A TIME DOMAIN MODEL FOR WAVE INDUCED MOTIONS ...... Figure C.4 Surge Added Mass and Damping Convergence in Free Surface Resolution.
A TIME DOMAIN MODEL FOR WAVE INDUCED MOTIONS COUPLED TO ENERGY EXTRACTION by James G. Bretl
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Naval Architecture and Marine Engineering) in The University of Michigan 2009
Doctoral Committee: Professor Robert F. Beck, Co-Chair Professor Guy A. Meadows, Co-Chair Associate Professor Bogdan Epureanu Assistant Professor Ryan M. Eustice
“Whatever have been thy failures hitherto, be not afflicted my child, for who shall assign to thee what thou has left undone” Henry David Thoreau “On Walden Pond”
© Jim Bretl All rights reserved 2009
Acknowledgements
I would like to thank the Office of Naval Research and the American Society for Engineering Education for providing support for the majority of this work through the National Defense Science and Engineering Graduate Fellowship. I owe special thanks to my advisors, Professors Guy Meadows and Bob Beck for their support of this work. They provided contrasting but equally valuable support to my research effort. They have my admiration for the daily efforts put forth for the education of students. They are asked to guide a diverse population of people through the various challenges presented at this great university. I am grateful for their efforts in guiding this aging student through this work. I am also grateful to Professors Eustice and Epureanu for their insightful participation and support. I appreciate the friendship, discussions and support given to me by my colleagues, Piotr Bandyk and Christopher Hart. The experimental work could not have been accomplished without the help of many people at the Michigan Marine Hydrodynamics Lab. In particular, I thank and am glad to know Ed Celkis, Joe Wild, Nick Wild, Dave Parsons, and Hans Van Sumeren. Finally, I could not have done this without the love and support of my family. Many times, it was the thought of them in my mind‟s eye that helped me forward. To my wife Liz and children, Anna, Isabelle and Max, Thank you.
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Table of Contents Acknowledgements ............................................................................................................. ii List of Figures ......................................................................................................................v List of Symbols ....................................................................................................................x List of Tables ................................................................................................................... xiii List of Appendices ........................................................................................................... xiv Chapter 1 Introduction .........................................................................................................1 1.1 Overview .......................................................................................................1 1.2 Motivation and Background ..........................................................................3 1.3 The Hydrodynamic Problem .......................................................................10 Chapter 2 A Simple Model to Assess Conversion Capacity..............................................15 2.1 Description of the Problem .........................................................................15 2.2 A Simplified Two-dimensional Model .......................................................17 2.3 Nonlinear Dynamics....................................................................................23 2.4 Conclusion ..................................................................................................26 Chapter 3 Rigid Body Motions ..........................................................................................27 3.1 Equations of Motion....................................................................................27 3.2 Euler vs. Quaternion Orientation Representation .......................................33 3.3 Euler Angles ................................................................................................34 3.4 Quaternion Algebra .....................................................................................36
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3.5 Rigid Body Motion Validation ...................................................................40 Chapter 4 Hydrodynamic Model – A Desingularized BIM with Weak Scatterer Free Surface Boundary Condition..............................................................................................42 4.1 Overview .....................................................................................................42 4.2 Hydrodynamic Formulation ........................................................................42 4.3 The Boundary Integtral ...............................................................................50 4.4 Hydrodynamic Forcing and Rigid Body Motion ........................................51 4.5 Numerical Method ......................................................................................53 4.6 Initial Numerical Validations ......................................................................55 Chapter 5 Experiment ........................................................................................................60 5.1 Introduction .................................................................................................60 5.2 Experimental Setup .....................................................................................61 5.3 Free Surface Characterization .....................................................................63 5.4 Buoy Motions Experimental Setup .............................................................67 5.5 Energy Extraction........................................................................................70 5.6 Error Reporting ...........................................................................................72 5.7 Experimental Results – No Pendulum ........................................................72 5.8 Pendulum Runs ...........................................................................................80 5.9 Experimental vs. Numerical Results ...........................................................86 Chapter 6 Summary, Conclusions and Future Work .........................................................91 Appendices .........................................................................................................................96 Bibliography ....................................................................................................................133
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List of Figures Figure 1.1 Various Data Buoys (NDBC 3 meter discus, Aanderaa DB4280, ALWAS). ... 2 Figure 1.2 Normalized Solar and Wind Energies vs. Latitude. .......................................... 4 Figure 1.3 Normalized Solar and Wind Energies vs. Month. ............................................. 4 Figure 1.4 Schematic of Energy Conversion Concept. ....................................................... 6 Figure 2.1 Coordinate Definitions for the: a) Buoy Hull and b) Pendulum. .................. 18 Figure 2.2 Power Conversion (Mass of pendulum=1kg, r=0.25m). ................................ 21 Figure 2.3 Bifurcation Diagram Describing Pendulum Rotation vs. Wave Frequency. .. 22 Figure 2.4 Oscillation and Phase Plane map displaying multiple solutions for different initial conditions.................................................................................................... 23 Figure 2.5 Parametric Pendulum...................................................................................... 24 Figure 3.1- Buoy Hull Coordinate System Definitions. ................................................... 28 Figure 3.2 - Pendulum Coordinate System Definition...................................................... 28 Figure 3.3 - Seven DOF Motion Validation - Body Frame Moment ( F5=cos(5.2*t) ). . 41 Figure 4.1 Boundary Value Problem Definition. .............................................................. 43 Figure 4.2 Vertical Diffraction Force on a Unit Sphere. .................................................. 56 Figure 4.3 Surge Diffraction Force on a Unit Sphere. ..................................................... 57 Figure 4.4 Heave Added Mass and Damping for a Unit Sphere. ..................................... 58 Figure 4.5 Surge Added Mass and Damping for a Unit Sphere. ...................................... 59 Figure 5.1 IR Vision Based Tracking Experimental Setup............................................... 62 Figure 5.2 Low Density Floating IR Reflective Markers. ................................................ 64
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Figure 5.3 Screen Shot of Surface Characterization. ....................................................... 65 Figure 5.4 Theoretical Second Order Drift vs. Measured Drift Speed. ............................ 66 Figure 5.5 Vision System vs. Acoustic Wave Probe. ...................................................... 67 Figure 5.6 Schematic for buoy physical dimension references. ....................................... 69 Figure 5.7 Nondimensional Drift Speed versus Encounter Frequency............................. 74 Figure 5.8 Rigid Body Motion – Wave Frequency 0.65 Hz. ............................................ 76 Figure 5.9 Rigid Body Motion - Wave Forcing Frequency 0.75 Hz. ............................... 77 Figure 5.10 Rigid Body Motion - Wave Forcing Frequency 0.80 Hz. ............................. 78 Figure 5.11 Rigid Body Motion - Wave Forcing Frequency 0.85 Hz. ............................. 78 Figure 5.12 Rigid Body Motion - Wave Forcing Frequency 0.90 Hz. ............................. 79 Figure 5.13 Rigid Body Motion – Wave Frequency 0.95 Hz. .......................................... 80 Figure 5.14 Rigid Body Motion – Wave Frequency 1.0 Hz. ............................................ 80 Figure 5.15 Run 28 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ................................ 82 Figure 5.16 Run 13 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ............................... 83 Figure 5.17 Run 22 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ............................... 84 Figure 5.18 Run 23 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ............................... 84 Figure 5.19 Experimental Power versus Wave Frequency and Pitch/Roll Response (R2=.54)................................................................................................................. 85 Figure 5.20 Numerical Calculation for Rigid Body motion – K=2.8. .............................. 87 Figure 5.21 Numerical Calculation for Rigid Body motion – K=3.6. .............................. 87 Figure 5.22 Drift Speed vs. Wave Frequency – Simulation and Experimental. ............... 88 Figure 5.23 Drift Speed vs. Encounter Frequency – Simulation and Experimental. ........ 89 Figure 5.24 Power versus Frequency -- Numerical and Physical Results. ....................... 90 Figure A.1 Vision System Calibration Screen Shot.......................................................... 96
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Figure A.2 Regression For Clutch Torque vs. DC Voltage ............................................ 97 Figure A.3 Instruments and Associated Errors. ................................................................ 97 Figure B.1 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ........................................... 100 Figure B.2 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ........................................... 100 Figure B.3 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ........................................... 101 Figure B.4 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ............................................. 101 Figure B.5 1.0 Kg Pendulum – Wave Frequency 0.85 Hz. ........................................... 102 Figure B.6 1.0 Kg Pendulum – Wave Frequency 0.85 Hz. ........................................... 102 Figure B.7 1.0 Kg Pendulum – Wave Frequency 0.8 Hz. ............................................. 103 Figure B.8 1.0 Kg Pendulum – Wave Frequency 0.75 Hz. ........................................... 103 Figure B.9 1.0 Kg Pendulum – Wave Frequency 0.7 Hz. ............................................. 104 Figure B.10 1.0 Kg Pendulum – Wave Frequency 0.65 Hz. ......................................... 104 Figure B.11 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ......................................... 105 Figure B.12 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ......................................... 105 Figure B.13 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ........................................... 106 Figure B.14 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ........................................... 106 Figure B.15 1.0 Kg Pendulum – Wave Frequency 1.0 Hz. ........................................... 107 Figure B.161.0 Kg Pendulum – Wave Frequency 1.0 Hz. ............................................ 107 Figure B.17 1.0 Kg Pendulum – Wave Frequency 0.95 Hz. ......................................... 108 Figure B.18 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. ........................................... 108 Figure B.19 1.0 Kg Pendulum – Wave Frequency 0.85 Hz. ......................................... 109 Figure B.20 1.0 Kg Pendulum – Wave Frequency 0.8 Hz. .......................................... 109 Figure B.21 1.0 Kg Pendulum – Wave Frequency 0.8 Hz. ........................................... 110 Figure B.22 1.0 Kg Pendulum – Wave Frequency 0.7 Hz. ........................................... 110
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Figure B.23 1.0 Kg Pendulum – Wave Frequency 1.0 Hz. .......................................... 111 Figure B.24 1.0 Kg Pendulum – Wave Frequency 1.0 Hz. ............................................ 111 Figure B.25 0.25 Kg Pendulum – Wave Frequency 0.75 Hz. ....................................... 113 Figure B.26 0.25 Kg Pendulum – Wave Frequency 0.80 Hz. ....................................... 113 Figure B.27 0.25 Kg Pendulum – Wave Frequency 0.85 Hz. ...................................... 114 Figure B.28 0.25 Kg Pendulum – Wave Frequency 0.9 Hz. ......................................... 114 Figure B.29 0.25 Kg Pendulum – Wave Frequency 0.95 Hz. ....................................... 115 Figure B.30 0.25 Kg Pendulum – Wave Frequency 1.0 Hz. ......................................... 115 Figure B.31 Rigid Body Motion – Wave Frequency 1.0 Hz. ......................................... 116 Figure B.32 Rigid Body Motion – Wave Frequency 0.95 Hz. ....................................... 117 Figure B.33 Rigid Body Motion – Wave Frequency 0.95 Hz. ....................................... 117 Figure B.34 Rigid Body Motion – Wave Frequency 0.9 Hz. ......................................... 118 Figure B.35 Rigid Body Motion – Wave Frequency 0.85 Hz. ....................................... 118 Figure B.36 Rigid Body Motion – Wave Frequency 0.85 Hz. ....................................... 119 Figure B.37 Rigid Body Motion – Wave Frequency 0.8 Hz. ......................................... 119 Figure B.38 Rigid Body Motion – Wave Frequency 0.75 Hz. ....................................... 120 Figure B.39 Rigid Body Motion – Wave Frequency 0.69 Hz. ....................................... 120 Figure B.40 Rigid Body Motion – Wave Frequency 0.65 Hz. ....................................... 121 Figure C.1 Heave Added Mass and Damping Convergence in Time. ............................ 122 Figure C.2 Heave Added Mass and Damping Convergence in Body Resolution. ......... 123 Figure C.3 Surge Added Mass and Damping Convergence in Body Resolution. .......... 123 Figure C.4 Surge Added Mass and Damping Convergence in Free Surface Resolution. ............................................................................................................................. 124 Figure C.5 Rigid Body motion – K (Wave No.) =2.0. .................................................. 125
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Figure C.6 Rigid Body motion – K=2.4. ....................................................................... 126 Figure C.7 Rigid Body motion – K=2.8. ....................................................................... 126 Figure C.8 Rigid Body motion – K=3.2. ....................................................................... 127 Figure C.9 Rigid Body motion – K=3.6. ....................................................................... 127 Figure C.10 Rigid Body motion – K=4.0. ..................................................................... 128 Figure C.11 Numerical Model Seven D.O.F. – 0.7 Hz.................................................. 129 Figure C.12 Numerical Model Seven D.O.F. – 0.75 Hz................................................ 130 Figure C.13 Numerical Model Seven D.O.F. – 0.8 Hz.................................................. 130 Figure C.14 Numerical Model Seven D.O.F. – 0.85 Hz................................................ 131 Figure C.15 Numerical Model Seven D.O.F. – 0.9 Hz.................................................. 131 Figure C.16 Numerical Model Seven D.O.F. – 0.95 Hz................................................ 132 Figure C.17 Numerical Model Seven D.O.F. – 1.0 Hz.................................................. 132 .
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List of Symbols The following table defines the symbols used within this document unless explicitly noted within the body of the text. Bold face is used to represent vectors and matrices. The first two time derivatives of an arbitrary variable x may be represented as x, x.
Amp Wave amplitude. a Wave amplitude. B Body coordinate frame. c Damping coefficient for pendulum equations of motions. cg Wave group velocity. eˆ px
Unit vector in the direction parallel to the pendulum arm.
eˆ py
Unit vector in the direction normal to the pendulum arm.
F
Force vector on the body F1 , F2 , Tp ,
Fp1
Force vector on the pendulum.
Fp1
Force vector on the pendulum.
Fp 2
Moment vector on the pendulum.
i I k mp n O p
F1 X , Y , Z
T
T
-1. Equilibrium inertial coordinate frame. Wave number. Mass of the pendulum. Normal vector pointing into the fluid. Earth fixed inertial coordinate frame. Rotational velocity in 4 direction.
q Rotational velocity in 5 direction.
x
F2 K , M , N . T
r Rotational velocity in 6 direction. b
rm Position of pendulum mass in the body frame.
p
rm Position of pendulum mass in the pendulum frame.
Ri j
3x3 Transformation matrix from frame i to frame j.
Ri a Transformation matrix about axis i through an angle a. s Rotational velocity in 7 direction. Sb
Surface representing the bottom in the boundary integral problem.
Sf
Surface representing the incident wave field in the boundary integral problem.
Surface representing the instantaneous hull wetted surface in the boundary integral problem. St Surface representing the truncation surface at infinity in the boundary integral problem. T Draft. u Velocity in 1 direction. U t Body surface velocity. Sh
v Velocity in 2 direction. ν Velocity of the body expressed in the body frame, x ν1 , ν 2 , s . T
ν1
Translational velocity u, v, w .
ν2
Rotational velocity p, q, r .
T
T
V Volume enclosing the fluid domain. w Velocity in 3 direction. x Earth frame coordinate. x Coordinates expressed in the body frame x1 , x2 , x3 , x4 , x5 , x6 , x7 . T
X Displacement in inertial frame X 1 , X 2 , with X 1 x, y, z T
T
X 2 , , . T
X1
Translational displacement in inertial frame , x, y, z .
X2
Orientation displacement , Euler Angles , , , Quaternions q0 , q1 , q2 , q3 .
T
T
y Earth frame coordinate. z Earth frame coordinate. Z CG Z coordinate for the buoy center of gravity relative to the baseline. ZP
Z coordinate for the pendulum relative to the buoy baseline.
ZT
Z coordinate for the top IR marker for vision experiments.
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T
Greek Symbols
0 1 5 p
A small paramter. Euler roll angle. Incident wave potential. Disturbance wave potential due to the presence and motion of the body. Phase lag of pitch relative to the free surface slope. Orientation of the pendulum plane relative to body coordinate frame.
DIF
Diffraction component of the disturbance potential. x , t Scalar velocity potential defined in the fluid domain.
Memory part of the radiation potential. Damping ratio for pendulum equations of motions. η Coordinates expressed in the body frame 1 , 2 ,3 , 4 ,5 ,6 ,7 . T
1 1 2 3 4 5 6 7
Displacement in the body frame x direction. * Free surface elevation of the disturbance potential. * Displacement in the body frame y direction. Displacement in the body frame z direction. Rotational displacement(linear) about body frame x axis. Rotational displacement(linear) about body frame y axis. Rotational displacement(linear) about body frame z axis. Rotational displacement of the pendulum relative the body frame.
Fluid density. Nondimensional time. Euler pitch angle or pendulum displacement. * Radial frequency. Body referenced rotational velocity vector. Rescaled frequency for pendulum equation of motion. Euler yaw angle or the impulsive potential. * The impulsive component of the radiation potential. * * Indicates a symbol which may represent more than one quantity.
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List of Tables Table 5.1 Buoy Particulars (3 Configurations). ............................................................... 68 Table 5.2 Test Matrix and Summary Results: Case III. .................................................. 73 Table A.1 Comparison of Second Order Drift to Measured Drift. .................................. 98 Table B.1 Test Matrix and Summary Results - Case I 1.0 Kg Pendulum. ...................... 99 Table B.2 Test Matrix – Case II 0.25 Kg Pendulum. .................................................... 112 Table B.3 Test Matrix - Case III No Pendulum. ............................................................ 116
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List of Appendices
Appendix A Experimental Details .....................................................................................96 Appendix B Test Matrices and Graphical Results .............................................................99 Appendix C Numerical Results .......................................................................................122
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Chapter 1 Introduction
1.1 Overview The development and deployment of autonomous data buoys for monitoring the marine environment has been an active area of research for a number of years. A significant limiting problem facing the development of such systems is that, in many locations, solar power alone is not always sufficient to satisfy power requirements. Figure 1.1 shows three typical data buoy configurations used at present. Recently there has been significant interest in the development of buoy scale sensing platforms that have the capability to stay on station for extended periods of time or even indefinitely. One of the chief obstacles to development of such systems is the power requirement for the station keeping task. Environmental energies are available for harnessing in the form of solar radiation, wind kinetic energy, wave energies and others. An order of magnitude analysis quickly demonstrates that current solar panel technologies are unlikely to have the capacity to supply all of the energy needed for even moderate station keeping demands. Therefore an effort has begun that is focused on harnessing wave energy to augment environmental scavenging abilities in order to maximize the spatial and temporal operability range of such systems.
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Figure 1.1 Various Data Buoys (NDBC 3 meter discus, Aanderaa DB4280, ALWAS). The focus of this research is numerical modeling and experimental validation of the coupled dynamics of water wave induced motions and energy extraction of a free floating body. Of interest is extraction of wave energy as a means to augment the powering requirements of a data buoy. A physics based simulation is developed for the purpose of exploring the design space for such a device. In contrast to conventional seakeeping analysis, the parameter range of interest does not constrain wave amplitude or wavelength by the characteristic body length scale. The specific application of a planar pendulum contained within a floating hull and coupled to an energy extraction device is suggested for the purpose of validation. A simplified model of the application is developed to gage reasonability of the application. The fully nonlinear coupled equations of motion for the seven degree of freedom system are explicitly written and implemented into the panel code. A potential flow panel code is developed that is designed to capture the nonlinearities associated with the problem of a body that has characteristic length that is an order of magnitude smaller than the amplitude of ambient waves. The Weak Scatterer nonlinear free surface boundary conditions are instituted in order to satisfy the demands of this paramter range. The hydrodynamic problem is solved using a desingularized
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boundary integral method. The body exact instantaneous wetted surface is determined by consideration of the time dependent position and orientation of the body relative to an ambient wave field. Experiments are performed that measure the response of the system to regular wave excitation. An infra-red vision system is used to characterize the free surface flows and to track rigid body motions in waves. The free surface characterization experiments agree well with Stokes second order drift. Experimental results for rigid body motions demonstrate large amplitude pitch as well as a nonlinear drift. Chaotic behavior of the pendulum is indicated by experimental results for the buoy with energy extraction. The numerical computations agree well with the nonlinear drift phenomenon and demonstrate similar chaotic behavior
1.2 Motivation and Background The idea of a self-powered buoy is by no means new. The integration of solar energy into the powering systems of buoy applications is a natural fit and has attracted a great deal of research and development. The US Coast Guard and NOAA both commonly equip their buoys with solar panels. It can be considered a mature research field. A significant limitation of such systems is illuminated by Ageev[1]. He presents a summary of the dependencies between wind and solar energy based on latitude and season. The data is extracted from wind and solar energies which he gives in kWh/m2. The wind energy is calculated based on the kinetic energy associated with the average wind velocity. In Figure 1.2 and Figure 1.3, energy is averaged over twelve months for each latitude. The normalization is based on the maximum of each measure.
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Figure 1.2 Normalized Solar and Wind Energies vs. Latitude.
Figure 1.3 Normalized Solar and Wind Energies vs. Month.
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Interestingly, these maximums are of similar magnitude, about 10 kWh/m2 per day for wind and 6 kWh/m2 per day for solar. Similarly, in Figure 1.3, the average of the energy for each month is taken across the latitudes from 0 to 60 degrees. Wave energy is primarily a manifestation of wind energy. Therefore, the complementary nature of solar and wind energies motivates exploration of wave energy as a natural source in order to extend operability conditions. This work was embarked upon with the intent to quantify the capacity for converting the wave induced motions of a buoy into energy for augmenting the power requirements of an autonomous data buoy. The broad focus was to consider the motions of the buoy hull as inputs to a mechanical system which then converts these motions into some form of energy. The unique parameters present in the buoy scale applications present challenges that are not present for conventional seakeeping analysis in naval architecture. The most distinguishing facet of the problem is that the floating body is generally small relative to the amplitude of a significant range of the wavelengths present. Yet the dynamics of interest may be coupled to forcing from these wave components as well as the shorter wavelengths riding over this surface. The resulting research has evolved toward the validation of a computational tool that captures the physics of this parameter range and could be used for similar endeavors. A specific application is explored in the research but the resulting simulation may be applied to different mechanical energy extraction applications with little added effort. The simple mechanical system of a planar pendulum, mounted within the body and whose relative motion to the buoy was used to extract energy was suggested as the method to explore. From the outset it will be suggested that a device for energy conversion is completely contained by the buoy hull.
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Figure 1.4 Schematic of Energy Conversion Concept. A first simplified model, described in the subsequent section, is explored in order to evaluate the pragmatism of applying this concept to the intended applications. This model decoupled the buoy and pendulum motions and then imposed motions of the buoy hull which could reasonably be expected in the marine environment. Evaluation of the performance of the system was thus simplified to modeling the dynamics of a pendulum undergoing prescribed base motions. As a result of decoupling motions of the buoy and the pendulum, the design space for the buoy hull parameters and the pendulum parameters were constrained. For instance, the restoring coefficients of the hull constrain the maximum mass and moment arm for the pendulum. That is, decoupling requires that the calm water equilibrium orientation of the hull would change only slightly by addition of the pendulum. This initial modeling was relatively straightforward and showed that there was energy available on a scale that would make further investigation reasonable. The history of our interest in harnessing wave energy dates back to Archimedes, but dwindling oil reserves, global warming and political unrest has refocused our interest in this area of research. A review of attempts to harness wave energy on differing scales and the lack of successes therein informs us of the difficult nature of the enterprise. The
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designers of any of these projects would have benefitted greatly from a tool set that allowed an efficient and reasonable exploration of the design space. A good deal of research exists regarding the challenges and solutions to design of systems for wave energy conversion. Research in this area dates back to the 1800‟s[11]. It remains an area that attracts significant attention because of the sheer magnitude of the energy present in waves. The 1980‟s through the present saw a great deal of work on the analytical solutions to wave energy absorption and optimal control for extracting device motions. References [11; 12] contain a reasonable statement of the work produced during this period in terms of analytical modeling of linear wave energy research. It is not the intent of this work to extend the bounds of research focused on converting energy from waves on the same order as that contained within them. As will be seen, efficiency on this order is not necessary for the application being suggested. The continued research and development in the area of design for wave energy harvesting holds great promise to continue increasing the efficiency and robustness of such systems [12; 37; 8]. Schlick[39] showed that one could use gyros to effect the pitch and roll response of a ship to the wave environment. The goal was to incorporate the gyros in such a manner as to provide roll damping. This concept has also been used to extract energy from the pitch and roll motions of a buoy by coupling gyros to an electric generator[36]. The fate of these devices is unpublished. The relatively small amplitudes of motion provided by such a device would be problematic in efficiently converting them into electrical energy. Also, the power consumption of the gyros creates an initial deficit in the energy balance sheet. Certain aspects of the problem proposed herein parallel those encountered in the dynamics of a floating crane, a problem that has received considerable attention over the years [4; 9]. The focus of these works was on minimization of the amplitude of the crane load motions and control. The early approaches assumed a prescribed motion for the hull. The resulting model was then reduced to parametric excitation of a pendulum. These models assumed that interaction between the load and the hull motions were negligible. In so doing, their applicability to open sea operation was limited. The ratio of the load inertia to the hull inertias was also limited. Ellermann[9] followed these analyses with a
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more rigorous treatment of the nonlinear dynamics of the multi-body system. The focus of this work was determination and characterization of bifurcation points. Kinematics of the barge and load were constrained to plane motions. The angle that the load makes with vertical was constrained to be small. The method of multiple scales and a Taylor series expansion of the load angle were thus justified in the presence of small amplitude waves. No such approximation is permitted for the pendulum being modeled. It can be expected that certain conditions will result in small oscillatory motions. However, our solution must include constant rotation of the pendulum. Xu[48; 47] has explored the behavior of a parametrically excited pendulum whose axis of rotation is parallel to the horizontal plane. The analysis is limited to pendulum behavior in response to harmonic excitation in the vertical direction. Asymptotic methods are used to characterize the undamped, linearized, and nonlinear oscillation regimes of such a system. Also, a perturbation analysis is applied to the damped nonlinear oscillations and analytically determined resonant boundaries are shown to agree well with numerical results. Finally, a perturbation method is applied to the fully nonlinear problem and a claim is made to a solution for full rotation of such a pendulum for the first time. At first glance this work appears to have left little to explore within the model proposed herein. It will be shown, however, that the character of the governing equations for the two respective problems precludes direct application of results from one to the other. The small parameter necessary for Xu‟s analysis is dependent on the simplified two degree of freedom nature of that model. The field of nonlinear dynamics generally seeks a small parameter, within the dynamic system of equations, that can be perturbed in order to explore nonlinear behaviors. On an intuitive level, the possibility of exploiting such techniques for application in real waves would have required unnecessary constraints on the parameter space for design. For instance, it might be necessary to bound certain motion amplitudes. Indeed, experimental results presented herein indicate nonlinear interactions that significantly affect encounter frequency. So much so that resonant behavior is generated by incident waves having a nonresonant frequency.
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Upon finding the above results, a review of the literature regarding wave drift was conducted. A great deal of literature exists regarding analytical exploration of drift forces on bodies in waves. Stokes is generally associated with some of the earliest second order analysis. Numerous references are available regarding second order forcing analysis[29; 14; 32] but most are based on linear, frequency domain analysis. The velocities measured in this experiment, however, departed drastically from what second order theory would predict. A few recent works explore nonlinear drift of two-dimensional and threedimensional bodies in waves [44; 45; 26; 25]. An experimental study of capsize stability was performed and compared to theoretical predictions based on a nonlinear development[6]. An important assumption within the development was that drift velocity was much lower than wave celerity and was thus ignored. The shift in resonant behavior noted in the experiments was therefore mistakenly attributed completely to a softening system, ignoring the frequency shift that can result from the mean body drift. Kuroda and Tanizawa[45; 25] both explore similarly nonlinear drift phenomenon. They use empirical formulas to solve for the balance between the wave drift force and drag due to drift (Eqs. 4.3). Also noted in both of these works is the correlation between pitch amplitude and the drift velocity. Kuroda achieves some success in producing a scheme for calculating the drift speed and roll amplitude but admits that the method lacks some nonlinear interactions due to the large roll motions. It is evident then that simplifications imposed for the sake of analytic solution may lead to significant limitations in the efficacy of the model. A time domain approach is thus pursued for solution of both fluid and rigid body dynamics. To go further with the research requires two main areas of development for a simulation. First, a hydrodynamic model for the simulation must be selected. Second, the coupled dynamics of a seven degree of freedom (DOF) rigid body system will need to be developed. The selection of the hydrodynamic model to be used requires an assessment of a number of competing considerations. The model needs to be able to resolve free surface fluid dynamics and the fluid structure interaction consistent with those expected of a
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buoy hull in a seaway. The model also needs to be computationally efficient in order to allow evaluation of a large design space. The model should capture as much of the nonlinearity of the problem as possible because of the nonlinearities expected from the rigid body dynamics.
1.3 The Hydrodynamic Problem This problem differs significantly from typical applications of seakeeping or offshore design. It is generally the goal of design in these areas to minimize motions or loads. In contrast, the suggested applications demands exploration of parameter regimes which may encourage large or even extreme motions and loads. The resulting system of coupled equations of motion is therefore more complicated than the typical rigid body system used in seakeeping analysis. Tackling the solution of the conventional six degree of freedom system of equations is generally further simplified by assuming planes of symmetry, enforcing geometric linearization(thin ship or slender body), and linearizing motion amplitudes. In contrast, the system of interest must admit extreme motions and is strongly nonlinear in all seven degrees of freedom. The modeling of wave induced motions and loads on ships and offshore structures is critical to early stage design as well as late stage design. It is therefore understandable that this area has received and continues to receive attention in research. The ability to fully solve, most real problems in seakeeping is limited because of the many scales present in the fluid and fluid/structure dynamics. Fortunately, careful consideration of the nature of most applications provides the basis for making a choice to model only the minimum complexity necessary to capture the physics important to the design goals. Therefore, there are many choices available for computational analysis, with each approach carrying its inherent strengths and limitations. The literature review in this area is well documented by Beck. He gives a taxonomy of hydrodynamics problems for seakeeping and their associated solution techniques. At the highest level, the modeling branches off into either viscous or inviscid. The viscous regime is broken down into the seven main areas, four of which constitute what is commonly associated with Computational Fluid Dynamics (CFD).
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These models attempt, in different ways, to include the viscous term from the NavierStokes equations. The most complete, and consequently computationally expensive, are Direct Numerical Simulation (DNS), Smoothed Particle Hydrodynamics (SPH), Large Eddy Simulation (LES) and Reynolds Averaged Navier-Stokes (RANS). DNS attempts to solve the Navier-Stokes equations directly, without relying on a simplifying model for turbulence. LES and RANS both model turbulence, but to differing limits. LES models the smaller scales while RANS attempts model turbulence across all scales. Stokes‟ flow and empirical approximations, such as Morrison‟s equation, make up the other end of the spectrum in terms of complexity of modeling viscous effects. Inviscid modeling is then broken down into three primary groups; rotational, irrotational infinite fluid, and irrotational free surface flows. This last category comprises the area where most early design modeling for seakeeping is performed. St.Denis and Pierson[43] gave early modelers in this area a very effective tool by laying the foundations for the statistical approach to evaluating seakeeping characteristics. The method exploited spectral methods already being applied in other fields. The basic assumptions put forth are that the motions of the body as well as that of the fluid can be reduced to discrete frequency components and that these components did not interact with each other. The sea surface could be described as a Gaussian, ergodic process with constant mean. Then, given the linear response amplitude operator for the response of a given hull to the spectral components of interest, the overall response of ship could be described for a seaway. They are given credit for providing the foundation for development of frequency domain methods in seakeeping, which account for a large cross section of the tools used to study the seakeeping problem. The frequency domain techniques employ a range of linearizations in order to model varying levels of complexity in the fluid structure problem. Consistent with our experience thus far, frequency domain techniques have been developed over time with judicious linearizations or simplifying assumptions and an ever increasing ability to capture the physics of experimental and analytical results. Thin ship and flat ship theory gave way to slender body and strip theory[38; 31; 3]. Strip theory remains the most widely used computational seakeeping method.
11
The ever increasing power of computers made possible the exploitation of Boundary Integral Methods (BIM) for solution of the fluid structure problem. Boundary integral problems are posed by solving the LaPlace equation in the fluid domiain and setting up conditions that must be satisfied on all or some of the boundaries of the fluid domain. Initially, the Neumann-Kelvin linearization is used where the mean wetted surface is used to simplify the problem, that is solved on a linear free surface. These methods can be separated into two distinct varieties. First, wave Green functions[46] may be used to satisfy boundary conditions over the fluid boundary everywhere except on the body surface. These methods are most popular in zero speed applications because the formulation of the Green functions with forward speed can be difficult to determine. The second variety of BIM distributes singularities over the free surface and any solid surface necessary to model the problem at hand. A number of methods for treating truncation boundaries are available. Work continues in this area because a truly universal method that fits the needs of all problems has yet to be found[49]. Virtually all of the BIM modeling takes advantage of at least some of the fundamentals demonstrated in the pioneering work of Hess and Smith[17]. Addition of a free surface to their problem results in the basis for what are commonly referred to as panel methods. The constant strength flat panels used in the original method have since been improved upon by developing a number of numerical methods to more accurately define the geometry of the problem as well as improving numerical accuracy and efficiency[35; 27]. Some 100 years after the first computational work of Froude and Krylov[16; 23], the fully nonlinear problem of a time dependent wetted surface in the presence of a time dependent free surface was begun. Palowski suggested the weak scatterer free surface boundary conditions[33] in order to formulate a consistent method to calculate hydrodynamic forces over the wetted surface defined by the intersection of a solid body and an incident free surface. The formulation linearizes the problem about the incident free surface based on some physical arguments about the nature of the waves caused by the presence and motion of the body. This approach has been explored and applied in the commercial code, SWAN 4[21; 41; 19] with some promising results. Sclavounos[42] has
12
continued along this line of research by developing the fully nonlinear problem followed by a careful exploration of linearizations from the completely defined problem. A number of numerical approaches have been employed in the design phase of Wave Energy Conversion (WEC) systems [10; 30; 18; 7; 34; 8]. Cruz[8] reviews the various types of boundary integral methods used to explore the WECs. He presents case studies with some details on the modeling of some of the more prominent full scale projects being undertaken, including Pelamis, the Archimedes Wave Swing and Wave Dragon. Commercial codes used in early design stages of noted projects include WAMIT, AQUADYN, and PEL. Of these, the PEL suite of software used to analyze Pelamis is the only nonlinear code. However, the nonlinear modeling relies on frequency domain results for the hydrodynamic aspects of the simulation. The nature of the nonlinearities captured by this model are apparently restricted to the mechanical power takeoff physics. All of the aforementioned projects have been prototyped and tested. The lack of electric grid scale energy generating systems deployed speaks clearly to the success in developing a design that can perform in service and are cost effective to manufacture, deploy and maintain. Recent estimates[8] indicate that the cost challenge for utility electric grid scale wave energy power is to reduce current lifetime costs by 50%. As with offshore design in other areas, the primary challenge is to arrive at a truly multidisciplinary optimized design that considers the lifetime extremes in loading and fatigue. Thus, the final product is a design with appropriate safety margins and minimum costs. It is suggested that improved early design tools may have been very useful in most of these projects. Following is a list of the nonlinearities that may have been neglected in the modeling of these systems. 1
The mean wetted surface versus the time dependent real wetted surface determined by the position and orientation of the hull.
2
Bluff body hydrodynamics can be expected to depart significantly from predictions of slender body theory.
13
3
Hulls that are not wall sided near the intersection of the free surface.
4
The change in the wetted surface of the hull due to the incident waves.
5
Interaction between the disturbance waves and the incident wave field. This work uses the weak scatterer approximation to formulate a consistent
mathematical model to solve the hydrodynamic forcing of a body in the presence of an ambient wave field. It is a pragmatic approximation for many applications and hull geometries. Its validity for many hull forms and wave conditions can be argued by visual evidence of common test conditions in waves. The hull motions and changing wetted surface may be significant while the disturbance waves are discarded by the eye as chafe over the ambient free surface. The weak scatterer formulation enables a mathematically consistent formulation that can capture significant nonlinearities associated with wave/body interactions while taking advantage of existing panel code techniques. The formulation captures the instantaneous wetted surface of the body. It also allows for implementation of a nonlinear ambient wave climate. The equations of motion for the seven degree of freedom system are integrated in time using a fourth order Adams Bashforth scheme.
14
Chapter 2 A Simple Model to Assess Conversion Capacity
2.1 Description of the Problem
Of interest is a reasonable estimate for the energy conversion capacity for coupling a pendulum to the wave induced motions of a free floating body. This upper bound can then be compared to the powering requirements of a data buoy. The typical powering requirement for data buoy systems is on the order of single watts. Drifter buoys designed and deployed by the University of Michigan Marine Hydrodynamics Lab (MHL) range in power consumption from approximately 1 to 5 watts. Power consumption depends on instrumentation and communication demands for the application. In order to provide scale for the energy available in water waves an example can be used for contrast. Consider, for example, two-dimensional, linear, regular waves with amplitude of one half meter and a period of three seconds. The power passing through a line parallel to the wave crests and 1 meter in length is given as:
1 (2.1) Wave Power ga2cg 3kilowatts 2 With the density of water, g gravity, a the amplitude and cg the group velocity of the incident wave. The magnitude of this number, in comparison to the powering requirements of the buoy, indicates that even modest efficiency of wave energy conversion is not necessary. The work being suggested can be thought of as looking at existing buoy applications and asking whether we can reasonably expect to power these buoys by adding some mechanical device and paying attention to buoy parameters that affect its response to wave forcing.
15
A brief discussion of the relevant parameter space for consideration in this problem follows. One area for application for this research would be to retrofit or improve the designs of existing data buoys. This perspective would encourage wave energy converter (WEC) designs that would not affect the existing performance characteristics of the buoys, including wave induced motions. In this case, parameter selection is constrained by requiring that the pendulum motions be decoupled from the buoy motions. Another area of application would be buoy designs whose functions are insensitive to hull motions. There is also the possibility of using hull motions to directly assist in propelling the hull in a desirable direction. Either of these situations allow for parameters that admit more significant interaction forces between buoy and pendulum. The magnitudes of the forces could approach those associated with a mooring system in light seas. The magnitudes of the allowed forces are not on the same order as would be associated with a mooring system in heavy seas. Parameters that allow coupled forces on the same order as that of a mooring system in heavy seas are the focus of research on energy absorbers for high efficiency or electric utility grid scale energy conversion. Such systems require complex sensing, mechanical, and control systems[13] that allow the system to tune itself to the ambient wave climate. An analogy to a self-winding watch is appropriate at this point. Self winding watch mechanisms are simply a pendulum whose relative motion to the watch case is transferred to a coil spring or directly into stored electrical energy. The inertial properties of these mechanisms could be sized in order to extract virtually any amount of energy from the motions that the normal human arm makes. This approach, unfortunately, would result in a pendulum with a large inertia, ignoring the ability of the normal human arm to make these motions. The inertia of the self winding mechanism is designed to provide the energy sufficient to drive the watch. Conveniently, the required inertia is undetectable by the user. Therefore, the parameter space suggested for exploration for this application begins with no coupling between the energy extraction device and buoy motions. The space would reasonably end where the interactive forces between buoy and pendulum approach the same order of magnitude as wave induced forces. At the uncoupled
16
extreme, the wave induced motions of the rigid hull alone can be interrogated for the bounds on the acceleration available for excitation of an energy extraction device.
2.2 A Simplified Two-dimensional Model
The initial modeling of this application treats energy conversion as an ideal process. Power take-off is modeled as converting rotary motion to usable form through a linear damping term. Adjustment of the energy extraction model to reflect a real application is left as an application specific task. The model explores the response of a pendulum subjected to periodic motions in three degrees of freedom with specified motion amplitudes, frequencies, and phasing that could reasonably be expected in the marine environment. Following are the coordinate definitions along with a reference schematic in Figure 2.1. X2
x X1 y z
Inertial :
X1 X , X2
Body :
x x1 , x2 , x3 , x4 , x5 , x6 , x7
T
u ν1 ν x , ν1 v , ν2 w
17
(2.2) p ν 2 q r
x3 Z x2 Y
r x, y, z
x1 b
O
Z
x4
X
iga
eit kx ekz
a) Buoy coordinates
b
m
)
Pendulum Plane View r
x1 x7
Side View b) Pendulum Coordinates Figure 2.1 Coordinate Definitions for the: a) Buoy Hull and b) Pendulum.
The simplified model studies the behavior of the pendulum given prescribed motions of the hull. An important assumption of this model is that the motions of the pendulum and body are completely decoupled. It is understood that this assumption puts strong constraints on the parameter space. In particular, this model neglects the dynamics
18
associated with the torque transferred from the pendulum to the hull as energy is transferred out of the system. Therefore, care is taken to ensure that the energy extraction parameters are such that the torque generated is small relative to the moments necessary to move the buoy under the given motions. This can be reasonably achieved by choosing inertial properties for the buoy and pendulum that can be argued to decouple their motions without energy extraction. The benefit of choosing this model is that the assumptions lead to a relatively simple system that can be readily analyzed. Also, it is suggested that results may inform as to the reasonable expectations for the magnitude of energy that might be extracted from the complete problem. The model considers plane motions of heave, pitch and surge. It will be useful for evaluation to describe body motions using typical linear wave kinematics where the incident free surface is described by the velocity potential for incident waves: I
iga
e
i t kx kz
(2.3)
e
The in-plane accelerations (orthogonal to the pendulum rotation axis, x3) represent the only excitation significant to the pendulum. The relevant accelerations are gravity, surge, and heave projected onto the body fixed coordinate frame, and centripetal acceleration due to the pitch angular velocity. The imposed motion is associated with a response amplitude operator of unity for heave and surge. The pitch motion amplitude is set equal to the slope of the incident wave. Heave and surge are in phase with the wave kinematics, while the pitch phase is allowed to lag the free surface slope by an angle ϕ5. Under these conditions, equation 2.4 gives the equation of motion for the pendulum, dimensioned as described in Figure 2.1.
G.E.: x7
x1 gravity surge heave centripital cx7 x1 sin x7 0 2 m r g sin x5 x cos x5 z sin x5 r cos x7 q
(2.4)
Equation 2.4 describes a parametric pendulum with damping, excited by a complex acceleration. It is not possible to simplify sin(x7in terms of a Taylor series, as is usually the case, because thex7term spans the real numbers. Regardless, certain
19
insights can be gained by simplification and inspection of the coefficients of the terms that constitute the body fixed surge acceleration. Small angle approximations for the wave slope are used to arrive at a simplified form for the surge acceleration transformed into the pendulum plane. x1 gka sin t 5 a 2 sin t a 2 ka cos t sin t 5 ka a 2 sin t 5 sin t sin 2t 5 sin 5 2
(2.5)
An interesting result is that the first two terms of the body-fixed surge acceleration equation cancel each other if pitch is in phase with the wave slope. That is, the components of gravity and surge projected onto the body-fixed x1 axis are equal and opposite under linear wave theory when there is no phase difference between the free surface kinematics and those of the buoy. Under these conditions, the resulting in-plane acceleration experienced by the pendulum would reduce to that contributed by the heave acceleration and pitch centripetal accelerations. Centripetal accelerations can be shown to be negligible if we enforce the condition that the pendulum radius is much smaller than the wavelength. Comparison of the magnitudes of the three main components of the acceleration shows that the heave component differs by a factor of ka and is twice the frequency. It is evident then that response in pitch must have some phase difference in order for the pendulum to experience significant excitation. Therefore, the initial model was probed in a brute force manner by integrating over a reasonable parameter range and considering the above result. A program (MATLAB) was written that integrated the motions of the pendulum based on prescribed motions of the hull. The pendulum was assigned a radius of 0.25 meters and a mass of 1 kilogram. Duration of the integration was set to 10 wave periods. A time step of 1/1000 of the wave period was used. The parameter space consisted of variations in wave steepness, wave frequency, damping coefficient, and pitch phase. It is understood that dependence on initial conditions exists for steady state as well as transient behavior. The analysis sought a qualitative understanding of the system‟s behavior.
20
Energy dissipated through the damping term was accumulated, the average power calculated, and mapped versus pitch phase and radial frequency. Figure 2.2 shows the resulting power versus variation in pitch phase and wave frequency. The motion of the pendulum associated with the peak power dissipation is that of continuous rotation with a mean angular frequency equal to that of the forcing frequency.
Figure 2.2 Power Conversion (Mass of pendulum=1kg, r=0.25m). A bifurcation analysis was performed over a reasonable range of parameters. Under regular forcing, the response of the pendulum will either be periodic, quasiperiodic, sub-harmonic, or chaotic. The bifurcation analysis provides maps of these regions. These regions can be associated with certain characteristics with respect to power dissipation. Figure 2.3 shows a bifurcation plot that characterizes local extrema in the pendulum rotational velocity versus the excitation frequency. Wave steepness,
21
damping, and pitch phase response are fixed. It can be seen that the power is maximum where constant rotation is exhibited. The presence of multiple solutions for differing initial conditions is demonstrated as well. The steady state oscillations and phase plane map for parameters generating multiple solutions are shown in Figure 2.4.
Figure 2.3 Bifurcation Diagram Describing Pendulum Rotation vs. Wave Frequency.
22
Figure 2.4 Oscillation and Phase Plane map displaying multiple solutions for different initial conditions. As seen in the bifurcation diagram, chaotic and regular response may exist in close proximity to each other. High energy conversion is associated with the solution that generates the larger amplitude motions. It is hoped that further characterization of the operational space for this system will lead to insights into optimal design parameters for the pendulum and hull geometry and inertial properties. Further investigation will include variation of the orientation of the pendulum axis with respect to the body fixed vertical axis.
2.3 Nonlinear Dynamics An application with similarities to that being explored herein has been examined by Xu[47]. The behavior of a parametrically excited pendulum, oriented in the conventional orientation, Figure 2.5, is explored. The analysis focuses on pendulum behavior in response to parametric base excitation in the vertical direction. The nondimensional governing equation for such a system is,
23
(2.6) 1 pcossin 0 2 with c/mn , p a / g, /n , n g / l and nt , and p is a small parameter.
z a cos(t)
g
m
Z Figure 2.5 Parametric Pendulum. Figure 8 - Parametric Pendulum Linearization of (2.6) and neglecting the damping term reduces to the well studied Mathieu equation[48];
1 p cos 0
(2.7)
Xu[48], reports the first analytic solution for full rotations through perturbation methods. It is not straightforward to apply the same nonlinear analysis techniques to our problem. The following derivation attempts a similar scaling of the governing equation for the problem at hand (2.5). The assumptions, again, are that the hull undergoes prescribed motions that can be described by the kinematics of a monochromatic free surface governed by linear wave theory. Vertical plane motions have the same amplitude and are 24
in phase with the free surface. The inclination, or pitch angle, of the body is periodic and has the same amplitude but lags the free surface slope by an angle 5. Note the sin(ϕ5) term in (2.5) represents a constant acceleration in the plane of the pendulum. If this term is treated similar to gravity, then the governing equation can be rescaled in a similar manner to the approach used in the analysis of a pendulum in its more common orientation. sin 2 5 G.E.: 1 sin sin 5 sin 0 sin 5
(2.8)
with, g
a 2 ka 2
sin 5 , n t , n g / r , c / mn ,
2
sin 5
, ka,
n
Note that equation 2.8 is singular if the pitch response is in phase with the forcing. The analysis must then exclude this regime. This is a reasonable assumption even for a less simplified analysis. If we linearize, ignore the dissipative term, and consider the phase to be - radians we can write:
2 1 cos 2 sin cos 0
(2.9)
Somewhat problematic is the presence of terms having order and Rescaling time so that
nt equations 2.9 becomes
1 cos 2 2 sin cos 0
(2.10)
Noting that the small parameter is a coefficient of both the temporal and constant coefficient of the stiffness term for this model, it is apparent that an analogy to the Mathieu equation is not achievable for the given problem statement. Considering that the above model is constrained by a number of reasonable and intuitive assumptions, it seems prudent to first pursue solutions to real behaviors for the fully described system and then reconsider a nonlinear analysis based on time domain results. Of course it would be possible to reconsider certain assumptions about the parameters and seek another small 25
parameter. But the actual equations of motion for the system are strongly coupled in all seven degrees of freedom. The motions generated experimentally and numerically will show that the fully nonlinear model provides a much richer response regime than the constrained model used for the nonlinear analysis could permit.
2.4 Conclusion This simplified pendulum analysis suggests an initial point buoy design that encourages a large response in pitch to the ambient wave field. Also, the phase response in pitch with respect the surface slope should be maximized. With this in mind we can propose initial hull geometry and inertia resulting in resonance at the incident wave frequency. An initial attempt at isolating a small parameter, that could be exploited using perturbation techniques for a nonlinear analysis, was carried out. Even considering the considerably constrained (simplified) case of prescribed motion for the hull, such a parameter is not immediately evident. Time domain analysis will then proceed with variation of parameters about this initial point design. Insights gained can be used to constrain parameters within an analysis that considers real seas. Indications from the model presented here, at the very least, encourage further investigation. The numerical and experimental results will demonstrate that the energy conversion estimates predict well the ability to convert wave excited, transient motions into energy.
26
Chapter 3 Rigid Body Motions
3.1 Equations of Motion The focus of this section is the derivation of seven coupled equations for the motions of a rigid body that is subjected to generalized forcing and is also coupled to a planar pendulum. The pendulum will be modeled as a point mass. The pendulum is allowed to undergo full rotation and can dissipate energy through an appropriate damping model. The orientation of the pendulum plane is allowed to vary from the quiescent horizontal plane, but is fixed relative to the body fixed coordinate system. External forcing of the rigid body will be represented by a general force vector. Solution for the actual values of this forcing vector represents the hydrodynamic problem. The following coordinate definitions will be used (Figure 3.1). Frame „O‟ is an earth fixed frame with the „z‟ axis positive in the upward direction. Frame „I‟ is parallel to the inertial frame but instantaneously coincident with some specified position within the body geometry. Frame „B‟ is the body fixed frame, which is initially coincident with frame ‟I‟. Frame „p‟ is a frame fixed to the pendulum. Its origin is located on the body fixed z-axis at some height hp. The orientation of the frame „p‟ z-axis relative to frame „B‟ is a parameter that can be specified as an angle p about an axis parallel to the body fixed x-axis (Figure 3.2).
27
𝒆𝒑𝒛 , 𝜂7
𝜂6 , 𝜂3 𝜙𝑝
𝒆𝒑𝒙
𝒆𝒑𝒚 ′𝑝′
𝜂5 , 𝜂2
𝑍 [𝑥, 𝑦, 𝑧]
′𝐵, ′𝐼′
𝑌
′𝑂′
𝜂4 , 𝜂1
𝑋
Figure 3.1- Buoy Hull Coordinate System Definitions.
Pendulum Plane
𝑒𝑝𝑦
𝑟𝑝
𝑒𝑝𝑥 𝜂6 , 𝜂3
View 𝑚𝑝
7 𝜂4 , 𝜂1 𝜂6 , 𝜂3
p ℎ𝑝
Side View ′𝑝′
′𝐵′
𝜂5 , 𝜂2
𝐶𝐺 Figure 3.2 - Pendulum Coordinate System Definition.
28
The following notation represents orientation and forces commonly associated with a rigid body seakeeping analysis.
X 2
x X1 y z
Inertial :
X X 1 , X2
Body :
η 1 ,2 ,3 ,4 ,5 ,6 ,7
T
ν1 u ν ν 2 , ν1 v s w
Body Force : F F1 , F2 , T
p ν 2 q r
X F1 Y Z
(3.1)
K F2 M N
To this definition we must add the seventh degree of freedom 7 and a generalized force, Tp, acting on the pendulum. The equations of motion will be developed in the body frame, so as to avoid the necessity of tracking time dependent inertia properties Let M represent the mass of the body without the pendulum and let mp represent the mass of the pendulum. I 0 represents the moment of inertia of the body with respect to its center of gravity. The pendulum mass will be modeled as a point mass. In so doing, the
moment of inertia about its own center of gravity is assumed to be negligible. Let m ' represent an inertial point coinciding with the instantaneous position of mp. The velocity and acceleration of the pendulum mass, mp, can be represented as the velocity and acceleration of
the point m ' added to that of the pendulum relative to the body coordinate system and the equations of motion may be written;
F1 P M I aCG mp I am' Bam 2ν2 B vm
29
(3.2)
F2 H I0 ν 2 ν2 I0 ν2 MrCG ν 1 ν2 ν1 mp r m/ b I am' B am 2 I ν2 B vm
𝑇𝑝 = −𝑐𝑑𝑎𝑚𝑝 𝜂7 / 𝜂7 − 𝑚𝑝 𝑅𝐼𝑝 𝑔 × 𝑟𝑝 = 𝑚𝑝 𝑟𝑝 𝑅𝐵𝑝 𝐼 𝑎𝑚′ ⋅ 𝑒𝜂 7 + 𝜂7 𝑟𝑝
(3.3) (3.4)
The resulting matrix representation of the equations of motion is:
F M mp CRB cRB
(3.5)
The 7x7 matrices, M, CRB , are the customary six degree of freedom rigid body mass and coriolis matrices augmented with a seventh row and column of zeroes. Whereas mp , cRB represent the mass and coriolis matrices for the pendulum mass. The dynamical system can be computationally solved using the above representation. However, the existence of parameter ranges for which nonlinear dynamics could find application must be admitted. For this reason, the development of the dynamic simulation sought to first explicitly write the coupled seven degree of freedom equations of motion in terms of the rigid body coordinates. In order to simplify the following expressions, let the notation 𝑠𝜙𝑝 and 𝑐𝜙𝑝 represent the sine and cosine, respectively, for the angle that the pendulum plane makes with the buoy calm water x-y plane. Also, let 𝑟𝑝𝑥 , 𝑟𝑝𝑦 , 𝑟𝑝𝑧 represent the time dependent position of the pendulum mass in the buoy coordinate frame. Given these simplifying notations, the matrices for the equations of motion are presented on the following pages. Note that symmetry is maintained within the mass matrix and that its representation is unique as is expected. Also note that the parameterization of the coriolis and centripetal matrix is not unique. The terms corresponding to the conventional six degrees of freedom follow those used by Fossen[15].
30
𝑀 0 0 𝐌 = 0 𝑀𝑧𝐺 −𝑀𝑦𝐺 0
0 𝑀 0 −𝑀𝑧𝐺 0 𝑀𝑥𝐺 0
0 0 𝑀 𝑀𝑦𝐺 −𝑀𝑥𝐺 0 0
0 −𝑀𝑧𝐺 𝑀𝑦𝐺 𝐼𝑥𝑥 −𝐼𝑥𝑦 −𝐼𝑥𝑧 0
𝑀𝑧𝐺 0 −𝑀𝑥𝐺 −𝐼𝑥𝑦 𝐼𝑦𝑦 −𝐼𝑦𝑧 0
−𝑀𝑦𝐺 𝑀𝑥𝐺 0 −𝐼𝑥𝑧 −𝐼𝑦𝑧 𝐼𝑧𝑧 0
0 0 0 0 0 0 0
𝐦 = 𝑚𝑝 1 0 0
31
= 0 𝑟𝑝𝑧
0 1 0
0 0 1
−𝑟𝑝𝑧
𝑟𝑝𝑦
0
−𝑟𝑝𝑥
0 −𝑟𝑝𝑧 𝑟𝑝𝑦 𝑟 2 𝑝𝑦 + 𝑟 2 𝑝𝑧 −𝑟𝑝𝑥 𝑟𝑝𝑦
𝑟𝑝𝑧 0 −𝑟𝑝𝑥
−𝑟𝑝𝑦 𝑟𝑝𝑥 0
−𝑟𝑝𝑥 𝑟𝑝𝑦
−𝑟𝑝𝑥 𝑟𝑝𝑧
𝑟 2 𝑝𝑥 + 𝑟 2 𝑝𝑧
−𝑟𝑝𝑦
𝑟𝑝𝑥
0
−𝑟𝑝𝑧 𝑟𝑝𝑥
−𝑟𝑝𝑦 𝑟𝑝𝑧
−𝑟𝑝 𝑠𝜂7
𝑟𝑝 𝑐𝜙𝑝 𝑐𝜂7
𝑟𝑝 𝑠𝜙𝑝 𝑐𝜂7
𝑟𝑝𝑥 𝑟𝑝𝑦 𝑠𝜙𝑝 − 𝑟𝑝𝑧 𝑐𝜙𝑝
𝑟𝑝 −𝑟𝑝𝑥 𝑐𝜂7 𝑠𝜙𝑝 − 𝑟𝑝𝑧 𝑠𝜂7
𝐂𝐑𝐁
0 0 0 = −𝑀𝑧𝐺 𝑟 𝑀 𝑥𝐺 𝑞 − 𝑤 𝑀 𝑥𝐺 𝑟 + 𝑣 0
0 0 0 𝑀𝑤 −𝑀 𝑧𝐺 𝑟 + 𝑥𝐺 𝑝 −𝑀𝑢 0
0 0 0 𝑀 𝑧𝐺 𝑝 − 𝑣 𝑀 𝑧𝐺 𝑞 + 𝑢 −𝑀𝑥𝐺 𝑝 0
𝑟𝑝𝑥 𝑟𝑝𝑦 𝑠𝜙𝑝 − 𝑟𝑝𝑧 𝑐𝜙𝑝
−𝑟𝑝𝑦 𝑟𝑝𝑧 𝑟
𝑀𝑧𝐺 𝑟 −𝑀𝑤 −𝑀 𝑧𝐺 𝑝 − 𝑣 0 𝐼𝑥𝑧 𝑝 − 𝐼𝑧𝑧 𝑥6 𝐼𝑦𝑦 𝑞 0
−𝑟𝑝 𝑠𝜂7 𝑟𝑝 𝑐𝜂7 𝑐𝜙𝑝 𝑟𝑝 𝑐𝜂7 𝑠𝜙𝑝
2
𝑝𝑦
+𝑟
𝑟𝑝 −𝑟𝑝𝑥 𝑐𝜂7 𝑠𝜙𝑝 − 𝑟𝑝𝑧 𝑠𝜂7 2
𝑟𝑝 𝑟𝑝𝑥 𝑐𝜂7 𝑐𝜙𝑝 + 𝑟𝑝𝑦 𝑠𝜂7
𝑝𝑥
𝑟𝑝 𝑟𝑝𝑥 𝑐𝜂7 𝑐𝜙𝑝 + 𝑟𝑝𝑦 𝑠𝜂7
−𝑀 𝑥𝐺 𝑞 − 𝑤 𝑀 𝑧𝐺 𝑟 + 𝑥𝐺 𝑝 −𝑀 𝑧𝐺 𝑞 + 𝑢 −𝐼𝑥𝑧 𝑝 + 𝐼𝑧𝑧 𝑟 0 𝐼𝑥𝑧 𝑟 − 𝐼𝑥𝑥 𝑝 0
−𝑀 𝑥𝐺 𝑟 + 𝑣 𝑀𝑢 𝑀𝑥𝐺 𝑝 −𝐼𝑦𝑦 𝑞 −𝐼𝑥𝑧 𝑟 + 𝐼𝑥𝑥 𝑝 0 0
r𝑝2
0 0 0 0 0 0 0
𝐜𝐑𝐁 = 𝑚𝑝
0 0 0 − 𝑟𝑝𝑦 𝑞 + 𝑟𝑝𝑧 𝑟
0 0 0 𝑤 + 𝑟𝑝𝑦 𝑝 − 2𝑟𝑝 𝑠𝜙𝑝 𝑐𝜂7 𝜂7
𝑟𝑝𝑥 𝑞 − 𝑤 + 2𝑟𝑝 𝑠𝜙𝑝 𝑐𝜂7 𝜂7 𝑟𝑝𝑥 𝑟 + 𝑣 − 2𝑟𝑝 𝑐𝜙𝑝 𝑐𝜂7 𝑟𝜂7 𝑟𝑝𝑥 𝑐ϕ𝑝 𝑟 − 𝑠ϕ𝑝 𝑞
𝑟𝑝𝑦 𝑞 + 𝑟𝑝𝑧 𝑟 − 𝑤 + 𝑟𝑝𝑦 𝑝 − 2𝑟𝑝 𝑠𝜙𝑝 𝑐𝜂7 𝜂7
− 𝑟𝑝𝑧 𝑟 + 𝑟𝑝𝑥 𝑝 −𝑢 + 𝑟𝑝𝑦 𝑟 − 2𝑟𝑝 𝑠𝜂7 𝜂7 𝑟𝑝 𝑠𝜂7 𝑟 + 𝑠ϕ𝑝 𝑐𝜂7 𝑝
− 𝑟𝑝𝑥 𝑞 − 𝑤 + 2𝑟𝑝 𝑠𝜙𝑝 𝑐𝜂7 𝜂7 𝑟𝑝𝑧 𝑟 + 𝑟𝑝𝑥 𝑝
0 0 0 −𝑣 + 𝑟𝑝𝑧 𝑝 + 2𝑟𝑝 𝑐ϕ𝑝 𝑐𝜂7 𝜂7 𝑢 + 𝑟𝑝𝑧 𝑞 + 2𝑟𝑝 𝑠𝜂7 𝜂7 − 𝑟𝑝𝑥 𝑝 + 𝑟𝑝𝑦 𝑞 −𝑟𝑝 𝑠𝜂7 𝑞 + 𝑐ϕ𝑝 𝑐𝜂7 𝑝 − 𝑟𝑝𝑥 𝑟 + 𝑣 − 2𝑟𝑝 𝑐𝜙𝑝 𝑐𝜂7 𝜂7 − −𝑢 + 𝑟𝑝𝑦 𝑟 − 2𝑟𝑝 𝑠𝜂7 𝜂7
32
− −𝑣 + 𝑟𝑝𝑧 𝑝 + 2𝑟𝑝 𝑐𝜙𝑝 𝑐𝜂7 𝜂7 −𝑟𝑝𝑥 𝑟𝑝𝑧 𝑞 + 2𝑟𝑝 𝑟𝑝𝑦 𝑐𝜙𝑝 𝑐𝜂7 𝜂7 𝑟𝑝𝑥 𝑟𝑝𝑧 𝑝 − 𝑟 2 𝑝𝑥 𝑟 − 2𝑟𝑝 𝑟𝑝𝑥 𝑐𝜙𝑝 𝑐𝜂7 𝜂7 𝑟 2 𝑝𝑥 𝑞 − 𝑟𝑝𝑥 𝑟𝑝𝑦 𝑝 − 2𝑟𝑝 𝑟𝑝𝑥 𝑠ϕ𝑝 𝑐𝜂7 𝜂7
− 𝑢 + 𝑟𝑝𝑧 𝑞 + 2𝑟𝑝 𝑠𝜂7 𝜂7 2 −𝑟𝑝𝑦 𝑟𝑝𝑧 𝑞 − 𝑟𝑝𝑧 𝑟 + 2𝑟𝑝 𝑟𝑝𝑦 𝑠𝜂7 𝜂7 𝑟𝑝𝑦 𝑟𝑝𝑧 𝑝 − 𝑟𝑝𝑥 𝑟𝑝𝑦 𝑟 − 2𝑟𝑝 𝑟𝑝𝑥 𝑠𝜂7 𝜂7 −𝑟 2 𝑝𝑦 𝑝 + 𝑟𝑝𝑥 𝑟𝑝𝑦 𝑞 − 2𝑟𝑝 𝑟𝑝𝑦 𝑠ϕ𝑝 𝑐𝜂7 𝜂7
𝑟𝑝𝑥 𝑝 + 𝑟𝑝𝑦 𝑞 2 𝑟𝑝𝑦 𝑞+𝑟𝑝𝑥 𝑟𝑝𝑦 𝑝 + 𝑟𝑝𝑦 𝑟𝑝𝑧 𝑟 + 2𝑟𝑝 𝑟𝑝𝑧 𝑠𝜂7 𝜂7 2 𝑟 𝑝𝑧 𝑝 − 𝑟𝑝𝑥 𝑟𝑝𝑧 𝑟 − 2𝑟𝑝 𝑟𝑝𝑧 𝑐𝜙𝑝 𝑐𝜂7 𝜂7 −𝑟𝑝𝑦 𝑟𝑝𝑧 𝑞 + 𝑟𝑝𝑧 𝑟𝑝𝑥 𝑞 − 2𝑟𝑝 𝑟𝑝𝑥 𝑠𝜂7 𝜂7
𝑐ϕ𝑝 𝑟𝑝𝑥 𝑟𝑝𝑥 𝑞 − 𝑟𝑝𝑦 𝑝 − 𝑠ϕ𝑝 𝑟𝑝𝑥 𝑟𝑝𝑧 𝑝 + 𝑟𝑝𝑥 𝑟
𝑟𝑝 𝑠𝜂7 𝑟𝑝𝑥 𝑞 + 𝑟𝑝𝑦 𝑝 + 𝑠ϕ𝑝 𝑟𝑝𝑥 −𝑟𝑝𝑦 𝑟 + 𝑟𝑝𝑧 𝑞
−𝑟𝑝 𝑠𝜂7 −𝑟𝑝𝑥 𝑟 + 𝑟𝑝𝑧 𝑝 − 𝑐ϕ𝑝 𝑟𝑝𝑥 𝑟𝑝𝑦 𝑟 − 𝑟𝑝𝑧 𝑞
−𝑟𝑝𝑥 𝜂7 −𝑟𝑝𝑦 𝜂7 −𝑠𝜂7 𝑠𝜙𝑝 𝜂7 𝜂7 𝑟𝑝 −𝑟𝑝𝑦 𝑠𝜂7 𝑠𝜙𝑝 + 𝑟𝑝𝑧 𝑠𝜂7 𝑐𝜙𝑝 + 2𝑟𝑝 𝑟𝑝𝑧 𝑠𝜙𝑝 𝑐𝜂7 𝑝 𝜂7 𝑟𝑝 𝑟𝑝𝑥 𝑠𝜂7 𝑠𝜙𝑝 − 𝑟𝑝𝑧 𝑐𝜂7 + 2𝑟𝑝 𝑟𝑝𝑧 𝑠𝜙𝑝 𝑐𝜂7 𝑞 𝜂7 𝑟𝑝 𝑟𝑝𝑥 𝑠𝜂7 𝑐𝜙𝑝 + 𝑟𝑝𝑦 𝑐𝜂7 + 2𝑟𝑝 𝑟𝑝𝑦 𝑐ϕ𝑝 𝑐𝜂7 𝑟 0
T
At each time step, the generalized force vector, F X , Y , Z , K , M , N , Tp is determined by solving a mixed boundary value problem for the six hydrodynamic forces. The torque on the pendulum is determined by the orientation of the pendulum axis in the presence of gravity and the torque given by the energy extraction model. It is important to note at this time that a stability problem exists for this problem statement. The hydrodynamic forces used to determine the body acceleration are dependent on the acceleration of the body. This inconsistency will be handled in the fluid forcing problem development to follow. For the moment, it is suggested that the hydrodynamic added mass is known and separable from the fluid forcing at each step.
3.2 Euler vs. Quaternion Orientation Representation When beginning a dynamic motion analysis, it is important to consider the bounds on the expected motions. This consideration is the basis for choosing between the two most common methods for describing body orientation, Euler angles and quaternions. Euler angle representation of body orientation is adequate for most applications in naval architecture and marine engineering. Indeed, it is almost the norm. One of the primary advantages of this method, which uses three consecutive elementary rotations about coordinate axis, is that the connection between the transformation matrix and the physical motion between frames of reference is readily envisioned and written. That is, any motion can be broken down into a sequence of elementary rotations. The visualization of which, although somewhat challenging, is straightforward to recall and communicate graphically. A common sequence for marine applications is to rotate the body in yaw, then pitch and finally roll. The primary drawback of this method is the presence of a singularity for pitch angles that approach±90°. Quaternions, on the other hand, use four real parameters to describe the orientation of one frame relative to another. The primary advantage of quaternions over Euler angles is the absence of singularities. There exists a one to one mapping between quaternions to Euler angles. Also, numerical methods exist that allow for interpolating smoothly between two quaternions and which have been shown to improve integration accuracy.
33
It is reasonable to expect that pitch motions exceeding ±90° may occur in buoy applications. Therefore quaternions were chosen to model the orientation of buoy hull. Orientation of the pendulum can be described by a single rotation relative to the body frame. The development of both approaches is discussed below. First discussed is the Euler angle representation, followed by the quaternion. Both approaches result in a 3x3 transformation matrix. Transformation of a vector expressed in one coordinate system to the other, rotated coordinate system, is accomplished by pre-multiplication of the vector by the transformation matrix. The transformation matrix from frame i to frame j will be represented by Ri j .
3.3 Euler Angles Although the 3x3 transformation matrix is unique, its description by Euler angle rotations is not. That is to say, reorienting a vector to a rotated coordinate system can be accomplished by differing the orders and/or combinations of three elementary rotations. The generation of the transformation matrix depends on this given convention for the order of rotation about the reference coordinate axis. Knowledge of the amplitudes of the expected motions for a given application allows modelers to select an ordering of the rotations that reduces the likelihood of encountering a singularity in generating the rotation matrix. The current formulation uses the common 3-2-1, yaw-pitch-roll Euler angle ordering, that encounters a singularity when pitch nears pi/2. In order to simplify the following expressions, let the notation 𝑠θ, 𝑐θ and tθ represent the sin, cosine, and tangent, respectively, of the arbitrary angle θ.
34
Transformations between inertial and the body frames are: c c c s s R s c c s s c c s s s c s , frominertial to body. s s c s c c s s s c c c B I
T
RBI RIB from body to inertial. (3.6)
Transforms between body and pendulum coordinates are: c7 c p s7 s p s7 R s7 c p c7 s p c7 , 0 s p c p p B
R pB RBp
T
The orthonormality of the fundamental transformation matrices ensures that inversion of any transformation matrix may be achieved by taking its transpose. Conversion of the body rotation rate to the reference frame Euler rotation rates is not as straight forward however. The problem can be envisioned by considering the nonorthonormal directions of the Euler rate vectors. In the current case a rotation first about the Z axis followed by a non-zero rotation about the Y axis will orient the X axis so that it is no longer orthogonal to the axis about which yaw has occurred, the Z axis. The transformations between body referenced angular rotation rates and Euler angle rates can be derived in a number of ways. A straight forward way is to consider infinitesimal changes in the Euler angles and transform them to the body frame individually. That is, 0 0 R R 0 R 0 0 0
(3.7)
Determination of the inverse of the rotational matrix can be accomplished by applying the definition for an inverse matrix.
35
Rotational rate transforms between inertial and the body frames: t c 1 t s B 0 c s , from body to inertial frame Euler rates. 0 s / c c / c s 1 0 B BI 0 c c s 0 s c c ν B B X I B
2
I
(3.8)
2
The singularities present in the above transforms make evident the problem for this approach. In general seakeeping analysis, this is not an issue due to the stiffness of the restoring coefficient in pitch. In analyses for geometries and inertias associated with buoy hulls, the possibility of extreme motions approaching this singularity must be admitted.
3.4 Quaternion Algebra The singularity present in any Euler formulation can be avoided by using Quaternion algebra. The solution, however, comes at the cost of adding one dimension to the problem. William Hamilton is responsible for developing quaternions and their associated algebra in the 1840‟s. Their application and use has shown a marked increase that has paralleled three-dimensional computational application developments. A detailed exploration of their development and application can be found in [24]. A brief summary of their development and application follows. A quaternion consists of four real parameters. In this development, the first parameter represents a scalar value associated with the magnitude of a rotation and the remaining three are associated with a three-dimensional vector about which the rotation takes place. Hamilton Quaternions cannot be treated with standard vector algebra. Their usefulness stems from the special rules of quaternion algebra, that Hamilton developed for this purpose.
36
Let p=(p0,p) and q=(q0,q) represent two quaternions. Here, p0, q0 represent the scalar component of each quaternion while p,q represent the vector components. Addition and multiplication are defined as follows;
𝑝 + 𝑞 = 𝑝0 + 𝑞0 , 𝐩 + 𝐪 𝑝𝑞 = 𝑝0 𝑞0 − 𝐩 ⋅ 𝐪, 𝑞0 𝐩 + 𝑝0 𝐪 + 𝐪 × 𝐩
(3.9)
Note that the product rule for quaternions defined above is not commutative. The complex conjugate, or adjoint, is defined such that the scalar component is unchanged and the vector points in the opposite direction. Multiplying a quaternion by its complex conjugate will result in a scalar value. The square root of this scalar product is defined as the norm, N(q), of a quaternion and can be considered a length. The unit quaternion, defined as a quaternion of length one, is an important concept and will be briefly discussed below. The unit quaternion is important in simplifying the development of the mapping for the time derivative of the quaternion and its rates and either the body or reference frame rotation rates. Further details can be found in [40]. Normalization of a quaternion can be accomplished by dividing the quaternion by its norm. 𝑞 ∗ = 𝑞0 , −𝐪 𝑞𝑞 ∗ = 𝑞0 𝑞0 + 𝐪 ⋅ 𝐪, −𝑞0 𝐪 + 𝑞0 𝐪 − 𝐪 × 𝐪 = 𝑞02 + 𝑞12 + 𝑞22 + 𝑞32 = 𝑞 𝑁 𝑞 =
2
(3.10)
𝑞𝑞 ∗
The rotation operator in quaternion algebra consists of the triple product qxq* where the quaternion, q, holds the transformation parameters and x = 0, 𝑥 is the vector to be rotated. Let x be a vector in the reference frame and x‟ be the same vector in a frame with different orientation from the reference. By carrying out the operations and then regrouping terms, it is possible to derive an expression for a matrix-vector product from which the rotation matrix may be taken. The absence of singularities in (1.8) is evident.
37
𝐱 ′ = 𝑞𝐱𝑞 ∗ = 2𝑞0 𝐪 × 𝐱 + 2 𝐪 ⋅ 𝐱 𝐪 + 𝑞02 − 𝐪 𝑞02 + 𝑞12 −
1 2
𝑇𝐱 = 2 𝑞1 𝑞2 + 𝑞0 𝑞3 𝑞1 𝑞3 − 𝑞0 𝑞2
𝑞1 𝑞2 − 𝑞0 𝑞3 𝑞02 + 𝑞22 −
1 2
𝑞2 𝑞3 + 𝑞0 𝑞1
2
𝐱 = 𝑇𝐱
𝑞1 𝑞3 + 𝑞0 𝑞2 𝑞2 𝑞3 − 𝑞0 𝑞1 𝐱 𝑞02 + 𝑞32 −
𝐱 = 𝑇 −1 𝐱′ = 𝑇 𝑇 𝐱′
(3.11)
1 2
The equations of motion are normally written in the body fixed frame and accelerations are determined at each time step in this frame. It is then required to find a way to relate these accelerations to incremental changes in the quaternion representing the body orientation. Conveniently, there are direct relationships between the body frame angular accelerations and velocities and the quaternion and its time derivatives. Let the prime (′) notation represent the body frame. Using quaternion multiplication as outlined above, and enforcing the unit quaternion constraint, the various mappings between quaternion rates, 𝑞 and 𝑞 , and angular rotation rates, 𝝎 and 𝝎 are; 𝝎 = 2𝑞 𝑞 ∗ and 𝝎′ = 2𝑞 ∗ 𝑞 1 1 𝑞 = 𝝎𝑞 and 𝑞 = 𝑞𝝎′ 2 2 𝝎 = 2𝑞 𝑞 ∗ + 2𝑞 ∗ 𝑞 and 𝝎′ = 2𝑞 ∗ 𝑞 + 2𝑞 ∗ 𝑞 1 1 𝑞 = 𝝎 − 2𝑞 ∗ 𝑞 𝑞 and 𝑞 = 𝑞 𝝎′ − 2𝑞 ∗ 𝑞 2 2
(3.12)
Thus, given the quaternion along with its temporal rates of change or the rotation rates of the body in either the reference or body frames, the rotational velocity in either coordinate system is readily available. It is possible to rewrite the rotational equations of motion in terms of quaternions. In order to do so, however, the mass matrix as well as the coriolis and centripetal matrix would need to be reshaped to have a row and column of zeros. Also, the imposition of the unit quaternion constraint in this development would require additional computational 38
overhead[40]. It is more straightforward and computationally efficient to solve the equations of motion for the body frame rotation acceleration and then convert this to quaternion acceleration using the above transforms. The unit quaternion constraint must be enforced following each iteration. Strictly speaking, a model that uses quaternions to track orientation does not need a mapping between the quaternion and Euler angles. The fact remains however that many of the tools in existence for dynamic modeling employ Euler angle representation. Indeed until recently Euler angles were ubiquitous in the fields of modeling vehicle dynamics in all but space vehicle modeling. Therefore, given below is the mapping from Euler angles to quaternions. Evident again is the singularity present when mapping from quaternions to Euler angles. Thus, there exist orientations that are expressible using quaternions, but which are not available to the more conventional Euler representation. 𝑞0 = cos 𝑞1 = cos 𝑞2 = cos 𝑞3 = sin 𝜓 = atan
𝜓 𝜃 𝜙 cos cos 2 2 2 𝜓 𝜃 𝜙 cos sin 2 2 2 𝜓 𝜃 𝜙 sin cos 2 2 2 𝜓 𝜃 𝜙 cos cos 2 2 2 2 𝑞0 𝑞3 + 𝑞1 𝑞2 1 − 2 𝑞2 2 + 𝑞3 2
𝜓 2 𝜓 −sin 2 𝜓 +sin 2 𝜓 −cos 2 +sin
𝜃 2 𝜃 sin 2 𝜃 cos 2 𝜃 sin 2 sin
𝜙 2 𝜙 cos 2 𝜙 sin 2 𝜙 sin 2 sin
(3.13)
𝜃 = asin 2 𝑞0 𝑞1 + 𝑞1 𝑞3 2 𝑞0 𝑞1 + 𝑞2 𝑞3 𝜙 = atan 1 − 2 𝑞1 2 + 𝑞2 2 Instituting these developments, the equations of motions are solved using the following notation.
39
Hull Inertial Coordinates (6 DOF) : T X X1 , X 2 ,
x X1 y , z
q0 q X2 1 q2 q3
Body Frame Coordinates (7 DOF) : η 1 ,2 ,3 , 4 ,5 ,6 ,7
T
ν1 u ν ν 2 , ν1 v s w Generalized Force :
F F1 , F2 , T
p ν 2 q r
X F1 Y Z
T
K M F2 N Tp
(3.14)
T
The hydrodynamic force and body accelerations are calculated in the body frame of reference. These accelerations are converted to quaternion accelerations and the time evolution of the orientation is carried in quaternion form.
3.5 Rigid Body Motion Validation
The above formulation was validated against the dynamics simulation software system Adams View™. The primary objective of the validation is to achieve a level of confidence in the seven degree of freedom motion evolution under general forcing. The model considers periodic torque applied to the body about the body frame y axis. The tools available within Adams did provide a straightforward means to model energy extraction. The model does provide strongly nonlinear behavior in the orientation of the bodies. The frequency of excitation is 5.2 radians per second. The maximum difference in displacement between the models occurred in the pendulum displacement. A comparison of the angular displacement time series is given in Figure 3.3. The maximum difference between the Adams View model and the calculation being evaluated is 0.02 radians.
40
41 Figure 3.3 - Seven DOF Motion Validation - Body Frame Moment ( F5=cos(5.2*t) ).
Chapter 4 Hydrodynamic Model – A Desingularized BIM with Weak Scatterer Free Surface Boundary Condition
4.1 Overview This chapter presents the detailed formulation for the fully nonlinear time domain simulation of the body motions in steep waves. The size of the body is unconstrained by the magnitude of the wave amplitude. Given this relaxed constraint, application of methods that commonly linearize about the calm water free surface are not allowed. The hydrodynamic problem is solved as a potential flow problem using the weak scatterer approximation for development of the free surface evolution equations. A proven desingularized boundary integral method (BIM) is used to solve a potential flow problem. It is possible to superpose other forcing models, such as thrust or viscous effects on the hull and appendages, but these problems will not be dealt with specifically in this work. Rather, the focus is on exploring the applicability of the model to conditions that were experimentally demonstrated to display highly nonlinear behavior. Solution of the hydrodynamic problem results in a time dependent force that is input as the generalized force vector for the fully non-linear coupled equations of motion that are derived for the seven degree of freedom (DOF) system developed in Chapter 3.
4.2 Hydrodynamic Formulation This section develops the ideal fluid potential flow boundary value problem used to solve the hydrodynamic problem. We consider an arbitrary hull geometry floating in a seaway ( Figure 4.1). The fluid domain, 𝑉, is defined by four surfaces: Sh defines the wetted surface of the hull, Sf defines the fluid free surface, Sb the bottom and St a truncation boundary assumed to be far enough from the body to be considered at infinity. The fluid velocity within this domain may be represented by the gradient of a scalar
42
velocity potential Φ 𝑥, 𝑡 . Continuity must be satisfied in the fluid domain. This constraint reduces to Laplace‟s equation for potential flow. 2 0
x V
(4.1)
Weak Scatterer Free Surface Conditions 2 Boundary 1 1 20 0 0 1 1 0 t z z z RAD DIF U t 0 0 g 1 U t 0 t z t t With : U t 0 t t 0
Sf
St
0 x, t
Sh
1 0 as R
U n n DIF 0 n n
2 0 x V
x, t 0 x, t 1 x, t 1 x, t DIF RAD 0 O 1 , U t O 1
DIF O , RAD O , O
Sb Figure 4.1 Boundary Value Problem Definition.
Conservation of momentum under potential flow leads to the Navier-Stokes equations and then to the unsteady Bernoulli‟s equation,
p
1 2 gz Ct t 2
x V
43
(4.2)
A useful feature of this formulation is that the problem may be solved independently of the pressure problem. The approach to solving this problem then becomes one of setting up a number of boundary value problems that satisfy 4.1 along with some boundary conditions on the surfaces enclosing the fluid domain, 𝑉. Initial conditions must be specified that correspond to the physical problem being described. Solid boundaries, such as the body and the bottom, require that the fluid not flow across the boundary. Potential flow describes the fluid as inviscid, so that the tangential flow to a solid boundary is unconstrained. Thus, a solid boundary condition requires that, VB n n
on Sh and Sb
(4.3)
where VB represents the normal velocity of the boundary. For conditions that justify deep water assumptions, the appropriate boundary condition on the bottom requires the gradient of the disturbance to vanish. Similarly, a radiation condition on the free surface is required for outgoing waves at infinity. In deep water, the gradient of the disturbance potential vanish as 𝑧 tends to -∞. Truncation of the free surface to a computationally manageable domain is necessary. A simple truncation of the free surface at an arbitrary distance from the body would result in reflected waves and would violate mass conservation. This work uses a numerical beach to gradually absorb energy before the outgoing waves reach the boundary [28]. On the free surface, both kinematic as well as dynamic conditions must be imposed. Surface tension is assumed to be insignificant for the parametric domain significant to this problem. The kinematic conditions require that a particle on the surface remain on the surface. The dynamic condition requires the pressure to be equal to the ambient, customarily given as zero. Defining the surface elevation as 𝜂 𝑥, 𝑡 , these boundary conditions can be stated as:
44
0 t z on S f 1 g 0 t 2
(4.4)
The numerical exploration of this problem will be based on a well documented method[3; 49]. We employ simple desingularized Rankine sources to model the flow on the free surface and panels of constant strength[17] to model the flow on the body surface. It is at this point that we depart from most previous potential flow formulations. It is common to linearize the free surface boundary condition about z=0. This is not reasonable for our purpose, primarily because the size of our body may be of an order of magnitude smaller than that of the wave amplitude. Given the free motions of this body, such an approach could result in evaluating pressures on a completely submerged or dry body. Thus, we require a reasonable method of modeling hydrodynamics given the actual motions of the body while capturing as closely as possible, the resulting wetted surface due to the ambient wave climate. Pawlowski[33] first suggested the weak scatter approximation in order to extend the capacity for potential flow boundary value problems with respect to ship-wave interaction. The resulting theory promised the ability to allow treatment of motions in steep ambient waves. Sclavounos[41] explored the weak scatterer approximation further by investigating application of this model to a slender body. Again, the emphasis is placed within the analysis on motions in steep ambient waves. The problem being studied here must accommodate motions of the body in long waves of amplitude comparable to the body dimension. The primary assumptions of the weak scatterer approximation may be summed up by the statement that the waves generated by the body are much smaller than the ambient waves. This is a reasonable approximation for a wide range of marine problems. The suggested formulation allows for the independent modeling of the ambient wave environment. That is, the disturbance waves do affect the ambient wave field. The conditions of interest are bodies of characteristic length that are similar to ambient wave
45
amplitudes. If the characteristic length is much larger than the largest wave amplitude present then we would expect reasonable results from a linear formulation that calculates hydrodynamics on the mean wetted surface. On the other hand, if the characteristic length is much smaller than the smallest wavelength present, the body would be expected to move with the free surface, generating virtually no disturbance. It is convenient to decompose the velocity potential within the fluid domain into physically significant components. The initial decomposition divides the potential into two components, 0 1
(4.5)
where 0 represents the ambient wave potential and 1 represents the disturbance wave potential. The disturbance wave problem will be started from rest. That is,
1 0,
1 0 t
t 0, in the fluid domain.
(4.6)
The weak scatterer approximation assumes that the disturbance, caused by the presence and motions of the body, is an order of magnitude smaller than the ambient wave climate over its free surface.
1 0 1 0
(4.7)
Next, assume an arbitrary reference frame with time dependent velocity U t , and let the operator
represent U t 0 . Expand equations 4.5 with the t t
scaling for of equations 4.7, retaining terms to first order, 1 0 1 0 1 0 t z z t 1 1 U t 0 0 0 g0 g1 t 2 t
46
(4.8)
Equations (4.8) are valid, to first order, on z . In the same way that the free surface is commonly linearized about z 0 , the formulation proceeds with the approximation that these conditions apply on z 0 with small error. Taking the Taylor series expansion results in equations (4.9), which are valid, to first order, on 0 . 1 0 0 1 t z t z 2 0
U t 0 0 1 1 0 2 z z
(4.9)
1 1 0 0 0 g0 t t 2
0 U t 0 1 g t z z
Noting that the first three terms on the right hand side each of equations (4.9) are, by definition, zero within the assumptions of this analysis, the following equations represent the free surface boundary conditions that must be integrated in time. 0 0 1 1 0 z on Z 0 0 1 g 1 t t z 2 1 1 20 t z z
(4.10)
The formulation thus far would support simulations for prescribed motions of the body. A numerical stability problem arises, however, for simulations requiring time evolution of the body motions. It is known that numerical stability depends on isolating the highest order derivative of a differential equation[41]. Recall that the equations of motion, (3.5) equate the hydrodynamic forces with the time rate of change of momentum
47
of the rigid bodies. The proposed formulation for the hydrodynamic force is, unfortunately, dependent on the acceleration of the body.
F F1 , F2 F p T
,X , t 1 gz p p X, X t 2
(4.11)
Therefore, a formulation that allows for a separation of the impulsive component of the pressure should be sought. Sclavounos suggests the following decomposition. The disturbance potential is separated into two components, one diffracted and one radiated. The diffracted potential has the customary body boundary condition, which invokes the Haskind relation on the instantaneous position of the hull. The radiated potential is further divided into separate components, one that carries memory effects and one carrying impulsive effects. The impulsive potential satisfies the kinematic body boundary condition and is set equal to zero over the free surface. The memory component carries the free surface effects, but its body boundary condition is set equal to zero. Thus, the sum of the two radiated components satisfies the usual boundary conditions.
x , t 0 x , t 1 x , t 1 x , t RAD DIF and RAD RAD x , t 0 x , t DIF RAD
(4.12)
Following is a summary of the potential ordering and the boundary conditions associated with each. 0 O1 ,
RAD O , DIF O
0 x, t : Ambient wave potential, known a priori.
1 x, t : The total disturbance potential.
48
1 v n on Sh. n
DIF x,t : Customary treatment of the diffracted waves. DIF 0 on instantaneous Sh, n n
DIF over 0 is initially zero and time stepped for each iteration thereafter. : Impulsive part of the disturbance potential due to accelerating body. v n on Sh n
0 over 0 . The impulsive potential is absorbed by the radiation body
boundary condition.
RAD : Memory part of the radiated disturbance potential responsible for outgoing waves.
RAD 0 on SB. n RAD is initially zero over 0 and time stepped for each iteration thereafter. An important facet of this decomposition is that it isolates the impulsive hydrodynamic force that allows the equations of motion to be written in a form that allows for stable numerical integration. It is straightforward to solve boundary integral problems for the three individual potential values described above. However, computational savings can be realized by combining the diffraction and radiation memory conditions into a single boundary integral problem. This is the method used to
49
solve the hydrodynamics in this work. Given the above ordering and potential separation, the free surface boundary conditions can be shown to reduce to:
1 20 1 2 0 0 1 1 0 t z z z
on Z 0
RAD DIF 0 g 1 t t t z
(4.13)
4.3 The Boundary Integtral At each time step, the unknown potentials or their gradients can be determined using well established boundary integral numerical methods given the above stated boundary conditions. The perturbation potential anywhere in the fluid domain can be represented as an integral of the form:
X1 X1D G X1 ; X1D d 1 X1
with
G X1 ; X1D
1 X1 X1D
1
x xD y y D z z D 2
2
is the integration surface X1 is the real surface enclosing the fluid boundary
2
(4.14)
X1D is the integration surface
X1D is the unknown source strength
The normal velocity of the disturbance potential is known on the body. The value of the potential over the free surface is also known from the previous time step. If the boundary is appropriately discretized, the source strength in the resulting boundary integral equations may be numerically determined by solving a system of equations defined by the following boundary integral equations,
50
X G X ; X d X 1
1
1D
1
X1 Free surface
1
X n G X ; X nd v X n 1
1
1D
1
X1 Hull surface(4.15)
where
is the normal derivative with respect to the solid boundary. n
Discretization of these boundaries into their source representations results in a system of linear equations in the discrete source strength. Solution of the system gives the individual source strengths over the surfaces of integration. This process must be carried out for each of the components of the disturbance potential, the memory (diffracted and radiated) and the radiated impulsive potentials. The total potential is then available discretely over the integration surfaces. The time evolution of the free surface boundary conditions is integrated using an Adams Bashforth 4th order scheme. Many numerical methods for approximating the values between the discrete points are available. When necessary, the potential will be approximated using cubic splines. Solution of the boundary values problems provides discrete values for the potential and its derivatives so that pressure can be calculated over the surface of the body and the free surface can be evolved in time.
4.4 Hydrodynamic Forcing and Rigid Body Motion The pressure over the body surface can be evaluated, to second order in terms of the potential decomposition as follows:
51
F1 pds, Sb
F2 p r n ds Sb
t 0 DIF RAD p 1 g z 0 DIF RAD 0 DIF RAD 2 pHS g z
(4.16)
1 pFK 0 0 0 t 2 DIF RAD pMEM 0 DIF RAD t pIMP 0 t
Kring and Sclavounos[22] show the necessity to isolate all dependence on accelerations to one side of the equations of motion. Thus, the acceleration dependent component of the impulsive force must be moved to the right hand side of equation (3.5). This is accomplished by determining the analytic, infinite fluid added mass for the instantaneous wetted geometry of the body. Beck, et al.[2] showed that
𝜕𝜙 𝜕𝑡
can be solved
for directly by setting up a similar boundary value problem to those already discussed. The right-hand side of their equation (21) can be split into components that are velocity and acceleration dependent. Similarly, Newman[31] exploits the decomposition of the impulsive potential into six individual components that are associated with the six degrees of freedom. The acceleration dependent terms can then be used as the boundary conditions to solve for the instantaneous added mass matrix for the wetted hull geometry as (4.17). The rotational velocity dependent terms become the body boundary condition for a velocity coupled matrix as in (4.18)
𝑚𝑖𝑗 = 𝜌
𝜙𝑗 𝑆𝐵
𝜕𝜙𝑖 𝑑𝑆 𝜕𝑛
52
(4.17)
𝜌
𝜙𝑗 𝜔 × 𝑛1:3 𝑑𝑆 𝑆𝐵
𝑏𝑖,𝑗 = 𝜌
(4.18) 𝜙𝑗 𝜔 × 𝑛4:6 𝑑𝑆
𝑆𝐵
where 𝜙𝑖 represents the velocity potential for body motion of unit velocity in the ith mode of motion. The 𝑏𝑖𝑗 term is velocity dependent but should not be confused with wave damping. It represents the impulsive force due to the time rate of change of the body normal caused by the rotation rate of the body frame. This representation lends itself readily to solution provided the influence matrix already determined for the boundary integral equations. Finally, the equations of motion may be written with the forcing terms made up of the memory and ambient components as well as a velocity dependent impulsive term. The terms representing the impulsive part of the added mass term are moved to the right-hand side. The resolution of stability issues in the hydrodynamic problem does not affect the kinematic solution for the seventh degree of freedom so that the seven equations may be written,
F1 pHS pFK pMEM nds b1:3,1:6v P m1:3,1:6v Sb
F2 pHS pFK pMEM r n ds b4:6,1:6v H m4:6,1:6v
(4.19)
Sb
𝑇𝑝 = 𝑚𝑝 𝑟𝑝 𝑅𝐵𝑝 𝐼 𝑎𝑚′ ⋅ 𝑒𝜂 7 + 𝜂7 𝑟𝑝
4.5 Numerical Method The discretization of the integral boundary equations results in fourteen individual problems (diffraction, impulsive, added mass (6) and rotational velocity coupled (6) ) that have identical geometries. There are now a total of fourteen boundary integral equations that must be solved based on a single influence matrix. This is important in choosing a method for solving the problems. The method used for solution depends on the size and condition of the coefficient matrix. Residual methods for solving systems of equations
53
may have advantages for very large grids because the solution requires O(n2) operations. However, each of the eight integral problems must be solved independently using such methods. On the other hand, LU decomposition of the influence matrix once (O(n3) operations) results in a direct means to solve any of the eight problems in O(n2) operations. Simple Rankine sources will be used to model the free surface boundary. The ability to employ simple sources, as opposed to constant strength panels, reduces the computational overhead in generating the influence matrix. They are located a small distance Ld above the ambient free surface in order to avoid singularities in the integration.
Ld Dm , with Dm equal to the local mesh dimension.
(4.20)
A method to account for the truncation boundary has yet to be defined. This work employs numerical damping as described by Lee[28] . In this method the free surface is divided into an inner and outer (numerical beach) domains. The grid spacing within the inner domain was shown to require 30 nodes per wave length by Zhang[49]. A gradual increase in the spacing of the free surface nodes in the outer region numerically damps out the outward wave propagation, minimizing reflection. The formulation of the free surface boundary conditions (Eq. 4.13) allows for updating the free surface given the instantaneous translational velocity of the hull. Thus, the simulation updates the horizontal position of the free surface nodes using the instantaneous horizontal plane position of the hull. The changing water plane has a velocity contribution in the radial direction at each time step and is appropriately added to this body velocity. Vertical position of the sources is determined by the ambient free surface elevation along with the desingularized distance.
54
4.6 Initial Numerical Validations The individual components of the numerical solution can be validated separately as a first step in building confidence in the model. It is customary and natural to separate the hydrodynamic force into the diffracted and the impulsive, or radiated force. As discussed above, the total velocity potential is divided into separate components that represent these individual problems. Φ x, 𝑡 = ϕ0 + ϕRAD + ϕDIFF
(4.21)
The behavior of the overall solution was evaluated by evaluating the force for the individual problems associated with radiation and diffraction. A unit sphere whose center is initially located at the calm water z=0 plane is used for this validation. The diffraction problem is compared to the results of Cohen [5] in Figure 4.2 and Figure 4.3. Good agreement is demonstrated for both heave and surge.
55
Figure 4.2 Vertical Diffraction Force on a Unit Sphere.
56
Figure 4.3 Surge Diffraction Force on a Unit Sphere.
A comparison of the added mass and damping coefficients for sinusoidal prescribed motions of a sphere in both heave and surge was also conducted. Heave results are compared to the analytical results of Hulme [20] in Figure 4.4 and Figure 4.5. Good agreement is demonstrated again, but it should be noted that wave damping is over predicted in heave and under predicted in surge at higher frequencies. Convergence studies in time, body panel, and free surface panel resolution were also conducted, the results of which can be found in Appendix C.1. Consistent with the findings of Zhang[49], the following values will be used in discretizing the numerical computations. The time step is set to Tp/100, where Tp is the wave period. The body is
57
divided into 16 radial sections, each having nine sections and the free surface resolution within the inner region is divided into thirty nodes per wave length.
Figure 4.4 Heave Added Mass and Damping for a Unit Sphere.
58
Figure 4.5 Surge Added Mass and Damping for a Unit Sphere.
59
Chapter 5 Experiment
5.1 Introduction The purpose of these experiments was to generate a data set that characterizes the free motions of a buoy subject to forcing from incident waves and simultaneously coupled to an energy extraction device. The intended use of the data set is to provide validation for a numerical simulation of the same. Validation of the proposed numerical simulation requires four time series. First, the incident wave climate must be known for the duration of the experiment. Second, the rigid body motion of the buoy hull must be tracked for the duration of the experiment. Third, the power take off system must be characterized in a manner that allows for calculation of energy extraction. Finally the coupling between the power take off and the body motions must be captured. An experiment was proposed that would rely primarily on an infra red (IR) vision system to measure both wave climate and rigid body motions. The University of Michigan Marine Hydrodynamics Laboratory has recently acquired a six dimensional (6D) motion capture system. Standard output from the system facilitates direct measurement of the position and orientation of the buoy hull as well as that of the pendulum. However, use of the system for characterizing the free surface needed to be evaluated. Therefore, the initial part of the experimentation set out to explore the utility of using buoyant IR markers to characterize the free surface. The intention was to use this type of measurement to characterize the free surface at the time dependent horizontal position of a freely floating body. It was assumed that the mean translational velocity of the body in the direction of wave propagation would not differ greatly from that of the second order drift associated with the waves.
60
The power take off for the experiment is a planar pendulum oriented to rotate about the calm water vertical axis of the buoy. Power takeoff is to be achieved through the differential angular motions of the buoy and the pendulum.
5.2 Experimental Setup Experimental setup and configuration of the sensors is shown in Figure 5.1. This figure is applicable to both the fee surface characterization and the rigid body motion experiments. The experiments were conducted in the large towing basin at the University of Michigan. The basin measures 360x22 feet with a depth of 10.5 feet. The incident wave height was measured using acoustic as well as capacitive wave probes. The analog signal from the probe was sampled at 50Hz with a 4Hz low pass filter. Rigid body motion capture was accomplished using an infra red vision system, the specification of which, are given in Appendix A. The motion capture system employs four (4) high speed infra red cameras to capture the time dependent position of discrete markers, or rigid bodies defined by a minimum of four distinct markers. The vision system is calibrated by moving a wand through the measurement space. The wand is defined by two markers separated by a known distance. Simultaneously present within the space is a calibration device that is defined by four precisely positioned markers. Results of the calibration include the standard deviation of the error in the measured length of the wand determined during the calibration. Calibration results for the vision system are presented in Appendix B.
61
X
X c(t)
V
IR Motion Capture System X 62
b(t)
Z v
Accoustic z
Y V 𝑋𝑏 (𝑡) v z z z v
X
Wave Probe
Z
z y b
0
xz
O
z
X
0
𝜂 𝑡 z Figure 5.1 IR Vision Based Tracking Experimental Setup.
z z
5.3 Free Surface Characterization Characterization of the incident wave climate is a critical component of any experiment in seakeeping. The first experiment sought to explore the capacity for the IR vision system to capture the wave climate. The vision system is capable of simultaneously tracking the individual trajectories of many discrete objects. The idea for the assay was to employ low density reflective markers floating on the free surface. The time dependent trajectories of the markers could be splined at each time step. The spline coefficients could then be used to estimate the conditions at any arbitrary position along the spline. In this manner it was hoped that the free surface characteristics could be accurately measured for some distance in the direction of wave propagation. Further, it was hoped that the markers and buoy would have similar drift velocities, allowing for the direct measurement of the incident wave climate at the time dependent position of the buoy. An ideal marker would exactly follow the trajectory of the water particles on the free surface. Thus, an experiment was performed that both measured the incident free surface while simultaneously tracking the trajectories of the IR markers. The markers needed to satisfy two conditions. They needed to present a profile that could be accurately detected by the vision system while also moving with the free surface flow. It was thus assumed that the marker must be a small low density body so that a sufficient profile is presented to the cameras for recognition. A design trade-off was recognized at this point. That is, the same profile necessary for recognition also presents a geometry that can interact with the relative motion of air to the local water particles. The air drag resistance generated by this interaction could affect the motion of the marker. Additionally, the low density markers would displace little water and thus present a low drag geometry to the water. Therefore, it was proposed to attach a keel to the spheres to encourage the accurate tracking of the free surface particle trajectory (Figure 5.2). In effect a nearly neutrally buoyant structure is added to each ball so that the profile present to the water is similar in scale to that presented to air. In this way, the effect of air resistance on the motion of the markers is minimized.
63
A number of markers were distributed in the direction parallel to wave propagation. The distance separating adjacent markers would be determined by the minimum wavelength to be resolved. The experiments were performed with eight markers distributed on the free surface. A list of test parameters may be found in Appendix B. Figure 5.3 shows a post processing image of the marker positions along with a portion of the trajectories from an arbitrary run.
Figure 5.2 Low Density Floating IR Reflective Markers.
64
Figure 5.3 Screen Shot of Surface Characterization. A study of the horizontal displacement of the markers was conducted. The time series of the horizontal position was plotted and the mean drift velocity of the markers was deteremined over a minimum of 10 wave periods. The theoretical second order drift velocity was determined using the mean amplitude of the waves and their frequency(5.1). A comparison of the exerimentally determined drift versus the analytic prediction (Figure 5.4) shows good agreement. (5.1) Stokes Second Order Drift =𝜔𝑘𝑎2 𝑒 2𝑘𝑧 The results of the study show that the vision system could be used to characterize the free surface in terms of elevation as well as velocity gradient with little error to second order accuracy. Unfortunately, it was found during the subsequent testing that for certain parameter ranges, the buoy motions greatly exceeded second order drift. This result precluded the utility of using floating markers to characterize the free surface during the intended experiments. This is not a statement that the assay has no value. The results do show that the procedure would be valuable in characterizing free surface characteristics including second order effects.
65
Figure 5.4 Theoretical Second Order Drift vs. Measured Drift Speed. Although the method was abandoned for the ultimate purpose of these experiments, the resulting measurements were interrogated further for possible future applications. A comparison of the surface elevation measurements by the vision system versus those of an acoustic wave probe was performed. The results of an arbitrary test are shown in Figure 5.5. The error referenced has two components. The first is the error associated with the accoustic wave probe. The second component is from the standard deviation of a dynamic calibration process for the camera system setup. The dynamics of the markers are thus unaccounted for. The wave amplitude determined by the vision system is higher than that of the acoustic wave probe. While the study did not pursue the source for this error further, two likely sources are suggested. First, the reflection of the markers off of the surface at the crest and trough could result in spurious results due to the maximums in curvature. A study of the effects of camera placement could yield
66
guidance in minimizing light reflection issues. Second, the dynamics of the markers themselves may affect the measurements. A frequency based study would isolate behavior of this nature.
Figure 5.5 Vision System vs. Acoustic Wave Probe.
5.4 Buoy Motions Experimental Setup The simple hemispherical (0.5m diameter) hull design of Figure 5.6 was conceived for the purposes of this experiment. A hysteresis clutch was employed to facilitate the transfer of energy from the relative motion between the buoy and the pendulum. The pendulum was located inside the buoy hull but rigidly attached to a shaft that was coupled to the clutch and extended through the top of the buoy. The vision system requires a minimum of four points to define a rigid body. As the body orientation and position within the measurement volume changes, individual markers can become obscured by the body or each other. For this reason, better recognition can be expected as the number of points defining the rigid body increases. The rigid body definition of the pendulum consisted of marking two points along the vertical and three other points rigidly attached to the shaft. The definition of the buoy rigid body consisted of seven markers total, five markers placed around the top circumference along with the two markers used to define the vertical pendulum shaft. The model was tested in three
67
different conditions consisting of; Case I - a 1.0 Kg pendulum, Case II - a 0.25 Kg pendulum and Case III - no pendulum.
Table 5.1 Buoy Particulars (3 Configurations). The test conditions were chosen in order to generate nonlinear response of the hull to the wave forcing. The overall purpose of the experiment, again, was to provide precisely the data set necessary to explore the efficacy of the proposed numerical model, or indeed any similar candidate model for the hydrodynamics in a wave energy extraction application. The natural frequency in pitch or roll of the buoy in the Case III configuration was experimentally determined to be 0.79±0.01Hz. The heave natural frequency was determined to be 0.99±0.02Hz. The excitation for the pendulum is expected to be a function of both the accelerations of the hull as well as the time dependent influence of gravity due to pitch and roll motions. The testing conditions were therefore designed to generate response of the buoy in the neighborhood of the pitch/roll natural frequency, 0.7-1.0 Hz.
68
ZVT
RB
RP ZP T ZCG
BL
Figure 5.6 Schematic for buoy physical dimension references.
Initial testing resulted in a somewhat surprising behavior. The buoy drift velocity was much higher than the drift velocity of the ambient waves and the low density markers. The difference in velocity was so great that it became evident that using the free floating markers to characterize the free surface would be impractical for this experiment. Therefore, it was decided to rely entirely on conventional single point methods for acquiring incident wave characteristics.
69
The distance from the wave probe to the imaging system coordinate frame was determined after calibration of the vision system. This was accomplished by physically locating a reflective marker so that the vision system location of the marker in the vision system global coordinate frame was zero. The distance from this position to the wave probe was then physically measured. This physically located the wave probe within the vision system coordinate system. In this manner the distance from the wave probe to the buoy could be determined as long as the vision system was able to resolve the buoy rigid body position Rigid motion capture for the experiment requires the position and orientation of the hull along with the orientation, in yaw, of the pendulum. The output of the vision system gives the six degree of freedom position of the hull and pendulum. There are, primarily, four measures of interest from the time series collected by the instrumentation: 1. The instantaneous position and orientation of the buoy hull and the pendulum. These measures allows for the calculation of velocities and accelerations for both bodies. 2.
The instantaneous relative velocity between the yaw velocity of the hull and the pendulum. This measure multiplied by the torque associated with the hysteresis clutch is a measure of power being dissipated through it.
3. The earth fixed position in time of the hull CG. This measure gives the drift velocity of the body. 4. The ambient wave climate in the vicinity of the buoy. This will be used to guide generation of the input wave climate for the numerical simulation.
5.5 Energy Extraction A hysteresis clutch was used to extract energy from relative rotational motion between the pendulum and the buoy hull. The hysteresis clutch provides a torque that resists rotational motion and is independent of rotational velocity. This departs from the model used for energy extraction in the simplified model explored in Chapter 1. The complexity and cost of the experiment are considerably simplified by using the hysteresis
70
clutch because a simple DC voltage is all that is required to control a steady torque. The torque standard deviation was a maximum of 0.8% of the mean across the voltage range tested during the calibration of the clutch. A velocity dependent torque system would require complex control systems and hardware. The difference in dynamics for the two approaches can be addressed with numerical experiments upon validation. It is suggested that the difference in the gross dynamics will be small as long as the torque values allowed are small relative to the expected inertial excitation. A calibration of the clutch was performed by measuring the torque output from the clutch using a load cell. During the calibration testing, a constant rotational velocity was input to the clutch. Torque was transferred from the clutch output to the load cell by wrapping a cable around a 2.54 cm diameter pulley and connecting it to the load cell so that the line of action was parallel to the line action of the load cell and tangent to the pulley to within measurable tolerance. The load cell was sized by considering this setup as well as the likely range of testing torques desired. The prescribed motion analysis described in Chapter 1 indicated that optimal energy extraction might be expected with a torque value in the neighborhood of 0.15 Newton-meters. Therefore, a load cell (Sensotec Model 41BN) with a range of 220 Newtons and a hysteresis clutch (MAGTROL HC500-3) with adjustment sensitivity in this range were selected. Voltage across the clutch was measured using the data acquisition system. This in turn was validated against a handheld Ohm meter (TekPower DT9602R) to 0.01 volts. The meter was used during testing to check the voltage across the clutch prior to each test. Data was gathered for voltages ranging from 0 to 6.2 Volts. The results were regressed in both the linear and quadratic sense. Figure A.2 shows the test results and the regression used to convert the DC voltage measured prior to each test into the torque output from the hysteresis clutch. The linear regression results were used for this analysis. The measured voltage during testing translates into a hysteretic torque of 0.17 Nm for the 1 Kg pendulum and 0.06 Nm for the 0.25 Kg pendulum.
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5.6 Error Reporting The measurements of interest presented here include, position and orientation of the buoy and pendulum as well as the power dissipated through the hysteresis clutch. Errors in the position and orientation of the body have two sources: the measurement of the horizontal position of the carriage, and the measurement by the vision system of the position of the rigid bodies relative to the carriage. The carriage position is generated by integrating a time series of the carriage velocity during testing. Also available from the carriage control system is a digital position indicator from a quadrature encoder attached to the carriage drive wheel. The indicator has a resolution of 0.3 mm. This measure was used as truth in a series of tests, performed to determine the accuracy of integrating position from the carriage velocity time series. The standard deviation of the integrated position error was found to be 34mm for the range of operating parameters used during these experiments. As noted earlier in Section 5.2, the standard deviation of the vision system motion capture is determined by the Qualisys software suite during the calibration procedure for the camera system. The calibration instruments are dimensionally similar to the markers used to define the buoy and pendulum. Therefore, the calibration results are used directly as a measure of the expect error within the experiments.
5.7 Experimental Results – No Pendulum
It will be helpful to first consider results for the simplified system modeled in Case III. That is, the buoy configured so that there is no pendulum. In the presence of unidirectional waves, the resulting motions can be expected to have only three degrees of freedom; surge, heave and pitch. A list of the testing conditions and summary of results is given in Table 5.2.
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Table 5.2 Test Matrix and Summary Results: Case III. The leading and trailing waves in a wave group have larger amplitudes than the waves traveling within the body of the group. The wave parameters in this experiment resulted in these transient amplitudes being 10-20% higher than the waves in the steady section of the envelope. The wave amplitude listed in Table 5.2 is the value associated with the amplitude measured in the regular portion of the generated wave train. The free surface elevation in this section of the wave group had a mean elevation of 3.1 cm with a standard deviation of 0.4 cm. The remaining data are then likewise associated with the values or average values measured concurrent to this part of the experiment. The pitch response amplitude operator (RAO) is the ratio of the average pitch amplitude divided by the linear wave slope. Velocity is calculated from the time series for the position of the body center of gravity. The encounter frequency is determined using linear wave theory and taking advantage of the symmetry of the problem. Given the measured velocity of the body, U, and ambient wave frequency, 𝜔0 , the encounter frequency, 𝜔𝑒 is (5.2) 𝜔𝑒 =𝜔0 +Uk cos 𝛽 The analytical stokes drift is calculated from 5.1 using the measured amplitude and reference frame frequency. Finally, a nondimensional form of the velocity given by the ratio of the measured velocity to the analytical second order drift velocity is shown. This last value is plotted versus the encounter frequency in Figure 5.7. The peak response
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occurs at approximately 0.76 Hz. This is slightly lower than the measured natural frequency of 0.79 Hz. The pitch response for the runs displaying fast drift also had large amplitudes, ~0.6 radians. The remarkable physics of this response lies in the fact that the body moves down stream at a velocity that is an order of magnitude greater than that expected, or predicted by second order theory.
Figure 5.7 Nondimensional Drift Speed versus Encounter Frequency. Upon finding the above results, a review of the literature regarding wave drift was conducted. A great deal of literature exists regarding analytical exploration of drift forces on bodies in waves. Stokes is generally associated with some of the earliest second order analysis. Numerous references are available regarding second order forcing analysis[29; 14; 32] but most are based on linear, frequency domain analysis. The velocities measured in this experiment, however, departed drastically from what second order theory would predict. A few recent works do explore, specifically, the drift velocity of two-dimensional and three-dimensional bodies in waves [44; 45; 26; 25].
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Kuroda and Tanizawa[45; 25] explore similarly nonlinear drift phenomenon. They use empirical formulas to solve for the balance between wave drift force and drag due to drift (5.3). Also noted in both of these works is the correlation between pitch amplitude and the drift velocity. Kuroda achieves some success in producing a scheme for calculating the drift speed and roll amplitude in short waves, but admits that the method lacks some nonlinear interactions due to the large roll motions. Tanizawa shows a frequency dependent linear correlation between drift speed and steepness for short waves. Their definition for short waves is less than five (5) times the characteristic body length. 1 𝜌𝐴𝐶𝐷 𝑉 2 2 1 𝐹𝑊 = 𝜌𝑔𝐶𝑅 𝑎2 2 𝐹𝐷 =
(5.3)
The experiments conducted for this work demonstrate a rapid acceleration from zero to the mean drift speed noted. The final velocity is very nearly established within two wave forcing periods. The force necessary for this acceleration averages over double that of the drag force necessary for steady translation. Thus, it is suggested that further study of the transient forcing of these experiments will yield important insights into the nature of the physics that drives the fast drift. Figure 5.8 to Figure 5.14 show details of the time series gathered from the vision system for certain of the above runs. The sequence shows response from low to high frequency relative to the measured natural frequency in pitch. The top graph in each figure shows the heave, roll and pitch during the experiment. Of course very little roll is evident for the runs without a pendulum. However, small asymmetries in the inertia about the z axis can be expected to contribute to some roll. Also, asymetries in the tank geometry and initial conditions can be expected to contribute to some roll perturbations due to cross-wave excitation. The vertical dashed lines bracket the time period used to generate the summary results in Table 5.2. The bottom plot in each figure shows the translational time series for the carriage position along with the position of the buoy within the vision system
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coordinate frame. The two taken together give the position of the buoy in the earth fixed frame. The amplitude of response is small for excitation much lower than the natural frequency. The beating phenomenon visible in Figure 5.8 is caused by the response at the natural frequency, .85 Hz, that combines with the forcing frequency, 0.65 Hz. The drift velocity for this response is double that predicted for the fluid particles on the free surface. This velocity does not significantly change the encounter frequency.
Figure 5.8 Rigid Body Motion – Wave Frequency 0.65 Hz. Run numbers 45, 43 and 41 employ waves (0.75, 0.80 and 0.85 Hz respectively), that bracket the pitch natural frequency of the hull. Figure 5.9 to Figure 5.11 all show that the response is initially very high, indicating forcing that is close to resonant. The response diminishes in each of the three cases and then stabilizes at a lower value, but the physics behind this response differs in each case. Consider this response in terms of the time dependent velocity of the buoy and thus the encounter frequency. The buoy, initially at rest, is excited at a frequency that is near pitch resonance. In Run #45, the incident frequency is lower than resonance and therefore any mean drift will result in the encounter frequency moving further away from resonance. Run #43 shows response to incident waves that match the measured natural
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frequency in pitch. The initial pitch response amplitude is large, as expected, but quickly diminishes. Post processing showed the drift velocity had shifted the encounter frequency to 0.76 Hz. Next, in Run #41, the incident frequency is higher than pitch resonance at 0.85 Hz. The plot shows the pitch response grow and then diminish as the drift speed shifts the encounter frequency to resonance and past to a steady state encounter frequency of 0.76 Hz. Two things are evident in these graphics. First, the down-stream velocity correlates very closely with the amplitude of the pitch response in these tests. Second, apparently, the steady state amplitude for these tests depend largely on how far away from resonance, this drift velocity can shift the encounter frequency into and then past resonance. Next, in Figure 5.12, the frequency of the incident wave set is 0.9 Hz. As would be expected, the initial response is small relative to that exhibited in waves closer to resonance. Again however, the velocity imparted on the buoy by this drift response shifts the wave forcing closer to resonance. This positive feedback loop drives the response to its peak, and in this case the eventual speed again appears to bring the encounter frequency slightly past this high amplitude resonance.
Figure 5.9 Rigid Body Motion - Wave Forcing Frequency 0.75 Hz.
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Figure 5.10 Rigid Body Motion - Wave Forcing Frequency 0.80 Hz.
Figure 5.11 Rigid Body Motion - Wave Forcing Frequency 0.85 Hz.
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. Figure 5.12 Rigid Body Motion - Wave Forcing Frequency 0.90 Hz.
Finally, in Figure 5.13 and Figure 5.14, the incident wave frequency is high enough that the drift velocity imparted by the buoy response is unable to shift the encounter frequency into the neighborhood of resonance. Notice that there is an initial smaller amplitude response that builds and then tapers off in the time frame from 15 to 20 seconds. Analysis of the heave time series confirms that this correlates to forcing from the leading waves in the group. Further, the velocity measured in the time frame from between 20 to 25 seconds is approximately 15% higher than that measured from 25 to 60 seconds. This indicates that the initially higher velocity was imparted on the body by the leading waves in the group. The initially higher velocity shifts the encounter frequency closer to resonance encouraging a larger response. The lower amplitude waves in the steady portion of the wave packet cannot sustain this velocity and the response tapers to a steady state response described by the encounter frequency and the natural frequency of the hull in pitch-roll.
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Figure 5.13 Rigid Body Motion – Wave Frequency 0.95 Hz.
Figure 5.14 Rigid Body Motion – Wave Frequency 1.0 Hz.
5.8 Pendulum Runs This section discusses the testing conducted with a pendulum integrated into the buoy hull as described in the experimental setup section. The test matrices and summary details are provided for Case I & II in Appendix C. Again, initial conditions of the geometry were the same for each test. That is, position of the bouy in the tow tank and the orientation of the pendulum relative to the hull were carefully set at the beginning of
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each test run. The free surface elevation in the regular part of the wave group had a mean elevation of 3.5 cm with a standard deviation of 0.3 cm. The graphical presentations for this section are similar to those of the previous section. Summary data for each plot is constrained to the time window indicated by vertical dashed lines within the plots. This time window was chosen as consistently as possible. The start for each test coincides with the arrival of the main body of excitation wave train. The duration of the time frame was limited in length by a number of factors. Primarily, the duration of the time window was chosen to capture what could be considered transient behavior of the pendulum. Other factors that affected the time window included first the ability of the vision system to acquire the rigid bodies. Occasionally this was limited by a large sway displacement. Also, contact by the buoy with the side of the tank was noted during testing and the time of contact was used to truncate the time series. A third graphic is added to each figure. This top graph shows the yaw displacement of both the pendulum and the buoy hull. The work done by the pendulum on the clutch over each time step is equal to the differential in this orientation multiplied by the torque. The voltage supplied to the clutch is used to calculate the energy dissipated within the selected time window and a power value is calculated given that time frame. This value is integrated in time for each plot. Figure 5.15 shows results of an arbitrary experimental run. This same graphic is provided for each run in Appendix C.
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Figure 5.15 Run 28 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz. As expected the behavior of the system provided a rich body of responses within the experimental test parameters chosen. The hull motions for Case II, the lighter pendulum, closely paralleled the responses of Case III, no pendulum. In this respect, the pendulum forcing for this case is arguably decoupled from the buoy-wave forcing problem. This set of time series are given in Appendix C. This set of data provides a baseline to set parameters to explore response in a numerical simulation for applications that have interest in decoupling motions of the pendulum and hull. The responses for Case I, on the other hand, demonstrate behavior that visibly depart from those without a pendulum present. Figure 5.16 to Figure 5.18 show results for similar test conditions to those in of Figure 5.15. In two of these four cases, the pendulum ultimately tends toward rotation in a single direction, but that direction is opposite for these two runs. In contrast, the other two tests show the pendulum starting to revolve in one direction and then reversing direction multiple times. This behavior does
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have similar characteristics to the higher energy oscillations that are demonstrated in the prescribed motion analysis of Chapter 2.
Figure 5.16 Run 13 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
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Figure 5.17 Run 22 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
Figure 5.18 Run 23 - 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
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Evaluating power takeoff relative to test parameters is the pragmatic focus of the application being evaluated. Two methods of evaluating this performance measure are suggested; power vs. incident frequency and power vs. pitch/roll amplitude. Figure 5.19 shows a graphical summary of these test results. The power versus frequency shows consistantly low output for low frequency excitation. Also, it shows extreme variablilty of performance at the same frequency. This is evidence of the highly nonlinear behavior of the system and its sensitivity to small changes in test parameters including initial conditions. Power versus the amplitude of the response is also shown. A linear regression of this metric shows a positive correlation. The low R2 value is again evidence of the nonlinear nature of the system being examined.
Figure 5.19 Experimental Power versus Wave Frequency and Pitch/Roll Response (R2=.54).
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5.9 Experimental vs. Numerical Results
The numerically calculated pitch natural frequency was 0.81 Hz as compared to the experimentally measured value of 0.791±0.014 Hz. The comparison of the numerical versus experimental measures for heave natural frequency are 0.95 to 0.987±0.036 Hz, respectively. The sensitivity of this measurement relative to the inertial properties of the hull was explored. It was found that the numerical pitch natural frequency could be made to match the experimental value by relatively small changes in the pitch gyration radius. The calculations were, nonetheless, conducted with the inertial values as calculated. With this in mind, certain frequency discrepancies will be expected. Numerical computations were conducted for similar parameter ranges to those explored in the physical experiments for both the rigid body and pendulum cases. Figures showing the resulting time series for these simulations are given in Appendix C. The gross physical behaviors described earlier in this section are clearly present. As an example, Figure 5.20 would be expected to correlate to the parameters for the experimental run shown in Figure 5.10. In these experiments, the incident waves are very near pitch resonance. Thus, the response is initially very high, but the downstream drift imparted to the hull causes the encounter frequency to shift away from resonance. The final drift velocities compare very well. The numerical calculations depart from the physical experiments at higher frequencies. Figure 5.21 correlates to parameters of Figure 5.13. It appears that the downstream velocity is higher in the numerical experiment. The resulting shift in encounter frequency brings the excitation closer to resonance and thus the higher response. The cause for this discrepancy cannot be completely accounted for by errors in the inertial property measurement. It is suggested without evidence that nonlinear viscous effects are largely responsible.
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Figure 5.20 Numerical Calculation for Rigid Body motion – K=2.8.
Figure 5.21 Numerical Calculation for Rigid Body motion – K=3.6.
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Figure 5.23 shows a summary of results of numerical simulations compared to experimental results for case III, no pendulum. The fast drift associated with higher frequencies in the numerical method is evident. The same data are presented in Figure 5.23, but the drift velocity is shown versus the encounter frequency. The peaks in drift velocity occur, similarly, at lower frequencies than pitch resonance. This behavior indicates a strong correlation with pitch natural frequency.
Figure 5.22 Drift Speed vs. Wave Frequency – Simulation and Experimental.
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Figure 5.23 Drift Speed vs. Encounter Frequency – Simulation and Experimental.
Finally, simulation parameters for the full seven degree of freedom system are explored and results compared to the physical experiment. The results from the experiment indicate that different solutions are possible for small perturbations in the initial conditions. Therefore, repeated numerical calculations are performed considering small variations in the initial conditions. The numerical simulation parameters are set to those of Case I, the 1.0 Kg pendulum with hysteretic resistance torque of 0.17 Nm. Response to regular waves of constant amplitude is calculated. Excitation is generated from linear monochromatic waves of amplitude is 3.2cm. Viscous effects are modeled using a drag coefficient of 0.5 for translation and 0.016 for rotation. The constant torque used to model the hysteresis clutch is 0.17 Nm. The wave frequencies are varied from 0.7 to 1.1 Hz. The response at each frequency is calculated for small variation of the initial orientation of the pendulum relative to the wave propagation direction. The initial
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orientation relative to the wave propagation is varied, in 0.1 radian increments, from 1.0 to 2.0 radians. The relative rotation rate between buoy and pendulum is initially zero. Results of the numerical experiment are given in Figure 5.24. Experimental results are reproduced for convenience in comparison.
Figure 5.24 Power versus Frequency -- Numerical and Physical Results. The graphic shows very good agreement in the behavior of a highly nonlinear system. A visual comparison of the computations to physical experiments for wave frequencies that show large power variations reveals similar behaviors. Low power conversion is associated with conditions for which the pendulum rotation rate is near zero and has a coincidental orientation that is aligned with the wave propagation direction. In such a situation, the pitching and accelerations of the hull have no moment to act on the pendulum and excitation is essentially removed. There are other possible sources for perturbations in conditions between the experiments and the computations. These may include but are not limited to asymmetries in reflected waves from the tank sides as well as inertia measurement errors and asymmetries. The exploration of initial pendulum displacement performed here is suggested as a convenient and realistic perturbation parameter.
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Chapter 6 Summary, Conclusions and Future Work The extraction of wave energy as a means to augment the powering requirements of a data buoy is the primary motivating factor for this research. The specific application of a planar pendulum contained within a floating hull and coupled to an energy extraction device is suggested for the purpose of validation. The work presented herein has contributions from four primary components. First, the simplified model in Chapter 2 is developed to gage reasonability of the application. The model imposes prescribed motions on the body that greatly simplified the free floating, fully coupled seven degree of freedom system. Computations from this simplified model gave an estimate of the energy extraction capacity for a specified parameter range. It is shown that the energy conversion capacity for this model are on the order of that demanded by data buoy applications. An attempt is made to rescale the resulting dynamical system for the purpose of nonlinear analysis but the small parameter necessary for this end is not evident. The possibility remains that a small parameter may be available for certain physical parameters. Secondly, the fully nonlinear equations of motion for the model are then developed. The dynamical system can be computationally solved without an explicit representation of the equations of motion. However, there may exist parameter ranges that simplify the system so that a rigorous nonlinear dynamical analysis could find application. For this reason, the development of the dynamic simulation sought to first explicitly write the coupled seven degree of freedom equations of motion in terms of the rigid body coordinates. The choice between Euler angle and Quaternion orientation is reviewed and the integration model for dynamic evolution of the system is given. A validation of the numerical model is conducted by comparing simulation results against the commercial code Adams View.
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Third, the numerical solution of the hydrodynamic problem is developed. The size of the body is unconstrained by the magnitude of the wave amplitude. Traditional methods that commonly linearize about the calm water free surface are not applicable because the incident wave amplitudes may be an order of magnitude larger than the body size. Therefore, development of a boundary integral method (BIM) that institutes the Weak Scatterer free surface boundary conditions is presented. This method allows for a consistent solution for the fluid potential over the exact wetted surface of the hull in the presence of an ambient wave field. Prescribed motion analysis show good agreement for the hydrodynamic force coefficients compared to previous analytical results. Finally, experiments are performed at full scale in the large modeling basin of the Marine Hydrodynamics Lab at the University of Michigan. The experiments use an infrared vision system to characterize the free surface and to track the rigid body motion of a wave energy conversion buoy. The results of the free surface characterization experiment show excellent agreement with the analytic solution for Stokes second order drift. The rigid body motion experiment is used to validate a time-domain numerical model for the wave energy conversion application. The rigid body motion experiments provide an excellent data set for validation of both the hydrodynamic model as well as the fully nonlinear seven degree of freedom wave energy conversion application. Experimental results for rigid body motions demonstrate large amplitude pitch as well a nonlinear drift. Wave excitation near the pitch natural frequency cause the hull to move in the direction of wave propagation with a velocity that exceeds analytic estimates by an order of magnitude. Chaotic behavior is indicated by experimental results for the buoy with energy extraction. The numerical computations are demonstrated to agree well with the nonlinear drift phenomenon. Also, chaotic behavior is reproduced in the simulation through perturbations in initial conditions. The character of the time series for pitch and surge are useful tools for evaluating the numerical solution. The coupling of the fast drift response to encounter frequency produces an easily detectable response shift in the pitch amplitude. Waves with a
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frequency slightly faster than pitch resonance excite a strong pitch response and downstream drift. The downstream drift has the effect of shifting the encounter frequency toward resonance in a positive feedback loop. This process displays a limit that evidently selects an encounter frequency just below resonance (Figure 5.23). The physical source of the nonlinear drift can be explored with the computational model. Such insights could be directed toward research within station keeping applications. Numerical computations should be performed to investigate the drift response versus changes in natural frequencies for both pitch and heave. Experimental data show a strong correlation between energy extraction and pitch amplitude. The nature of the response to excitation by regular waves is demonstrated to be nonlinear and chaotic for the length of excitation achievable in the wave basin. Good agreement with these experimental results is shown numerically through perturbation of initial conditions. The lower power conversions found for certain initial conditions would be expected to be less frequently encountered in real seas, for which excitation would be expected from differing orientations. There would be some value in experiments of longer duration that might lead to steady state behaviors. However, the utility of such data is questionable. A more fruitful direction for this research is suggested as following a path toward evaluating the model for use with irregular and/or directional seas. Validation of the model in irregular and directional seas would require research efforts in the following areas:
Physical experiments for a buoy model, scaled to a realistic ambient wave spectrum should be carried out. The results of such testing would again be used to validate the numerical model. The seastate spectral parameters should be determined based on the response of the system in regular waves. The buoy is expected to primarily respond to excitation by the shorter frequencies of the
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spectrum. This would encourage focus on a bimodal spectrum having a short wave dominant period near pitch resonance.
A rational approach and numerical validation will be required for setting the free surface node spacing. The intuitive metric is to use the shorter of the characteristic body length and the shortest wavelength contained in the incident wave spectrum. The performance of such spacing while simultaneously modeling the long wave components should be evaluated. An initial approach would be to model a two component spectrum containing a limiting long wave component. Successful validation would then lead to a gradual expansion of the high frequency component to a full spectrum.
It is possible that computational savings may be achieved by switching to the linearized free surface boundary conditions in the outer domain or even in an intermediate domain. This approach could simplify computation of the influence matrix, a significant expense, at each time step.
The numerical beach spacing used within this work was based on a single wave length. Therefore, a numerical beach must be evaluated for whatever free surface discretization is used.
Investigation of viscous effects should be investigated.
In conclusion, the work presented here explores a specific model for wave energy extraction. It goes on to develop and validate a nonlinear physics based model that can be applied more generally to applications for wave energy extraction. Finally, the results of testing and numerical computations presented here inform us as to the capacity for energy conversion on a buoy scale. The power conversion for the application reaches efficiencies of approximately 30% of the power present in monochromatic ambient waves passing through a line parallel to the wave crest and equal to the buoy diameter. These relatively high efficiencies are only demonstrated for higher wave frequencies. Energy extraction from longer waves is very low.
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In real waves, the higher frequency components can be envisioned as small waves traveling over a larger set of ambient waves. Measured wave spectra are restricted too these larger waves because virtually all of the energy present is in the longer wave components. The shorter wave lengths are generally filtered from measured wave spectra electronically or by the response amplitude operator of the measuring platform. This finding illustrates, in some sense, that a wave energy conversion system should be physically scaled to the wavelengths for which energy extraction is desired. Within a data buoy application that requires relatively little energy conversion, wave energy conversion by a buoy scale hull is a reasonable area for research and development. However, application requiring energy conversion for utility grid scale energy conversion demand scaling of characteristic hull lengths to the long wave, higher energy components of the expected ambient wave climate.
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Appendix A Experimental Details
A.1 Vision System Details
Figure A.1 Vision System Calibration Screen Shot.
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Figure A.2 Regression For Clutch Torque vs. DC Voltage (Quadratic 0.02, R2=0.99 , Linear 0.002, R2=0.99).
Figure A.3 Instruments and Associated Errors.
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A.2 Dynamic Wave Probe Analysis Details
Table A.1 Comparison of Second Order Drift to Measured Drift.
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Appendix B Test Matrices and Graphical Results B.1 Case I – 1.0 Kilogram Pendulum
Table B.1 Test Matrix and Summary Results - Case I 1.0 Kg Pendulum.
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Figure B.1 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
Figure B.2 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
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Figure B.3 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
Figure B.4 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
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Figure B.5 1.0 Kg Pendulum – Wave Frequency 0.85 Hz.
Figure B.6 1.0 Kg Pendulum – Wave Frequency 0.85 Hz.
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Figure B.7 1.0 Kg Pendulum – Wave Frequency 0.8 Hz.
Figure B.8 1.0 Kg Pendulum – Wave Frequency 0.75 Hz.
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Figure B.9 1.0 Kg Pendulum – Wave Frequency 0.7 Hz.
Figure B.10 1.0 Kg Pendulum – Wave Frequency 0.65 Hz.
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Figure B.11 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
Figure B.12 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
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Figure B.13 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
Figure B.14 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
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Figure B.15 1.0 Kg Pendulum – Wave Frequency 1.0 Hz.
Figure B.161.0 Kg Pendulum – Wave Frequency 1.0 Hz.
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Figure B.17 1.0 Kg Pendulum – Wave Frequency 0.95 Hz.
Figure B.18 1.0 Kg Pendulum – Wave Frequency 0.9 Hz.
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Figure B.19 1.0 Kg Pendulum – Wave Frequency 0.85 Hz.
Figure B.20 1.0 Kg Pendulum – Wave Frequency 0.8 Hz.
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Figure B.21 1.0 Kg Pendulum – Wave Frequency 0.8 Hz.
Figure B.22 1.0 Kg Pendulum – Wave Frequency 0.7 Hz.
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Figure B.23 1.0 Kg Pendulum – Wave Frequency 1.0 Hz.
Figure B.24 1.0 Kg Pendulum – Wave Frequency 1.0 Hz.
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B.2 Case II – 0.25 Kilogram Pendulum
Table B.2 Test Matrix – Case II 0.25 Kg Pendulum.
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Figure B.25 0.25 Kg Pendulum – Wave Frequency 0.75 Hz.
Figure B.26 0.25 Kg Pendulum – Wave Frequency 0.80 Hz.
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Figure B.27 0.25 Kg Pendulum – Wave Frequency 0.85 Hz.
Figure B.28 0.25 Kg Pendulum – Wave Frequency 0.9 Hz.
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Figure B.29 0.25 Kg Pendulum – Wave Frequency 0.95 Hz.
Figure B.30 0.25 Kg Pendulum – Wave Frequency 1.0 Hz.
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B.3 Case III – No Pendulum
Table B.3 Test Matrix - Case III No Pendulum.
Figure B.31 Rigid Body Motion – Wave Frequency 1.0 Hz.
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Figure B.32 Rigid Body Motion – Wave Frequency 0.95 Hz.
Figure B.33 Rigid Body Motion – Wave Frequency 0.95 Hz.
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Figure B.34 Rigid Body Motion – Wave Frequency 0.9 Hz.
Figure B.35 Rigid Body Motion – Wave Frequency 0.85 Hz.
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Figure B.36 Rigid Body Motion – Wave Frequency 0.85 Hz.
Figure B.37 Rigid Body Motion – Wave Frequency 0.8 Hz.
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Figure B.38 Rigid Body Motion – Wave Frequency 0.75 Hz.
Figure B.39 Rigid Body Motion – Wave Frequency 0.69 Hz.
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Figure B.40 Rigid Body Motion – Wave Frequency 0.65 Hz.
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Appendix C Numerical Results
C.1 Convergence Studies
Figure C.1 Heave Added Mass and Damping Convergence in Time.
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Figure C.2 Heave Added Mass and Damping Convergence in Body Resolution.
Figure C.3 Surge Added Mass and Damping Convergence in Body Resolution.
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Figure C.4 Surge Added Mass and Damping Convergence in Free Surface Resolution.
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C.2 Numerical Time Series For Rigid Body Motions The following figures show the calculated time series for a parameter range that is similar to that used in the rigid body experiments. Excitation is generated from monochromatic waves of amplitude is 3.2cm. Viscous effects are modeled using a drag coefficient of 0.5 for translation and 0.016 for rotation. The projected area used for translational drag calculation is based the projected area normal to the instantaneous wetted surface.
Figure C.5 Rigid Body motion – K (Wave No.) =2.0.
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Figure C.6 Rigid Body motion – K=2.4.
Figure C.7 Rigid Body motion – K=2.8.
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Figure C.8 Rigid Body motion – K=3.2.
Figure C.9 Rigid Body motion – K=3.6.
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Figure C.10 Rigid Body motion – K=4.0.
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C.3 Numerical Time Series With Pendulum The following figures show the resulting time series for a parameter range that is similar to that used in the pendulum experiments. Excitation is generated from monochromatic waves of amplitude is 3.2cm. Viscous effects are modeled using a drag coefficient of 0.5 for translation and 0.016 for rotation. The constant torque used to model the hysteresis clutch is 0.17 Nm.
Figure C.11 Numerical Model Seven D.O.F. – 0.7 Hz.
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Figure C.12 Numerical Model Seven D.O.F. – 0.75 Hz.
Figure C.13 Numerical Model Seven D.O.F. – 0.8 Hz.
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Figure C.14 Numerical Model Seven D.O.F. – 0.85 Hz.
Figure C.15 Numerical Model Seven D.O.F. – 0.9 Hz.
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Figure C.16 Numerical Model Seven D.O.F. – 0.95 Hz.
Figure C.17 Numerical Model Seven D.O.F. – 1.0 Hz.
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