches on the radio control hand-held unit. 14 4.1.5 DISPLAY LIGHTS A set of six LED indicator lights was mounted on the model to monitor at a distance the computer's activity mode. Since the LEDs operated off the A D C - 1 ' s controlled outputs, signals were sent by software control to the ADC-1 which activated the appropriate indicator lights. These lights indicated whether the computer was sampling data, writing to the cassette tape or ready to sample. 4.2 RADIO CONTROL SYSTEM The radio control system used was a.UDISCO 5-channel FM proportional control system manufactured by FUTABA. The transmitter portion is a hand-held unit which, by means of its various switches and joy sticks, was set to control the boat speed and the rudder angle and to activate the throw switch. The receiving unit mounted in the model had 3 of its output channels connected to servo motors which'controlled the following: - channel 1 controlled the motor's speed controller. - channel 2 controlled the rudder. - channel 5 controlled the throw switch. The receiving unit and its servos initially operated off its own battery pack (4 A A cells). The pack was found to last for only 20 minutes. For longer operating time, the unit was connected to the joystick port of the computer which possessed a 5 Volt supply. The transmitting unit was powered by its own batteries. 15 4.3 SENSORS Collected data originates from sensors which provide a measurable voltage output proportional to given stimuli. In the case of a ship, these stimuli are represented by it's motions. The motions measured in the experimental part of this work were roll, yaw, and pitch angles, heave and surge accelerations along with the rudder angle. The instruments used in measuring these quantities were first calibrated as described in Appendix A . A brief description of these instruments is presented below. 4.3.1 ROLL AND PITCH SENSOR The roll and pitch angles were provided by a HUMPHREY vertical axis gyroscope. This device required separate powering for its signal circuitry and for its motor. The signal circuitry was powered by the two 1.5 Volt C-cel ls which powered the throw switch. The motor was powered by the two 12 V batteries that powered the accelerometers. The output signal was trimmed by means of resistors in order to comply with the voltage range of ±0 .4095 V required by the A D C - 1 . 4.3.2 YAW SENSOR The yaw sensor was composed of a high performance low power consumption and high holding accuracy autopilot unit designed for work boats of up to 50 ft and sailing yachts of up to 75 ft. It was coupled with a 5 inch fully gimballed magnetic compass. Yaw was measured as the deviation from a preset course charted along the test tank. The magnitude of this measurement was slightly affected by circuit noise ( ± 0 . 5 degrees). Therefore, filtering of the data was required. A smoothing routine without fast Fourier transform, as published by Aubanel and Oldham [27], was used. 16 Power for the device is obtained from two 9 Volt batteries connected in series. These batteries did not power any other device in the electronic system since the autopilot unit required negative voltage levels. 4.3.3 SURGE AND HEAVE SENSORS Surge and heave accelerations were measured by two SCHAEVITZ LSM-2 servo-accelerometers with a range of 2g. The accelerometer that measured surge accelerations was mounted with its axis parallel to the model's longitudinal axis and the one that measured heave accelerations was positioned with its axis coinciding with a vertical line. They were installed on a common wooden platform which was located at about the model's center of gravity. Power was obtained from two sets of two A A batteries connected in series with the two 12 Volt batteries supplying the propulsive motor ( ± 15 Volt and a zero were required). 4.3.4 RUDDER SENSOR The rudder angle was measured with a precision potentiometer mounted on top of the rudder shaft. As the rudder turned, the potentiometer's voltage output changed a value proportional to the rudder angle. Powering of the potentiometer was supplied by the two C-cel ls that powered the throw switch. 4.4 PROPULSION SYSTEM The model was driven by an HECTOPERM GT 500 Watt maximum power electric motor made by MARX. This motor was capable of operating in the range of 6 to 24 Volts. The power supply was provided by a Y U A S A battery of 12 Volts and 15 Amp-hr . 17 Model speed was initially controlled by a 20-Ampere speed controller. Experience showed this speed controller to be inadequate since, in reversing the propeller's rotational direction to stop the model, a large transient current was drawn. As a result, failure of the circuitry occured. Speed changes were inevitable due to the limitations in the length of the tank. A new speed controller is now developed. 4.5 TRANSFER OF D A T A When data collection was completed, data was transferred from cassette to diskette by using a transfer program. Files were transfered to cassette in binary form.Transfer options were as follows; - Specific file. - Multiple file. Error messages generally indicated cassette-deck volume problems. 4.6 DATA DEMULTIPLEXING The file output by the transfer routine contained uncalibrated information for all channels sampled and was in binary format. A demultiplexing programme was used to convert the binary values into calibrated ASCII format. Calibration files were those obtained from the calibration routine as shown in Appendix A . In this manner, a separate file was created for each channel sampled. These files were named after the file being demultiplexed with the channel number at the end of it. After demultiplexing data was transferred to be analized in the V A X system of the Mechanical Engineering Department at U.B.C.. 5. EXPERIMENTS 5.1 OBSERVATIONS Oakley et. al. [13] consider three different modes of capsizing in a longitudinal seaway.These modes are; - Broaching. - Pure loss of stability. - Low cycle resonance. Du Cane and Goodrich [14] reported that when a ship is travelling in very steep waves of large amplitude in a longitudinal seaway, the probability of sudden yaw with an increased difficulty in steering is considerably high. This motion is known as "broaching" and may lead to capsizing. This phenomenon may be simulated in a wave tank by using breaking waves that strike the model by the stern. However, the possibility of the model becoming ungovernable in a tank of limited width is not desirable. For this reason broaching was not studied in the in-tank phase of this project. "Pure loss of stability" requires a number of events to occur together. Oakley et. al. [13] list them as; a - relatively high ship speed. b - ship speed close to wave speed. c - wave length about the same as the ship length. d - wave height sufficient to immerse the deck with the model upright. 18 19 Under these conditions, the wave crest remains almost stationary with respect to the midship section and, with little preliminary roll motion, suddenly overturns as a result of total loss of stability. For the reasons described below, it is believed that this mode of capsizing is unlikely to occur for a fishing vessel with the characteristics of a B.C. seiner; 1 - The prototype of the model under study possesses maximun economical speeds ranging between 9 and 10 knots. The corresponding model speed range is 4.21 ft /sec to 4.68 f t /sec. The waves in the tank with phase speeds contained in this interval have lengths 56%to 27%shorter than the model's length. 2 - The wave of length equal to the model length has a frequency of 0.975 Hz and a phase speed of 5.23 ft/sec . This phase speed is much higher than the model's maximum speed. In addition, the maximum height that a wave can attain without breaking in finite and uniform depth, can be estimated from the maximum wave steepness expression proposed by Miche [15] as fol lows: H = 0.142 X tanh(kd) (V-1) When this expression is applied to a wave frequency of 0.975 Hz the resulting wave height is 9.1 inches. This wave height is not large enough to immerse the model's deck. This reinforces the belief that capsizing caused by pure loss of stability only is not probable for this type of fishing vessel under the conditions tested here. "Low cycle resonance" received most of the attention in this study. Fraser et. al. [16] showed that the decrease in metacentric height was most pronounced with the wave crest positioned between 10% aft and 5% forward of amidships. The increase in stability was observed to be greater when the wave trough was 20 positioned about amidships. Figures 4a to 4g show the righting arm curves for the prototype in still water. These figures also show the effect on the GZ-curves of a wave of equal length to the ship length between perpendiculars with its crest and trough positioned amidships. The drastic change in the GZ-curves can be realized. Consequently, when a ship is sailing in a longitudinal seaway, as shown in Figure 8, there is a periodic variation of its static stability curves corresponding to the relative position of the wave along the ship's length caused by the vertical displacement of the transverse metacenter. As a result, the initial transverse stability will vary with a period equal to the period of wave encounter. Low cycle resonance can be qualitatively visualized by considering the two extreme positions of the wave, that is with the crest amidships and with the crest at the ends. For conventional ship forms with flared sections at the ends and wal l -sided sections amidships it can be seen that the inertia of the waterplane will change in each case. It is smaller than the still water value with the wave crest amidships and greater than the still water value with the wave crest at the ends. If the ship is now given an initial roll angle 6Q caused by some small disturbance, it will oscillate at its natural period until the energy contained by the disturbing moment is dissipated by damping. However, if the period of the stability fluctuation and the natural roll period are in the right ratio, it is possible that the initial roll angle be sustained and even increase to considerable proportions. This is shown later in the solution of the equations of motion as well as by the experimental results. This process may result in constantly increasing roll angles that may eventually culminate in capsizing. All capsizes reported in the experimental part of this work are considered to belong to this particular mode. 21 5.2 TESTING PROCEDURE During testing the model was held at the end of the tank where the wavemaker was located. Following the formation of transient waves, it was released. As forward thrust was applied, the model was steered using the hand-held remote control unit (RCU). Data acquisition was started by moving one of the switches on the RCU, thus allowing the throw switch to toggle. The computer then started recording data for about 54 seconds. The model speed was about 4.3 ft/sec and this recording period was found to be sufficient to complete the run. In spite of the inconvenience of having a single speed, observations suggested that, for the range of wave frequencies of interest for this experiments, speeds slower than 4.3 ft /sec would have resulted in a higher value of the encounter frequency and probably in less severe motions. 5.3 EXPERIMENTAL RESULTS The model was tested at each of the frequencies and amplitudes listed in Table 2. This procedure was repeated for each GM value and load condition. TABLE 2 : EXPERIMENTAL W A V E CHARACTERISTICS Freq. (Hz.) Wave Length (ft.) Amp. # 1 (in.) Amp. # 2 (in.) Amp. # 3 (in.) Amp. # 4 (in.) 0.700 10.50 2.43 3.25 4.32 0.800 8.000 3.06 4.55 4.93 0.850 7.100 3.05 3.85 4.27 0.900 6.310 3.03 3.85 4.120 4.22 0.950 5.680 2.88 3.58 3.480 3.40 0.975 5.390 3.02 3.39 2.750 2.74 1.000 5.120 3.27 3.15 2.270 2.43 22 A non-dimensional parameter S was used in order to define zones of stability and instability [19].This parameter was defined as; s = A " R (V-H) AREA Table 3 gives the number of capsizes, the number of runs that might have resulted in capsizing and the number of safe runs. TABLE 3 : CAPSIZING STATISTICS Observation Number Parameter S Capsizes 10 10 or more Possible Capsizes 31 5 to 10 Safe 182 below 5 Figure 9 and Figure 10 respectively show values of the parameter S versus ship length to wave length ratios for GM # 1 and GM # 2. It was found that values of S below 5 represented a safe condition. Values of S between 5 and 10 corresponded to conditions where capsizing might occur. Values of S above 10 depicted zones of unstable behaviour due to low cycle resonance. All capsizes corresponded to a value of the parameter S greater than 10. Roll motions in light conditions were observed to be more violent than in heavy conditions. This follows what could be related to the diminished waterplane coefficient shown in Table 1 and the more pronounced changes in ship geometry as the draft is decreased. Figure 11 shows an example of capsizing caused by low cycle resonance. The sequence of events leading to this type of capsizing can be described as fol lows. The model is overtaken by the incoming waves as it moves along the tank. As a 23 wave crest reaches the model's stern, there is a marked pitch by the bow. Simultaneously, it imparts a forward surge acceleration to the model that causes an increase in model speed. This produces a reduction in the wave encounter frequency and as a result the wave crest progresses along the length of the model at a reduced relative velocity. Consequently, the model seems to "surf" on the wave crest for a short period of time. As the model's increased speed due to forward surge dissipates, it is overtaken by the wave. This in turn causes the model to virtually fall off the back of the wave. This abrupt descent initiates a roll motion. Since the waves are regular, this process is cyclically repeated and results in larger roll angles. The existing roll motion is agravated by a decrease in stability as the wave crest is in the proximity of amidships. When the trough approaches the midship section the righting moment is suddenly increased and the model is then able to counter-act the rolling motion and rights itself. At this point, the compounded effect of the model's momentum in roll and the presence of another wave crest in the neighborhood of the midship section contributes to an even larger roll motion to the other side. This process continues until the model capsizes at the 15th second after only three cycles. For this run the rudder was not activated and was kept at about a zero angle. A s a consequence, the roll motion amplitude continuously increased up to the point where capsizing occurs. In contrast to Kerwin's [9] conclusions, a large number of roll cycles is not found to be required for large roll angles and capsizing to occur. Figure 12 shows another case of capsizing caused by low cycle resonance. In this case the intentional use of the rudder was substantial. It can be seen that at the 12th second the rudder was applied to correct the model's course. As a result, the roll motion increase was severe due to the additional heeling moment contributed by the rudder. From then on the roll motion gradually became larger. At about the 35th second , again a large rudder movement caused capsizing after only seven cycles. From this figure the low values of surge acceleration resulting from the low 24 encounter frequency can be realized. This will subsequently be considered as one of the assumptions for the numerical simulation. Rudder usage, as it seemed to increase the likelyhood of capsizing, was kept at a minimun level whenever possible. It was observed that sudden use of the rudder could cause the low cycle resonance process to accelerate greatly, as shown in Figure 12. The continuous increase in the roll motion amplitude is evident from the very beginning of the run. One can conjecture that this run would have resulted in large roll motions even without the use of the rudder. However, the number of cycles required to achieve capsizing would have been much larger. Such a high number of roll cycles would not likely have a practical interest. Wave frequency and wave amplitude also showed to be of capital importance in the capsizing process. Table 4 shows the number of capsizes obtained at each wave frequency tested. TABLE 4 : CAPSIZING STATISTICS Wave Frequency Number of Capsizes (Hz.) Heavy Light 0.700 None None 0.800 2 3 0.850 1 1 0.900 1 1 0.950 None 1 0.975 None None 1.000 None None It can be seen that about 50% of the capsizes occured at a wave frequency of 0.8 Hz. Figure 13 shows a record of this wave. While this was not the steepest wave used, it caused the most severe motions. Observed capsizes with GM values other than # 1 and wave amplitude other than # 4 were noted with this particular wave 25 frequency only. One of these capsizes is the one shown in Figure 12, where GM # 2 configuration was tested. For this case, the encounter period corresponding to the wave frequency of 0.8 Hz. was 3.81 seconds. The model's roll natural period in waves was about 3.5 seconds. Paul I ing [11] noted that "the maximum possible roll angle is attained when the wave encounter period is about equal or greater than the natural period of roll" .The proximity of the values of these two periods and a severe rudder usage are'believed to have been the cause of the only capsize in the full load condition with GM # 2. Most capsizes occured with a wave amplitude # 4. These corresponded to waves whose crests were sufficiently high to reach the model's deck. Waves of lesser amplitude did not cause capsizing in the full load condition. However, wave amplitudes lower than # 4 did cause capsizing in the light load condition. 6. NUMERICAL SIMULATION A realistic numerical simulation of a fishing vessel motions requires that especial attention be placed on the computation of the hydrodynamic coefficients contained in the equations of motion. The nonlinear nature of these coefficients coupled with the interaction among them renders a heretofore unsolved problem. Instead, assumptions must be introduced in order to make the problem mathematically tractable. It has been pointed out [13] that, at low encounter frequency values in a longitudinal seaway, motions may be assumed to be mainly governed by hydrostatic forces. Hydrodynamic forces may then be calculated without extreme accuracy. Emphasis is therefore placed in the accurate computation of hydrostatic forces for the "exact" position of the ship in waves. These forces are used as the right-hand side of the equations of motion which in turn can be numerically integrated in the time domain. 6.1 FORMULATION OF THE PROBLEM The vessel is assumed to have a plane of symmetry as regards the shape of its lines plan. Further, it is assumed to behave as a rigid body having, initially, six degrees of freedom. Newton's second law may then be written in its linear momentum form in the following manner: ^-(m v ) dt = w (VI-1) Conservation of angular momentum gives; ^-(l 3) = M (VI-2) dt 26 27 The right hand side of Eqs. VI — 1 and VI-2 are the sums of the forces and moments resulting from the interaction of the ship and waves in addition to the gravitational force. Effects such as wind and currents are not considered. The motion variables involved are the instantaneous position, velocity and acceleration of the ship. The high degree of nonlinearity of the problem arises from the fact that forces and moments are nonlinear functions of the motion variables and the rate of change of momentum in VI-2 contains nonlinear terms. 6.2 SYSTEMS O F COORDINATE A X E S For the determination of the spatial position of the moving vessel , three coordinate systems are introduced as shown in Figure 14 and Figure 15. These are: 1 - O j - ^ r j i S , Newtonian system of coordinates fixed in space and so oriented that i; 1?71 plane lies on the undisturbed water surface. 2 - G-%r}$ attached to the center of gravity of the vessel and parallel to the system O i - ^ r j ^ i . during the entire duration of the motion. It defines the mean translational position of the vessel with respect to 3 - G - X Y Z system of Cartesian coordinates attached to the center of gravity of the vessel . The motion of G - X Y Z with respect to G-if7j$ gives the angular desplacements of the ship. The motion of the vessel is then composed of the translational motion of its center of gravity G and the rotational motion of the axes G - X Y Z about the axes G-£f?$ . Therefore, the position of the ship is determined by three coordinates G£g , G?7g, G$g as shown in Figure 15 and three Eulerian angles 6, \p and 0 as shown in Figure 14. Then, two vectors, translation 3 and rotation T, can be defined as fol lows: 28 €3} = (Jg Vg Sgl (VI-3) {T} = {0 ^ <j>} (VI-4) 6.3 EULER ANGLES Euler angles may be chosen in diverse ways. A s first published by Scientia Navalis of St. Petersburg in 1749 and as adopted in theoretical mechanics, Euler angles proved to be inconvenient in the study of ship motions. Kriloff [20] proposed a new "nautical" system of Euler angles free from their initial inherent problems. This nautical system of Euler angles can in turn be chosen in different ways which, although having no effect on the final results, permit a greater or lesser degree of clarity in the visualization of the motions. The system adopted in this work was first introduced by Blagoveshchensky [21], and as mentioned above, is shown in Figure 14. The line GN, called nodal line, is defined by the intersection of the coordinate planes £GT ] and YGZ. The positive direction of GN is obtained by the counterclockwise rotation of the axis $ toward the X-axis through the smallest angle looking from this positive direction. The plane defined by XG$ is always perpendicular to the nodal line. Euler angles are defined as follows: - Roll angle 0 lies in the plane YGZ. Positive values of this angle are measured counterclockwise from the line GN to axis GY, looking from the positive end of the X-ax is . The angle of roll is thus the rotation about the X-axis and positive to starboard. The roll angular velocity vector 0 is oriented along the X-ax is . - Pitch angle \p lies in the plane $GX. Positive values of pitch are measured by counterclockwise rotation of the moving frame G - X Y Z about the nodal line looking 29 from the positive direction of GN. The value of \pAs obtained by substracting 90 degrees from the angle formed by G$ and GX. The angular velocity vector \[/ is directed along the nodal line. - Yaw angle <j> lies in the plane | G 1 7 . Positive values are measured counterclockwise from Gr? to GN looking from the positive direction of G$. Yaw angular velocity <j> is directed along the axis G $. If yaw angular motion is absent, the pitch angle is measured as the rotation about the axis G17. 6.4 COORDINATE SYSTEM TRANSFORMATION Transformation from the stationary coordinate system to the ship coordinate system is attained by means of the following expressions,Blagoveshchensky [21]: and visceversa: £ 1 = £ g + a t x + b,y + c, z (Vl-5a) r?! = 7 } g + a 2 x + b 2 y + c 2 z (Vl-5b) $1 = Sg + a 3 x + b 3 y + c 3 z (Vl-5c) x = a ^ - ^ g ) + a2(r?,-7? g) + a 3 («J 1-5 g) (Vl-6a) Y = M S i - S g ) + b2(771-7?g) + b 3 ($ 1 -S g ) (Vl-6b) z = c x (Si-5g) + C 2 (T7 , -17g) + C 3 (S . -Jg) (VI-6C) The instantaneous position of the vessel is given by the three coordinates of the center of gravity £g , rjg and $g and the three Eulerian angles 6, \p and 0. The relationship between the system of axes attached to the ship is defined by the direction cosines aj, bj and Cj with i = 1,2,3. The equations defining the direction cosines are given in Appendix B. 30 6.5 ANGULAR VELOCITIES The vessel's angular velocity vector U may be decomposed in three orthogonal components p,q and r.That is: 8 (VI-7) These vectors are the projections of 0 onto the movable axes x, y and z respectively. Their expressions, derived in Appendix B, are as follows; p = 6 - <t> sin \p q = <j> sin 6 cos ^  + \jj cos 6 r = 0 cos 6 cos \^  - \p sin 6 (VI-8a) (Vl-8b) (Vl-8c) Equations Vl-8a, Vl-8b and Vl-8c can be written in matrix form in the following manner; 1 0 -sin \}j 0 cos 6 sin 6 cos \p .0 -sin 0 cos 0 cos (VI-9) Then; &} = [T] _ {T} dt (VI-10) where; 31 1 0 - s in \j/ [T] = 0 cos 6 sin 6 cos \p .0 -s in 6 cos 5 cos \[/ is the coordinate transformation matrix. 6.6 EQUATIONS OF MOTION The equations of motion of the ship, assumed to behave as a rigid body, can be divided in two groups. One group defines the translation of the center of gravity of the ship. A second group describes the rotations of the vessel about the fixed axes passing through its center of gravity. On the basis of Newton's second law the translational equations of motion are as fol lows: V d 2 - - «g = E 0/1-1.18) g dt V d 2 „ _ - - 7j = Z F (Vl -11b) g dt 2 a " V d 2 - - S g = E F . (VI-11C) g dt 2 y b i The right-hand side members of Equations VI-11a, VI — 11b and VI-11c are the sums of all the forces acting upon the vessel projected onto the stationary axes O ^ . O ^ a n d 0^,. The equations of motion for the rotational degrees of freedom were introduced in vectorial form in Equation VI-2. This equation can be decomposed into three equations representing the moments about the moving axes. 32 The rate of change of angular momentum expressed in the moving frame G - X Y Z is given by, [22]; —c ^) = i — + o x in = H (vi-12) dt 9t where: xx 'xy 'yx 'yy 'zx ~'zy 'xz 'zy <zz (VI-13) and: l x x = L m ' ( y 2 + z 2) (Vl-14a) lyy = L m' (x2 + z 2) (Vl-14b) l z z = I m'(x 2 + y 2) (Vl-14c) 'xy = 'yx = 2 m ' x y (Vl-14d) l y 2 = 'zy = 2 m ' y z (Vl-14e) 'xz = 'zx = L m ' z x (Vl-14f) where the sumations are applied to all particles of mass m' comprising the ship. The partial derivative of the angular velocity vector with respect to time is defined as; I0?'} = (P q r} (VI-15) 33 The ship is assumed symmetric with respect to the centerplane, therefore; l X y = lyx = "yz = >zy = ° ( V ' " 1 6 ) Bearing in mind the ship's symmetry, substitution of Equations VI—9 and VI-13 into VI — 12 gives the equations of motion corresponding to the rotational degrees of freedom. These equations are: Z M X = l x x p + ( l z z - l y y ) q r - l z x p q - l x z r (Vl-17a) Z M y = l y y q + ( l x x - l z z ) pr + l z x (p'-r>) (Vl-17b) Z M Z = l z z r - l z x p + ( l y y - l x x ) pq + l x 2 r q (Vl-17c) In order to solve Eqs. 17, values of the moments of inertia were estimated as described in Appendix C. 6.7 COMPUTATION OF FORCES AND MOMENTS It was mentioned that forces and moments acting on the ship may be assumed to be predominantly hydrostatic provided that a low value of the encounter frequency is maintained. Hydrostatic forces are modelled by accurately integrating the wave pressure field around the wetted surface of the hull up to the wave surface. Estimation of the hydrodynamic coefficients is presented in Appendix C. Hydrodynamic coefficients for pitch and heave are obtained by using a c lose- f i t method. Roll motion, as one of the most important responses of a ship, demands a correct prediction of roll damping in order to ensure safety as well as to attain a fair understanding of the ship motions in waves. The vessel under study was observed to possess a very high roll damping with marked nonlinear behaviour. For this reason a more detailed component analysis of the roll damping was performed as outlined in 34 Appendix C following Himeno [23]. Forces and moments due to wave action in the G - X Y Z system of coordinates are: Fx.w =SSS | £ dv (Vl-18a) F y > w = - / / / | £ dv (Vl-18b) ?z,w =SSf dv (Vl-18c) M x , w = - J ' / / ( z fy -V fz ) dv (Vl-19c) M y > w = - / / / ( x F z ~z H > d v (VI-19C) M z > w=-///(y I; " x g~y ) d v (Vl-19c) Gauss Theorem is used to transform the volume integrals in Equations 18 and 19 into surface integrals more amenable to numerical analysis. At this stage the computer program uses the undisturbed pressure fields derived from two-dimensional linear wave theory and nonlinear Stokes second order wave theory as given by [28]. Relevant formulae are readily available from this reference. It is apparent that the flow field is modified by the motions of the ship. These motions, which are dependent on the ship geometry and mass distribution, are also affected by the modifications of the flow field itself. A three dimensional flow pattern developes with flow separation and turbulence in particular towards the ends of the ship. However, the assumption of an undisturbed two-dimensional flow field under the condition of a low encounter period yields reasonably good agreement 35 between experimental and numerical results. The numerical part of this work is restricted to ship response in a pure longitudinal seaway. For this reason, sway and yaw motions are not considered. In case of having any obliquity between the ship's centerplane and the wave crest, consideration must be given to these two degrees of freedom as well as to a wave exciting term in the roll equation of motion. Under the assumption of a low encounter frequency value, surge may be neglected. The ship speed is then considered to be constant. The equations to be solved are then VI — 11c, Vl - I7a and VI-17b. The total heave force is obtained in the following manner; C n = C h w - B h $ g - R n $ g (VI-20) where; = pg J* 2 y(x)dx The value of C n is calculated at each time step and introduced into equation Vl-23a in place of W. Moments due to wave action are affected by damping and restoring moments in a similar fashion as in Eq. VI-20. That is; L M X = M x w - B r 0 (VI-21) I M y = M y , w " B p * " R p * ( V ' " 2 2 ) The restoring moment coefficient in pitch in VI-22 was approximated as V GM|_ [25]. Equations VI-21 and VI-22 are calculated at each time interval and introduced in equation Vl-24a in place of ft to carry out the numerical integration. The equations are referred to the G - X Y Z system of coordinates. 36 6.8 NUMERICAL INTEGRATION OF THE EQUATIONS OF MOTION The numerical integration in the time domain of the equations of motion was made using a fourth-order Runge Kutta process as developed by Gill [26]. The equations of motion Vi — 1 and VI-2 are re-written as first order simultaneous ordinary differential equations in the following manner: dv (Vl-23a) dt m dt v (Vl-23b) ^ = l - i p - J J x i n ] (VI-24a) dt & = [T]- 1^} (Vl-24b) dt Eqs. VI-23 and VI-24 give a system of six coupled simultaneous highly nonlinear first order differential equations which are solved by numerical integration in the time domain. 7. COMPUTER PROGRAM AND NUMERICAL RESULTS 7.1 COMPUTER SOLUTION The computer program has been written in VAX-11 FORTRAN language. The program is approximately 1800 lines long and is compiled on the FORTRAN 77 V-4.4 compiler attached to the VAX-750 computer of the Department of Mechanical Engineering at U.B.C.. The flow chart shown in Figure D-1 of Appendix D represents the basic structure of the computer program. The main program contains all input/output information whereas all relevant calculations are made in separate subroutines in a block form. This block-structure was chosen so as to facilitate potential changes and/or additions to the program. 7.2 TIME STEP AND NUMBER OF HULL ELEMENTS A series of runs was performed in order to study the economical aspect of the computer program. The number of hull elements was gradually increased and the CPU time in seconds required for a fixed maximum simulation time to be achieved was measured. Figure 16 shows plots of the results. An almost linear upward increasing tendency in computer dollars can be realized as the hull is defined in a more accurate manner. From this figure, an estimate of the CPU time, and hence of the computer dollars required for a given run, can be obtained once the number of hull elements and number of time steps are defined. It was found that, for the ship studied in this work, a waterline spacing of 0.5 ft., i.e. 676 hull elements, gave accurate results at minimun cost (0.591 CPU sec/number of At). 37 38 7.3 VALIDITY OF THE NUMERICAL METHOD In order to assess the correctness of the numerical simulation, a series of still water runs at zero forward speed were made and compared with experimental values. Test runs at zero speed in waves were not performed due to the lack of experimental data to be compared with. The natural periods of the ship were calculated for a number of GM configurations. Figure 17a and 17b show plots of roll natural periods for the heavy and light load conditions. Agreement was found to be very good. The other set of test runs in still water included the measurament of the roll decay characteristics of the vessel. Figure 18 shows comparison of experimental and numerical results for the extreme GM configurations in heavy load condition. Again agreement was very satisfactory. From this results, it was concluded that the roll damping scheme reallistically predicted the energy loss during the roll motion. Furthermore, exact knowledge of the initial conditions resulted in a very acceptable prediction of the ship motions. 7.4 ANALYSIS OF INITIAL CONDITIONS Initial conditions are, in general, of fundamental importance in the behaviour of a system represented by an initial value problem. The actual initial conditions of the experimental runs were not known and this was expected to lead to disagreement between experimental and numerical results. For this reason, a brief analysis of the roll angle as an initial condition was performed in order to study its effect on the roll motion record. The study was limited to the initial roll angle due to the numerous possible combinations of the six initial conditions. A detailed study of the initial conditions is highly advisable if a higher matching level between numerical and experimental values is to be attained. 39 Figure 19 shows motion records with a zero initial roll angle. As expected, roll motion was not developed. Thus a non-zero value of the initial roll angle is required for roll to be initiated. Heave and pitch records show almost equal periods of oscillations and a beat-type of behaviour. This phenomenom is found in a number of vibratory systems in which the frequency of the forcing function w e is very close to the system's natural frequency ( J N . In this case, the pitch and heave motions could be considered as sinusoidal functions of frequency (a>e + w n)/2 with peaks having a phase angle of 90 degrees. In the experimental part of this work, roll oscillations begun due to the combined effect of the heeling moment created by the rudder motion and the deviations of the model from a straight course along the tank - yawing. Consideration of both the steering mechanism and yaw motion in the numerical procedure pose a hydrodynamic problem of compounded complexity. The abundance of terms which must be considered in this case give the problem a formidable aspect. For this reason, at the present stage of this work, the initial roll motion was started by introducing an initial roll angle 9Q. Figures 20 through 22 show the effect of the initial roll angle whose value was progressively increased from 0.5 to 5 degrees. A transient behaviour during the first 25 seconds of the roll records is apparent, particularly in the ones with initial roll angles of 3 and 5 degrees. From this time onwards, roll records have similar features with the same capsizing time of about 185 seconds. Initial roll angles greater than 5 degrees caused capsizing at earlier times. Since the wave and GM configurations for these runs proved to be one of the most dangerous during the experimental part of this work, initial roll angle values were specified as less than 5 degrees for subsequent runs unless otherwise stated. 40 7.5 LINEAR WAVE ANALYSIS The computer program was run for the prototype scale, whose characteristics are shown in Table 5, with the same speed as in the experimental procedure. TABLE 5 ; PROTOTYPE CHARACTERISTICS Light Heavy Length overall (LOA) 77.00 ft. Length btn. perpendiculars (LBP) 70.00 ft. Beam (B) 23.00 ft. Depth (D) 15.00 ft. Draft (T) 9.500 ft. 10.50 ft. Displacement (V) 220.3 ton 254.2 ton Block Coefficient (Cb) 0.500 0.531 Midship-Section Coefficient (Cm) 0.756 0.775 Waterplane Coefficient (Cwp) 0.850 0.862 KM 13.780 ft. 12.820 ft. GM # 1 (1.11 % B) 0.2560 ft. 0.2560 ft. GM # 2 (2.22%B) 0.5110ft. 0.5110ft. GM # 3 (5.00% B) 1.1520 ft. 1.1520 ft. G M # 4 (9.27% B) 2.1330 ft. 2.1330 ft. The model speed of 4.3 ft /sec then corresponded to an operational speed of 9.2 knots. Heavy and light weight conditions were run under a sinusoidal wave system corresponding to the frequencies and amplitudes shown in Table 6. This table shows the wave characteristics scaled-up from Table 2. The experimental condition for wave frequency of 1 Hz was not investigated numerically. A new wave condition, unattainable experimentally, was used to simulate the case of equal wave and ship speed. The resulting wave had a length of 47.06 ft (model scale frequency of 1.249 Hz). The greatest amplitude (Amp. # 4) was obtained from Miche's maximum wave steepness expression given by Equation V - 1 . Amplitudes # 2 and # 1 were obtained by reducing amplitude # 4 by 15% and 30% respective I ly. 41 7.5.1 PARAMETER S The stability parameter S developed in Section 5.3 was obtained for the wave conditions shown in Table 6. The maximum experimental run time resulted in a maximum simulation time of about 190 seconds. During experiments, the model was initially held vertically until it was released and data acquisition started. This testing procedure did not introduce an initial roll angle. The computer program, however, requires a non-zero value of the initial roll angle. In order to counteract this arbitrary initial condition, the average roll amplitudes R for the calculation of the parameter S were calculated by only taking into account the roll peak values higher than the initial roll angle 60, TABLE 6 : PROTOTYPE-SCALE W A V E CHARACTERISTICS Freq. Wave Length Amp. # 1 Amp. # 2 Amp. # 3 Amp. # 4 (Hz.) (ft.) (^.) (ft.) (ft.) (ft.) 0.194 136.5 2.63 3.52 _ _ 4.680 0.222 104.0 3.32 4.93 5.340 0.236 92.30 3.31 4.55 4.633 0.250 82.03 3.28 4.17 4.470 4.570 0.263 73.84 3.12 3.88 3.770 3.680 0.270 70.00 3.27 3.68 2.980 2.965 0.346 47.06 2.34 2.84 3.340 Figure 23 and Figure 24 show numerically obtained parameter S values for GM # 1 and GM # 2. As in the experimental part, values of S below 5 showed regions of stable ship behavior. Values of S above 10 corresponded to capsizing cases. The only exception was the case for L/X = 0.513, light load condition, GM # 1 and wave amplitude # 4 where capsizing occured and the S value" was 9.83. Values of S between 5 and 10 corresponded to runs with large roll motions. These were considered unsafe conditions. 42 The effect of rudder usage and yawing on the roll motion obviously influenced the value of the experimental stability parameter. Values of S obtained numerically were found to be closer to the experimental ones in cases with minor rudder usage and with a small model deviation from a straight course. One case is the run of Figure 11 whose ship length to wave length ratio was L/X = 0.758. Agreement between numerical and experimental values of S in this case is quite good. 7.5.2 LOW CYCLE RESONANCE Examples of low cycle resonance were presented and discussed in Section 5.3. In this section, the wave conditions used are the same as those in 5.3 and the motion records are compared with the experimental ones. The initial conditions for each run of the model were unknown, and therefore a predetermined initial roll angle was used in the simulation. Figure 25 shows real time motion records for the scaled-up wave conditions used in Figure 11 where a minor usage of the rudder was observed. The characteristic build-up in the roll angle for this type of capsizing mode is apparent from the start of the record. At t = 0 , due to the wave crest presence amidships, the ship rolls to starboard until the restoring moment is sufficient to counteract the rolling moment. At this point, the ship rolls in the other direction but the proximity to the midship section of the wave trough does not allow the roll to port to develope. In this situation, the increased restoring moment causes the vessel to roll to starboard with a higher roll angle than the previous positive peak. This regular pattern in the record is maintained up to about 20 seconds when a more erratic roll motion sets in. Due to the wave trough position in the neighborhood of the midship section, a more dynamic behavior is observed to force a violent change in the roll direction. However, the quick build-up nature of the roll motion is again apparent until 43 capsizing takes place at about 48 seconds with the wave crest positioned amidships. This time corresponds with a capsizing time of 13.31 seconds in the model scale.The time required by the experimental model to capsize was 15 seconds. It can be seen that, as obtained experimentally, only a small number of cycles is required for capsizing to occur. The encounter period to roll period ratio for this run seemed to roughly vary from one to three halves. The periods of the roll motion have relatively good agreement with the corresponding periods of the experimental run during the first 20 seconds of the record. The difference in roll amplitude could be attributed to a number of reasons. The most important one is believed to be the absolute lack of information on the actual initial conditions in all degrees of freedom. These conditions were assumed to be zero with the exception of the initial roll angle 9Q. Minor error is expected to arise from the small deviation of the rudder from a zero-angle position and the 2-degree model departure from a straight course shown in Figure 11. Other sources of disagreement could be ascribed to the assumptions made in the formulation of the problem. The basic assumption of strip theory is that the time dependent flow bordering thin strips of the ship is two-dimensional. That is, the fore-and-aft component of the flow is neglected. This approximation leads to questionable results at the ends of the ship where pronounced sectional changes in the longitudinal direction occur, - the so called end-effect problem. The Froude-Kriloff hypothesis states that the ship motions do not alter the particle motions in the surface wave although the wave particles do influence the ship motions. The violent nonlinear nature of the motions experienced by the type of vessels studied in this work could make this assumption a source of error even with a low encounter frequency value. However, analysis of the actual fluid-structure interaction in nonlinear ship dynamics in the presence of forward speed poses a tremendously laborious hydrodynamic problem, heretofore, without solution. 44 For the wave conditions shown in Figure 12, capsizing was not achieved. The program was run for an unusually long maximum simulation time of 400 seconds. The low cycle resonance build-up seemed to occur during times in which the right phase between roll motion and wave encounter set in. However, it damped out as soon as this phase was disturbed. This phenomenon is shown in Figure 26 where, for reasons of clarity, only part of the record has been plotted. The highly detrimental effect of the rudder usage can then be attributed a major role in the capsizing shown in Figure 12. A wave condition of particular interest for most ships is the one of equal ship and wave length. The maximum wave height that was attained experimentally for this condition was 5.93 ft. However, it was believed that greater heights could occur in a seaway. A higher wave was used according to Equation V-1 with a resulting height of 9.8 ft. Figure 27 show motion records for this case. The initial roll angle used was 1 degree. At the onset of the run, the ship roll angle increased to about 5 degrees due to the wave crest presence amidships. Following the roll angle decrease, the wave trough causes the roll motion attenuation until the wave crest is again located amidships. A severe roll angle of about 20 degrees takes place at about 20 seconds. Subsequently, the wave trough causes the roll motion to almost die out. Large roll again developes as the wave crest progresses from about 9% aft station 5 with two roll peaks occuring at about the 43th second and the 46th second. The first peak develops as a result of a negative heave value that, with the wave crest amidships, causes the vessel to lose waterplane area significantly. In the beginning of the second peak, the wave has progressed past station 5 and a large positive pitch peak of about 10 degrees is experienced. This causes great bow emergence and a corresponding loss of stability at a relatively high roll angle. As a result, the vessel undergoes the second roll peak. As the pitch angle decreases, a higher value of the water plane area is rapidly gained and the roll motion decreases. The wave trough, 45 positioned between 20% aft.and 20% forward of station 5, tends to decrease the roll motion to about 18 degrees at t = 5 2 seconds and 15 degrees at t = 5 5 seconds. Then, the vessel rapidly swings to port and the presence of a wave crest causes capsizing at t = 6 0 seconds with a roll period of about half the value of the encounter period. For a low cycle resonance capsize to occur, there must be a relationship between the wave encounter period and the vessel's natural roll period. The encounter period may be expressed for deep water as a function of the ship speed, the wave length and the ship's heading angle by the following equation [22]: T - 2 7 1  6 ~~ w w 0 - S v c o s a /9) (VH-1) Figure 28 shows wave encounter periods at different angles of heading and various wave lengths for a ship speed of 9.2 knots. For heading angles below 50 degrees, there is an apparent condensation of the curves. This means that for quartering and overtaking seas situations, the vessel will encounter a large number of different waves at almost the same encounter period. The consequences could be disastrous if this encounter period to the vessel's natural roll period ratio took on one of the critical values predicted by the Mathieu instability chart. For heading angles greater than 60 degrees, the encounter periods are smaller than 9 seconds. Thus for beam and head seas situations, the encounter periods could be close to the roll periods corresponding to GM # 3 and GM # 4 configurations. These were found to be very stable configurations throughout all testing. Therefore, in view of the high stability observed in these configurations and the more dangerous nature of the longitudinal seaway, stability problems are not expected to arise in head or beam seas situations. 46 Figure 29 shows encounter period values as a function of the wave length for the case of zero angle of heading, i.e. pure longitudinal seaway. It can be seen that for waves in the length range betwen 75 ft and 125 ft the encounter period varies from 18 seconds to 13 seconds. These values are extremely close to the vessel's natural roll periods for GM # 1 and GM # 2 configurations. For wave lengths greater than 125 ft the encounter period shows very little variation with the wave length. 7.5.3 PURE LOSS OF STABILITY It was mentioned in Section 5.1 that capsizing due to pure loss of stability only was not likely to occur for this type of fishing vessel. However, this was not experimentally proven. In this section this conclusion is discussed under some of the conditions required for pure loss of stability to occur. These conditions are wave speed equal to ship speed and relatively high ship speed. It should be emphasized that coexistence of all requirements listed in 5.1 is very unlikely for B.C. fishing vessels. Figure 30 shows, for the heavy condition, the prototype's roll motions with a wave whose crest permanently travels amidships. The wave length is 47.06 ft and the wave height is the maximum possible according to Miche's expression (6.68 ft). Heave and pitch motions were found to be very small. An initial roll angle of 10 degrees was used to assess the amount of stability possessed by the ship under this wave condition. It can be seen that the roll motion decays abruptly in only a few cycles as a result of the high value of the roll damping. Furthermore, in this condition the ship possesses a very reduced GM value and still shows a considerable surplus of stability to counteract a potentially large external force, such as wind, existing roll motion or unsymetrical loading, that could cause the heeling action. Figure 31 shows results for the light load condition. The decay tendency of the roll motion is more pronounced in this relatively more disfavorable load 47 condition and the vessel still remains stable. 7.6 NONLINEAR WAVE ANALYSIS Linear waves have been proven to be most useful in the analysis of engineering situations. However, it is an observed fact that actual waves have steeper crests and flatter troughs than linear sinusoidal waves. Waves with these characteristics resemble the profile of finite amplitude waves and are nonlinear in nature. Figure 32 shows free surface elevation profiles for linear and nonlinear waves. The considerable greater steepness of the nonlinear wave is apparent from this figure. The hydrodynamic theory of finite amplitude waves, not infinitesimal as assumed by linear wave theory, was developed by Stoke [30]. The solution of the wave problem by Stoke's theory consists of assuming that wave properties can be represented by means of perturbation series. Therefore, the precision of the solution can, in principle, be extended to any degree by considering a sufficient number of terms in the series. For practical purposes, Stokes' second order waves present a satisfactory approximation. One important characteristic of this theory is that the linear dispersion relationship is still valid. In this case, the wave celerity and the wave length remain independent of the wave height. This greatly facilitates the theory's application. Higher order waves present a difficulty in that the wave celerity depends on the wave height and therefore, their use is more complicated. Second order theory, however, presents serious limitations when applied to shallow water situations. The range of applicability is given by the conditions H/d < < (kd)2 for kd < 1 and H/X < < 1 [31]. All cases studied here satisfy the conditions for second order wave theory to be applicable. The effect on the fishing vessel's stability due to Stokes 11 order waves is analized in the following sections. 48 7.6.1 PARAMETER S Parameter S values were again numerically obtained for a nonlinear wave system with the characteristics shown in Table 6. The maximum simulation time and the procedure to obtain the average roll amplitudes R were the same as explained in section 7.5.1. Figure 33 shows numerically obtained values of the parameter S as a function of the the ship length to wave length ratio for nonlinear waves. The tendency of the stability parameter concerning regions of stability and instability followed that reported in Table 3 and in section 7.5.1. The magnitude of the parameter, however, was frequently larger for every wave amplitude and frequency as well as for every GM configuration. This effect can be attributed to the more severe roll motions experienced by the vessel due to the steeper surface profile of the nonlinear wave crest. This caused the stability reduction to be even more pronounced than that produced by the linear wave crest. 7.6.2 LOW CYCLE RESONANCE Figure 34 shows motion records for the same wave conditions used in Figure 27. Capsizing was again observed to occur at an encounter period to roll period ratio of about two. However, there is important differences between these two figures that should be noted. Heave and pitch records have higher amplitudes for the nonlinear wave case due to higher crest steepness. This effects become of particular importance for positive pitch values and negative heave values. The ensuing relatively higher loss of waterplane area causes the stability of the vessel to reduce further. For large positive heave values, water-on-deck conditions and free surface effects could be more frequent, causing greater virtual loss of stability by raising the position of the center of gravity of the vessel. It is believed that the higher pitch and heave values caused the low cycle resonance process to start at an earlier time, 49 producing a reduction, for this case, of about 30% in the capsizing time. Figure 35 shows motion records for the same configuration studied in Figure 26 using a nonlinear wave of equal characteristics. The wave crest is significantly steeper. A s a result of this, large roll occurs between t = 0 and t = 5 seconds. The roll motion decreases thereafter until the wave crest presence amidships compounded with negative heave causes the roll motion to peak at almost 40 degrees at about t = 17 seconds. Subsequently, the wave crest progresses past station 5 causing a gain in water plane area as the heave value becomes positive. Then, the roll motion rapidly decreases to small values. At about t = 50 seconds, there is some build-up in the roll motion which again decreases at about t = 80 seconds. From this time onwards the roll motion becomes insignificant. The energy contained by the seaway is then basically absorbed by heave and pitch. In this situation, heave and pitch records resemble those shown in Figure 19. Though capsizing for this GM configuration was not achieved, larger roll motions than the ones observed in Figure 26 for a linear wave occured. However, it is believed that the use of only one initial condition 9Q causes a transient behaviour that enhances the ship's dynamic response at the begining of the record. A method to counteract the effect of starting transients could have been used. Such a method could be a linearly increasing step function varying from 0 to 1 that multiplies all forces applied on the vessel during the first 30 seconds of the run. However, the resulting initial motions would be unrealistic. As mentioned before, an in-depth study of the initial conditions should improve the results and reduce the starting transients. 7.6.3 PURE LOSS OF STABILITY The conditions studied in 7.5.3 were repeated for a nonlinear wave with the same characteristics as in that section. The resulting wave had a considerably 50 steeper crest of 4.1 ft as opposed to 3.34 ft of the linear wave. Figure 36 show motion records for this wave condition for the light load configuration with a very low GM value. Pitch motions were again observed to be very small. Heave, however, had a higher intensity than for the linear wave case. The vessel's roll motion again showed a rapid decay tendency. This tendency, however, tended to be more sluggish than for the linear case. A s a result, the roll period increased by about 20%. Linear and nonlinear wave analysis of pure loss of stability showed that the vessel remained stable under extreme wave conditions and greatly reduced GM values. 8. CONCLUSIONS The dynamic behavior and stability of a typical B.C. coast fishing vessel in regular waves have been experimentally and numerically studied in a longitudinal seaway. Extensive experimental in-door tests showed that very large roll motions could develope within a small number of cycles and easily lead to capsizing. It is important to note that for small vessels, such as fishing boats and tugs, the concept of a small number of cycles for capsizing to occur is comparatively more important than for large vessel types. Large cargo or passenger ships possess roll periods that may range from 30 to 60 seconds. Even when a small number of cycles is required for large roll motion to develope, the large vessel would probably take a considerable period of time. The small vessel, however, will require substantially less time for significant roll build-up to take place. The ensuing short period of time to attain large roll might not be sufficient for seaman action to be taken. Severe roll motions and a large number of capsizes were found to be caused by low cycle resonance. Motions were noticeably larger in light load conditions. Capsizes were closely related to the portion of the wave system corresponding to waves of about 100% to 150% of the ship length. This relatively broad wave length range indicates that capsizing could occur in a fairly large number of sea conditions. As recorded by this author from fishermen oral reports, the wave pattern in the B.C. coast is generally regular with a wave length system of about 100 ft to 140 ft. Therefore, the wave conditions analysed in this work could certainly be assumed as realistic for the B.C. coast. Unfortunately, the lower length band of the B.C. wave range corresponds with the upper limit of the capsizing wave range where 50%of the total number of capsizes occured. Wave amplitudes so that the wave crest was close to the model's deck were required for capsizing to occur in the full load condition. In the light load condition, 51 52 however, smaller wave amplitudes were sufficient to cause capsizing. Sudden or excessive applications of the rudder were observed to accelerate greatly the low cycle resonance process. Furthermore, this process could be enhanced in such a way that initially safe conditions could be transformed into a situation where large roll angles, 20 - 30 degrees, occured. For a roll angle of 20 degrees, the deck of this type of ship would already be under. The additional effect of wind or further usage of the rudder, as in Figure 12, could lead to the loss of the vessel. A numerical model to predict the vessel's coupled nonlinear motions has been performed. A linear and nonlinear wave analysis has been carried out. The method is quite efficient and gives results which compare favorably over the whole range of wave frequencies with experimental results within the limits of the implied assumptions of the numerical procedure. The model also showed to be very useful in predicting capsizing conditions. The most significant limitation of this model seems to be its inability to account for the rudder effect on the ship motions. Accurate values of the initial conditions should be obtained if the motion records are to be in better agreement with the experimental ones. A time dependent analysis of the roll damping has been implemented. The high nonlinearity of the roll damping coupled with its strong dependence on the ship forward speed has been taken into account. The method showed to be very effective in predicting the roll motion characteristics of the stationary fishing vessel in calm water. Numerical results in the presence of forward speed and waves, although not compared with experimental results, are expected to be satisfactory. The effect of nonlinear waves on the fishing vessel stability has been assessed and compared with linear wave results. Linear and nonlinear wave analysis of pure loss of stability showed that the vessel remained fairly stable under most unfavorable wave conditions and very 53 reduced GM values in both heavy and light load conditions. It could be concluded that capsizing due to pure of stability only is unlikely for this kind of fishing vessel under the wave conditions used in this work. Pitch and heave motions were observed to have an important role in the low cycle resonance capsizing process, particularly for waves in the lower length band where H/X ratio was higher. These motions were observed to be more severe in the presence of nonlinear waves. In this case, relatively higher loss of waterplane area, more severe water-on-deck conditions and/or free surface effects can occur causing greater ship instability. The amount of stability possessed by a ship is generally defined in calm water conditions and for a general type of ship regardless of its overall geometrical characteristics. In this work, a parametric study of the fishing vessel stability in regular linear and nonlinear waves has been performed. Ship geometry is implicitly accounted for. It has been attempted to define a value of the dynamic stability of the vessel by considering its dynamic response in regular waves. Analysis of the parameter S values gives an insight of the roll energy generated by a given regular sea state. The value S = 5 represented the limit between safe and unsafe conditions. Therefore, estimation by an appropiate technique of the numerator A-R affected by the minimum S value of 5 yields an estimation of the required dynamic stability in regular waves. Conversely, for a given dynamic stability value, a limit value of the roll angle or the maximum safe wave amplitude could be found. Since some of the most severe stability and rolling problems a ship may encounter are those associated with a longitudinal seaway, the dynamic stability calculated by the present method should be sufficient for most heading conditions. REFERENCES 1. KRILOFF, A . "A New Theory of the Pitching and Heaving Motions of Ships in Waves and the Stresses Produced by this Motions", Trans. IN A , vol . 37, pp. 326-359, 1896. 2. KRILOFF, A . "A General Theory of the Oscillations of a Ship in Waves" and "On the Stresses Experienced by a Ship in a Seaway", Trans. INA, vol. 40, pp. 135-196, 1898. 3. WEINBLUM, G. and ST. DENIS M. "On the Motions of Ships at Sea", Trans. S N A M E , vol . 58, pp. 184-231,1950. 4. ST. DENIS, M. "On Sustained Sea Speed",Trans. S N A M E , vol . 59, pp. 758-763, 1951. 5 .ST .DENIS ,M. "On the Motions of Ships in Confused Seas", Trans. SNAME, vol. 61, pp. 280-357,1953. 6. KORVIN-KROUKOVSKY, B.V. "Investigation of Ship Motions in Regular and Irregular Seas", International Shipbuilding Progress, vol. 2, No. 6, pp. 81-95, 1955. 7. KORVIN-KROUKOVSKY, B.V. and JACOBS W.R. "Pitching and Heaving Motions of a Ship in Regular Waves", Annual Meeting, S N A M E , pp. 590-632, November 1957. 54 55 8. GRIM,0 . "Rollschwingungen, Stabilitat and Sicherheit im Seegang", Schiffstechnik, Heft 1, 1952. 9. KERWIN, J.E. "Notes on Rolling in Longitudinal Waves", International Shipbuilding Progress, vol . 2, pp. 597-614, 1955. 10. PAULLING,J.R.and ROSENBERG,R.M. "On Unstable Ship Motion Resulting from Nonlinear Coupling", Journal of Ship Research, vol. 3, No. 1, pp. 36-46, June 1959. 11. PAULLING, J.R. "The Transverse Stability of a Ship in a Longitudinal Seaway", Journal of Ship Research, pp. 37-49, March 1961. l 2 .SALVENSEN,N . ,TUCK,E .0 .and FALTINSEN O. "Ship Motions and Sea Loads",Trans. S N A M E , pp. 250-287, 1970. 13. OAKLEY, O.H., PAULLING, J.R. and WOOD, P.D. "Ship Motions and Capsizing in Astern Seas", Tenth Symposium in Naval Hydrodynamics, ACR-204 Office of Naval Research, Dept. of The Navy, pp. 297-350,1974. 14. DU C A N E , P. and GOODRICH, G.J. "The Following Sea, Broaching and Surging", Trans. RINA, vol. 104,1962. 56 15. MICHE, R. "Mouvements Ondulatoires des Mers en Profondeur Constante on Decroissante", Annales des Ponts et Chaussees, 1944. 16. FRASER.D.J. , JONES,D.I. and VAN DER NET, G.A. "Cost of Stability for Fishing Vessels", Marine Technology, pp. 64-73, January 1976. 17. LEFRANCOIS, M. "Battery Powered Data Acquisition System for Self-Propelled Boat Models", University of British Columbia, Dept. of Mechanical Engineering Report, 1985. 18. U.S. C O A S T GUARD "Stability of Fishing Vessels", U.S. Coast Guard Navigation and Vessel Inspection Circular No. 6-68, 1968. 19. ALLIEVI, A.G. , C A U S A L , S.M. and ROHLING, G. "Motions and Stability of a Fishing Vessel in Longitudinal and Tranverse Seaways", Procs. of the 11th Ship Technology and Research (Star) Symposium, SNAME, Oregon, U.S .A., pp. 13-31, 1986. 20. KRILOFF, A.N. "Collected Works of A.N. Kriloff ", vol. XI, Ship Motions,USSR, 1951. 21. BLAGOVESHCHENSKY, S.N. "Theory of Ship Motions", Leningrad Shipbuilding Institute, USSR, 1954. Translated from russian,Dover Publications, 1962. 57 22. C A U S A L , S.M. "Dynamics of Marine Vehicles",Unplubished Class Notes,Dept. of Mechanical Engineering, University of British Columbia, 1985. 23. HIMENO.Y. "Prediction of Ship Roll Damping - State of the Art", University of Michigan, Report No. 239, September 1981. 24. K A T O . H . "On the Approximate Calculation of Ship's Rolling Period", J . of the Society of Naval Architects of Japan, 1956. 25. BHATTACHARYYA, R. "Dynamics of Marine Vehicles", John Wiley & Sons, New York, pp. 75-77, 1978. 26. G I L L , S. "A Process for the Step-by-Step Integration of Differential Equations in an Automatic Digital Computing Machine", Proc. Cambridge Philosophical Society, vol. 47, pp. 96-108, 1951. 27. AUBANEL, E.E. and OLDHAM, K.B. "Fourier Smoothing Without the Fast Fourier Transform", B Y T E Magazine, pp. 207-218,February 1985. 28. M C C O R M I C K M I C H A E L E . "Ocean Engineering Wave Mechanics", John Wiley & Sons, New York, 1973. 58 29. COMPTON,R.H. "The Resistance Wire Gauge", Proceedings of 14th General Meeting of American Towing Tank Conference, Webb Institute of Naval Architecture, Glen Cove, New York, 1965. 30. STOKES G.G. "On the Theory of Oscillatory Waves", Trans, of Cambridge Philosophic Society, 1847. 31. PEREGRINE D.H. "Equations for Water Waves and the Approximation Behind Them", Waves on Beaches and Resulting Sediment Transport, ed. R.E. Meyer, Academic Press, New York, 1972. 59 Figure 1. Body plan. Figure 2. Sealing arrangement and rods positioning. 61 Figure 3 Inclining experiment set-up. GZ-CURVES: light condition, GM # 1, wave height 6ft 2 a: < o o on -2 -5 \ X \ \ v . . . X \ s \ \ x \ \ ' X \ Legend A S T I L L W A T E R X W A V E C R E S T • W A V E T R O U G H 20 40 60 80 100 ANGLE OF HEEL (DEG) Figure 4a. Righting arm curves for light condition and GM # 1. o> Figure 4b. Righting arm curves for light condition and GM # 3. 00 GZ-CURVES: light condition, GM # 4, wave height 6ft 0 20 40 60 80 100 ANGLE OF HEEL (DEG) Figure 4c. Righting arm curves for light condition and GM # 4. Figure 4d. Righting arm curves for heavy condition and GM # 1. GZ-CURVES: full load, GM # 2, wave height 6ft 2 - i : : : : 1 Figure 4e. Righting arm curves for heavy condition and GM # 2. GZ-CURVES: full load, GM # 3, wave height 6ft DC < IE o ----K * 1 \ i \ • i Legend A S T I L L W A T E R X W A V E C R E S T O W A V E T R O U G H ANGLE OF HEEL (DEG) Figure 4f. Righting arm curves for heavy condition and GM # 3. GZ-CURVES: full load, GM # 4, wave height 6ft < X o on \ t // \ \ \ Legend A S T I L L W A T E R X W A V E C R E S T O W A V E T R O U G H ANGLE OF HEEL (DEG) Figure 4g. Righting arm curves for heavy condition and GM # 4. 69 Figure 5a. Testing session. Figure 5b. Testing session. INFRA RED LINK ^ M. OPTICAL EYE RIBBON CORDLESS KEYBOARD _r— CABLE 18V AUTOPILOT UNIT ADC1 IBM PC jr. 3 | II o o o D CONNECTION CABLE ADC1 LED DISPLAY o o o o o o CASSETTE RECORDER POWER © RADIO CONTROL RECEIVER RUDDER POTENTIOMETER 1.5V 1.5V, GYROSCOPE SPE GND ^- 1.5V Z X L . O O - TO RECEIVERS # © and (D RADIO CONTROL ©a SERVO / THROW SWITCH C0NTR0L<=fU SERVO u + TOADC1 INPUTS RUDDER SERVO SPE ACCELEROMETERS =£1 GND. + CONTROLLER DC MOTOR + Figure 6. Schematic of the electronic system. Figure 7a. Electronic system. Figure 7b. Electronic system. Figure 7c. Electronic system. Figure 8. Longitudinal seaway. SINGLE CHINE SEINER. GM # 1. CO oc < OC 30 25-20-15-10-5---TT • • • • 0.5 0.6 0.7 0.8 0.9 1 SHIP LENGTH/WAVE LENGTH x Legend 6 LIGHT. WAVE AMP. f 1 X LIGHT. WAVE AMP. f 2 • LIGHT. WAVE AMP. # 4 HEAVT, WAVE AMP. I 1 B HEAVY, WAVE AMP. I 2 M HEAVY. WAVE AMP. # 4 1.1 Figure 9. Parameter S as a function of ship length to wave length ratio for GM # 1. Experimental results. SINGLE CHINE SEINER. GM # 2. CO oc Ul i— U J < oc < Q_ 12 1 0  6  A ^ A Legend A WAVE A M P . # 2 H WAVE A M P . # 4 0.5 0.6 0.7 0.8 0.9 1 SHIP LENGTH/WAVE LENGTH 1.1 Figure 10. Parameter S as a function of ship length to wave length ratio for GM # 2. Heavy condit Experimental results. ion. WAVE CONDITIONS: Freq.: 0.85 Hz Amp. # 4 •50 0 H E A V Y A R O L L A N G L E 5 10 TIME sec 15 Figure 11. Low cycle resonance capsizing. GM # 1. WAVE CONDITIONS: Freq.: 0.8 Hz Amp. § 4 100 50 HEAVY A ROLL ANGLE X PITCH ANCLE A A A A A 7 A A i 1 •J f \ ! y I / r i 1 1 -50 10 15 20 25 30 35 0.5 HEAVY A HEAVE ACCL. X SURGE ACCL. -0.5 T 10 15 20 25 TIME sec 30 35 40 Figure 12. Low cycle resonance capsizing with rudder usage. GM # 2 . 6-r 4i\ 2H UJ Q ZD ct o UJ I -2--4-- 6 5 10 15 20 TIME sec. 25 30 35 40 Figure 13. Wave record for wave frequency 0.8 Hz and amplitude # 4. 81 Figure 15. Coordinate systems and linear translation of C G . 82 3.5 0 500 1000 1500 2 0 0 0 2 5 0 0 3 0 0 0 3500 NUMBER OF HULL ELEMENTS Figure 16. CPU time in seconds to number of time steps ratio as a function of number of hull elements. 83 ROLL PERIOD IN STILL WATER 18-1 : : : : '— 4 _ | j j 1 j 1 10 10.5 11 11.5 12 12.5 13 DISTANCE OF CG FROM BASE LINE ft Figure 17a. Comparison of numerically and experimentally obtained roll natural periods in still water.Heavy condition. 84 ROLL PERIOD IN STILL WATER 20 15-o (A Q O cn UJ O 10 5-LIGHT A EXPERIMENTAL X NUMERICAL 10 11 12 13 DISTANCE OF CG FROM BASE LINE f i 14 Figure 17b. Comparison of numerically and experimentally obtained roll natural periods in still water.Light condition. 85 ROLL DECAY: HEAVY CONDITION GM # 1 A EXPERIMENTAL X NUMERICAL \ f \ / \ / \ 1/ v / \ I \ I \J '• 50 100 150 150 TIME sec. Figure 18. Comparison of numerically and experimentally obtained roll decay curves in still water. Heavy condition. 86 I N I T I A L R O L L A N G L E 0 D E G . 15-1 ;  500 0 100 200 300 400 500 0 100 200 300 400 500 TIME sec Figure 19. Motion records for zero-angle roll initial condition. Heavy condit ion,GM # 1,wave length 104 ft, wave height 10 ft. 87 I N I T I A L R O L L A N G L E 0.5 D E C 40 20 Q 3 0_ < Legend A WAVE AMP. X HEAVE AMP. W W V A - / V / V A . A W v A , i r v r v v CD (1? "D UJ Q 3 100 TIME sec 200 200 Figure 20. Motion records, initial roll angle 0.5 deg. Heavy condit ion,GM # l .wave length 104 ft,wave height 10 ft. I N I T I A L R O L L A N G L E 3 D E G . 40 Legend A WAVE AMP. X HEAVE AMP. w w w / W V A , A -A-A A / W W /-v\r\r \rV\j-\) Arvl 20 20 50 100 150 200 100 TIME sec 200 Figure 21. Motion records, initial roll angle 3 deg. Heavy condition,GM # 1,wave length 104 ft,wave height 10 ft. 89 I N I T I A L R O L L A N G L E 5 D E G . 15-1 : : • 200 50 - 5 0 4 -100 . A A A / \ A A A A A / W v I Legend A PITCH ANGLE X ROLL ANGLE I 50 100 TIME sec 150 200 Figure 22. Motion records, initial roll angle 5 deg. Heavy condit ion,GM # l .wave length 104 ft,wave height 10 ft. SINGLE CHINE SEINER. GM # 1. NUMERICAL RESULTS. ex 2 < OH 30 25 20 15-10 o-f • • X A * — 1 1 I I I 0.5 0.6 0.7 0.8 0.9 t I I I I 1.1 1.2 1.3 1.4 1.5 SHIP LENGTH/WAVE LENGTH Legend A HEAVY, WAVE AMP. # 1 X HEAVY, WAVE AMP. # 2 • HEAVY, WAVE AMP. | 4 LIGHT, WAVE AMP. # 1 H LIGHT. WAVE AMP. # 2 X LIGHT, WAVE AMP, # 4 gure 23. Parameter S as a function of ship length to wave length ratio for GM # 1.Numerical result SINGLE CHINE SEINER. GM # 2. NUMERICAL RESULTS 2 < 2 3-X X X X A A A A X A A X X A 0.5 0.6 0.7 0.8 0.9 1 SHIP LENGTH/WAVE LENGTH 1.1 1.2 1.3 1.4 1.5 HEAVY A WAVE AMP. # 2 X WAVe AMP. § 4 Figure 24. Parameter S as a function of ship length to wave length ratio for GM # 2.Heavy condit Numerical results. 92 WAVE CONDITIONS: LENGTH 92.3ft AMP # 4 50 Figure 25. Low cycle resonance capsizing. GM # 1. 93 WAVE CONDITIONS: LENGTH 104ft AMP # 4 20 10H - , o -10H HEAVY, GM # 2 A WAVE AMPLITUDE ft X ROLL AMPLITUDE deg •20 T 50 100 T 150 200 TIME sec 250 300 Figure 26. Roll motion record. GM # 2. 94 -10 0 10 20 30 40 50 60 70 50 -100 30 40 TIME sec Figure 27. Low cycle resonance capsizing. GM # 1. 95 HEADING A N G L E d e g Figure 28. Wave encounter period as a function of wave length and heading angle for a given ship speed. 96 S H I P S P E E D = 9.2 K N O T S 100 80-60 40 20-I (£> q II SHIP SPEED a AVE SPEED = 80 16 0 240 WAVELENGTH ft 320 400 Figure 29. Wave encounter period as a function of wave length in overtaking waves for a given speed. 97 I N I T I A L R O L L A N G L E 1 0 D E G . 15 10 Q Z) _ l Q_ < 5-- 5 --10 / 20 HEAVY A WAVE AMPLITUDE ft. X ROLL AMPLITUDE deg. 40 60 TIME sec 80 100 Figure 30. Roll record for equal ship and wave speed. Heavy condition. GM # 1. 98 I N I T I A L R O L L A N G L E 1 0 D E G . 15 10 UJ Q 0_ < 5H 5H -10 + 0 5 10 L I G H T A WAVE AMPLITUDE ft X ROLL AMPLITUDE deg. -A X 15 20 25 TIME sec Figure 31. Roll record for equal ship and wave speed. Light condition. GM # 1. FREE SURFACE ELEVATION VS. PHASE ANGLE PHASE ANGLE deg Figure 32. Free surface elevation for linear and nonlinear waves. SINGLE CHINE SEINER. HEAVY CONDITION. NONLINEAR WAVE. 35-30-25-10 CC 20-2 < an , c £ 1 5 " 10-5-C 6 A • A • A • Legend A GM # 1, WAVE AMP. f 1 X GM J I. WAVE AMP. § 4 O GM H 2. WAVE AMP. # 1 GM # 2, WAVE AMP. f 4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 SHIP LENGTH/WAVE LENGTH 1.3 1.4 1.5 Figure 33. Parameter S as a function of ship length to wave length ratio. Heavy condition. Nonlinear wave. Numerical results. 101 WAVE CONDITIONS: LENGTH 70 FT HEI6. 9.8 FT 10 T : : : : Figure 34. Low cycle resonance capsizing. GM # 1. Nonlinear wave. 102 WAVE CONDITIONS: LENGTH 104 FT A M P L # 4 200 40 20- i ,i\ (\J\ / A A A„ T Wv \J HEAVY, GM # 2 A PITCH AMPLITUDE X ROLL AMPLITUDE -40 50 100 TIME sec 150 200 Figure 35. Motion records. GM # 2. Nonlinear wave. 103 HEAVY, GM # 1, INITIAL ROLL ANGLE 10 DEG. 4 4r Q Z> 2H 0 20 40 60 20 40 60 TIME sec Legend A WAVE AMPLITUDE X HEAVE AMPLITUDE 80 100 100 Figure 36. Roil record for equal ship and wave speed. GM # 1. Light condition. Nonlinear wave. 104 APPENDIX A SENSOR'S CALIBRATION A calibration program was used to calibrate the sensors to give scalar values correponding to their A / D values. This procedure was executed on shore where the magnitude of the motions and accelerations could be accurately measured. The routine performs a straight-line fit. Figure A-1 shows calibration curves for the various sensors. The abscissae correspond to the A / D values and the ordinates give the scalar value of the desired measurement. The program outputs a calibration file containing the channel number, the slope and the intercept. These values are subsequently used in data demultiplexing. 105 CALIBRATION CURVES CD CD TJ L d _ l o z: < -3000 -2000 -1000 2000 3000 40 CD <D "O L d _ J o < 20--20 Legend A ROLL X PITCH • "*7ZS 0 500 1000 1500 2000 cn o cz o o o < -1 Legend A HEAVE X SURGE -4000-3000 -2000 -1000 0 1000 2000 3000 4000 CALIBRATION READING Figure A-1. Calibration curves. APPENDIX B DIRECTION COSINES Table B-1 gives the relationship between the movable and stationary coordinate systems in terms of the direction cosines. The translational coordinates of the center of gravity have been omitted. TABLE B-1 X Y Z b, C i a 2 b 2 c 2 Si a 3 b 3 <= 3 where: = cos (XG£) = cos 0 cos \^  a 2 = cos(XGrj) = sin 0 cos \// a 3 = cos(XG$) = - sin \p bi = cos(YG£) = sin 0 cos 0 sin i/> - cos 6 sin 0 b 2 = C O S ( Y G T } ) = cos 6 cos 0 + sin 6 sin 0 sin $ b 3 = cos(YG$) = sin 9 cos \p C i = cos(ZG£) = cos 0 cos 0 sin \/> + sin 6 sin 0 = COS(ZGT7) = cos 6 sin 0 sin i// - sin 6 cos 0 c 3 = cos(ZGS) = cos d cos \// 106 107 ANGULAR VELOCITIES Projection of the angular velocity vector $2 onto the X-axis gives; p = 69 + 0 cos(XGS) + </ cos (XGN) Since GX is perpendicular to GN and cos(XG$)can be readily obtained from Eqs. B-1 we have; p = 9 - 0 sin \p (B-2a) Projection of U onto the Y-axis gives: q = 6 cos (XGY) + ^ cos (NGY) + 0 cos (YG$) where the cosine terms are defined as follows; cos(XGY) = 0 cos(NGY) = cos 9 cos(YG$) = b 3 = sin 6 cos i// Then: q = 0 sin 6 cos y\i + cos 9 (B-2b) Finally, projection of U onto the Z-axis, gives; r = 9 cos(XGZ) + \p cos(NGZ) + 0 cos($GZ) 108 and as before we have; cos (XGZ) = 0 cos (NGZ) = cos(90 + 8) = - sin 8 cos ($GZ) = c 3 = cos 8 cos \p Then; r = <j> cos 8 cos \p - ^ sin 8 (B-2c) APPENDIX C ESTIMATION OF l x x Kato [24] gives for ordinary merchant ships: ( ^ « ) » = f [ C b C u + 1.10 C u (1 -C b ) (jje - 2.0) (C - l ) where: f = 0.177 - 0.2 for fishing vessels Comparison with a fishing vessel of similar characteristics gave; A d = 110 ft 2 H e = 16.59 ft. C u = 0.914 Then the values of k x x are: 0.37563 B (Light Condition) 0.36680 B (Heavy Condition) The mass moment of inertia including the added moment of inertia about the X-axis is then given by: "xx= k xx V /g (C-2) 109 110 ESTIMATION OF I and l z z The virtual mass moment of inertia about Y-axis lyy can be roughly estimated from the moment of inertia of the area under the ship's sectional area curve plus the contribution of the added mass in heave. Sectional added mass and damping coefficients in heave were obtained using a subroutine from the Naval Architecture library of the Deptartment of Mechanical Engineering at U.B.C. belonging to Dr. S.M. Calisal. Then lyy is: •yy = p f A(x)x 2 dx + J a n x 2 d x (C-3) The added mass in heave is of the same order of magnitude as the ship's mass, therefore the added inertia term must be taken into consideration in the calculation O f lyy. The radius of gyration k z z is given by Bhattacharyya [25] as 0.24-0.26 L. The mass moment of inertia in the Z-direction is then given by: 'zz = k z z V / g (C-4) ESTIMATION OF PITCH AND HEAVE HYDRODYNAMIC DAMPING The hydrodynamic damping in heave is obtained by integrating the sectional heave damping coefficients b n along the length of the ship.That is: B h = / b n dx (C-5) Pitch damping was obtained in a similar manner. The heave damping coefficients were integrated along the length of the ship length as fol lows: 111 B p = f b n x > d x (C-6) ESTIMATION OF ROLL DAMPING A component analysis of the roll damping was developed following Himeno [23]. The formulae can be readily obtained from this reference and are not reproduced here. The total roll damping is estimated as the sum of equivalent linear damping coefficients.That is: Figures C - 1 , C-2 and C-3 respectively show non-dimensional roll damping components as a function of Froude number, roll natural period and roll amplitude. Mutual interaction among different components has not been considered. The effect of appendages such as the rudder is included in the main configuration of the ship hull. Figures C - 4 , C-5 and C-6 show curves of component to total roll damping as a function of the same variables as Figure C-1 through C - 3 . All figures are for heavy load condition. It can be seen that the keel roll damping contributes most of the ship roll damping and its effect becomes of particular importance at large angles of roll. B r = Bp + B E + B | _ + B W + B B K (C-7) NON-DIMENSIONAL ROLL DAMPING: roll period & amplitude: 6 . 4sec , 30deg. 0.06 0.05-0.00 H f Legend A FRICTION DAMP. X WAVE OAMP. O EDDY DAMP. 8 LIFT DAMP. 0.2 0.3 FROUDE NUMBER Figure C - 1 . Roll damping as a function of Froude number. NON-DIMENSIONAL DAMPING: roll amplitude 30deg, Froude number 0.34 0.25 0.20-O LZ 0.15 u_ U J O C_> O Z D_  0.10-< 0.05-O . O O ^ h \ \ \ Legend A FRICTION X WAVE • EDDY  B LIFT O KEEL X TOTAL * 4 6 8 ROLL PERIOD (SEC) 10 Figure C -2 . Roll damping as a function of roll period. NON-DIMENSIONAL ROLL DAMPING: roll period & Froude number: 6.4sec; 0.34 o 0.14 0 . 1 2 0 . 1 0 0 . 0 8 O <_> ^ 0 . 0 6 bZ Q 0 . 0 4 -0 . 0 2 -0 . 0 0 Legend A FRICTION X WAVE O EDDY  8 LIFT B KEEL _ K TOTAL / / a-S / \ -I 1 1 1 1 1 1 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 ROLL AMPLITUDE (DEG) Figure C-3. Roll damping as a function of roll amplitude. RATIO OF ROLL DAMPING COMPONENTS TO TOTAL DAMPING < O z < Q 0.4 0.2 " " " " • — « t • — • " - " " " ^ \ \ i I' 1 * 9 0.0 0.1 0.2 0.3 0.4 FROUDE NUMBER 0.5 0.6 LEGEND A FRICTION/TOTAL X WAVE/TOTAL • EDDY/TOTAL H LIFT/TOTAL O KEEL/JOTAL Figure C-4. Component to total roll damping ratio as a function of Froude number. COMPONENT TO TOTAL ROLL DAMPING: roll amplitude 30deg, Froude number 0.34 < o z < 0 . 8 0 . 6 -0 . 4 -0 . 2 Legend A FRICTION/TOTAL X WAVE/TOTAL • EDDY/TOTAL B LIFT/TOTAL S KEEL/TOTAL A 4 6 ROLL PERIOD (SEC) 1 0 Figure C-5. Component to total roll damping ratio as a function of roll period. COMPONENT TO TOTAL ROLL DAMPING: roll period & Froude number: 6.4sec;0.34 o.a 0.6-< o z D_ < 0.4-0.2-0.0-Legend A FRICTION/TOTAL X WAVE/TOTAL D EDDY/TOTAL 8 LIFT/TOTAL a KEEL/TOTAL 10 20 30 40 50 60 ROLL AMPLITUDE (DEG) ••V / A —r— 70 80 Figure C-6. Component to total roll damping ratio as a function of roll amplitude. APPENDIX D USER GUIDE FOR THE COMPUTER PROGRAM "MOTION" INTRODUCTION The program MOTION was developed to simulate, in the time domain, the motion of a ship in a longitudinal seaway. The numerical method used is described in CHAPTER 6. The input required to run the program has not been formatted (free format) so as to facilitate its use. Data input can be done interactively from the terminal or assembled in a data file (MOTION.DAT) to be run as a batch job. Information concerning access to this program may be obtained by contacting: Dr. S.M. Calisal Department of Mechanical Engineering University of British Columbia Vancouver, B.C. Canada V6T 1W5 REQUIRED INPUT. FILE: MOTION.DAT The input data file is read in UNIT 7 and conforms to the following lines: LINE 1: NSTAT N S T A T : Number of stations including station number zero. LINE 2: S T A S P A S T A S P A : station spacing. 118 119 LINE 3: HEAVIN, HEIDOT HEAVIN: initial heave value. HEIDOT; initial heave velocity. LINE 4: TETAIN, TEDOTI TETAIN: initial roll angle. TEDOTI: initial roll velocity. LINE 5: TSIINI.TSDOTI TS11NI: initial pitch angle. TSDOTI; initial pitch velocity. LINE 6: INPUT INPUT: data file containing table of offsets. LINE 7: OUTPUT OUTPUT: output data file containing input data and ship characteristics in UNIT 2. The program can easily be transformed to give a more detailed output file as explained further on in this Appendix.Units 8,9,10 and 11 are specified for the following output files which can be readily plotted: -OLAMOT (8) wave record HIVMOT (9) heave record ROLMOT (10) roll record PICMOT (11) pitch record 120 LINE 8: XLBP, B E A M , DRAFT XLBP: length between perpendiculars. B E A M : ship beam. DRAFT; ship draft. LINE 9: X C G X C G : distance of center of gravity from the midship section. LINE 10: X K G . X K M XKG: distance of center of gravity from baseline. X K M : distance from metacenter to baseline. LINE 11:RHO,NUE RHO: water density. NUE: kinematic viscocity. LINE 12: NPTS NPTS: number of offset points for calculation of added mass and damping coefficients in heave and pitch. LINE 13: COEFF COEFF: name of datafile containig offsets for calculation of hydrodynamic coefficients in heave and pitch. Offsets must be entered starting at bottom keel. LINE 14; VSHIP VSHIP: ship speed in knots. 121 LINE 15: WAVLEN, HEIGHT, DEPTH WAVLEN: wave length. HEIGHT: wave height. DEPTH: water depth. LINE 16: TPI TPI; approximate value of tons per inch of immersion. LINE 17: T IMAX, DELTIM T IMAX: maximum simulation time. DELTIM: time step. LINE 18: DTETA(I) DTETA(I): Lewis-form coefficients for calculation of wave roll damping in still water starting at station number zero. LINE 19: EBBK, EXBK1, EXBK2 EBBK: keel equivalent width. EXBK1: station number at aft-end of keel. EXBK2: station number at fore-end of keel. LINE 20; WAVPAR WAVPAR; parameter to define wave theory. WAVPAR = 1: linear wave theory. WAVPAR = 2: nonlinear wave theory. FORMAT FOR THE TABLE OF OFFSETS Station # Breadth 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.371568 0.743326 1.106152 1.444201 1.787847 2.184882 2.616004 3.037748 3.461234 3.913203 4.375398 4.847211 5.345247 5.876709 6.444003 7.042548 7.518448 7.809599 8.100750 8.391900 8.683051 8.974202 8.994000 0.000000 0.276527 1.028784 1.703205 2.273019 2.780542 3.220813 3.625290 3.982264 4.268934 4.512154 4.740543 4.972618 5.229560 5.490157 5.750977 6.030540 6.336005 6.643367 6.937064 7.246970 7.598289 7.961390 8.307389 Dist. to B.L. 9.500000 10.50000 11.00000 11.50000 12.00000 12.50000 13.00000 13.50000 14.00000 14.50000 15.00000 15.50000 16.00000 16.50000 17.00000 17.50000 18.00000 18.50000 19.00000 19.50000 20.00000 20.50000 21.00000 21.50000 21.53400 3.87.4000 4.000000 4.500000 5.000000 5.500000 6.000000 6.500000 7.000000 7.500000 8.000000 8.500000 9.000000 9.500000 10.00000 10.50000 11.00000 11.50000 12.00000 12.50000 13.00000 13.50000 14.00000 14.50000 15.00000 123 1.000000 8.660149 15.50000 1.000000 9.041756 16.00000 1.000000 9.451247 16.50000 1.000000 9.874976 17.00000 1.000000 10.28910 17.50000 1.000000 10.58396 18.00000 1.000000 10.62357 18.50000 1.000000 10.66317 19.00000 1.000000 10.70278 19.50000 1.000000 10.74238 20.00000 1.000000 10.78199 20.50000 1.000000 10.81700 20.94200 10.00000 0.000000 7.040000 10.00000 3.250014 7.500000 10.00000 6.850007 8.000000 10.00000 10.45686 8.500000 10.00000 10.54071 9.000000 10.00000 10.62455 9.500000 10.00000 10.70840 10.00000 10.00000 10.79225 10.50000 10.00000 10.87609 11.00000 10.00000 10.95994 11.50000 10.00000 11.04379 12.00000 10.00000 11.12763 12.50000 10.00000 11.21148 13.00000 10.00000 11.29532 13.50000 10.00000 11.37917 14.00000 10.00000 11.43359 14.50000 10.00000 11.43525 15.00000 10.00000 11.43691 15.50000 10.00000 11.43857 16.00000 10.00000 11.44023 16.50000 10.00000 11.44189 17.00000 10.00000 11.44200 17.03300 8888 9999 DETAILED OUTPUT FILE FROM PROGRAM MOTION A detailed output file can be obtained by removing the comment cards from the WRITE statements. These cards were placed to minimize the output volume. The output file will then have information on the initial conditions, ship characteristics, a reduced table of offsets, cross sectional areas, hydrodynamic coefficients and 124 motion records. It is also possible to obtain records of velocities, accelerations, forces and moments. USER GUIDE FOR THE COMPUTER PROGRAM "S" The program S was developed to calculate the parameter S value. The input required to run the program is unformatted. Data input is done interactively from the terminal. REQUIRED INPUT FOR PROGRAM S The roll data to be used in the calculation of the parameter S is contained in the file ROLMOT previously mentioned. The input data is read in UNIT 5 and conforms to the following lines: LINE 1: NUMBER NUMBER: number of points in file ROLMOT. LINE 2; WAVHIT WAVHIT: wave height for the run. LINE 3: AREAGZ A R E A G Z : area under the GZ-curve in still water. 125 CALCULATE WAVE PROFILE ALONG THE SHIP CALCULATE SECTIONAL HEAVE AND PITCH HYDRODYNAMIC COEFF. CALCULATE SECTIONAL AREAS AND DISPLACEMENT READ INPUT DATA BEGIN NO CALCULATE SECTIONAL FORCES AND MOMENTS DUE TO WAVE CALCULATE SECTIONAL ROLL HYDRODYNAMIC COEFF. INTEGRATE FORCES AND MOMENTS ALONG THE SHIP INTEGRATE EQUATIONS OF MOTION STOP Figure D - 1 . Flow chart for computer program. 

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