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Dynamic response of a floating structure with a flexible air chamber Shiratani, Hiroshi 1998

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D Y N A M I C S T R U C T U R E  R E S P O N S E  W I T H  A  O F  A  F L E X I B L E  F L O A T I N G A I R  C H A M B E R  by HIROSHI SHIRATANI  B. E., Osaka University, Japan, 1986 M . E., Osaka University, Japan, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA February, 1998 © Hiroshi Shiratani, 1998  In presenting this thesis in partial fulfillment 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 the 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 Civil Engineering The University of British Columbia 2324 Main Mall, Vancouver, B.C., Canada V6T 1Z4  11  Abstract Althoughfloatingstructures have been widely used for many purposes, their dynamic response in waves limits the extent of their usefulness. Thus, approaches to reducing their response in waves would extend the range of prospective applications of such structures. In this context, a new type of floating structure is proposed in the present study, intended to achieve a notable reduction of the dynamic response to waves. The proposed structure is equipped with a flexible air chamber bounded by a rubber membrane attached on the bottom of a rectangular body, and a piston between the body and the air chamber. The objective of the present study is to assess whether the proposed structure is practically effective for the reduction of the dynamic response. First, a preliminary examination was conducted to understand the basic characteristics of the proposed structure. In this examination, two methods to predict the hydrostatic states of the proposed structure were introduced. An experimental investigation of the dynamic response of the proposed structure was carried out in the wave flume of the Hydraulics Laboratory of the Department of Civil Engineering, the University of British Columbia. In the free condition, the motions of the experimental model under waves were recorded with a video camera. Then, the dynamic responses were obtained from the records.  The performance of the proposed structure was evaluated through a  comparison with that of the equivalent rectangular structure. In addition, for heave mode, a closed set of solutions of simplified governing equations was introduced, and the predicted heave responses were compared with the measured responses.  The video images of the  experimental model under waves were examined to understand qualitatively the dynamic behavior. Furthermore, the dynamic response characteristics in the heave mode were examined analytically.  Ill  A primary conclusion of this study is that the proposed structure with a suitably proportioned air chamber could have the ability of reducing the dynamic response, and a properly designed piston could enhance the structure's performance if its roll motion is restrained externally.  iv  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  List of Symbols  xi  Acknowledgments  xv  Chapter 1. Introduction  1  1.1 General  1  1.2 Features of the Floating Structures Proposed in the Present Study  2  1.3 Literature Survey  3  1.3.1 Prediction of Dynamic Response  3  1.3.2 Reduction of Dynamic Response  4  1.4 Objective and Scope of the Present Study  Chapter 2.  Preliminary Assessment  6  8  2.1 General  8  2.2 Dimensional Analysis  9  2.3 Equilibrium Configuration of Air Chamber  11  2.3.1 Ellipse Approximation  12  2.3.2 Lumped-Mass Method  13  2.3.3 Comparison of the Two Methods  15  2.4 Examination of Hydrostatic Restoring Moment  16  2.4.1 Ellipse Approximation  16  2.4.2 Lumped-Mass Method  17  2.4.3 Results  18  2.4.4 Reduction Ratio of Restoring Moment in Ellipse Approximation  19  2.4.5 Effect of Membrane Properties on Restoring Moment  19  Chapter 3. Theoretical Formulation 3.1 General 3.2 Governing Equations 3.2.1 Equations of Motion for the Body and Piston 3.2.2 Equations of Motion for Membrane Elements 3.2.3 Variation of Air Chamber Volume and Pressure 3.3 Simplified Model for Heave 3.3.1 Model 3.3.2 Governing Equations Chapter 4.  Experimental Investigation  21 21 21 22 23 24 24 24 26 28  4.1 General  28  4.2 4.3 4.4 4.5  28 29 31 32  Wave Flume and Generator Experimental Model Test Conditions Measurements  Chapter 5.  Results and Discussion  33  5.1 Response Amplitude Operators of the Proposed Structure 5.1.1 Roll 5.1.2 Heave 5.1.3 Sway 5.2 Behavior of the Proposed Structure 5.2.1 Free Condition 5.2.2 Vertically Moored Condition  33 34 34 35 36 36 38  5.3 Parametric Examination of Dynamic Characteristics  39  5.3.1 Calculation Conditions 5.3.2 Effects of Air Chamber Properties 5.3.3 Effects of Piston Properties 5.4 Scale Effect 5.5 Summary and Supplemental Remarks Chapter 6.  Conclusions  39 40 42 43 44 47  References Appendix A Solutions of the Simplified Governing Equations for Heave Response  List of Tables Table 4.1  Summary of experimental model properties  Table 4.2  Wave conditions  Table 4.3  Summary of test series  Vlll  List of Figures Fig. 1.1  Schematic of the floating structure proposed in the present study  Fig. 1.2 Concept of the proposed structure to reduce response (roll) Fig. 1.3 Concept of the proposed structure to reduce response (heave) Fig. 1.4 Schematic of the floating structure presented by Iwata et al. (1986) and Ikeno etal. (1988) Fig. 1.5  Schemes of floating structures with active control techniques ( Committee for Development of Offshore Cities in Japan , 1992 )  Fig. 2.1 Sketch of general configuration of air chamber Fig. 2.2 Definition sketch Fig. 2.3 Definition sketch for the lumped-mass method Fig. 2.4 Comparison of membrane tensions obtained by the lumped-mass method and the ellipse approximation Fig. 2.5 Comparison of air chamber configurations obtained by the lumped-mass method and the ellipse approximation Fig. 2.6 Example of calculation results for configuration of air chamber of the proposed structure in heel condition Fig. 2.7 Dimensionless restoring moments as a function of external to internal pressure ratio Fig. 2.8 Reduction ratios of restoring moments as a function of external to internal pressure ratio Fig. 2.9 Dimensionless restoring moments with membrane properties as parameters Fig. 3.1 Definition sketch for motion of the proposed structure Fig. 3.2 Definition sketch for the simplified analytical model for heave Fig. 4.1 Experimental set-up Fig. 4.2 Sketch of the experimental model Fig. 4.3 Relationship between displacement and force of the pistons for the experimental model Fig. 5.1 Roll response amplitude operator of the proposed structure (Test series S-ll) Fig. 5.2 Roll response amplitude operator of the proposed structure (Test series S-21)  IX  Fig. 5.3 Heave response amplitude operator of the proposed structure (Test series S - l l and S-13) Fig. 5.4 Heave response amplitude operator of the proposed structure (Test series S-12 and S-14) Fig. 5.5 Heave response amplitude operator of the proposed structure (Test series S-21) Fig. 5.6 Sway response amplitude operator of the proposed structure (Test series S-ll) Fig. 5.7 Sketch showing tracks of the center of gravity of body under waves Fig. 5.8 Behavior of the experimental model under wave at particular time instants (T=1.10 sec, free condition) Fig. 5.9 Behavior of the experimental model under wave at particular time instants (T= 1.50 sec, free condition) Fig. 5.10 Sketch showing transmitted force from membrane to side panels Fig. 5.11 Behavior of the experimental model under wave at particular time instants (T=1.50 sec, vertically moored condition) Fig. 5.12 Exciting forces of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll Fig. 5.13 Added masses of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll (d) Sway - Roll Fig. 5.14 Damping coefficients of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll (d) Sway - Roll Fig. 5.15 Response amplitude operators in heave with air chamber properties as parameters (free condition) (a) M/pB = 0.20 (b) M/pB = 0.30 2  2  Fig. 5.16 Non-dimensionalized mooring forces with air chamber properties as parameters (fixed condition) (a) M/pB = 0.20 (b) IVL/pB = 0.30 2  2  Fig. 5.17 Response amplitude operators in heave with piston spring stiffness as a parameter (free condition) Fig. 5.18 Response amplitude operators in heave with piston width as a parameter (free condition) Fig. 5.19 Non-dimensionalized mooring forces with piston spring stiffness as a parameter (fixed condition) Fig. 5.20 Non-dimensionalized mooring forces with piston width as a parameter (fixed condition)  Fig. 5.21 Response amplitude operators in heave with piston mass as a parameter (free condition) Fig. 5.22 Non-dimensionalized mooring forces with piston mass as a parameter (fixed condition) Fig. 5.23 Dynamic responses of piston and air chamber (fixed condition) Fig. 5.24 Response amplitude operators in heave for the experimental model with length scale as a parameter (free condition) Fig. 5.25 Non-dimensionalized mooring forces for the experimental model with length scale as a parameter (fixed condition) Fig. 5.26 Mooring arrangements considered to be suitable to the proposed structure  xi  List of Symbols A  =  Length of piston  a  =  Length of horizontal axis of ellipse  B  =  Beam of body  b  =  Length of vertical axis of ellipse  cm  =  Damping coefficient of membrane  cp  =  Damping coefficient of piston  cx  =  Hydrodynamic damping coefficient in sway  cxe  =  Hydrodynamic damping coefficients in sway - roll  cz  =  Hydrodynamic damping coefficient in heave  Ce  =  Hydrodynamic damping coefficient in roll  D  =  Reduction ratio in ellipse approximation method  Dm  =  Fraction of critical damping of membrane  Dp  =  Fraction of critical damping of piston  d  =  Water depth  Em  =  Modulus of elasticity of membrane  Fjn  =  Force acting on membrane element from inside of air chamber  Fm  =  Mooring force  Fout  =  Force acting on membrane element from outside of air chamber  Fx  =  Exciting force in sway  Fz  =  Exciting force in heave  Fe  =  Exciting force in roll  f  =  Vertical exciting force per unit area on the surface of structure  fi  =  Tension in i-th element of membrane  fm  =  Tension in membrane in the ellipse approximation method  Tension in membrane at the center of membrane calculated by the ellipse approximation method Tension in membrane at the center of membrane calculated by the lumped-mass model Relative self weight of membrane element Gravitational constant Wave height Draft of body Height of air chamber in rest condition Water depth of i-th element of membrane Total draft of the structure Total draft of the structure in rest condition Moment of inertia of body Stiffness of mooring system in vertical direction Spring stiffness of piston Length scale factor Wave length Unstretched length of membrane Unstretched length of i-th element of membrane Total mass for heave ( real mass plus added mass ) Mass of body Mass of piston Hydrostatic restoring moment Added mass in sway Added mass in sway - roll Added mass in heave Added mass in roll  Air chamber pressure (gage pressure) Atmospheric pressure Dynamic variation of air chamber pressure Air chamber pressure in rest condition Radius of curvature of membrane Coefficient relating free surface elevation to vertical exciting force tangent to each element of membrane Wave period Time Thickness of membrane Total submerged volume of the floating structure Volume of air chamber in motion Volume of air chamber in rest condition Horizontal coordinate, taken on the still water level Motion of body in x direction Motion of i-th element of membrane in x direction Vertical coordinate, taken from the still water level, upward as positive Motion of body in z direction Displacement of body to give mooring force in rest condition Elevation of center of buoyancy Elevation of center of gravity of body Motion of i-th element of membrane in z direction Elevation of center of ellipse Motion of piston Displacement of piston to give piston spring force in rest condition Water surface elevation under the body  xiv a  =  Amplitude of motion of body in sway  (3  =  Amplitude of motion of body in heave  %  =  Amplitude of dynamic air chamber pressure  Alj  =  Elongation of i-th element of membrane  8j  =  Angle between membrane and s axis of i-th element  Y  =  Constant  T)  =  Free water surface elevation  X  =  Amplitude of motion of piston  r>  =  Inclination of body  0b  =  Dynamic rotation of body  p  =  Fluid density  pm  =  Membrane density  Tj  =  Angle between z axis and s axis of i-th element  co  =  Wave angular frequency  C,,  =  Amplitude of water surface elevation under the bottom of body  7=1.4 for adiabatical variation of air  XV  Acknowledgments  The author would like to thank his supervisor Dr. Michael Isaacson for his guidance and encouragement throughout the preparation of the present thesis. The author would like to express his gratitude to Mr. Kurt Nielson for his help in constructing the experimental model; and to Mr. John Baldwin for his help relating to the analytical investigation. The author's colleagues, Dr. Shankar S. Bhat, Mr. Sundaralingam Premasiri, and Mr. Gang Yang are thanked for their advice, help and encouragement throughout the research. The author would like to thank Dr. Noboru Yonemitsu of University of Northern British Columbia, for his help throughout the author's life and study in Canada. The author also thank his wife, Hatsuho Shiratani, for her support and encouragement. Finally, Taisei Corporation in Japan, to which the author belongs, is gratefully acknowledged for the grant of a study leave and financial support.  1  Chapter 1 Introduction 1.1 General In the field of civil engineering, floating structures have been widely used for many purposes, such as floating jetties, platforms, breakwaters, foundations of floating bridges, and working crafts. Much research on the performance of floating structures has been carried out to the present, ranging from the theoretical development of techniques to predict accurately the performance of general types of floating structures, to the investigation of particular types of floating structures for specific projects. One of the major disadvantages of floating structures is their compliance with wave motion. Although considerable research and development has been carried out to reduce the dynamic response to waves, this disadvantage has still limited the application of floating structures to a wider range of projects and facilities than would otherwise be the case. In this context, a new type of floating structure is proposed in the present study, intended to achieve a notable reduction of the dynamic response to waves. Figure 1.1 shows a schematic of the floating structure proposed in the present study. It has an air chamber bounded by a rubber membrane attached on the bottom of a rectangular body; and it includes a piston between the body and the air chamber, which is able to move upward or downward in accordance with the variation of the air pressure in the chamber. The present study is concerned with the dynamic characteristics of the proposed structure. The proposed structure may have applications to such facilities as floating jetties and the foundations of floating bridges, mainly in deep water, and to working crafts, for which the proposed structure could enhance workability.  2  1.2 Features of the Floating Structures Proposed in the Present Study Concepts of the proposed floating structure to reduce the dynamic response to waves are expected to be as outlined below.  Roll When the proposed structure is subjected to a wave pressure distribution at the phase of maximum overturning moment, associated with the water surface elevation, as shown in Fig. 1.2 (a), the air chamber may deform asymmetrically in response to the wave pressure. The deformation of the air chamber would reduce the force transmitted from the air chamber to the body, if the air chamber is very flexible. The mechanism of this wave force transmission to the body may be conceptually explained with a simple model shown in Fig. 1.2 (b); in this model the spring stiffness would represent the air chamber flexibility, depending on the air chamber pressure and membrane properties. From dynamical consideration, the transmitted force to the body would be obtained as shown in Fig. 1.2 (c), as a function of the natural period of the structural system and wave period. If the natural period is much longer than the wave period, a reduction of the transmitted force could be expected. Furthermore, the deformation of the air chamber may induce an eccentricity of the buoyancy force acting on the air chamber. If the overturning moment due to the buoyancy is out of phase with the wave exciting force in roll, a reduction in the exciting force could be expected. In addition, the hydrostatic restoring moment of the proposed structure is much smaller than that of the equivalent rectangular structure in general, because of the deformation of the air chamber in heel condition. This could make the natural period in roll longer, and contribute to the reduction in the dynamic response to relatively short waves in the free cases. Heave When the proposed structure is subjected to a wave pressure distribution, associated with a  3 wave crest passing the structure, as shown in Fig. 1.3, the air chamber pressure would increase, causing the piston to be lifted. This induces a decrease of the air chamber volume and a resulting decrease of the buoyancy; as a result the body would tend to go down. This behavior of the air chamber and the piston could reduce the dynamic response. In addition, as in the case of roll, the deformation of the air chamber could reduce the transmitted force from the air chamber to the body. As described so far, the main feature of the proposed floating structure is that the exciting force is reduced to some extent by taking advantage of the deformability of the flexible air chamber. Unlike floating structures equipped with active control techniques (which reduce a structure's dynamic response using a computer-controlled system, as described in the next section), the proposed structure may be feasible without the complexity of such a feedback system. In considering the proposed concept, it should be pointed out that the performance of rubber products used in ocean structures has been improved considerably in recent years. For example, a flexible mound breakwater which has been in practical use since 1992 in Japan is constructed of rubber sheets that are assured to have a service life of over 30 years in sea water (Tanaka et al., 1992). Therefore the durability of rubber is not a major problem, even if the proposed structure would be used on a permanent basis. Furthermore, the air chamber may possibly be attached to existing rectangular floating structures with appropriate attachment devices.  1.3 Literature Survey 1.3.1 Prediction of Dynamic Response Several texts, including that by Sarpkaya and Isaacson (1981), describe predictions of the response of floating structures under waves using potential flow theory. Although the multipole expansion method and the strip method have been largely employed as suitable  4  numerical techniques for the case of ships, the dividing region method and the wave source distribution method have been mainly developed in the context of civil engineering applications. These techniques are summarized concisely by Aoki (1990). Detailed examinations of the behavior of floating structures, including those with rectangular or circular sections, have been conducted by many authors. Of these studies, those in the early years such as by Ijima et al. (1972), for rectangular sections, or Yamamoto et al. (1982), for rectangular and circular sections, mainly focused on the development of numerical models based on potential flow theory. On the other hand, the importance of viscous damping for the dynamic response of floating structures has been suggested by several authors (e.g. Stewart et al, 1981, and Fugazza and Natale, 1988).  Further  developments of numerical models taking account of viscous damping have been carried out by such authors as Isaacson and Byres (1988), and Wang and Katory (1993) for rectangular section bodies, and Isaacson et al. (1994) for circular section bodies. On the basis of such studies, the dynamic response of floating structures can generally be predicted with reasonable accuracy.  1.3.2 Reduction of Dynamic Response Methods of reducing the dynamic response of floating structures can be roughly categorized in the following four ways: • Appropriate mooring line arrangements; • Improvements of shape of floating structures; • Improvements of mooring system by the use of certain devices such as hanging weights and dashpots; and • Active control techniques. A suitably designed mooring arrangement is a common way to reduce the dynamic response, although there are many practical cases for which severe response remains problematic. The  5 effects of the shapes of floating breakwaters on the reduction of the dynamic response are discussed briefly in a report of the Ocean Engineering Research Centre of University of Newfoundland (1994). In this discussion, the catamaran (twin-pontoon) type is presented as one of the types of floating structures which could be somewhat effective for the reduction of motion. Bhat (1998) has indicated that the twin-pontoon type can provide a significant reduction of the dynamic response, especially in the roll mode, due to its relatively large moment of inertia. Semi-submersible structures, which generally consist of superstructures above the sea level, submerged floaters, and columns connecting the superstructures and floaters, have been found to be effective in reducing the wave exciting force. Furthermore, when the vertical motion of a semi-submersible structure is restrained by a tight mooring system, changes in the mooring force due to draft changes induced by the tide is generally not as large as those of other types of structures, which have larger waterplane areas. Because of these advantages, the semi-submersible type is a superior floating structure for the reduction of mooring force in tightly moored cases. However, because of the complexity of design and relatively high construction costs, the application of this type has been limited to very specific projects to date. In studies by Iwata et al. (1986) and Ikeno et al. (1988), a floating structure with an openbottom air chamber, as shown in Fig. 1.4, has been examined both analytically and experimentally. Although this type appears similar to the catamaran type, the distinctive feature of this type of floating structure is that its natural period in roll is changeable only by changing the air chamber pressure. It has been reported that this function makes it possible to shift the natural period away from the dominant wave period; therefore this type is effective for a reduction of the dynamic response, especially in roll. Improvement of the dynamic performance of floating structures can be achieved to some extent by the pretension of mooring systems using hanging weights as suggested by Wang  6  and Katory (1993), and Isaacson et al. (1994). They have indicated that the hanging weights sometimes contribute to a reduction of the total tension in mooring lines (initial plus dynamic tension) as well as a reduction of motion. The effect of dashpots attached to mooring lines has been examined by Sawaragi et al. (1983), who have found that they are effective for waves with relatively high frequencies. Recently "active control" techniques, involving various kinds of thrusters to generate resisting forces against wave forces, together with computer controlled systems to control the thrusters, have been examined as one of the newer ways to reduce the dynamic response of floating structures. Some concepts of active control are presented by the Committee for Development of Offshore Cities in Japan (1992). Figure 1.5 shows some schematics of floating structures using possible active control techniques.  Although active control  techniques may be applicable to massive floating facilities such as offshore cities or floating airports, they are not yet practical for more modest structures because of high construction costs and problems relating to maintenance and reliability of the control system.  1.4. Objective and Scope of the Present Study The main objective of the present study is to assess whether the proposed floating structure shown in Fig. 1.1 may be useful in the reduction of the dynamic response to waves. The present study is a first step towards the development of this type of floating structure. Therefore, only a two-dimensional structure and regular wave conditions are considered. Rather than a comprehensive examination of the hydrodynamic characteristics of the proposed structure, the main focus of this study is on the verification of whether a proposed structure which is proportioned appropriately can achieve a significant reduction of the dynamic response. First, some basic characteristics of the proposed structures are examined analytically. Next, the structure's dynamic response characteristics are investigated experimentally. A scale  7  model for the experimental investigation is designed on the basis of a preliminary examination, such that it can be expected to produce some reduction of the dynamic response. In the experiments in the wave flume, the dynamic responses of the model are measured for free conditions. In addition, the behavior of the structure in the vertically moored cases are also observed in the wave flume and examined analytically with a set of solutions of simplified governing equations, which is also introduced in the present study. The performance of the proposed structure is evaluated through a comparison with that of conventional rectangular floating structures.  8  Chapter 2 Preliminary Assessment 2.1 General In this chapter, a simplified analytical assessment is carried out in order to understand the basic characteristics of the proposed floating structure. A dimensional analysis is a helpful means of identifying important parameters of the problem; therefore this is conducted as a first step of this examination. Second, two methods to predict the configuration of a submerged air chamber in the rest condition (no wave and no external eccentric load) are introduced. They include a numerical model based on the lumped-mass method and a simplified method. Finally, the hydrostatic restoring moment against a rotation of the body, one of the primary parameters for the performance of floating structures, is examined analytically. Alternative configurations of the submerged air chamber in the rest condition are considered as shown in Fig. 2.1, depending on such parameters as the length of the membrane, the air chamber pressure, and the mass of the body. In Fig. 2.1, the full tension condition means that the air chamber pressure is relatively large and a nearly uniform tension occurs throughout the membrane. In the slack condition, the air chamber pressure is relatively small so that the configuration of the air chamber is nearly flat except near both ends, where the membrane may protrude horizontally or be irregularly slack. If the membrane is relatively short, the configuration of the air chamber may be as shown in Case-1 of Fig. 2.1. On the other hand, in cases where the length of the membrane is significantly larger than the beam of the body, the configuration may be as in Case-2, with horizontal protrusion near the both ends, even in the full tension condition. In the present study only Case-1 is treated, because it has been found, in the process of the preliminary study, that the application of Case-2 is limited to  9  special situations with relatively large body mass. In addition, only the full tension condition is treated in this study, because the behavior of a membrane in the slack condition cannot be easily treated analytically with simple models.  2.2 Dimensional Analysis Figure 2.2 shows a definition sketch of the proposed floating structure and the sea state. For simplicity, the piston is assumed to be connected to the body by a linear spring with certain values of stiffness and damping coefficient. In a two-dimensional problem, the performance of the proposed structure in a non-moored condition is influenced by variables which include the following: Fluid properties  Wave properties  Structural properties  Density  p  Dynamic viscosity  p  Wave height  H  Wave period  T  Water depth  d  Mass of the body  Mb  Moment of inertia of the body  lb  Elevation of the center of gravity of the body z (above the still water level)  Air chamber membrane properties  g  Beam  B  Total draft of the structure  h  Air chamber pressure (gage pressure)  p  Thickness  t  Unstretched length  L  m  Modulus of elasticity  E  m  t  m  10  Piston properties  Other properties  Density  pm  Damping coefficient  c  Mass  M  Length  A  Spring stiffness  K  Damping coefficient  c  p  Atmospheric pressure  p  a  Gravitational constant  g  m  p  p  The masses of the body and piston, the moment of inertia of the body, the damping coefficients of the piston and membrane, and the spring stiffness of the piston are all taken as two-dimensional variables in the present study. The mass of the body may include external vertical static loads. The unstretched length of the membrane L and the total draft h are m  t  interdependent if a certain value of the air chamber pressure p is specified; therefore one or the other should be omitted from the dimensional analysis. Of the above parameters, the effect of the fluid viscosity may reasonably be neglected. In addition, in practice the effect of the atmospheric pressure could be omitted from the dimensional analysis, although a scale effect due to the atmospheric pressure remaining invariable cannot be avoided in experiments with scale models. Other parameters neglected in the present study may include the surface tension and compressibility of water, and the surface roughness of the structure. Although only two-dimensional properties are treated in the present study, the transmission of the membrane tension in the transverse direction to the body should also be considered in experiments and in practical structures, as discussed in Chapter 5. In moored cases, the effects of the mooring system should be incorporated. Using the Buckingham 7t-Theorem, the amplitude of the body motion in a specified mode, denoted A, may be expressed in the form:  11  2  A  = f  H B T 'gT 'gT 2  H  g  m  2  m  r  m  2  , f.  Pgh pB  2  t  p B ' B v p ; B pgB 4  A K M — — - D —B ' pgB' ' M by pn  m  pgB ' pB 2  pn  (2.1)  P  A is defined for the three modes of the motion: sway, heave and roll. For roll, the amplitude of the motion A may be expressed as f}B/2, where f> is the roll angle. The functions f , f , w  f  m  s  and f account for the effects of wave properties, structural properties, membrane p  properties, and piston properties, respectively. D and D denote the fractions of critical m  p  damping of the membrane and piston respectively. For heave, the second and fourth terms in the function f , V p B  and z /B, may be neglected.  3  s  g  In examining hydrostatic characteristics, including the restoring moment against a forced inclination, the effects of waves and the parameters associated with dynamic characteristics such as the moment of inertia of the body and the damping coefficients can be omitted. In addition, empirically it can be assumed that the effects of the membrane properties are negligible unless the membrane is considerably thick (this assumption is verified in Section 2.4.5). Furthermore, the inclination of the body hardly influences the behavior of the piston. Therefore, the hydrostatic restoring moment M against the inclination of the body -& can be r  expressed in the form: M. M L Pgh, •=f M gBsin'r3 pB B i p J B pgB b  n  2  (2.2)  b  2.3 Equilibrium Configuration of Air Chamber Two methods to predict the air chamber configuration in the rest condition are introduced in this section. The first is a simplified approach in which the air chamber configuration is approximated as an arc of an ellipse. The other is a numerical model based on the lumpedmass method, which has often been used to predict the behavior of flexible thin structures.  12 2.3.1 Ellipse Approximation In general, taking a small element of the membrane in the full tension condition and neglecting its mass, the equilibrium of the forces acting on the element is expressed as Ap = — R  (2.3)  where Ap is the difference of the pressure acting on both surfaces of the element, f is the m  tension in the element, and R is the radius of curvature of the element. In the case of the proposed structure, the air chamber pressure inside, p, is constant along the membrane, and the tension can be assumed constant throughout the membrane unless the mass of the membrane is significantly large. On the basis of Eq. 2.3, the radius of curvature can then be expressed as R-  —  ^  (2.4)  —  p + pgz  where the vertical coordinate z is measured upward as positive from the still water level. The following discussion is made with reference to the definition sketch in Fig. 2.2, with the effects of mass, thickness and stiffness of the membrane being neglected. The configuration of the air chamber in the rest condition is approximated as an arc of an ellipse, as follows: 2  Vay  /  _  • _  \2  +| ^ | = 1  (2.5)  where a and b are the lengths of the horizontal (major) and vertical (minor) axes of the ellipse respectively, and zn is the elevation of the center of the ellipse. The conditions to determine the properties of the ellipse are z = -h  at x = 0  t  z = -h  at x = B/2  b  (2.6)  where hb is the draft of the body. Now, the radius of curvature of an ellipse is given as 3  (a sin t + b cos t) R =^ '— ab where x = a cos t, z = b sin t. 2  2  2  2  2  (2.7)  13 At z = -h the radius of curvature is also given, deriving from Eq. 2.4, as t  f R  —  (2.8) m  P-Pgh, Therefore, another condition at x = 0 (z = -h ) is obtained from Eqs. 2.7 and 2.8 as t  p-pgh,  (2.9)  b  Another equation derives from the equilibrium between the self load of the body and the buoyancy acting on the entire structure:  - =v +v i  b  c  = Bh +ab(5-^sin2f -^ b  2  (2.10)  b  2  J  where t is the angle made by the horizontal axis of the ellipse with the line which connects b  the center of the ellipse (0, zn) and either corner of the body (±B/2, -hb), given as z +h B n  t = tan b  K  (2.11)  Once any four of the five variables Mb, p, h , hb and f are specified, the unknowns a, b, ZQ t  m  and the remaining unknown of the five variables above can be obtained by solving these simultaneous equations.  2.3.2 Lumped-Mass Method As shown in Fig. 2.3, the elastic membrane is discretized into small elements. The relative self load of the element f , and external forces acting on the surface of the elements, F s  o u t  (due to the hydrostatic pressure outside) and Fj (due to the air pressure inside), are assumed n  to be concentrated at the center of the element. Both F  o u t  and Fj are always normal to the s n  axis (tangent to each element). The equations of equilibrium for each element of the membrane in the x and z directions are expressed in the form: x direction (F  out  - F )cosx. + f, sinCx, + 8,) - f sin(T, - 8,) = 0 in  M  (2.12)  14  z direction -(F  out  - F J s i n T j -l-.ficosdi +8 )-f _ cos(T - S ^ - f , =0 i  i  1  i  (2.13)  where F =pgh (Al +l )  (2.14)  F =p(Al +l )  (2.15)  f,=t E (^)  (2.16)  f. = ( P « " P ) U i  (2.17)  0Ut  i  iB  1  i  m  li  1  i  m  : unstretched length of element  Ali : elongation of element due to tension fj hj  : water depth at center of element  Tj  : angle made by z axis with s axis  8;  : angle made by s axis with membrane element  From geometric considerations, the following relationship can be obtained: x =x _ +8 _ +8 i  i  1  i  1  (2.18)  i  The calculation procedure is as follows : (1) The following parameters are taken as specified: Mass of body  Mb  Beam  B  Air chamber pressure  p  Total draft  h,  Thickness of membrane  t  m  Modulus of elasticity of membrane  E  m  Density of membrane  p  m  (2) The calculation starts from the element located on the center line of the body as shown in Fig. 2.3, taking the total draft h as a known variable. The calculation is conducted for one t  15 side only, from the center to either corner of the body. Providing initial value of the tension f[ to the first element, 81 is obtained from either Eq. 2.12 or 2.13. Since the configuration of the air chamber is symmetric, at this first element f_i =f\, 8.1 =-8], and Ti=-7t/2  for the left side and n/2 for the right side. For subsequent elements, Eqs. 2.12  and 2.13 are simultaneously solved for the tension fj and the angle 8j by the NewtonRaphson method. When the horizontal component of the total stretched length reaches B/2, the total submerged volume and resulting total buoyancy are calculated. This procedure is repeated with a change in the value of fi until the difference between the total vertical force and the buoyancy converges. Then the body draft hb and the unstretched length of the membrane L are finally obtained m  2.3.3 Comparison of the Two Methods Figure 2.4 shows an example comparison of the calculation results obtained by the two methods mentioned previously. In the present examination, the calculations by the lumpedmass method are conducted first, then, using the body draft hb obtained by the lumped-mass method, the calculations based on the ellipse approximation are conducted for the specified body mass Mb, air chamber pressure p and total draft h . Since the calculations have been t  carried out so that the air chamber volumes calculated by the two methods are to be identical, the difference of the two methods may result in the difference of the membrane tensions. In Fig. 2.4 the difference of the tensions obtained by the two methods is indicated. f (L) and m  f (E) respectively denote the tensions at the center of the membrane obtained by the lumpedm  mass method and by the ellipse approximation. As the relative external to internal pressure ratio, characterized by pgh /p, increases, the difference in the tensions becomes large. t  The air chamber configurations obtained by the two methods are compared for two typical cases in Fig. 2.5. For Case-1, the air chamber pressure and the resulting tension are relatively large, and the unstretched length of the membrane is relatively short, while for Case-2 the opposite is true. In Case-1, the air chamber configurations obtained by the two methods are  16 very similar, and the ratio of the tensions f (L) / f (E) = 1.08. Even in Case-2, for which this m  m  ratio exceeds 1.5, the configurations obtained by both methods are similar. Overall, although the numerical model based on the lumped-mass method may be more accurate for predicting the properties of the air chamber, the simplified method based on the ellipse approximation also appears to provide reasonable estimates of the air chamber configuration.  2.4 Examination of Hydrostatic Restoring Moment In this section, the hydrostatic restoring moment of the proposed structure is examined analytically. First, the two methods introduced in the previous section to calculate the air chamber configuration in the rest condition, the simplified method based on the ellipse approximation, and the numerical model based on the lumped-mass method, are extended to predict the restoring moment against a forced inclination. Next, the restoring moment characteristics are examined with the structural properties as parameters.  2.4.1 Ellipse Approximation The restoring moment of a floating body, whose sides are vertical near the water surface in the rest condition and whose configuration below the water level is arbitrary, can be generally expressed as M =WGMsint>  (2.19)  r  where _ i ' G M = — ± CG =±2— + ( z - z ) V V ±  B  (2.20)  E  W is the self load of the body (= Mbg), r) is the inclination of the body, z and z are the g  c  elevations of the centers of gravity and buoyancy from the still water level respectively, and V is the submerged volume. Both z and z are taken as positive upward. g  c  17 In the case of the proposed structure, for which the air chamber is approximated as an arc of an ellipse, and if it is assumed that the air chamber is so stiff that it hardly deforms in the heel condition, the variables in Eq. 2.20 are given as 1  Z  1  ' "V =  V h + V (h h J b  V=V +V b  b  c  b +  c  = Bh +ab(|-|siri2f -f b  h  b  (2.21)  c  b  1 —cos3/ + cosf b  n  b  ~ ~  2  2 +  -(h + o) z  b  (2.23)  sin2f - 1 K b  b  t = tan" 2| ° > B Z  (2.22)  b  h  b  (2.24)  The effect of the air chamber deformation in the heel condition may be incorporated by defining a reduction ratio D as M =(l-D)WGMsint> r  (2.25)  2.4.2 Lumped-Mass Method Equations 2.12 - 2.18 are used as before to predict the air chamber configuration in the heel condition. In this case, the calculation is started at either end of the membrane. Because the coordinates of both corners of the body in the rest condition have already been determined in the previous stage, the coordinates of the both corners for a forced inclination can be specified. With the unstretched length of the membrane determined in the previous stage, the calculation is repeated with a change in the tension in the first element and the angle made by the underside of the body with the tangent of the first element, until the other end of the membrane reaches the other corner of the body. Finally, by taking the moment of the entire buoyancy and the self load about the center of rotation, the restoring moment is obtained.  18  In the above procedure, it is assumed that no vertical movement of the body is induced and the air chamber pressure remains constant when the body is subjected to a forced inclination. To verify this assumption, the air chamber volumes in the rest and heel conditions are compared during the calculation. It turns out that in most cases the differences in the air chamber volumes are so small that this assumption is adequate. In this calculation, the equilibrium of the forces acting directly on the body must be maintained, and the restoring moment can also be obtained by taking the moment of these forces. They include the self load, the air chamber pressure acting on the underside, the hydrostatic pressure on the sides, and the tensions at both ends of the membrane. The calculation results have indicated that the restoring moment calculated by both approaches are close in most cases, although sometimes a fine discretization is required to match both calculation results. The equilibrium conditions of vertical and horizontal forces are satisfied in almost all cases.  2.4.3 Results Figure 2.6 shows an example of calculation results by the lumped-mass method for a prototype structure subjected to a forced inclination. It can be seen that the air chamber deforms asymmetrically. Figure 2.7 exhibits the relationship between the restoring moments of the proposed structures and relevant dimensionless parameters. The restoring moments M are calculated using the r  lumped-mass method and non-dimensionalized with respect to the body mass M , the beam b  B, the inclination f>, and the gravitational constant g. In this examination, the center of gravity of the body is taken at the still water level for simplicity. For comparison, the dimensionless restoring moments of the equivalent rectangular structures which have the same body masses and beams are shown. As the total draft increases while the air chamber pressure is kept constant, the restoring  19 moment decreases, together with a decrease of the membrane tension. This is associated in part with the decrease of the difference between the hydrostatic pressure outside and the air pressure inside. It could be understood intuitively that if an unstretched length of the membrane is kept constant, the tension decreases with a decrease of the air chamber pressure. At the same time, the deformability of the air chamber becomes larger, and the effect of the deformation on the restoring moment becomes more significant; consequently the restoring moment decreases. The calculation results shown in Fig. 2.7 reflect this trend quantitatively.  2.4.4 Reduction Ratio of Restoring Moment in Ellipse Approximation The reduction ratio D for the restoring moment in Eq. 2.25 can be obtained by equating the restoring moment obtained by the lumped-mass method to the right-hand side of Eq. 2.25. Some examples of the calculation results for D are shown in Fig. 2.8. In Fig 2.8, the reduction ratio D is presented in terms of the same dimensionless parameters as in Fig. 2.7. In general, the restoring moments calculated by Eq. 2.25 with assuming D=0 decrease only a little from those of the equivalent rectangular floating structures; therefore the deformability of the air chamber is a primary factor which determines the reduction ratio D. It can be seen that the decrease of the restoring moment with the increase of pgh /p shown in Fig 2.7 almost t  completely corresponds, to the increase of D in Fig. 2.8. In Fig. 2.8, the ratio of the body weight to the air chamber pressure Mbg/pB are about 0.89 for the three solid lines, 0.82 for the dash lines, and 0.75 for the dotted lines. The trend of D to pgh /p seems similar for the same values of Mbg/pB, regardless of the mass. This implies t  the reduction ratio D could be expressed in terms of the parameters Mbg/pB and pgh /p. t  2.4.5 Effect of Membrane Properties on Restoring Moment Figure 2.9 exhibits an example of the calculation results conducted to examine the effects of membrane properties on the restoring moment. The lumped-mass method is employed for this calculation. Figure 2.9 (a) shows the relationship between the restoring moment and the  20 unstretched length of membrane, with modulus of elasticity as a parameter and thickness kept constant. Figure 2.9 (b) shows the same calculation results of the restoring moment as in Fig. 2.9 (a), as a function of pgh /p instead of L / B . From these figures, it can be seen that the t  m  stiffness of the membrane hardly influences the restoring moment while the unstretched length is varied. A similar trend has been found in the calculation results with the thickness as a parameter and the modulus of elasticity kept constant, although not illustrated. Figure 2.9 (c) shows the restoring moment as a function of L / B for three cases with the modulus of m  elasticity E and the thickness t being changed so that E t m  m  m  m  are kept constant. In these  three cases the restoring moment characteristics are very similar. From these results, it may be concluded that within the range that the mass of membrane is very small, the membrane properties influence only the unstretched length of the membrane.  21  Chapter 3 Theoretical Formulation 3.1 General This chapter is concerned with the theoretical development of the prediction of the dynamic response of the proposed floating structure, as a two-dimensional problem. First, a set of governing equations is introduced to express the dynamic response as accurately as possible. Since the entire behavior of the structure is nonlinear, a numerical model based on a timestepping analysis would likely be needed to solve these equations. However, with respect to the heave mode only, some simplification may be possible by assuming that the wave height and the resulting deformation of the air chamber are so small that it is possible to neglect the nonlinearity of the behavior of the structure. Thus, simplified governing equations for the heave mode are also presented, together with their solutions.  3.2 Governing Equations The governing equations are described with reference to the definition sketch in Fig. 3.1. Although the equations presented in the present section are intended to express the dynamic response as accurately as possible, the following premises are adopted for simplification: • The piston is located at the center of the body. • The piston moves, relative to the body, only perpendicularly to the underside of the body. • The piston is connected to the body with a linear spring. • Some minor forces arising with a rotation of the piston about the center of rotation of the structure are negligible. Such forces include a centrifugal force and an inertia moment whose arm would vary at each instant during the vertical oscillatory motion of the piston. To realize this assumption, at least one of the following conditions must be satisfied: (1) the mass of the piston is relatively small, (2) the motion of the piston relative to the body  22 is small, and (3) the roll amplitude of the structure is small. In many practical cases these conditions should be satisfied to some degree, as implied in Chapter 5. The membrane may be discretized into small elements, as shown in Fig. 2.3. The primary unknown variables are as follows: Displacements in the x and z directions and rotation of the body:  Xb, z , 8b  Variation of the air chamber pressure:  p  c  Displacement of the piston (Relative to the motion of the body):  z  p  Displacements of the membrane elements in the x and z directions:  x j, z j  Tensions in the membrane elements:  fj  b  m  m  3.2.1 Equations of Motion for the Body and Piston For simplicity, the rotation of the body 0 is assumed to be sufficiently small such that b  sin0b=0b and cos0b=l. The equations of motion are expressed as follows: • x direction Since the piston motion in the x direction is consistent with the body motion, they can be expressed together in one equation as ( M + M ) x = ' J p ( x , ( z ) , z : t ) d z - J p ( x ( z ) , z : t) dz + f, c o s T - f b  p  b  :  w  w  r  E L  -(B - A)p0 - ( K z + c z )9b - c x - F b  p  p  p  p  dx  b  r  COST,  x  er  (3.1)  • z direction Body M z = (B - A)p - M g + K ( z + z ) - f, sinT - f sinx b  b  b  p  p  pi  J p ( * i ( z ) , z : t) d z -  +  w  e]  r  er  J p ( x ( z ) , z : t) dz 0 - c d z z b + c p z p - F z w  r  b  (3.2)  Piston M ( z + z ) = Ap - M g - K ( z + z ) - c z p  b  p  p  p  p  pi  p  p  (3.3)  23 • Rotation r B r B ( I + I ) 6 = - J p ( x , ( z ) , z : t ) ( z - z + - e ) dz+ jp (x (z), z : t ) ( z - z - - 9 ) dz - w. ~ b, b  p  b  g  w  b  w  r  g  b  h  h  +f, {r, sin x + (h + z ) cos x } - f {r sin x + (h + z ) cos x } el  -c e -F d 9  where p  b  bl  g  el  r  r  er  br  g  er  (3.4)  e  water pressure along the submerged surface of the structure  w  T|  free water surface elevation  p  air chamber pressure p=Pi+p , where pi = air pressure in rest condition (gage pressure) c  f[, f  tensions at both ends of the membrane  r  xi(z), x (z)  x coordinates corresponding to z along both sides of the body  F , F , Fe  mooring forces in x, z and 6 directions  x ,Ti  angles between x axis and tangents of the membrane at both ends  hbi, h^  depths at both corners of the body  zi  displacement of the piston which gives piston spring force in rest condition  z  elevation of the center of rotation of the structure  r  x  z  er  e  p  g  Ip  moment of inertia of the piston about the center of rotation of the structure for rest condition  rj, r  horizontal distances from center of rotation to the connection  r  points of the body and membrane c<jx> c,j , Cde z  viscous damping coefficients in sway, heave, and roll  To obtain the water pressure p , the sum of the hydrostatic and hydrodynamic pressure, w  potential flow theory may be employed.  3.2.2 Equations of Motion for Membrane Elements The equations of motion for the membrane elements can be obtained by adding the terms accounting for the inertia and damping forces to Eqs. 2.12 and 2.13, as  24 PmUiXmi  = (F  out  Pmtm^mi  = - (  o u ,  F  - FJcosx, + f, sinCij + 8 , ) - f._, s i n ^ - 8 ) - c x S  "  F  i n )  ^  +  f  i COS(l, +  8; ) -  f C O S ^  -  m  (3.5)  mi  8,) -  - f,  (3.6)  where c is the damping coefficient of the membrane, and Eqs. 2.14-2.18 are also used for m  Eqs. 3.5 and 3.6, except that the external normal force F F  0Ut  out  is replaced by  =p (Al +l ) w  i  (3.7)  i  and the variables F , F; , out  n  Sj, fj, p, hj and Alj are all dependent on time in this case.  3.2.3 Variation of Air Chamber Volume and Pressure Assuming adiabatic conditions, the relationship between the air chamber volume and pressure may be expressed as Pa + Pi + Pc _  (3.8)  Pa+Pi  where V and V j are the volumes of the air chamber in motion and in the rest condition c  C  respectively, p is the atmospheric pressure, and y = 1.4 in general. a  3.3 Simplified Model for Heave 3.3.1 Model To simplify and linearize the aforementioned governing equations for the heave mode, the proposed structure is modeled as shown in Fig. 3.2. The air chamber configuration is modeled as a rectangle whose volume is the same as that of the real air chamber, and the membrane is not directly taken into account in the prediction of the response. The virtual total draft hj and air chamber height h j in the rest condition are respectively defined as t  c  h*=— Pg  (3-9)  h ,=^-  (3.10)  c  25 where pi and V j are the air pressure and volume of the real air chamber in the rest condition C  respectively. The configuration and volume of the real air chamber in the rest condition may be obtained by using either of the methods presented in the previous chapter, with specifying the relevant properties of the structure including the membrane properties. The effect of mooring system is taken into account as a linear vertical spring. In this model, the wave exciting force, and the hydrodynamic added mass and damping coefficient in heave are calculated separately. The vertical exciting force per unit area acting on the bottom of a floating structure is, assuming a uniform loading, given as f = pgSr, = ipgSHe  (3.11)  k0,  where i = V - l , t is time, r\ is the free water surface elevation due to the incident wave, H is the incident wave height, co is the wave angular frequency, and S is the coefficient which relates the free water surface elevation to the exciting force. In the present study, the coefficient S, added mass and damping coefficient are calculated by using a numerical model developed by Isaacson and Nwogu (1987). It is based on two-dimensional linear diffraction theory, and applicable to rigid floating bodies with arbitrary shapes. These hydrodynamic properties are calculated for the real configuration of the structure in the rest condition. Unknown variables are: Motion of the body:  z  b  Motion of the piston:  z  p  Water surface elevation under the body:  z  s  Variation of the air chamber pressure:  p  c  The motion of the piston and the water surface elevation under the body are relative to the motion of the body. Each variable is treated as a complex number, and regarded as the product of a complex amplitude and the harmonic function exp[icot]. In this model, the damping effects of the piston and membrane are neglected for simplicity.  26 3.3.2. Governing Equations The equations of motion for the body and piston are expressed as follows: • Body Mz + c z + K ( z + z ) = - M g + K ( z + z ) + (B - A)( + pc) b  z  b  m  b  bi  b  p  p  pi  (3.12)  Pi  where the mass M includes the real mass of the body Mb and the hydrodynamic added mass in heave M , z^ and z j respectively denote the body and piston displacements which give a  p  the mooring force and piston spring force in the rest condition, and c is the hydrodynamic z  damping coefficient in heave. Although the damping effects of the piston and membrane are neglected for simplicity, the effects of structural damping of mooring system and viscous damping of the water could be included in the damping coefficient c if necessary. z  Subtracting the terms associated with the static equilibrium, Eq. 3.12 is rewritten as M z + c z + K z = K z + (B - A)p b  2  b  m  b  p  p  (3.13)  c  • Piston M (z + z ) + K .(z + z ) = A ( + p ) - M g p  p  b  p  p  pi  P i  c  p  (3.14)  Subtracting the terms associated with the static equilibrium, M ( z + z ) + K z = Ap p  p  b  p  p  c  (3.15)  The equations of equilibrium for the water surface under the body and the air chamber pressure are expressed respectively in the forms: • Water surface under the body Pg(h„-Zb-Zs  + l ) = P +Pc Sr  1  (3-16)  Subtracting the terms associated with the static equilibrium, pg(-z -z +Sr,) = b  s  P c  (3.17)  • Variation of the air chamber volume and pressure Assuming adiabatic conditions once more, the relationship between the air chamber volume and pressure is expressed, in the same manner as in Eq. 3.8, as  27 h ,B B ( h - z ) + Az  Pa + Pi + Pc  -,1.4  (3.18)  c  Pa+P,  ci  s  D  If the variations of the air chamber volume and pressure are relatively small, Eq. 3.18 could be approximated as Pc = - l - 4 ( p + P i ) a  Az -B.z, h„ B p  (3.19)  ;  Since Eqs. 3.13, 3.15, 3.17 and 3.19 are all linear, a closed set of solutions for the unknown variables zt,, z , z and p can be obtained by solving these four simultaneous equations, after p  s  c  decomposing each variable to the complex amplitude and the harmonic function, and eliminating the harmonic function. The solutions are presented in Appendix A.  28  Chapter 4 Experimental Investigation 4.1 General An experimental investigation has been carried out to study the hydrodynamic performance of the proposed floating structure. The experiments were carried out in the wave flume in the Hydraulics Laboratory of the Department of Civil Engineering, the University of British Columbia. A scale model was constructed for the experimental investigation. The motions of the experimental model in the free condition were recorded for regular waves with a constant water depth. From the records, the response amplitude operators in sway, heave and roll were obtained for a wide range of wave periods. In addition, the behavior of the proposed structure in the vertically moored condition was also observed. The objective of the present study is to assess whether a proposed structure which is proportioned appropriately could achieve a significant reduction of the dynamic response. However, the performance of the proposed structure cannot be predicted at the present, except for the heave which can be obtained by the solutions of the simplified governing equations presented in Chapter 3. Thus the present experimental model was proportioned mainly on the basis of a simple preliminary examination, although the resulting model might not be optimum for the reduction of the dynamic response. Later in Chapter 5, the selection of suitable properties of the proposed structure are discussed briefly on the basis of the experimental results.  4.2 Wave Flume and Generator Figure 4.1 shows a schematic of the experimental set-up. The wave flume is 20 m long, 0.6 m wide, and 0.75 m deep. An artificial beach is located at the downwave end of the flume to  29 reduce wave reflection. Waves are generated by a single paddle wave actuator located at the upwave end. The generator is controlled by a DEC V A X station-3200 minicomputer using the GEDAP software package developed by the National Research Council, Canada (NRC). The experimental model was located at about 9 m downwave of the wave paddle. Through all of the tests, the water depth was maintained at 0.55 m.  4.3 Experimental Model Figure 4.2 shows a sketch of the experimental model. The body is constructed of acrylic panels with a thickness of 19 mm. The mass of the body is about 27 kg and the mass moment of inertia is about 1.0 kgm , including additional lead weights placed inside the 2  body. The center of gravity of the body is located at 0.08 m above the bottom of the body. The piston consists of a flat rubber sheet as a spring and lead weights on the sheet. The mass of the piston is about 9 kg. The effective draft to beam ratio M^/pB  2  is about 0.2. To  maintain the two-dimensional behavior of the air chamber as much as possible, the acrylic panels, whose configuration is similar to that of the air chamber in the rest condition, are placed at both sides of the air chamber. A manometer consisting of a transparent U-tube and colored water is placed inside the body to measure the air chamber pressure in the rest condition, although this is not illustrated in Fig. 4.2 for clarity. Details of the model properties are outlined in the following. Air chamber properties In order to determine the air chamber properties, such as the air pressure in the rest condition and the unstretched length of the membrane, a preliminary examination was conducted as follows. First, the configuration of the air chamber in the rest condition was calculated by the numerical model based on the lumped-mass method. Then, the hydrodynamic properties of the rigid structure with the calculated configuration were obtained using the aforementioned numerical model by Isaacson and Nwogu (1987). The hydrodynamic properties relating to the heave mode were used to predict the heave response by the solutions presented in Chapter  30 3, while for sway and roll the dynamic responses of the rigid body in the free condition were examined. This procedure was conducted for several cases with changes to the air chamber properties. As a result, it has been found that a large air chamber contributes to a reduction in the peak heave response in the free condition while it causes a significant increase of the exciting force in roll (details are presented in Section 5.3.2). On the basis of this preliminary examination, the air chamber properties were determined so that the heave response could be reduced significantly while the exciting force in roll does not increase drastically, from those of the equivalent rectangular structure. In addition, the air chamber pressure in the rest condition was determined so that the membrane tension is as small as possible so as to increase the deformability of the chamber while the hydrostatic restoring moment is not too small. The resultant unstretched length of the membrane is 0.548 m, the air chamber pressure is 1.24 kPa, and the total draft is 0.125 m. The calculated hydrostatic restoring moment is about 20% of that of the equivalent rectangular structure, and the calculated tension in the membrane is 45 N/m, with an axial strain of 0.001. Rubber sheets for the air chamber membrane and piston spring Two types of rubber sheets were used for the model. First, rubber sheets with a thickness of 1.6 mm and a modulus of elasticity of 3.0 MPa were used for the air chamber membrane and the piston spring. However, during the experiments it turned out that this type of sheet is too stiff for the air chamber membrane as well as for the piston spring. Therefore the sheets were subsequently replaced by another type of sheet with a thickness of 0.4 mm and a modulus of elasticity of 1.5 MPa. Besides the problem of stiffness, the unstretched length of the former membrane in the transverse direction was almost same as the body width. This also seemed to cause a deterioration in the performance of the air chamber ( as discussed in detail in Chapter 5). In addition, the former air chamber membrane was shorter than the planned length in the longitudinal direction because of inaccuracy in the construction. In this context, the latter air chamber membrane was made longer in both directions than the former one. As  31 a result, some slackness in the transverse direction has been provided; although it is still a little shorter than the planned length in the longitudinal direction. Hereafter, for brevity, the former model with the thick rubber sheets is referred to as M - l , and the latter model is referred to as M-2. Because of the shortness of the membrane lengths in the longitudinal direction, the air chamber configurations in the both model cases are also different somewhat from the planned configuration. Piston stiffness The stiffnesses of the piston springs were obtained from the measured relationship between the total vertical force due to the air pressure and the displacement of the piston at the center, as shown in Fig. 4.3. In Fig. 4.3, the origin corresponds to the rest condition. The principal properties of the two model cases are summarized in Table 4.1. In Table 4.1, the length of the piston A is a virtual length, estimated so that the product of the piston displacement at the center and the virtual length A provides the real change of the volume.  4.4 Test Conditions  Table. 4.2 lists the wave conditions adopted in the experimental investigation. Twelve wave periods ranging from 0.7 sec to 2.0 sec are adopted. Originally, the wave steepness was intended to be about 0.025 for all the wave cases; however, later the incident wave heights for a few short wave cases were increased because the motions of the models in these cases were too small to measure accurately. The beam to water depth ratio is 1.0 (B = d = 0.55 m) for all the test cases. Originally, several experimental cases were planned with the air chamber pressure as a parameter, while maintaining the unstretched length of the membrane. However, during preliminary tests it was found that the dynamic response changes very slightly with changes in the air pressure. This is considered to be partly because the acrylic side panels of the air  32 chamber, placed with an intention of maintaining the two-dimensional behavior of the air chamber (refer to Fig. 4.2), restrained the motion of the membrane too strongly. Consequently, the experiments were conducted with only one case of the air pressure for each model case. In this sense, it might have been better if the sides of the air chamber had been constructed of rubber sheets, although the model construction would have been more complicated. The test series are summarized in Table 4.3. In the series using the model M - l , the tests were conducted for two piston conditions. In the first condition the piston works with the spring stiffness as shown in Fig. 4.3, while the piston is restrained in the second condition. In addition, two externally restraining conditions of the structure are adopted for the test series with the model M - l : the completely free case and the case in which sway and roll are restrained by four vertical bars placed along the upwave and downwave sides of the body. In the test series with the model M-2, only one condition is adopted, in which the piston works and the model is free.  4.5 Measurements The motions of the model under waves were recorded with a video camera. From the records, the responses were measured against the grid placed on the transparent side of the flume. The motions in heave and sway were represented respectively by the vertical and horizontal displacements of a reference marker, which was attached on the side of the body at the center of gravity.  Roll angles were obtained from the inclinations of two reference  markers attached at both upper corners of the body. The video records were also used to examine qualitatively the behavior of the air chamber in the free and vertically moored conditions.  33  Chapter 5 Results and Discussion In this chapter, the dynamic characteristics of the proposed floating structure are discussed on the basis of the experimental and analytical results. First, the responses obtained from the experiments are evaluated through a comparison with those of an equivalent rectangular structure and a rigid structure with the same configuration as the experimental model. The response of these rigid structures are calculated by using the aforementioned numerical model by Isaacson and Nwogu (1987). In addition, for heave, the measured responses are compared with the predicted response of the proposed structure based on the solutions of the simplified governing equations presented in Chapter 3. Second, the behavior of the proposed structure under waves is qualitatively discussed on the basis of the video records taken during the experiments, for both the free and vertically moored conditions. Third, the effects of principal structural properties on the dynamic response are examined. Then, scale effect on the heave response, associated with the atmospheric pressure and which are therefore unavoidable in the experimental study, is discussed. These effects are examined analytically, for the free and fixed conditions. Finally, the assessment to the performance of the proposed structure is summarized, along with a brief discussion of the applicability to real structures.  5.1 Response Amplitude Operators of the Proposed Structure In this chapter, the dynamic response is presented as a function of beam to wave length ratio B/L. This section is focused mainly on the evaluation of the measured responses, through a comparison with the analytical responses. The causes of the particular performance of the proposed structure are discussed mainly in the next section.  34 5.1.1 Roll Figure 5.1 shows the measured response amplitude operator in roll for the test series S - l l (the model M - l with the thick rubber sheets), together with the calculation results for the equivalent rectangular structure and the rigid structure with the same configuration as the experimental model. On the whole, the measured response is considerably larger than these calculation results. The natural period of the model is close to that of the rigid structure, about 1.10 sec. However, as shown in Fig. 5.2, the response for the test series S-21 (the model M-2 with the thin rubber sheets) is significantly different from that of the test series S - l l . The natural period of the model M-2 is about 1.35 sec. The whole response is smaller than that of the model M - l except for the peak response. From these results, it can be concluded that the proposed structure with suitable air chamber properties has the ability to make the natural period longer, so that a reduction of the roll response could be achieved.  5.1.2 Heave Figure 5.3 shows the measured and calculated response amplitude operators in heave for the test series S - l l and S-13. The calculations are conducted using the method presented in Chapter 3 for the proposed structure, the equivalent rectangular structure, and the rigid stricture with the same configuration as the model. In both these test series, the piston is not restrained. For the test series S - l l , in which the structure is free, the measured response is close to that of the rigid structure with the same configuration. On the other hand, for the test series S-13, in which sway and roll are restrained, the measured response is close to the calculated response of the proposed structure, although the calculated responses of the rigid structure and the proposed structure are not significantly different because of the stiff piston spring.  35 During the tests, it was observed that the piston moved somewhat relative to the body when sway and roll were restrained, while in the free case the piston hardly moved. Figure 5.4 shows the measured response for the test series S-12 and S-14. In the both series the piston is restrained. The response for the two series are similar. This result indicates that the locking effect of the vertical bars, placed to restrain sway and roll, is negligible with respect to the examination of the heave response. Furthermore, these responses are also similar to the response for the test series S - l l . Figure 5.5 shows the measured response for the test series S-21, with the model M-2. Even with the soft piston spring, the measured heave response of the proposed structure is similar to that of the rigid structure, although the predicted response for the proposed structure is much smaller than that of the rigid structure. As discussed in Section 5.2, the piston moves significantly in the vertically moored cases, in which roll is restrained while sway is not. Therefore, it can be said that the piston functions properly only when the roll is restrained, and otherwise the heave response of the proposed structure is close to that of the rigid structure with the same configuration.  5.1.3 Sway Figure 5.6 shows the measured and calculated response amplitude operators in sway. Only the result of the test series S - l l is shown as the measured response, because the results of the free test series S - l l , S-13 and S-21 were all similar. The measured response, and the calculated responses for the equivalent rectangular structure and the rigid structure with the same configuration are all similar. This result indicates that the air chamber does not significantly influence the sway response. The tracks of the center of gravity of the body over time can be categorized in two ways as shown in Fig. 5.7. For relatively short waves (e.g. B/L > 0.3), the horizontal motions are dominated by the wave drift force, and there is difficulty in completely removing the drift component from the horizontal motion and obtaining the pure oscillatory amplitudes.  36 Therefore some error might be contained in the measured response for these cases. On the other hand, for relatively long waves the drift motion is relatively small.  5.2 Behavior of the Proposed Structure 5.2.1 Free Condition Figures 5.8 and 5.9 exhibit the behavior of the experimental model in the free condition under waves at particular time instants. The model case is M-2. Wave cases are W-5 (T = 1.1 sec, B/L = 0.304: close to the natural period of the equivalent rectangular structure) for Fig. 5.8, and W-10 (T = 1.5 sec, B/L = 0.189) for Fig. 5.9. Regardless of the wave period, the deformation of the air chamber is nearly in phase with the rotation of the body. For the falling side of the body, the membrane moves upwards relative to the body, and the opposite is true for the rising side. Consequently, the structure loses some portion of the restoring moment when it is tilted, and the reduction of the restoring moment contributes to making the natural period longer. The trend of the air chamber behavior of the model case M-1 (with the thick rubber sheet) is basically the same as that of the model case M-2 mentioned above. However, the roll response of the model M - l is much larger on the whole and its natural period is close to that of the rigid structure, as shown in Fig. 5.1. Although the true reason for this large difference would be more complicated, the possible reason, based on a simple physical consideration, may be explained as follows: (1) With reference to Fig. 5.10, because the transverse length of the membrane is nearly same as the body width for the model M - l , the relative displacement of the membrane to the side panels may give rise to the membrane tension in the transverse direction, which would then be transmitted to the side panels. As a result, the reduction of the restoring moment of the model from that of the rigid body is not as drastic as expected from the solely two-dimensional analysis (the reduction of about 80% is predicted analytically).  37  (2) Since the deformation of the air chamber is in phase with the body, the proposed structure cannot generate as large a radiated wave as that of the rigid structure; therefore its hydrodynamic added mass is smaller than that of the rigid structure. The resultant natural period of the model M - l could be close to that of the rigid structure, if the total mass (the real mass plus the added mass) decreases at the same rate as the restoring moment. (3) Since the air chamber membrane of the model M - l is stiff, the air chamber is not so flexible as to reduce the roll exciting force acting on the body, which would be transmitted from the membrane as indicated in Section 1.2. Therefore the roll exciting force may be nearly the same as that of the rigid body. (4) Assuming the roll response of the body as a single-degree-of-freedom system for simplicity, the response amplitude is expressed as (5.1) 1  (5.2)  where F is the amplitude of the exciting force, K is the hydrostatic stiffness associated e  r  with the restoring moment, (p is the damping ratio, CO is the wave angular frequency, and co is the natural frequency. In the case of the model M - l , the dynamic magnification n  factor Rd is not significantly different from that of the rigid structure except for the resonant range, because the natural periods are close. The exciting force is also close. However, the response of the model is larger than that of the rigid structure because the hydrostatic stiffness of the model M - l is smaller than that of the rigid structure. (5) In contrast, in the case of the model M-2, the restoring moment is reduced drastically enough to make the natural period longer; because the transversely transmitted force from the membrane to the side panels is much smaller than that of the model M-1, due to the slackness of the membrane provided in the transverse direction. In addition, the  38 membrane is more flexible; therefore the roll exciting force transmitted from the air chamber membrane to the body is significantly reduced as expected in Section 1.2. Therefore, in Eq. 5.1 both F and K are reduced, and the resulting static amplitude F / K e  r  e  r  may not be significantly different from that of the rigid structure. If this explanation were true, it would be necessary to increase the flexibility of the air chamber as much as possible, in order to reduce the roll exciting force acting on the body. In addition, a three-dimensional structural consideration for the air chamber would also be necessary so as not to increase the membrane tension in the transverse direction. Both in Figs. 5.8 and 5.9, the piston hardly moves relative to the body. In addition, from the observation of the manometer, it was found the dynamic air chamber pressure amplitudes were very small. On the other hand, the air pressure varied to a greater extent when the roll was restrained. It was also observed that in this restrained condition the deformation of the air chamber was not as drastically asymmetric as in the completely free condition. These observation results, and the fact that the air chamber deformation is nearly in phase with the body motion in the free condition, imply that the large asymmetric deformation of the air chamber in the free condition is caused mainly by the body rotation rather than by the hydrodynamic pressure; and this large asymmetric deformation somehow seems to lower the expected variation of the air pressure, leading to the measured heave responses which are similar to the calculated responses for the rigid structure.  5.2.2 Vertically Moored Condition Figure 5.11 exhibits the behavior of the experimental model in the vertically moored condition. The model case is M-2 and wave case is W-10 (T = 1.5 sec, B/L = 0.189). The deformation of the air chamber is nearly in phase with the free water surface elevation, although it is not drastically asymmetric. This deformation could reduce the exciting force transmitted from the air chamber membrane to the body, and also produce an eccentricity of  39  the buoyancy, as expected in Section 1.2. However, the present air chamber seems to be too small to reproduce these effects satisfactorily. The piston moves significantly, and its motion is also nearly in phase with the free surface elevation. According to the calculation results by the method presented in Chapter 3, for the vertically fixed condition the piston motion is nearly in phase with the free surface elevation and the exciting force for relatively long waves. This piston behavior in phase with the exciting force gives rise to the variation of the buoyancy out of phase with the exciting force, and this leads to some reduction of the mooring force. The observation mentioned above partly verifies the predicted behavior; therefore some reduction of the mooring force could be expected. In conclusion, a reduction of the mooring force could be achieved in the vertically moored cases with a suitable air chamber, although this has not been verified quantitatively in the present study.  5.3 Parametric Examination of Dynamic Characteristics 5.3.1 Calculation Conditions In this section, the dynamic characteristics of the proposed structure are examined with various structural properties taken as parameters. First, the exciting forces, added masses and damping coefficients of the rigid structures with the same configurations as the proposed structures are examined for the three modes of motion. Then, the predicted heave response characteristics, by the method presented in Chapter 3, are discussed with the air chamber and piston properties as parameters. In this discussion, the free and fixed cases are taken, based on the assumption that the method presented in Chapter 3 can appropriately predict the actual heave response, at least when the roll is restrained by a suitable mooring system. The following properties are taken as constant, after confirming that they have little effect on the heave response of the proposed structure.  40 Damping ratios of the piston and membrane:  Dp = D = 0.05  Thickness of membrane:  t /B = 0.002  Stiffness of membrane:  E /pgB = 40.0  m  m  m  Actually, these properties may affect the response significantly if the piston and/or the membrane resonate with waves. However, their resonance may not be desirable in terms of their stability and durability, although the motion of the body might be reduced well. In this examination the viscous damping is not considered. As discussed in Section 5.4, scale effect due to the atmospheric pressure affects the heave response; therefore a certain length scale must be specified. In this examination the beam is taken as 10.0 m, although all the parameters are expressed in non-dimensional form. The examination is carried out for the body length to water depth ratio B/d = 0.33.  5.3.2 Effects of Air Chamber Properties Hydrodynamic properties In the present examination the following six cases are taken, with the mass of the body and the air chamber properties as parameters: • M / p B = 0.20 2  b  Case-1  Pghti/Pi = 0.988,  Pi/pgB = 0.225  Case-2  Pghti/Pi = 0.930,  Pi/pgB = 0.225  Case-3  Pghti/Pi = 0.930,  Pi/pgB = 0.250  • M / p B = 0.30 2  b  Case-1  pghti/pi = 0.988,  Pi/pgB = 0.330  Case-2  Pghti/Pi = 0.930,  Pi/pgB = 0.330  Case-3  pghti/pi = 0.930,  Pi/pgB = 0.370  The configurations of the air chambers are obtained by the lumped-mass method. In both cases of M / p B , the air chamber is largest for Case-3 and smallest for Case-2, and the 2  b  tension in the membrane is largest for Case-2 and smallest for Case-1.  41 Figures 5.12 - 5.14 show the exciting forces, hydrodynamic added masses and damping coefficients calculated for the rigid structures with the configurations of the three cases of Mb/pB = 0.20. These results are non-dimensionalized with respect to the incident wave 2  height H, the wave angular frequency co, the beam B, the total submerged volume V j, the C  density of the water p, and the gravitational constant g. Notable characteristics of these figures are as follows: • With an increase of the air chamber volume, the exciting force in sway decreases while those in heave and roll increase. Especially, the exciting force in roll increases drastically. • The added masses of the proposed structures are smaller than those of the rectangular structure in general, except for the sway - roll coupling added mass and the sway added mass for relatively short waves. • The damping coefficients increase with an increase of the air chamber volume, except for sway. • On the whole, the effects of the air chamber on these hydrodynamic properties seem to affect the roll most significantly. In the case of Mb/pB = 0.30, the same characteristics were obtained qualitatively. 2  Heave response Figures 5.15 and 5.16 show the predicted heave responses, with the air chamber properties as the parameters. The response amplitude operators and non-dimensionalized mooring forces are presented for the free and fixed cases respectively. In these calculations the piston is not considered. For comparison, the responses of the equivalent rectangular structures are also shown. In the free case the proposed structure can reduce the response for the peak range even without piston, while the response increases for large B/L. In the fixed case the air chamber  42 seems to provide negative effects. This is considered to be because the exciting force of the proposed structure is larger than that of the rectangular structure, as shown in Fig. 5.12. The dynamic response characteristics seem to depend on the configuration of the air chamber resulting from the combination of the parameters pj/pgB and pgh /pj rather than on either of t  them. For example, in the fixed case, the response is smallest for Case-2, for which the tension in the membrane is largest and the air chamber volume is smallest, therefore the air chamber is considered to be least flexible. These results imply that the predicted responses are dominated by the hydrodynamic properties of the corresponding rigid structure, rather than the deformation of the air chamber. This has been verified in the experiments, at least for the free cases, as discussed in Section 5.1.  5.3.3 Effects of Piston Properties Figures 5.17 - 5.23 present examples of the calculation results with the piston properties as parameters. For all of these figures, Mb/pB = 0.20 and Case-1 is taken as the air chamber 2  condition. In Figs. 5.17 and 5.19 the piston spring stiffness is taken as a parameter with a constant piston width A/B = 0.20, while in Figs 5.18 and 5.20 the piston width is a parameter with a constant piston spring stiffness K /pgB = 0.50. In Figs. 5.17 - 5.20 the piston mass to p  body mass ratio M /Mb =0.01. p  It can be seen that the heave response of the proposed structure can be improved by using the piston. In both the free and fixed conditions, the response become smaller for a wide range with a decrease of the piston spring stiffness or an increase of the piston width. However, for the range of large B/L, the response is still larger than that of the rectangular structure. Therefore a very soft or wide piston may be needed in cases where the fatigue of structural elements dominates the design, or where the motions of the body due to ordinary small waves are of concern for the operation of the floating facility. Furthermore, as the piston spring stiffness becomes smaller or the piston width becomes larger, in the free condition the  43 peak response becomes larger. In this sense, and because of its long natural period in roll, the proposed structure is not appropriate for long waves in the free condition. The effects of the piston mass are indicated in Figs. 5.21 and 5.22. In the free condition a large mass provides a negative effect for the response, while in the fixed condition the effect of the piston mass is small. Figure 5.23 shows the dynamic response operators of the piston and water surface elevation under the bottom of the body, and the dynamic air chamber pressure amplitude in the fixed condition. The motions of the piston and the water surface are relative to the body motion. In these figures the two cases of K /pgB = 0.25, A/B = 0.20 and K /pgB = 0.50, A/B = 0.40 p  p  are compared. While the case of K /pgB = 0.50, A/B = 0.40 can provide better performance p  as shown in Figs. 5.19 and 5.20, the piston motion and the air pressure variation are smaller in this case. Therefore it may be said that unless large membrane motion causes any problem such as fatigue of the membrane or structural failure at the connection points, a wide piston is more desirable than a soft one in terms of the stability of the entire structure.  5.4 Scale Effect  The scale effect of the proposed structure arises due to the atmospheric pressure remaining invariable. To discuss the effect of the atmospheric pressure, the equation which expresses the relationship between the variations of the volume and air pressure, Eq. 3.8, is rewritten with a slight modification: Pa  + Pi + Pc  (5.3)  =  Pa+Pi  where V j is the initial volume and AV is the variation of volume. The air chamber pressure C  C  of the experimental model in the rest condition pi = 1.23 kPa (model case M-2) and the atmospheric pressure p = 101 kPa at a temperature of 15°. According to Eq. 5.3, the a  dynamic air pressure amplitude p required to change the volume 10% is about 15 kPa, which c  44 is over 10 times larger than pj. On the other hand, using a length scale factor kf = l / l = 40 p  m  as an example, the value of p required to change the volume 10% is now about 21 kPa, c  which is about 0.4pj. Therefore, with an increase of the length scale of the structure, the compressibility of the air may contribute to a reduction of the heave response. Calculations which examine the above scale effect have been conducted for two cases, in which the length scale factors kf are 20 and 40 with respect to the present experimental model, and all the non-dimensional hydrostatic, hydrodynamic, and structural parameters are retained. Since the beam of the experimental model is 0.55 m, the beams of these prototype structures are 11.0 m and 22.0 m for kf = 20 and 40 respectively. Figures 5.24 and 5.25 show the predicted dynamic responses for the free and fixed cases respectively. In these calculations, the piston is not taken into account, but instead the whole body is considered as the air chamber, as illustrated in the inset of the figures. The response decreases with an increase of the scale factor except for B/L > 0.6 in the free condition. Furthermore, the non-dimendionalized mooring force for the scale factor kf = 40 (B = 22.0 m) in the fixed condition seems to be smaller than that of the structure with B = 10 m and proper piston (refer to Fig. 5.20). From these results, in the case of the prototype structure whose beam exceeds the order of 20 m, the proposed structure could provide a significant reduction in heave response even without a piston. On the other hand, the response of the scale model is close to that of the rigid body. These results indicate that the heave response of the proposed structure cannot be predicted appropriately by a scaled experimental model.  5.5 Summary and Supplemental Remarks  From the experimental investigation, it has been found that the proposed structure has the ability to make the natural period in roll longer. In the free or loosely moored condition, this could be helpful for the reduction of the roll response in the resonant range, while the proposed structure is not suitable for long waves. In addition, a reduction of mooring force  45 could be expected in the tightly moored condition. The principal parameters which determine the effectiveness of the air chamber may include the air chamber volume and pressure, the stiffness of the membrane, and the transmitted force in the transverse direction from the membrane to the body. However, in the present study, the role of each parameter has not been investigated quantitatively. The air chamber of the present experimental model was designed to be relatively small, so that the exciting force in roll, calculated for the corresponding rigid body, is not drastically larger than that of the rectangular structure. However, if the exciting force could be reduced significantly by the deformation of the air chamber, as inferred in Section 5.2.1, a larger and more flexible air chamber would be desirable, subject to the requirement that the hydrostatic restoring moment does not become too small. One of the major advantages of the proposed structure is that it could reduce the dynamic response in both the loosely and tightly moored conditions. This feature could be especially advantageous if the proposed structure is used as a working craft which is to serve under various mooring conditions. However, the reduction of the roll response in the free condition depends on the reduced restoring moment. Therefore, the capacity of the proposed structure for eccentric loading is much smaller than that of rectangular structures. In addition, the air chamber might provide negative effects if the mooring lines are arranged such that the sway amplitude and the coupling effect between sway and roll are large. Furthermore, the roll motion must be restrained to make the piston function. In this context, suitable mooring arrangements may be as shown in Fig. 5.26. Because of scale effect, the proposed structure becomes more advantageous as the structure becomes larger. On the other hand, an experimental investigation for heave response with a scale model cannot appropriately predict the response of the corresponding prototype structure. In addition, considerable effort is needed for many tests involving changes to the air chamber properties. Therefore a suitable numerical model may be needed for further  46 study, although the three-dimensional behavior of the air chamber should somehow be taken into account.  47  Chapter 6 Conclusions The objective of the present study was to assess whether a proposed floating structure, equipped with a flexible air chamber and piston, has the ability to cause a significant reduction of the dynamic response to waves. Following a preliminary examination to understand the basic characteristics of the proposed structure, the dynamic response characteristics of the proposed structure were investigated experimentally and analytically. The primary results obtained from this investigation are as follows: • A numerical model based on the lumped-mass method is a useful means of predicting the configuration of the air chamber in the rest condition. On the other hand, the air chamber configuration can also be approximated reasonably well as an arc of an ellipse. • The numerical model can be extended to predict the hydrostatic restoring moment against a forced rotation. The deformability of the air chamber drastically reduces the restoring moment of the proposed structure, and this in turn increases the roll natural period of the structure. Therefore, a reduction of the roll response in the free condition may be achieved for wave conditions for which resonance would otherwise be problematic. • The primary parameters which determine the roll performance of the proposed structure include the air pressure and' volume of the air chamber, and the stiffness of the air chamber membrane.  In addition, the transmission of the membrane tension in the transverse  direction to the body is expected to play a significant role. • The heave response of the proposed structure in the free condition is similar to that of a rigid structure with the same configuration. Only when the roll motion of the body is  48  restrained externally, can the piston and air chamber function properly for a reduction of the heave response. • With respect to the heave mode, a closed set of solutions for the dynamic response of the proposed structure has been obtained by solving simplified governing equations. The solutions predict the heave response reasonably well, provided that the roll motion is restrained. An analytical examination based on the solutions has indicated that a properly designed air chamber and piston should enhance the heave performance. • For the vertically moored condition, a reduction of the mooring force is expected, although this could not be verified experimentally in the present study. • The constancy of the atmospheric pressure between model and prototype gives rise to a scale effect on the dynamic behavior. With an increase of the scale, the compressibility of the air improves the structure's performance. For prototype structures whose beams exceed about 20 m, the heave response may be notably reduced, even without a piston. In conclusion, the present study has demonstrated that the proposed structure has, to some degree, the ability of reducing the dynamic response in waves in free or loosely moored conditions, and also reducing the mooring force in a tightly moored condition. This feature enables the proposed structure to have potential applications to a wide range of floating facilities. However, it has a limitation of giving rise to eccentric loading in the loosely moored condition, and thus inappropriate mooring arrangements could diminish the dynamic performance. Although a large and very flexible air chamber seems to be suitable with respect to the dynamic performance, further investigation would be necessary in order to identify the optimum properties of the air chamber.  49  References Aoki, S. (1990). "Prediction and Attenuation of Wave-Induced Ship Motion in a Harbor", Ph. D. thesis, Department of Civil Engineering, Osaka University, Osaka, Japan (in Japanese) Bhat, S. S. (1998). "Performance of Twin-pontoon Floating Breakwater", Ph. D. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, BC Committee for Development of Offshore Cities (1992). "Active Control Technology", Nikkan-Kogyo, Japan (in Japanese) Fugazza, M . and L. Natale (1988). "Energy Losses and Floating Breakwater Response", Journal of the Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 114, No. 2, pp. 191 - 205 Ijima, T., M . Tabuchi and Y. Yumura (1972). "Scattering of Surface Waves and the Motions of a Rectangular Body by Waves in Finite Water Depth" , Proceedings of JSCE, No. 202, Japan, pp. 33 - 48 (in Japanese) Ikeno, M . , N . Shimada and K. Iwata (1988). "A New Type of Breakwater Utilizing Air Compressibility", Proceedings of the 21th International Coastal Engineering Conference, ASCE, Costa del Sol-Malaga, Spain, pp. 2326 - 2339 Isaacson, M . and O. Nwogu (1987). "Directional Wave Effects on Long Structures", Journal of Offshore Mechanics and Arctic Engineering, ASME, Vol. 109, No. 2, pp.126 - 132 Isaacson, M . and R. Byres (1988). "Floating Breakwater Response to Wave Action", Proceedings of the 21th International Coastal Engineering Conference, ASCE, Costa del Sol-Malaga, Spain, pp. 2189 - 2200 Isaacson, M . , N . Whiteside, R. Gardiner and D. Hay (1994). "Modelling of a CircularSection Floating Breakwater", Proceedings of the International Symposium: Waves Physical and Numerical Modeling, IAHR, Vancouver, BC, Vol. 3, pp. 1344 - 1353 Iwata, K., M . Oki, S. Kitaura, T. Okuoka and M . Ikeno (1986). "Dynamic Characteristics of Floating Structure with Pressurized Air-Chamber", Proceedings of the 34th Japanese Conference on Coastal Engineering, JSCE, Japan, pp. 531 - 535 (in Japanese) Ocean Engineering Research Centre, University of Newfoundland (1994). "The Development of a Design Manual for Floating Breakwaters in the Atlantic Environment", Memorial University of Newfoundland, St. John's, NF Sarpkaya, T. and M . Isaacson (1981). "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York, NY  50 Sawaragi, T., M . Kubo and S. Aoki (1983). "Reduction of Motion of Ships and Berthing Energy by Improvement of Mooring System" Proceedings of the 30th Japanese Conference on Coastal Engineering, JSCE, Japan, pp. 460 - 464 (in Japanese) Stewart, W. P., W. A . Ewers and P. J. Denton (1981). "Non-linear Marine Barge Motion Response", in Offshore Structures - The Use of Physical Models in Their Design, Eds. G. S. T. Armer and F. K. Garas, the Construction Press, UK, pp. 251 - 261 Tanaka, M . , T. Ohyama, T. Kiyokawa, T. Uda and A . Omata (1992). "Characteristics of Wave Dissipation by Flexible Submerged Breakwater and Utility of the Device", Proceedings of the 23th International Coastal Engineering Conference, A S C E , Venice, Italy, pp. 1613 - 1624 Yamamoto, T., A . Yoshida and T. Ijima (1982). "Dynamics of Elastically Moored Floating Objects", in Dynamic Analysis of Offshore Structures, Vol. 1, Ed. C. L. Kirk, Gulf Publishing Company, Houston, T X , pp. 106 - 113 Wang, D. and M . Katory (1993). "On the Reduction of Peak Mooring Loads and Motion responses of Moored Structures Subject to Beam-Sea Conditions", in Integrity of Offshore Structures - 5, Eds. D. Faulkner, M . J. Cowling, A . Incecik and P. K. Das, E M A S Scientific Publications, U K , pp. 63 - 84  51  Appendix A Solutions of the Simplified Governing Equations for Heave Response The solutions of the simultaneous equations 3.13, 3.15, 3.17 and 3.19 are given as follows: Motion of the body UB 1 pgSri K (1 — <p) + 2icp<> 2  Motion of the piston 1 G  p  = -  —[ST1-Z ] b  Q Q1 + G  L  J  Water surface elevation under the body z„ =  1 S rii -bz +G—z 1 + G| h  B  P  n  Variation of the air chamber pressure Q Pc =  -PgRTT7T[STl- b] 1 + Lr z  where G = 1.41 P a + P i Pgh c  -co M +K  A G  2  Q=  D  pgA  R = 1-  U=  D  B1 +G  1A G Q B1+G  1 + G pgQB co co„  B  co„ =  2A/KM  co M„ 2  K = K +pgUB m  | K p  pgQAL  -(B-A)APgG' B l + G.  p  M = M + M„ K  Table 4.1 Summary of experimental model properties  (kPa) » (m) t» (m) M /pB  Pi h  h  Model Case M - l (Rubber sheet t= 1.6 mm) 1.12 0.114 0.090  Model Case M-2 (Rubber sheet t=0.4 mm) 1.23 0.124 0.080  0.20 0.019 1.00 -0.018 0.31 0.53 0.33  0.20 0.019 0.99 0.0 0.31 0.17 0.33  2  b  Pgh / Pi z /B ti  g  A/B K /pgB M /M p  p  b  Mass Mb Moment of inertia lb  Center of gravity of body  Mass M Stiffness K p  p  Table 4.2 Wave conditions Wave case No.  Wave period T (sec)  Wave length L (cm)  Wave height H (cm)  B/L  H/L  W-l W-2  0.70 0.80  76.5 99.7  2.5 3.0  0.719 0.552  0.033 0.030  W-3 W-4  0.90  125.4  3.3  0.439  0.026  152.7 180.8  3.7 4.6  0.360 0.304  0.025  W-5  1.00 1.10  W-6  1.20  208.9  5.4  0.263  W-7  1.25  222.9  5.9  0.247  0.026  W-8  1.32  242.3  6.4  0.227  0.026  W-9  1.40  264.2  6.6  0.208  W-10  1.50  291.3  6.8  0.189  0.025 0.024  W-ll  1.70  344.3  8.3  0.160  0.024  W-12  2.00  421.5  10.0  0.131  0.024  0.025 •  0.026  Note : Beam of the body B=0.55 m Water depth is 0.55 m for all cases  Table 4.3 Summary of test series Series  Model  S-ll S-12  M-l M-l  S-13 S-14 S-21  Restraining condition  Piston condition  Wave cases  Free Free  Working Restrained  W-l - W-12 W-l - W-12  M-l  Sway and roll restrained  Working  W-l - W-12  M-l  Sway and roll restrained  Restrained  W-l - W-12  M-2  Free  Working  W-l - W-12  Spring  Fig. 1.1 Schematic of the floating structure proposed in the present study  55  Surface Elevation Due to a Wave S7  ^^^^^+  Center of Buoyancy  Vertical Wave Pressure Distribution  (a) Deformation of air chamber due to wave  ~r  Transmitted force Spring _ ^  1.0  * e  ft  Wave exciting force fe  1.0  Natural period / wave period  (b) Model explaining mechanism of wave exciting force transmission to the body  Fig. 1.2  (c) Transmitted force as a function of the natural period to wave period ratio  Concept of the proposed structure to reduce response (Roll)  Surface Elevation Due to a Wave  r  Vertical Wave Pressure Distribution  Fig. 1.3  Concept of the proposed structure to reduce response (Heave)  Floating Body  Pressurized Air  Fig. 1.4  Schematic of the floating structure presented by Iwata etal. (1986) and Ikeno et al. (1988)  (a) Air-balance method Exhaust pipe  Valve controller  a  ^  Air tank  """*  I  Valve  (b) Tension leg method Spring  •  1  Superstructure  Floating body Wave absorber  * * * * ^ Controller  *~0  Tension leg  (c) Weight-shift method  Controller  Weight  •••••••• •••ODD  Floating body Superstructure  Fig. 1.5 Schemes of floating structures with active control techniques (Committee for Development of Offshore Cities in Japan, 1992)  57  Case-1  Case-2  Full Tension Condition  Uniform tension  Fig. 2.1 Sketch of general configuration of air chamber  Fig. 2.2 Definition sketch  f  T  : Angle between z axis and s axis  ( T =0 at z axis)  8 : Angle between s axis and membrane  (8 =0 at s axis)  (Positive in clockwise direction)  Fig. 2.3 Definition sketch for the lumped-mass method  59  1.8  T  M  1.6 ••  PB  E  )  2  0.20  f*(L) 1-4 U  h  P PgB  0.25 0.275  1.2 |  0.30  1.0  0.28  0.8  0.30 0.35  -a—i0.75  0.80  0.85  0.90  0.95  1.00  Pg",  Fig. 2.4 Comparison of membrane tensions obtained by the lumped-mass method and the ellipse approximation  Lumped-mass method Ellipse approximation 0.20  (a) Case-1 (  (b) Case-2 (  PB  2  0.20  ' PgB  ' PgB  = 0.30  Pgh,  = 0.25  Pgh L  = 0.735 )  =- 0.950 )  Fig. 2.5 Comparison of air chamber configurations obtained by the lumped-mass method and the ellipse approximation  60  10 000  Unit: mm  Mass of body Thickness of membrane Stiffness of membrane Air pressure  .Mil •  PB  = 0.24  2  PgB  40.0  : 24.0 t/m : 15.0 mm : 4.0 MPa : 0.027 MPa  PgB B  = 0.275  Pgh,  :0.96  = 0.0015  Fig. 2.6 Example of calculation results for configuration of air chamber of the proposed floating structure in heel condition  61 0.35 0.30 0.25 M gBsinr3 0.20 0.15 0.10 0.05 0  Rectangular b o d y  T  P PgB 0.225 0.250 0.275  b  0.75  0.80  0.85  0.90  0.95  1.00  Pgh, P (a)  24L • = 0.20  PB  2  0.30 T 0.25 •• M. M gBsinr> 0.20 f 0.15 0.10 0.05 0  -^- = 0.002  PgB  = 40.0  PgB  Rectangular b o d y  0.275 0.30 0.325  b  0.75  0.80  0.85  0.90  0.95  1.00  Pgh, (b)  ML  —^ = 0.25 PB 2  t - = 0.002 L B L  m  P PgB  0.25 M gBsinu 0.20 0.15 •• 0.10 •• 0.05 •• 0 b  (c)  F  -^- = 40.0 pgB  Rectangular b o d y  0.75  M - ^ - = 0.30 PB 2  0.80  0.85 Pgh,  t - ^ = 0.002 L m  0.90  0.95  0.325 0.350 0.375 1.00  F  - ^ = 40.0 pgB  Fig. 2.7 Dimensionless restoring moments as a function of external to internal pressure ratio  62 1.00 P  D  0.80  PgB  0.60  0.225 0.250 0.275  0.40 0.20 0 0.75  0.80  0.85  0.90  0.95  1.00  Pgh, P (a)  ^ = 0.20 PB 2  1.00  D  P  0.80  PgB  0.60  0.275 0.30 0.325  0.40 0.20 0 0.75  0.80  0.85  0.90  0.95  1.00  Pgh,  (b)  PB  2  :0.25  1.00 P  D  0.80  PgB  0.60 ••  0.325 0.350 0.375  0.40 •• 0.20 •• 0 0.75  0.80  0.85  0.90  0.95  1.00  Pgh,  (0  ^  PB  2  = 0.30  Fig. 2.8 Reduction ratios of restoring moments as a function of external to internal pressure ratio  63 1 ^  Mb  0.30  PB  :0.24  2  PgB  0.30  PgB  ^ = 0.002 B  10.0  M. M gBsint} 0.20  40.0  b  80.0 0.10 0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  MaL  B  (a) Relationship between restoring moment and unstretched length of membrane, with membrane modulus of elasticity as a parameter -Mm.  ML = 0.24 PB  0.30 +  PgB  2  0.30  PgB  ^ - = 0.002 B  -  10.0 40.0  M gBsintf ° b  2 0  f  80.0  0.10 0  • f t "  0.80  0.85  0.90  0.95  1.0  pgh, P  (b) Relationship between restoring moment and external to internal pressure ratio, with membrane modulus of elasticity as a parameter  -Mfc. pB*  0.30 + M gBsint) ° -  =  o.24  — = 0.30 pgB  2 0  b  pgB  B  20.0  0.004  40.0  0.002  80.0 0.001  0.10 0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  Mm.  B  (c) Relationship between restoring moment and unstretched length of membrane, with membrane stiffness and thichness as parameters Fig. 2.9 Dimensionless restoring moments with membrane properties as parameters  64  In rest condition  Density p Thickness t Damping coef. c„  r  r  In motion  | —  .1 T  |  c  •  S ,L  Air pressure p=p. +p Volume V ri  r  r  Fig. 3.1 Definition sketch for motion of the proposed structure  B z  Mass Stiffness T  M K  Mass MH  p n  4 ri i i i i i i i i  J=  Stiffness K,  Exciting force per unit area pgSTi^pgSHe" ' 0  Unknown variables: Motion of the body Motion of the piston Water surface elevation under the body Variation of the air chamber pressure  z z z p  b  p  s  c  Fig. 3.2 Definition sketch for the simplified analytical model for heave  66  Wave generator  Model  T  9m  Beach  7m  Holding tank  20 m  Fig. 4.1 Experimental set-up  550  Side View 600 (Width of flume) 580 Bearing  Lead weight Side panel  500  Rubber sheet  Front View  Fig. 4.2 Sketch of the experimental model  Unit: mm  67  Fig. 4.3 Relationship between displacement and force of the pistons for the experimental model  68  Fig. 5.1 Roll response amplitude operator of the proposed structure (Test series S-l 1)  Fig. 5.2 Roll response amplitude operator of the proposed structure (Test series S-21)  Fig. 5.3 Heave response amplitude operator of the proposed structure (Test series S - l l and S-13)  Fig. 5.4 Heave response amplitude operator of the proposed structure (Test series S-12 and S-14)  70  Fig. 5.5 Heave response amplitude operator of the proposed structure (Test series S-21)  Fig. 5.6 Sway response amplitude operator of the proposed structure (Test series S-ll)  B/L > 0.3  B/L «  0.3  Fig. 5.7 Sketch showing tracks of the center of gravity of body under waves  71  t/T=0.00  Piston  t/T=0.25  t/T=0.50  t/T=0.75  Fig. 5.8 Behavior of experimental model under wave at particular time instants (T=1.10 sec, free condition)  t/T=0.00  Piston  t/T=0.25  t/T=0.50  t/T=0.75  Fig. 5.9 Behavior of experimental model under wave at particular time instants (T=1.50 sec, free condition)  73  Restoring moment  Side panel /  M  e  m  b  r  a  n  Transmitted force from membrane to side panel  e  Side View  Side panel  f  Tension T  LU*  *»si  Transmitted force  Wave force  Front View  Fig. 5.10 Sketch showing transmitted force from membrane to side panels  74  t/T=0.00  Piston  t/T=0.25  t/T=0.50  t/T=0.75  Fig. 5.11 Behavior of experimental model under wave at particular time instants (T=1.50 sec, vertically moored condition)  75  PB ~ 2  Pgh.i  0.20  R Case-1 Case-2 Case-3  0.988 0.225 0.930 0.225 0.930 0.250 Rectangular  (a)  ipgHB  Pi pgB  0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  B L (b) 1.00 0.80  ipgHB  0.60 0.40 0.20 0 0.1  0.2  0.3  0.4  B L (c) 0.08 0.06  7PgHB 1.  U T 3 2  0.04  0.02 +  0.1  0.2  0.3  0.4  B L Fig. 5.12 Exciting forces of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll  76 p£ .i  Pi  Pi  PgB  h  = 0.20  ^  pB  (a)  2  0.80  M  Case-1 Case-2 Case-3  j  0.988 0.225 0.930 0.225 0.930 0.250 Rectan gular  0.60 +  Y  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  B L  (b) 2.5  j  2.0 ••  .Ms  1.5 •• 1.0 •• 0.5 -• o  *•  0.1  0.2  0.3  0.4  B L  (C)  0.08  M  P  B  0.061  f l  V  j  c i 0.04-  0.020 0.1  0.2  0.3  0.4  B L  (d) 0.20  M P  i2_  BV  T  0.15 0.10 +  c  0.05 + 0 -0.05 0.1  0.2  0.3  0.4  B L Fig. 5.13 Added masses of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll (d) Sway - Roll  Pg ri h  i2  (a)  * pCOV c  ci  Pi  PgB  -0.20 Case-1 Case-2 Case-3  0.70 j 0.600.50-0.40 -0.300.200.10-  -  0.988 0.225 0.930 0.225 0.930 0.250 Rectan gular  o •*-  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  0.5  0.6  0.7  0.8  0.9  B  (b)  L  0.1  0.2  0.3  0.4 B  (c)  L 0.025 j 0.020 ••  pcoB V  0.015 ••  z  c  0.010 •• 0.005 -• 0 0.1  0.2  0.3  0.4 B  (d)  L  0.1  0.2  0.3  0.4 B L  Fig. 5.14 Damping coefficients of rigid structures with the same configurations as the proposed structures (a) Sway (b) Heave (c) Roll (d) Sway - Roll  78  (b)  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  B L Fig. 5.15 Response amplitude operators in heave with air chamber properties as parameters (free condition) (a) = o 20, (b) = n 30 pB ' pB ' 2  2  79  Fig. 5.16 Non-dimensionalized mooring forces with air chamber properties as parameters (fixed condition) (a)  = 0 20, (b) = 0 30 pB ' pB ' 2  2  80  0.988 K.  A  PgB  B  0.50 0.20 0.25 0.20 No piston Rectangular body  'H  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  B L Fig. 5.17 Response amplitude operators in heave with piston spring stiffness as a parameter ( free condition )  1.4  Free condition  pB  2  Pgh  1.2  =0.20 ti  0.988  PgB  = 0.225  1.0  H  K  A  PgB  B  0.50 0.20 0.50 0.40 No piston Rectangular body  0.8 0.6 0.4 •0.2 •0  -+-  -+-  0.1  0.2  0.3  •+0.4  -+-  -+-  0.5  0.6  •+-  0.7  0.8  0.9  B L Fig. 5.18 Response amplitude operators in heave with piston width as a parameter ( free condition )  81  —i  0.1  1  1  1  1  1  1  1  1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  B L Fig. 5.19  Non-dimensionalized mooring forces with piston spring stiffness as a parameter ( fixed condition )  Fig. 5.20  Non-dimensionalized mooring forces with piston width as a parameter ( fixed condition )  82  Fig. 5.21 Response amplitude operators in heave with piston mass as a parameter ( free condition )  Fig. 5.22 Non-dimensionalized mooring forces with piston mass as a parameter ( fixed condition )  83  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  B L (a) Response amplitude operators for the piston motion  (c) Non-dimensionalized dynamic air chamber pressure amplitudes Fig. 5.23 Dynamic responses of piston and air chamber ( fixed condition )  84 Free condition Experimental model k =20 f  k =40 f  Rigid body with same configuration as experimental model Rectangular body  H  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.!  B L Fig. 5.24 Response amplitude operators in heave for the experimental model with length scale as a parameter ( free condition )  B  h/B=0.36 Air chamber  Fixed condition 0.9  -r  Experimental model  0.8 ••  pgBH  k =20 f  0.7 ••  k =40  0.6 ••  Rigid body with same configuration as experimental model Rectangular body  0  5  f  •  0.4 •• 0.3 -• 0.2 •• 0.1 •• -+-  •+-  0 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  B L Fig. 5.25 Non-dimensionalized mooring forces for experimental model with length scale as a parameter ( fixed condition )  85  uu 3  u  03  u E C  o  E  #  C  o o  T3 U  03  ^  C  o +z  o  03  ° \s too  c/3  .5 5  "O  .1 §  >  o o E  2 o E .2  11 O  X  W  ID  o -=  p Vl  "O  g  rt  •a xi  M  C  o o E  S >.£? E  So  2^  — a)  E.I  ^ -a -o o x  u  u,  a)  p  3 T3  c  0)  ID  O  £  S3  00 03 toO  C  o o c o oo ID  £0§ (D  C  O  Vi  O &  E  ID  "D  3  a 03  ii  too  o  a-  c  '•5  e  o  •ou o  <D  Pi 1  i—H C3  (D  W  -a <D  oj  ^- "a, oc Ea  X  2 v,  o  c  <D  c  w  (D O  60  ID  03  o  C •  s  SB  (D  pe rfoi  (D  g f t M  ii  •a  (D  c  E  >  o  im rov em roll  o  c .ti *;  <D  3  c o  duct  3  t+Z  Ui  ID  w  O  ID  u too  pe  O  O  s  03  C  O <D  X> O  t3 a> u, tD  'c/3  e o o  a  o  B  a 03  t>0  o o J  CN  vi bb  Hi"  

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