analysis of a tensegrity system for ocean wave...
TRANSCRIPT
ANALYSIS OF A TENSEGRITY SYSTEM FOROCEAN WAVE ENERGY HARVESTING
By
RAFAEL ESTEBAN VASQUEZ MONCAYO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2011
c© 2011 Rafael Esteban Vasquez Moncayo
2
I dedicate this work to my love (G), my family and my friends
3
ACKNOWLEDGMENTS
I express my gratitude to my supervisory committee: Dr. Carl Crane, Dr. Jacob
Hammer, Dr. John Schueller, Dr. Warren Dixon and Dr. Julio Correa for their time,
effort, and invaluable contributions to my academic growth during my time at the
University of Florida.
This research was done with financial support from the U.S. Department of State,
through the Fulbright Program; the University of Florida, through the Center for Latin
American Studies and the Department of Mechanical and Aerospace Engineering;
the Colombian Administrative Department of Science, Technology and Innovation:
Colciencias; and the Universidad Pontificia Bolivariana, Medellin.
4
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Ocean Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.1 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.2 Wave Energy Resource . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.3 Wave Energy Technology . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Tensegrity Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 Advantages of Tensegrity Systems . . . . . . . . . . . . . . . . . . . 161.2.2 Motion Applications of Tensegrity Systems . . . . . . . . . . . . . . 181.2.3 Ocean Applications of Tensegrity Systems . . . . . . . . . . . . . . . 19
2 OCEAN WAVE MECHANICS . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Airy’s Linear Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Water Particle Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Wave Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Wave Mechanics Numerical Example . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Kinematic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5.2 Water Particles Motion . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.3 Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Wave-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.1 Heaving and Pitching Body Motions . . . . . . . . . . . . . . . . . . 322.6.2 Heaving Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 34
2.6.2.1 Mass and Added Mass . . . . . . . . . . . . . . . . . . . . 352.6.2.2 Radiation and Viscous Damping . . . . . . . . . . . . . . . 36
2.6.3 Wave-Induced Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6.4 Steady State Solution of the Heaving Equation . . . . . . . . . . . . 38
3 TENSEGRITY MECHANISM . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Morphology Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Position Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Forward Position Analysis . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Reverse Position Analysis . . . . . . . . . . . . . . . . . . . . . . . . 42
5
3.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Forward Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 Reverse Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 463.3.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4.3 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.4 Friction Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.5 Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 OCEAN WAVE ENERGY HARVESTING . . . . . . . . . . . . . . . . . . . . . 54
4.1 Electrical Generators for Wave Energy Harvesting . . . . . . . . . . . . . . 544.2 Sea State Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Wave Power and Wave Kinematic Properties . . . . . . . . . . . . . 564.3 Direct Drive Heaving System . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Tensegrity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6
LIST OF TABLES
Table page
1-1 Ocean energy global resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3-1 Mechanism parameters for kinematic chain . . . . . . . . . . . . . . . . . . . . . 40
3-2 Position analysis numerical example results . . . . . . . . . . . . . . . . . . . . . 43
4-1 Direct drive heaving float coefficients . . . . . . . . . . . . . . . . . . . . . . . . 57
4-2 Tensegrity harvesting system coefficients . . . . . . . . . . . . . . . . . . . . . . 60
7
LIST OF FIGURES
Figure page
1-1 Ocean energy conversion development . . . . . . . . . . . . . . . . . . . . . . . . 13
1-2 Global coastal wave power estimates . . . . . . . . . . . . . . . . . . . . . . . . 14
1-3 Tensegrity morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2-1 Notation for the linear wave analysis . . . . . . . . . . . . . . . . . . . . . . . . 22
2-2 Particle paths predicted by Airy’s linear wave theory . . . . . . . . . . . . . . . 28
2-3 Notation for the wave energy analysis . . . . . . . . . . . . . . . . . . . . . . . . 28
2-4 Example water particles motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2-5 Rigid body with six degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . 32
2-6 Floating bodies undergoing heaving or pitching motion . . . . . . . . . . . . . . 33
2-7 Floating bodies: pure heaving and pitching conditions . . . . . . . . . . . . . . . 33
2-8 Added mass and added-mass moment of inertia coefficients for a rectangular body 35
2-9 Non-dimensional radiation damping coefficient for a heaving rectangular section 36
3-1 Concept of a wave energy harvester based on tensegrity systems . . . . . . . . . 39
3-2 Kinematic diagram of the mechanism . . . . . . . . . . . . . . . . . . . . . . . . 40
3-3 Vector diagram of the mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3-4 Position analysis numerical example . . . . . . . . . . . . . . . . . . . . . . . . . 43
3-5 Velocity analysis numerical example . . . . . . . . . . . . . . . . . . . . . . . . . 48
3-6 Velocity analysis numerical example results . . . . . . . . . . . . . . . . . . . . . 48
4-1 Location and bathymetry of Isla Fuerte, Colombia . . . . . . . . . . . . . . . . . 56
4-2 Wave power variation for the sea state at 9.408◦ N, 76.180◦ W . . . . . . . . . . 57
4-3 Heaving body simulation: wave-induced force . . . . . . . . . . . . . . . . . . . 58
4-4 Heaving body simulation response . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4-5 Tensegrity simulation: wave-induced forces . . . . . . . . . . . . . . . . . . . . . 60
4-6 Tensegrity simulation: surging, heaving and pitching motions . . . . . . . . . . . 61
4-7 Tensegrity simulation: instant dissipated power . . . . . . . . . . . . . . . . . . 62
8
4-8 Tensegrity simulation: motion in generators and springs . . . . . . . . . . . . . 62
4-9 Variation of power dissipation with mechanism parameters . . . . . . . . . . . . 63
9
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
ANALYSIS OF A TENSEGRITY SYSTEM FOROCEAN WAVE ENERGY HARVESTING
By
Rafael Esteban Vasquez Moncayo
December 2011
Chair: Carl D. Crane IIIMajor: Mechanical Engineering
Tensegrity systems have been used in several disciplines such as architecture,
biology, aerospace, mechanics and robotics during the last fifty years. However, just
a few references in literature have stated the possibility of using tensegrity systems in
ocean or energy-related applications. This research addresses the analysis of a tensegrity
mechanism for ocean wave energy harvesting.
The mechanics of ocean waves is described using the linear theory developed by
George B. Airy. The kinematic properties of the waves, the motion of particles and wave
power calculation are addressed. Then, the fluid structure interaction is reviewed making
emphasis on the concepts of radiation damping and viscous damping.
A planar tensegrity mechanism is proposed based on a planar morphology known
as “X-frame” that was developed by Kenneth Snelson in 1960s. A geometric approach
is used to solve the forward and reverse displacement problems. The theory of screws is
used to perform the forward and reverse velocity analyses of the device. The Lagrangian
approach is used to deduce the equations of motion considering the interaction between
the mechanism and ocean waves.
The tensegrity configuration is compared to a purely heaving body that is commonly
used in ocean wave energy harvesting. The result shows that tensegrity systems could
play an important roll in the expansion of clean energy technologies that help the world’s
sustainable development.
10
CHAPTER 1INTRODUCTION
Due to sustainability concerns, a world race started several years ago to incentivize
the research, development and utilization of renewable energy sources [1–3]. The ocean
represents an enormous potential energy source [4, 5]; however, its exploitation is still
incipient compared to other well-established power harvesting technologies such as wind
and solar energies [6].
Ocean energy can be obtained from waves, tides, currents, thermal gradients and
salinity gradients. During the last 30 years the R&D works in all these resources have
increased considerably [7–17]. Several assessment studies to evaluate the amount of ocean
energy which is available at a particular region, and to determine the technology that is
convenient for the local conditions have been conducted in different countries [17–26].
Wave energy constitutes the most noticeable form of ocean energy, maybe because
of its (often impressive) destructive capabilities [27]. The U.S. Department of Energy
(DoE) developed the Marine and Hydrokinetic Technology Database [28], as a shared
resource for the marine/hydrokinetic industry and government. There are more than 160
different devices for ocean energy harvesting registered in the data base, with about 40%
corresponding to wave energy.
The word tensegrity is a combination of the words tension and integrity [29].
Tensegrity systems were introduced in the 20th century by Fuller [29], Emmerich [30]
and Snelson [31]. These systems are formed by a combination of rigid elements (struts)
under compression, and elastic elements (ties) under tension [32, 33].
Tensegrity systems have been used in several disciplines such as architecture, biology,
aerospace, mechanics and robotics during the last fifty years [34]. Applications in sciences
and engineering include, among others, development of structural domes and bridges
[35–40], deployable systems for space applications [41–46], description and modelling of
living organisms and biological systems [47–51], and applications in robotics [52–60].
11
Just a few references in literature have stated the possibility of using tensegrity
systems in ocean applications. Scruggs and Skelton [61] made a preliminary investigation
in the potential use of controlled tensegrity structures to harvest energy, and suggested
their suitability to harvest energy from ocean waves. They showed how a tensegrity
structure, with one active bar, can be used to effectively harvest energy when it is excited
at a single frequency. Jensen et al. [62] proposed tensegrity structures in the design of
wave compliant structures for offshore aquaculture; they studied different combinations of
pre-stress and determined how it influences the stiffness of the whole structure. Wroldsen
[63] developed analysis tools based on differential-algebraic equations (DAEs) of motion,
and extended the formulation to include the dynamics of relatively long and heavy cables
with increased computational efficiency. Tensegrity systems could play an important roll
in the expansion of clean energy technologies that would contribute to world’s sustainable
development. Therefore, this can be a promising field of development that is still on a
conceptual level and needs to be explored.
1.1 Ocean Energy
The oceans contain a large amount of renewable energy. Reports of the estimated
global resource, presented by AEA Energy & Environment [6] and the U.S. Department of
Energy [64], are summarized in Table 1-1.
Table 1-1. Ocean energy global resource.
Energy type Estimated global resourceWave energy 80000 TWh/yTidal energy 300+ TWh/yMarine current 800+ TWh/yThermal energy 10000 TWh/ySalinity gradient 2000 TWh/y
The International Energy Agency (IEA) [14] reported the global status of technology
development for ocean energy systems, Figure 1-1A, highlighting the United Kingdom,
the United States, Canada and Norway, as leaders in development. Figure 1-1B shows the
emphasis placed on the development of wave energy technologies.
12
0
5
10
15
20
25
30
35
40
Country
Fin
land
Ger
many
Isra
elIt
aly
New
Zea
land
Kore
aB
razi
lG
reec
ePort
ugal
Russ
iaM
exic
oFra
nce
India
Irel
and
Net
her
lands
Swed
enA
ust
ralia
Chin
aD
enm
ark
Japan
Norw
ay
Canada
US
UK
Num
ber
ofsy
stem
s
SalinityThermalWaveCurrentTidal
A Country participation.
0
5
10
15
20
25
30
Technology
Salinity
Ther
mal
Tid
al
Curr
ent
Wave
Num
ber
of s
yste
ms
Concept designPart-scale (Tank)Part-scale (Sea)Full-scalePre and commercial production
B Technology status.
Figure 1-1. Ocean energy conversion development [14].
Although environmental legislation for the deployment of ocean energy projects is
not clear yet in plenty of countries, several studies are being carried out around the world
to demonstrate sustainability and commercialization potentials of ocean energy, and to
fulfill legislative requirements that aim to protect the environment near exploitation areas
[65–67]. In this direction, the Department of the Interior of the United States [68] stated
policies about the use of renewable energy and alternate uses of existing facilities on
the outer continental shelf. Additionally, Det Norske Veritas (DNV) [69] developed the
Offshore Service Specification DNV-OSS-312, which presents the principles and procedures
with respect to certification of tidal and wave energy converters.
1.1.1 Wave Energy
Wave energy is an indirect form of solar energy. Temperature differences produced
by the solar radiation around the world create winds that blow over the ocean surface,
generating waves. Such waves can travel thousands of kilometers through deep waters with
minimal loss of energy, representing a source with higher power density than wind or solar
power [70].
13
1.1.2 Wave Energy Resource
Waves transport both kinetic and potential energy. The total energy of a wave
depends mainly on its height (H) and its period (T ), and is usually measured in Watt
per meter (W/m) of wave front. The global wave power potential is of the same order of
magnitude as the world’s electricity consumption, around 1-10 TW [4]. Cruz [71] states
that it is possible to extract 10-25 % of this energy, suggesting that wave power could
make a significant contribution to the renewable energy industry. The best wave climates,
whose annual average power levels are between 20 and 70 kW/m or higher, are found in
zones from 30 to 60 degrees latitude, Figure 1-2. Nonetheless, attractive wave climates are
found also within ±30 degrees latitude, where the lower power level is compensated by
smaller power variability [4].
Figure 1-2. Global coastal wave power estimates from the Topex altimeter [72].
As deep-water waves approach shallow waters, they slow down, their wavelength
decrease and their height grows, which leads to breaking. The major losses of energy are
due to breaking and to friction with the seabed; therefore, only a fraction of the resource
reaches the shore [71]. Not all sites are suitable for deployment of harvesting devices for
several reasons, including unsuitable geomorphologic conditions at the shoreline, excessive
tidal range and environmental impact [4]. Nonetheless, shoreline devices could provide a
substantial contribution to the electric energy demands in small islands or isolated coastal
regions where the energy consumption is small.
14
1.1.3 Wave Energy Technology
There are several technologies that result from different ways in which the energy can
be harnessed from waves, depending on the water depth and the location (i.e. shoreline,
near-shore, offshore) [27]. Several ways to classify wave energy devices have been proposed.
Falcao [27] recently proposed a classification based on the type, the deployment style, and
the deployment place as follows:
• Oscillating water columns (OWC) are devices that have a partially submergedstructure, open below the water surface which holds air inside. The incident wavesthat go through the OWC, create oscillating motion of the air inside the chamber,forcing the air to flow through a turbine which drives an electrical generator.Self-rectifying turbines which provide a unidirectional rotation for an alternating airflow, such as the Wells turbine [73, 74], are often used in such devices.
• Oscillating body systems are generally located offshore (water depth > 40 m), andcan be either floating or fully submerged [27, 75]. The main element of these systemsis an oscillating body that either floats or is submerged near the surface [4]. Severalconfigurations such as single-body heaving buoys, two-body heaving systems, fullysubmerged heaving systems, pitching devices and bottom-hinged systems, havebeen used to build this type of energy converters [27]. Several techniques havebeen proposed to harness the energy with oscillating bodies, using either linear(translational) or rotational electrical generators.
• Overtopping converters constitute another type of wave energy converters. Thewater in the wave crest is introduced by overtopping through a sloping wall or rampinto a reservoir located at a level higher than the surrounding water surface. Thepotential energy of the water is then converted into useful energy through an arrayof low-head hydraulic turbines (e.g. Kaplan turbines) [27, 76].
Dunnett and Wallace [77] proposed an alternative classification for wave energy
converters. This classification method is based on how the devices get the mechanical
energy from the waves:
• Point absorbers are devices whose surface area is very small in comparison to thewavelength of ocean waves.
• Attenuators are relatively long devices that are placed parallel to the generaldirection of wave travel.
• Terminators are placed perpendicularly to the waves and are intended to absorb alarge proportion of the energy of the wave.
15
1.2 Tensegrity Systems
Skelton and de Oliveira [34] recently defined a tensegrity configuration as follows:
“In the absence of external forces, let a set of rigid bodies in a specific configuration have
torqueless connections. Then this configuration forms a tensegrity configuration if the
given configuration can be stabilized by some set of internal tensile members, i.e. connected
between the rigid bodies.”.
Figure. 1-3 shows three different tensegrity morphologies that have been extensively
addressed in literature for different applications.
A X-frame [31]. B Octahedron. C 6-bar prism.
Figure 1-3. Tensegrity morphologies.
1.2.1 Advantages of Tensegrity Systems
Tensegrity systems offer a number of advantages as described by Skelton et al. [78]
and Wroldsen [63]. Such characteristics are summarized in the following paragraphs.
Stabilizing tension. A tensegrity structure gets a stable static equilibrium configuration
when all ties are in tension and all struts are in compression (in the absence of external
forces or torques) [63]. Compressive members lose stiffness as they are loaded, whereas
tensile members, possessing less weight, gain stiffness as they are loaded [78].
Geometry and structural efficiency. Traditionally, structures tend to be made with
orthogonally arranged elements [78]. However, this type of architecture does not usually
yield minimal mass designs for given sets of stiffness properties [79]. Tensegrity systems,
16
on the other hand, use longitudinal members arranged in unusual patterns, to achieve
strength conditions with less mass [78].
Deployability and small storage volume. Since the members in compression are either
disjoint or connected by simple joints, large displacement, deployability, and stowage in a
compact volume appear as advantages of tensegrity systems [46, 78].
Tunable stiffness. One remarkable property of tensegrity systems is the possibility
to change shape without changing stiffness and vice versa [80]. These structures usually
have low structural damping, leading to challenges/opportunities with respect to vibration
in some applications [63]. Skelton et al. [78] addressed that structures designed to allow
tuning would play an important role in the development of next generation mechanical
systems.
Reliability in modelling. Since most members in a tensegrity system are axially
loaded, the equations needed to represent the static and dynamic behavior requires, in
general, less simplifications, resulting into more reliable models [63, 78].
Active control. A single member of a tensegrity system can serve multiple functions:
as a load-carrying member, a sensor, an actuator, etc. This flexibility provides an
encouraging opportunity for integrating structural and control design processes [78, 81].
Motivation from Biology. Ingber [47, 49] and Huang et al. [50] stated that tensegrity
is a fundamental building architecture of life. Hence, if tensegrity is part of nature’s
building architecture, the capabilities of tensegrity could make the efficiency present in
natural systems transferable to man-made systems [78].
Modularity through cells. Tensegrity systems are often made using a large number
of identical building blocks, or cells. The modularity facilitates large-scale production of
identical units that can be later assembled [63].
Robustness through redundancy. Using more ties than strictly needed (redundancy)
increases the system robustness and helps to avoid infinitesimal mechanisms [63].
17
1.2.2 Motion Applications of Tensegrity Systems
The involvement of tensegrity systems in motion applications is relatively new,
and several works have been increasingly appearing during the last twenty years. At
the beginning of the 1990’s, Pellegrino [82] presented a theory for the matrix analysis
of kinematically indeterminate prestressed assemblies made from pin-jointed bars based
on a linear approach. Calladine and Pellegrino [83] discussed the analytical conditions
under which a pin-jointed assembly, which has independent states of self-stress and m
independent mechanisms, tightens up when its mechanisms are excited. Djouadi et al.
[84] described a numerical scheme of active nonlinear control of tensegrity systems for
space applications. Stern [41] developed generic design equations to find the lengths of
the struts and elastic ties needed to create a desired geometry. Sultan [85] worked on
modelling, design, and control of tensegrity structures for several applications. Duffy et al.
[42] presented a review of a family of tensegrity structures that self deploy from a stowed
or packed configuration. Knight [43] addressed the problem of stability of tensegrity
structures for the design of a deployable antenna. Pellegrino et al. [44] studied deployable
structures for small satellite missions. Sultan et al. [52] proposed the development of a
flight simulator based on a tensegrity structure.
Starting the decade of 2000s, Oppenheim and Williams [86] examined the dynamic
behavior of a simple elastic tensegrity structure. Skelton et al. [87] developed an explicit
analytical model of the nonlinear dynamics of a large class of tensegrity structures. Sultan
et al. [88] formulated the general prestressability conditions for tensegrity structures,
expressed as a set of nonlinear equations and inequalities on the tendon tensions.
Kanchanasaratool and Williamson [54] developed a passive nonlinear constrained particle
dynamic model for a class of tensegrity platform structures. Sultan et al. [89] derived
the linearized equations of motion for tensegrity structures around arbitrary equilibrium
configurations. Tibert [45] worked on the development of deployable tensegrity structures
for space applications. Tran et al. [53] performed the reverse displacement and compliance
18
analysis of a tensegrity based parallel mechanism. Sultan and Skelton [46] presented a
strategy for tensegrity structures deployment using sets of equilibria. Bossens et al. [90]
analyzed the dynamic behavior of a tensegrity structure by comparing a finite element
model with an experimental model. Chan et al. [91] used simple control strategies for
the active vibration control of a three-stage tensegrity structure. Marshall and Crane [56]
proposed a six-degree-of-freedom tensegrity-based parallel platform which combines rigid
and elastic elements. Sultan and Skelton [55] used the intrinsic properties of tensegrity
structures to construct a smart force/torque sensor.
More recently, Crane et al. [92] obtained the equilibrium position for a general
skew-prismatic structure with a variety of external loads and moments acting on the
structure, using the virtual work principle. Arsenault and Gosselin [57–59] addressed the
possibility of the tensegrity systems to be used in the development of new lightweight
mechanisms for motion applications where only rigid-link robots have been considered.
Bayat and Crane [93] presented a closed-form analysis of a series of planar tensegrity
structures to determine all possible equilibrium configurations for each device when
no external forces or moments are applied. Arsenault and Gosselin [94] used general
static balancing conditions adapted for the case of tensegrity mechanisms. Vasquez
and Correa [60] presented the kinematic and the dynamic analyses, and a nonlinear
control strategy for a planar three-degree-of-freedom tensegrity robot manipulator.
Crane et al. [95] performed the equilibrium analysis of a planar tensegrity mechanism
showing the complexity that results from non-zero free lengths in the compliant elements.
Arsenault and Gosselin [96] presented the direct and inverse static analyses for a new
spatial tensegrity mechanism minimizing its potential energy. Wroldsen et al. [97] used a
non-linear feedback control law for non-minimal realizations of tensegrity systems.
1.2.3 Ocean Applications of Tensegrity Systems
Scruggs and Skelton [61] made a preliminary investigation in the potential use of
controlled tensegrity structures to harvest energy. They presented an approach to use
19
linear regenerative actuators as active bars into the structure. They illustrated the
approach in a simulation example for a small scale one-actuator system, and suggested the
suitability of tensegrity systems to harvest energy from ocean waves.
Jensen et al. [62] and Wroldsen [63] proposed tensegrity structures in the design
of wave compliant structures for offshore aquaculture. They addressed the promising
properties with respect to control of geometry, stiffness and vibration, that could make
tensegrity an enabling technology for future developments in open ocean aquaculture
construction systems for high energy environments.
20
CHAPTER 2OCEAN WAVE MECHANICS
Ocean waves can be generated by several phenomena, such as motions of celestial
bodies (sun and moon), seismic disturbances (earthquakes), moving bodies (ships) and
winds. The wind produces waves of different types, from the short capillary wave to the
long swell, that can be classified as follows [98]:
• Linear waves (sinusoidal profiles).
• Nonlinear waves (non-symmetrical profiles).
• Random seas (predictable in the frequency domain under certain assumptions).
For this research, a linear model of wind-generated waves, developed by George B.
Airy in 1841 [99], is used.
2.1 Airy’s Linear Wave Theory
George B. Airy developed the first meaningful analysis of ocean waves [100]. It
involves the solution of the linear equation of continuity (conservation of mass) for an
irrotational flow subject to linearized boundary conditions. The wave properties derived
with this theory are good approximations for small values of the wave steepness (defined
as the ratio between the wave height H and the wave length λ, Figure 2-1) [98, 101], i.e.
waves with relatively small amplitudes.
The following parameters are defined for the linear wave analysis, Figure 2-1:
SWL: still water level.
h: water depth, measured from the floor to SWL.
H: wave height, measured from the trough to the crest.
λ: wave length, measured from crest to crest.
η: vertical free-surface displacement, measured from the SWL. This parameter is a
function of x and t.
c: wave celerity (phase velocity).
21
Figure 2-1. Notation for the linear wave analysis [98].
The conservation of mass theorem is expressed by the equation of continuity, whose
differential form is given by
− ∂ρ
∂t= ∇ · (ρV) , (2–1)
where the del operator is given in Cartesian coordinates by
∇ =∂
∂xi +
∂
∂yj +
∂
∂zk, (2–2)
and the fluid velocity vector is given by
V = ui + vj + wk. (2–3)
The flow beneath the free surface is assumed to be irrotational, i.e. ∇× V = 0. Then,
the velocity of water particles is a conservative vector field that can be represented by a
potential function, in this case, the velocity potential φ, as follows
V = ∇φ. (2–4)
Substituting (2–4) into (2–1), and assuming a steady and incompressible flow, gives
∇2φ = 0. (2–5)
22
Equation (2–5) is an elliptic partial differential equation (Laplace’s equation),
that can be solved, subject to a set of boundary conditions, to determine the velocity
potential φ. The boundary conditions are given by:
Kinematic free-surface condition: the velocity of a particle on the free surface must
equal the velocity of the free surface itself.
V|z=η ≈ ∂η
∂tk ≈ ∂φ
∂z
∣
∣
∣
∣
z=0
k. (2–6)
Sea-floor condition: adjacent particles to the sea floor can not cross the solid
boundary.
V · N|z=−h =∂φ
∂z
∣
∣
∣
∣
z=−h
= 0. (2–7)
Dynamic free-surface condition: the pressure on the free surface is zero.
η = −1
g
∂φ
∂t
∣
∣
∣
∣
z=η
. (2–8)
Combining (2–6) and (2–8), and eliminating η yields the linearized free-surface
boundary condition, which is given by
{
1
g
∂2φ
∂t2+∂φ
∂z
}∣
∣
∣
∣
z=η≈0
= 0. (2–9)
The solution of (2–5) can be found by separation of variables for a traveling wave in
the form
φ = X(x± ct)Z(z) = X(ξ)Z(z). (2–10)
Substituting (2–10) into (2–5) yields
1
X
d2X(ξ)
dξ2= − 1
Z
d2Z(z)
dz2= −k2, (2–11)
where k is a constant.
The general solution of the two ordinary differential equations is given by
X(ξ) = Cξ sin(kξ + α), (2–12)
23
Z(z) = Cz cosh(kz + β), (2–13)
where Cξ, Cz, α and β are arbitrary constants.
Since the origin of the x (and ξ) coordinate is arbitrary, α can be assigned a zero
value without loss of generality. Therefore, (2–12) becomes
X(ξ) = Cξ sin(kξ). (2–14)
Applying the sea-floor boundary condition (2–7) to (2–13), we get β = kh, which gives
Z(z) = Cz cosh[k(z + h)]. (2–15)
Substituing (2–14) and (2–15) back into (2–10) yields
φ = Cφ cosh[k(z + h)] sin(kξ), (2–16)
where Cφ = CξCz.
Now, using the linearized free-surface boundary condition (2–9), Cφ can be removed
from (2–16). Hence, the velocity potential for a traveling wave is given by
φ± = ±H2
g
kc
cosh[k(z + h)]
cosh(kh)sin[k(x∓ ct)], (2–17)
where the upper (+) sign in φ represents right-traveling waves, and the lower (-) sign
represents left-traveling waves.
2.2 Traveling Waves
Two-dimensional problems in wave mechanics generaly consider right-traveling
waves [70, 98, 101]. From the combitation of the linearized free-surface condition (2–9)
and the general solution for the velocity potential (2–16) (eliminating φ, and using the
definition of the horizontal coordinate ξ), it is possible to demonstrate that the free-surface
displacement caused by the wave is given by
η =H
2cos[k(x− ct)]. (2–18)
24
Since the free-surface displacement from the SWL is sinusoidal in both space and
time, the maximum value for η (the crest), occurs when
k(x− ct) = 0,±2π,±4π, . . . (2–19)
For t = 0, the distance between two successive crests is the wavelength λ. Then, from
(2–19) we can compute the wave number k, as follows
k =2π
λ. (2–20)
Now, for x = 0, the time lapse between two succesive crests is the wave period T .
From (2–19)
kc =2π
T= 2πf ≡ ω, (2–21)
where f is the wave frequency in Hz and ω is the circular wave frequency in rad/s.
From (2–20) and (2–21), we can compute the celerity or phase velocity as
c =λ
T. (2–22)
The velocity potential can be written in terms of the circular wave frequency as
φ =H
2
g
ω
cosh[k(z + h)]
cosh(kh)sin(kx− ωt). (2–23)
Eliminating the velocity potential from (2–9) and (2–23), the circular wave frequency
can be computed as
ω =√
gk tanh(kh). (2–24)
Combining (2–20) and (2–24), results in an expression for the wave length given by
λ =2π
k=
2πg
ω2tanh(kh) =
gT 2
2πtanh
(
2πh
λ
)
= cT. (2–25)
Equation (2–25) is transcendental because λ can not be isolated and solved
analytically. Therefore, a numerical technique is required for its solution.
25
Because of the behavior of the hyperbolic tangent (tanh(kh) → 1 as kh → ∞) in
(2–25), it is a common practice in ocean engineering to divide the infinite space of h/λ
into three regions which define the relative sea depth as follows [98]:
1. Shallow water: h/λ ≤ 1/20
2. Intermediate water: 1/20 ≤ h/λ ≤ 1/2
3. Deep water: h/λ ≥ 1/2
As a result, approximations are used in the computation of the wave length for
shallow and deep water conditions, because the difference between the values of kh and
tanh(kh) is about 3% or less for h/λ ≤ 1/20, whereas for h/λ ≥ 1/2 the difference between
tanh(kh) and unity is less than 0.4% [98].
The deep-water approximation of (2–25) is given by
λ ≡ λ0 ≈gT 2
2π≈ c0T, (2–26)
where the subscript 0 is used to identify the deep-water properties.
The shallow-water approximation of (2–25) is given by
λ ≈ gT 2
2πkh =
gT 2
2π
2πh
λ=gT 2h
λ≈ cT, (2–27)
then
λ =√
ghT ≈ cT. (2–28)
The result in (2–26) shows that the deep-water wavelength and celerity are functions
of the wave period only. On the other hand, the result in (2–28) shows that the shallow-water
wavelength is a function of both depth and period, and that the celerity is a function of
depth and independent of period. Consequently, we see that waves shorten and slow down
as they approach the shoreline.
26
2.3 Water Particle Motions
The Cartesian velocity components in a two-dimensional irrotational flow are given, in
terms of the velocity potential, by the Cauchy-Riemann equations
u =∂φ
∂x=∂ψ
∂z, (2–29)
w =∂φ
∂z= −∂ψ
∂x, (2–30)
where the scalar function ψ is called the stream function.
Then, the velocity components in traveling waves can be computed substituting
(2–23) into (2–29) and (2–30), which yields
u =Hω cosh[k(z + h)]
2 sinh(kh)cos(kx− ωt), (2–31)
w =Hω cosh[k(z + h)]
2 sinh(kh)sin(kx− ωt). (2–32)
The horizontal and vertical displacements (ξ, ζ) of a particle about a fixed mean point
(xo, zo), are found substituting (x, y) by (xo + ξ, zo + ζ), expanding the results in Maclaurin
series in ξ and ζ, and finally, integrating the Cartesian velocity components (2–31) and
(2–32) over time.
The result expressions are, to the first order, given by
ξ =
∫
udt|xo,zo= −H cosh[k(zo + h)]
2 sinh(kh)sin(kxo − ωt), (2–33)
ζ =
∫
wdt|xo,zo=H sinh[k(zo + h)]
2 sinh(kh)cos(kxo − ωt). (2–34)
Then, the position of a particle measured from the mean position (xo, zo) is given by
the position vector
r = ξi + ζk. (2–35)
Substituting (2–33) and (2–34) into (2–35) yields
r =H/2
sinh(kh){− cosh[k(zo + h)] sin(kxo − ωt)i + sinh[k(zo + h)] cos(kxo − ωt)k} . (2–36)
27
Figure 2-2 shows the particle paths predicted by by Airy’s linear wave theory, using
appropriate approximations for the hyperbolic functions in (2–36).
A Deep water. B Intermediate water. C Shallow water.
Figure 2-2. Particle paths predicted by Airy’s linear wave theory [98].
2.4 Wave Energy and Power
The mass of an element displaced above the SWL, Figure 2-3, is given by
δm = ρη(δx)b, (2–37)
where b is the width of the wave crest.
Figure 2-3. Notation for the wave energy analysis [98].
28
The center of mass is located at a distance η/2 above the SWL. So, the potential
energy of the element is given by
δEp = g(δm)η
2=
1
2ρgη2(δx)b. (2–38)
The total potential energy of the elements above the SWL is given by
Ep =ρgH2
8b
∫ λ
0
cos2(kx− ωt)dx =ρgH2λb
16. (2–39)
The kinetic energy of a submerged element is given by
δEk =1
2ρ(u2 + w2)bδxδz, (2–40)
where (u,w) are the particle Cartesian velocity components given by (2–31) and (2–32).
The total kinetic energy of the submerged elements is given by
Ek =1
2ρb
∫ 0
−h
∫ λ
0
(u2 + w2)dxdz =ρgH2λb
16. (2–41)
Finally, the total energy of a linear wave, which is equally divided between potential
and kinetic energy, is given by
E = Ep + Ek =ρgH2λb
8. (2–42)
Substituting (2–26) and (2–28) into (2–42) results into approximations for deep- and
shallow-water, which are respectively given by
E0 =ρg2H2T 2b
16π, (2–43)
E =ρg3/2H2h1/2Tb
8. (2–44)
The time-rate of change of energy transmission per unit area normal to the
flow direction, i.e. the energy flux, can be computed by using the equation of energy
conservation for an irrotational flow, i.e. a simplified form of Bernoulli’s equation. The
following assumptions were made to use the simplified expression: the pressure at the
29
free surface is zero, and the kinetic energy term (ρV 2/2) is negligible for waves with small
steepness (H/λ << 1). The simplified Bernoulli’s equation is given by
ρ∂φ
∂t+ ρgz + p = 0. (2–45)
The only time-dependent part in (2–45) is the first term, that represents both the
unsteady kinetic energy and the dynamic pressure. Therefore, the energy flux is given by
the product of the dynamic pressure and the fluid veloctiy, as follows
ρ∂φ
∂tV = ρ
∂φ
∂t∇φ. (2–46)
The total energy flux can be computed substituting (2–23) into (2–46) and integrating
over both the normal area (bh) and the wave period (T ). The resulting expression is
considered the wave power [98], and is given by
P = ρb1
T
∫ T
0
∫ 0
−h
∂φ
∂t∇φdzdt =
ρgH2cb
16
[
2kh
sinh(2kh)+ 1
]
i. (2–47)
Deep- and shallow-water approximations are respectively given by
P0 =ρg2H2Tb
32πi, (2–48)
P =ρg3/2H2h1/2b
8i. (2–49)
2.5 Wave Mechanics Numerical Example
Let us suppose that a wave is traveling in water that is 10 m deep (h = 10 m).
Measurements indicate that the wave has a period of 3 seconds (T = 3 s) and an
amplitude of 2 meters (H = 2 m).
2.5.1 Kinematic Properties
Iteratively solving Equation (2–25) yields
λ = 14.0367 m.
Then, equations (2–20), (2–22) and (2–21), can be used to compute the other
kinematics properties of the wave, which gives
30
k = 0.4476 rad/m,
c = 4.6789 m/s,
ω = 2.0944 rad/s.
The ratio h/λ = 0.7124, indicates that we have deep water conditions.
2.5.2 Water Particles Motion
The wave can be represented graphically using Equation (2–36). Figure 2-4 shows
a discrete plot of water particles for the given wave at a particular instant of time.
The small closed path is described by the motion of one surface particle. As predicted
by Airy’s wave theory, for deep water conditions such path is circular (compare to
Figure 2-2A).
0 5 10 15 20 25 30
−10
−5
0
Distance, x (m)
Dep
th,z
(m)
Figure 2-4. Example water particles motion.
2.5.3 Wave Energy
For the given wave, the total energy and power per unit of wave front, i.e. b = 1 m,
can be computed using equations (2–43) and (2–48). Substituting the corresponding values
in such equations yields
E = 70.8471 kJ,
P = 11.8078i kW.
31
2.6 Wave-Structure Interaction
2.6.1 Heaving and Pitching Body Motions
The motion of a rigid body in the three-dimensional space is characterized by six
components corresponding to six degrees of freedom (6-dof). In ocean engineering it is
common to define such motions as depicted in Figure 2-5 and named as follows:
1. Surge: displacement along x axis.
2. Sway: displacement along y axis.
3. Heave: displacement along z axis.
4. Roll: rotation about x axis.
5. Pitch: rotation about y axis.
6. Yaw: rotation about z axis.
For an axisymmetric (or another not-elongated) body, e.g. an sphere, the numbers 1
and 2 (and 4 and 5) are ambiguous (i.e. they can be arbitrarily interchanged). However,
this ambiguity is removed by defining the x direction as the wave propagation direction.
Figure 2-5. Rigid body with six degrees of freedom.
Since this work considers a two-dimensional linear model for ocean waves, only three
degrees of freedom (3-dof) are of interest. The motion is analyzed in the x − z plane,
hence, only surge, heave and pitch motions are considered. Figure 2-6 shows two of such
configurations in which a floating body can be excited by ocean waves.
32
A Purely heaving float. B Purely pitching float.
Figure 2-6. Floating bodies undergoing heaving or pitching motion [70].
Assuming a sinusoidal wave profile, we can see that when λ = L, no heaving motion
will occur since a crest and a trough of a wave act simultaneously over the body as in
Figure 2-7B. Therefore, there is not vertical force acting on the float and no heaving
motions when
L = Nλ, N = 1, 2, 3, · · · (2–50)
Conversely, when an extra crest (or trough) act over the body, there is a net vertical
force that creates heaving motions. Heaving can be expected when
L =Nλ
2, N = 1, 3, 5, · · · (2–51)
The body depicted in Figure 2-6B is allowed to rotate (pitch) about its center of
gravity. A minimum pitching moment is experienced by the body under the condition
described by (2–51) when N = 3. On the other hand, a maximum pitching moment occurs
when 2–50 is satisfied.
A Pure heaving condition. B Pure pitching condition.
Figure 2-7. Floating bodies: pure heaving and pitching conditions [70].
33
2.6.2 Heaving Equation of Motion
The law of conservation of linear momentum states that the rate of change of the
linear momentum of the system in an inertial reference frame, is equal to the total
external force acting on the system. For a purely heaving floating body the equation of
motion is given by [98]
md2z
dt2= − awz
d2z
dt2− brz
dz
dt− bvz
(
dz
dt
) ∣
∣
∣
∣
(
dz
dt
)∣
∣
∣
∣
N
− bpzdz
dt− ρgAwpz −Nksz + Fzo cos(ωt+ αz).
(2–52)
McCormick [98] defines each term in the right hand side of (2–52) as follows:
1. Inertial reaction force of the water, where awz is the added mass.
2. Radiation damping force, where brz is the damping coefficient.
3. Viscous damping force, where bvz is the viscous damping coefficient.
4. Damping due to power take off, where bpz is the power take-off coefficient.
5. Hydrostatic restoring force, where Awp is the waterplane area when the body is atrest.
6. Mooring restoring force, where ks is the effective mooring spring constant of eachline, and N is the number of lines.
7. Wave induced vertical force, where Fzo is the force amplitude, ω = 2π/T is thecircular wave frequency (T is the wave period), and αz is the phase angle betweenthe wave and the wave-induced force.
The power N on the viscous damping term depends on the flow regime, N = 0 for
laminar flow and N = 1 for turbulent flow. When N = 1 the equation of motion is
nonlinear. In this work, we assume that the flow is laminar and N = 0. Therefore, bvz is
replaced by a linear damping coefficient bvz. Reorganizing terms in (2–52), the linearized
equation of motion is given by
(m+ awz)d2z
dt2+ (brz + bvz + bpz)
dz
dt+ (ρgAwp +Nks)z = Fzo cos(ωt+ αz). (2–53)
ρ is the density of seawater (1030 kg/m3), g is the acceleration due to gravity (9.8 m/s2).
34
2.6.2.1 Mass and Added Mass
The mass of a float must be equal to the mass of displaced water. Hence, the mass for
a rectangular floating body can be computed as
m = ρdAwp, (2–54)
where the waterplane area is given by
Awp = LB. (2–55)
The added mass is the inertia added to a system due to the presence of a structure in
a moving fluid or due to the motion of a structure in a stationary fluid. The magnitude of
such mass is proportional to the inertial reaction force on the body [98]. For rectangular
solids such the ones shown in Figure 2-6, the added-mass and the added-mass moment of
inertia are given respectively by [70]
awz =KmπρLB
2
4, (2–56)
Aw =KIπρL
4B
16. (2–57)
The values for the coefficients Km and KI can be found graphically from Figure 2-8.
0 5 10 15 200.5
0.75
1
Km
B/d
A Added mass coefficient.
2 4 6 8 100.05
0.075
0.1
0.15
KI
L/d
B Added-mass moment of inertia coefficient.
Figure 2-8. Added mass and added-mass moment of inertia coefficients for a rectangularbody [70].
35
2.6.2.2 Radiation and Viscous Damping
Due to body motions near to the fluid free surface, waves are also created. Such
waves take energy away from the body. This energy loss is called radiation damping. For
rectangular solids such the ones shown in Figure 2-6, the radiation damping coefficient is
given by [98]
brz = b′
rL =ρg2
ω3R2
ZL, (2–58)
where b′
r is the radiation damping coefficient per unit length, and RZ is the ratio of the
radiated-wave amplitude and the body motion amplitude, and is given by
RZ = 2e−ω2
gd sin
(
w2
g
B
2
)
. (2–59)
Both the added mass and the radiation damping coefficients vary with frequency.
However, McCormick [98] states that a frequency-invariant value for the added mass given
by (2–56) and the expression for the radiation damping coefficient given in (2–58) match
measurements done by Vugts [102] for a body with a rectangular section. Figure 2-9 shows
the variation of radiation damping coefficient for a heaving rectangular section when the
draft is the half of the length, i.e. d = L/2.
0 0.5 1 1.50
1
brz
(ρd√
2Lg)
ω
√
L
2g
Figure 2-9. Non-dimensional radiation damping coefficient for a heaving rectangularsection.
The nonlinear viscous damping coefficient in (2–52) is given by
bvz ≈1
2ρCdAd, (2–60)
36
where Ad is the projected area, i.e. Awp for a rectangular body, and Cd is the drag
coefficient. For a rectangular body in heaving motion in laminar flow, the drag coefficient
is Cd = 1.2 [103].
McCormick [98] states that the nonlinear and the linear viscous damping coefficients
in (2–52) and (2–53) are related by the following expression
bvz =8
3
ω
πZobvz, (2–61)
where Zo is the heaving amplitude which is computed as in Section 2.6.4. Since Zo
depends on bvz, a nonlinear equation must be solved for Zo to compute the linear viscous
damping coefficient.
2.6.3 Wave-Induced Forces
The wave-induced heaving force on a rectangular float in a linear wave is given by [70]
Fz(t) = Fzo cos(ωt), (2–62)
where
Fzo =ρgHBλ
2π
(
e−2πdλ + 1
)
sin
(
πL
λ
)
. (2–63)
The wave-induced moment on a rectangular float (assuming the draft d to be
constant) in a linear wave is given by [70]
Mθ(t) = Mθo sin(ωt), (2–64)
where
Mθo =ρgHBλ
4π
(
e−2πdλ + 1
)
[
λ
πsin
(
πL
λ
)
− L cos
(
πL
λ
)]
. (2–65)
The phase angle between the wave and the wave-induced force/moment, αz, is equal
to zero (αz = 0) for a body that is symmetric about the x− z and y − z planes [70].
37
2.6.4 Steady State Solution of the Heaving Equation
Equation (2–53) is a second-order ordinary differential equation whose steady-state
solution can be found easily and is given by
zss(t) = Zo cos(ωt+ αz − εz), (2–66)
where
Zo =
Fzo
(ρgAwp +Nks)√
(
1 − ω2
ω2n
)2
+
[
2ω
ωn
(brz + bvz + bpz)
bcz
]2cos (ωt+ αz − εz) . (2–67)
In (2–66) the natural heaving frequency is given by
wnz =2π
Tnz
=
√
ρgAwp +Nks
m+ awz
, (2–68)
where Tnz is the natural heaving period; the critical damping coefficient is given by
bcz = 2√
(m+ awz)(ρgAwp +Nks); (2–69)
and the phase angle between the force and motion is given by
εz = arctan
[
2ω
ωn
(brz + bvz + bpz)
bcz
]
(
1 − ω2
ω2n
)
. (2–70)
38
CHAPTER 3TENSEGRITY MECHANISM
Salter et al. [104] stated that most wave energy harvesters must have, among others,
the following subsystems:
• Slow primary displacing element, such as a float or buoy.
• Connecting linkage to transmit the wave-generated forces (analyzed in this work).
• Electrical generators, electrical network, transformers and switchgear.
3.1 Morphology Definition
The proposed tensegrity mechanism is based on the two-dimensional “X-frame”
morphology proposed by Kenneth Snelson in [31], Figure 1-3A. The mechanism comprises
four members in tension and two members in compression. The members in compression
can be replaced by two bars connected by prismatic joints which represent the electrical
generators. Two of the ties have a very high modulus of elasticity (the mechanism base
and the element excited by ocean waves) with respect to the other two ties. Therefore,
the deformations of the base and the buoyant element are negligible and the lateral ties
are the two deformable members under tension that are necessary to keep the tensegrity
configuration. Figure 3-1 depicts the concept of a tensegrity-based wave energy harvester.
Figure 3-1. Concept of a wave energy harvester based on tensegrity systems.
39
The kinematic diagram of the proposed tensegrity mechanism is shown in Figures
3-2 and 3-3. The mechanism parameters for the selected kinematic chain are listed in
Table 3-1.
0 1
1 2
2 3
3 4
4 5
5 6
Figure 3-2. Kinematic diagram of the mechanism.
Table 3-1. Mechanism parameters for kinematic chain.
Link length, m Twist angle, deg Joint offset, m Joint angle, dega1 = 0 α1 = 90 0S1 = 0 0θ1 = variablea2 = 0 α2 = 90 1S2 = variable 1θ2 = 180a3 = L α3 = 0 2S3 = 0 2θ3 = variablea4 = 0 α4 = 90 3S4 = 0 3θ4 = variablea5 = 0 α5 = 90 4S5 = variable 4θ5 = 180a6 = L0 α6 = 0 5S6 = 0 5θ6 = variable
3.2 Position Analysis
3.2.1 Forward Position Analysis
The forward position analysis allows one to determine the position and orientation of
one of the links for a specified set of joint variables. The problem is stated as follows:
• Given: the constant mechanism parameters a3, a6 and Lpxz (position of point p onthe top platform); and the set of joint variables, 1S2, 4S5 and 0θ1.
• Find: the position and the orientation of the top platform, xm, zm and φ.
40
��
�
��
��
��
��
��
��
��
�
�����
��
������
��
Figure 3-3. Vector diagram of the mechanism.
Let us define the following vectors
S2 = 1S2 cos (0θ1)i + 1S2 sin (0θ1)k, (3–1)
a6 = a6i. (3–2)
From Figure 3-3, the following vector loop equation can be written
LB = S2 − a6. (3–3)
Equation (3–3) permits to evaluate LB and θB. The angle δ3 can be computed using
the cosine law
δ3 = arccos
(
L2B + 4S
25 − a2
3
2LB4S5
)
. (3–4)
Hence,
5θ6 = θB + δ3 + π. (3–5)
Then, the vector S5 is given by
S5 = 4S5 cos (5θ6)i + 4S5 sin (5θ6)k. (3–6)
41
From Figure 3-3, other two vector loop equations can be written as
LA = a6 − S5, (3–7)
a3 = LA − S2. (3–8)
Equation (3–8) permits to evaluate the orientation φ. There are two solutions for δ3,
however, only the one in the first quadrant is taken for this mechanism’s configuration.
Finally, the position of the point p = [ xm 0 zm ]T can be computed as
p = S2 +Lpxz
La3. (3–9)
The length of the springs denoted by vectors LA and LB can be computed in terms of
the position and orientation (φ = θ + π) of the top platform, when Lpxz = L/2, as follows
LA =[
(xm − L/2 cos θ)2 + (zm − L/2 sin θ)2]
1
2 , (3–10)
LB =[
(xm + L/2 cos θ − L0)2 + (zm + L/2 sin θ)2
]
1
2 . (3–11)
3.2.2 Reverse Position Analysis
The reverse position analysis allows one to determine the position and orientation of
one of the links for a specified set of joint variables. The problem is stated as follows:
• Given: the constant mechanism parameters a3, a6 and Lpxz; and the position and theorientation of the top platform, xm, zm and φ.
• Find: the set of joint variables, 1S2, 4S5 and 0θ1.
Let us define the following vector
a3 = a3 cos (φ)i + a3 sin (φ)k. (3–12)
From Figure 3-3, the following vector loop equations can be written
S2 = p − Lpxza3, (3–13)
LA = S2 + a3, (3–14)
42
S5 = a6 − LA. (3–15)
Equation (3–13) permits to evaluate 1S2 and 0θ1, and (3–15) permits to evaluate 4S5.
3.2.3 Numerical Example
Let the constant mechanism parameters be L = 3 m, L0 = 4.5 m and Lpxz = 1.5
m. To verify the position analysis, a desired path can be defined in the Cartesian space.
Then, the values of the joint variables that satisfy the path can be found using the reverse
kinematics. Finally, the forward kinematics can be used to compute the points in the
Cartesian space that are generated with each set of joint variables. Figure 3-4 shows the
mechanism in the final position and the path. Table 3-2 shows the numerical results which
indicate that the reverse and forward kinematics functions work as expected.
−1 0 1 2 3 4 5 6
0
1
2
3
4
5
6
Position in x (m)
Position
inz
(m)
Figure 3-4. Position analysis numerical example.
Table 3-2. Position analysis numerical example results.
Original path Reverse kinematics Forward kinematicsxmd (m) zmd (m) φd (deg) 1S2 (m) 4S5 (m) 0θ1 (deg) xm (m) zm (m) φ (deg)
1.50 2.00 170 3.4482 5.0155 30.2969 1.5000 2.0000 170.00001.75 2.25 175 3.8751 4.8664 33.1537 1.7500 2.2500 175.00002.00 2.50 180 4.3012 4.7170 35.5377 2.0000 2.5000 180.00002.25 2.75 185 4.7242 4.5695 37.5735 2.2500 2.7500 185.00002.50 3.00 190 5.1428 4.4267 39.3445 2.5000 3.0000 190.0000
43
3.3 Velocity Analysis
The velocity state is defined as a set of parameters from which the velocity of any
body/point of the linkage can be determined relative to a reference body [105]. Then,
using this concept, the velocity analysis allows one to obtain the relationships between the
velocity of any body/point of the mechanism and the velocities of the joint variables. Rico
et al. [106] and Crane et al. [105] presented the concept of velocity state using the theory
of screws, developed by Sir Robert Stawell Ball [107].
Since the position analysis is complete, the directions of the unit vectors isi+1 along
each axis as well the coordinates of one point ri on each joint axis are known. Then,
Plucker coordinates of the lines along the revolute joint axes are given by
{isi+1; isi+1OL} = {isi+1; ri × isi+1}. (3–16)
The coordinates of the line at infinity associated with the prismatic joints are given
by
{isi+1; isi+1OL} = {0; ri × isi+1}. (3–17)
Then, using (3–16) and (3–17), the Plucker coordinates of the lines along the joint
axes are given by
0$1 = {0s1; 0s1OL} = {s1;0 × s1}, (3–18)
1$2 = {0; 1s2OL} = {0; s2}, (3–19)
2$3 = {2s3; 2s3OL} = {s3; S2 × s3}, (3–20)
3$4 = {3s4; 3s4OL} = {s4; LA × s4}, (3–21)
4$5 = {0; 4s5OL} = {0; s5}, (3–22)
5$6 = {5s6; 5s6OL} = {s6; a6 × s6}. (3–23)
All the unit vectors in (3–18) to (3–23) are known from the position analysis and are
given by:
44
s1 = s3 = s4 = s6 = [ 0 −1 0 ]T ,
s2 =S2
|S2|,
s5 =S5
|S5|.
Since the mechanism is a closed-loop kinematic chain, bodies 0 and 6 are the same
(i.e. the ground of the mechanism).
3.3.1 Forward Velocity Analysis
The forward velocity analysis allows one to determine the velocity state of one of the
links for a given set of joint rates. The problem is stated as follows:
• Given: the constant mechanism parameters a3, a6 and Lpxz; the set of joint variables,
1S2, 4S5 and 0θ1; and the set of velocities of the joint variables 1v2, 4v5 and 0ω1.
• Find: the velocity state of the top platform[
0ω3 0v3O
]Tand xm, zm and φ.
The velocity for the closed kinematic chain can be written in screw form as follows,
[105, 106]:
0ω10$1 +1 v2
1$2 +2 ω32$3 +3 ω4
3$4 +4 v54$5 +5 ω6
5$6 = 0. (3–24)
Substituting (3–18) to (3–23) into (3–24) yields
0ω1
0
−1
0
0
0
0
+ 1v2
0
0
0
s2x
0
s2z
+ 2ω3
0
−1
0
s2z
0
−s2x
+ 3ω4
0
−1
0
LAz
0
−LAx
+ 4v5
0
0
0
s5x
0
s5z
+ 5ω6
0
−1
0
0
0
−a6x
= 0
(3–25)
Equation (3–25) can be written as a 3x3 system in matrix form as follows
45
−1 −1 −1
s2zLAz
0
−s2x−LAx
−a6x
2ω3
3ω4
5ω6
= −0ω1
−1
0
0
− 1v2
0
s2x
s2z
− 4v5
0
s5x
s5z
(3–26)
The solution of (3–26) gives the magnitudes of the angular velocities between
consecutive bodies. Then, the velocity state of the top platform can be computed as
0ω3
0v3O
=0 ω1
0$1 +1 v21$2 +2 ω3
2$3. (3–27)
Now the velocity of any point p on the top platform, whose position is represented by
rO→P , is given in terms of the velocity state by
0v3p = 0v3
O + 0ω3 × rO→P . (3–28)
Equations (3–26), (3–27) and (3–28) complete the forward velocity analysis and allows
one to compute the velocity of any point in the top platform of the mechanism for a given
set of velocities of the joint variables.
3.3.2 Reverse Velocity Analysis
The reverse velocity analysis allows one to determine the set of joint rates for a given
velocity state of one of the links. The problem is stated as follows:
• Given: the constant mechanism parameters a3, a6 and Lpxz; the set of joint variables,
1S2, 4S5 and 0θ1; and the velocity state of the top platform[
0ω3 0v3O
]Tin terms
of xm, zm and φ.
• Find: the set of velocities of the joint variables 1v2, 4v5 and 0ω1.
Since the the velocity of the point p (0v3p = xmi + zmk), and the angular velocity of
the top platform (0ω3 = φj) are known, the element associated with linear velocities in the
velocity state can be computed using (3–28) as
0v3O = 0v3
p − 0ω3 × rO→P . (3–29)
46
Equation (3–27) defines the velocity state of the top platform. Substituting (3–27)
into (3–24) yields
0ω3
0v3O
= −3ω4
3$4 −4 v54$5 −5 ω0
5$0. (3–30)
Since the velocity state of the top platform is known, the joint velocities of the
mechanism can be computed from (3–27) and (3–30) as follows
−1 0 −1
0 s2xs2z
0 −s2z−s2x
0ω1
1v2
2ω3
=
0ω3
0v3x
0v3z
, (3–31)
−1 0 −1
LAzs5x
0
−LAxs5z
0
3ω4
4v5
5ω6
= −
0ω3
0v3x
0v3z
. (3–32)
Equations (3–31) and (3–32) complete the reverse velocity analysis and allows one
to compute the velocities of the joint variables (and all the joint velocities) for a given
velocity state of the top platform.
3.3.3 Numerical Example
Let the constant mechanism parameters be L = 3 m, L0 = 4.5 m and Lpxz = 1.5 m.
Then, we define a circular path in the Cartesian space, centered in the point (xm, zm) =
2.5i + 3k with a radius of 0.5 meters, Fig 3-5A. We want the mechanism to follow such
path in 5 seconds with constant celerity and fixed orientation (φ = 190 deg). Fig 3-5B
shows the corresponding desired position and velocity of the point p (centered in the top
platform of the mechanism), which are computed using the following parametric equations
(parameter t0) of the circle
(xm, zm) = [x0 + r cos(t0)]i + [z0 + r sin(t0)]k. (3–33)
The velocity of p can be computed using the derivative with respect to the time of (3–33).
47
Finally, the reverse position and the reverse velocity analysis equations can be used
to compute the corresponding values of the joint variables that would make the point p
follow the described Cartesian path.
Figure 3-6 shows the results with the joint variables values and their corresponding
velocities.
−1 0 1 2 3 4 5 6
0
1
2
3
4
5
6
Position in x (m)
Pos
itio
nin
z(m
)
A Mechanism and Cartesian path.
0 1 2 3 4 50
2
4
Time (s)
Position
(m)
xz
0 1 2 3 4 5−1
0
1
Time (s)
Vel
oci
ty(m
/s)
xz
B Position and velocity of point p.
Figure 3-5. Velocity analysis numerical example.
0 1 2 3 4 53.5
4
4.5
5
5.5
Time (s)
Posi
tion
(m)
1S 2
4S 5
0 1 2 3 4 5−2
−1
0
1
Time (s)
Vel
oci
ty(m
/s)
1v2
4v5
A Position and velocity for 1S2, 4S5.
0 1 2 3 4 535
40
45
50
Time (s)
0θ1
(deg
)
0 1 2 3 4 5−0.2
0
0.2
Time (s)
0ω
1(r
ad/s)
B Position and velocity for 0θ1.
Figure 3-6. Velocity analysis numerical example results.
48
3.4 Equation of Motion
The differential equation of motion can be obtained using the Lagrangian approach.
This equation represents the dynamic behavior of the mechanism, and is given, in terms of
the generalized coordinates and its derivatives (q, q, q), by
M(q)q + N (q, q) + G(q) = Q, (3–34)
where:
M(q) is defined as the inertia matrix,
N (q, q) = V (q, q) + F (q) accounts for Coriolis/centripetal effects and friction,
G(q) is called the gravity vector, and
Q is the generalized force vector.
The Lagrange’s equation of motion is given, in terms of the generalized terms, by
[108, 109]
d
dt
∂EK
∂q− ∂EK
∂q+∂EP
∂q= Q, (3–35)
where:
EK is the kinetic energy,
EP is the potential energy,
Q is the generalized force vector,
q is the vector of generalized coordinates,
q is the vector of generalized coordinates derivatives (velocities).
3.4.1 Assumptions
The following assumptions are made for the tensegrity mechanism
• The links of the mechanism, except for the top platform, are massless. It is assumedthat they can be designed to have neutral buoyancy.
• The elastic ties are massless.
• The stiffness of each tie is constant, i.e. they behave as linear springs.
49
• The vector of generalized coordinates is defined as
q =
q1q2q3
=
xm
zm
θ
. (3–36)
3.4.2 Kinetic Energy
Since the links of the mechanism are considered massless, the kinetic energy of the
mechanism is all concentrated at the top platform. The kinetic energy of the top platform
due to surge, heave and pitch motions is given, in terms of the generalized coordinates, by
EK =1
2(m+ awx) q1
2 +1
2(m+ awz) q2
2 +1
2(Iy + Aw) q3
2, (3–37)
where m is the mass, Iy = m(L2 + Z2)/12 is the mass moment of inertia, awx = 1/4ρπd2B
and awz are the added masses due to surging and heaving, and Aw is the added-mass
moment of inertia due to pitching.
The first and term of (3–35) is obtained by taking the derivative of (3–37) with
respect to the generalized coordinates as
∂EK
∂q=
(m+ awx) q1
(m+ awz) q2
(Iy + Aw) q3
, (3–38)
and then, taking the derivative of (3–38) with respect to the time gives
d
dt
∂EK
∂q=
(m+ awx) q1
(m+ awz) q2
(Iy + Aw) q3
. (3–39)
Equation (3–39) can be written in matrix form as
d
dt
∂EK
∂q= Mq =
(m+ awx) 0 0
0 (m+ awz) 0
0 0 (Iy + Aw)
q. (3–40)
50
From (3–37) we see that the kinetic energy of the top platform does not depend on
the position. Hence, the second term of (3–35) is given by
∂EK
∂q= 0. (3–41)
3.4.3 Potential Energy
The potential energy due to heaving and pitching motions of the top platform are
described by McCormick [70] as
EPz =1
2ρgAwpq
22, (3–42)
EPθ =1
2Cq2
3, (3–43)
where Awp is the waterplane area and C is the restoring moment constant which is
defined, for a bottom-flat body in terms of the draft, as
C =gIyd. (3–44)
The potential energy of the elastic ties is given by
EPk =1
2kA (LA − LA0
)2 +1
2kB (LB − LB0
)2 , (3–45)
where LA0, LB0
; and kA, kB; are the free lengths and the stiffnesses of the ties respectively.
Substituting (3–10) and (3–11) into (3–45) gives
EPk =1
2kA
(
[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2 − LA0
)2
+1
2kB
(
[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2 − LB0
)2
.
(3–46)
The total potential energy of the mechanism, EP = EPz + EPθ + EPk, is given by
EP =1
2ρgAwpq
22 +
1
2Cq2
3
+1
2kA
(
[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2 − LA0
)2
+1
2kB
(
[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2 − LB0
)2
.
(3–47)
51
The last term of (3–35) is computed taking the derivative of (3–47) with respect to
each generalized coordinate as follows:
∂EP
∂q1= kA
(
[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2 − LA0
)
· (q1 − L/2 cos q3)[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2
+ kB
(
[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2 − LB0
)
· (q1 + L/2 cos q3 − L0)[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2
,
(3–48)
∂EP
∂q2= ρgAwpq2
+ kA
(
[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2 − LA0
)
· (q2 − L/2 sin q3)[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2
+ kB
(
[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2 − LB0
)
· (q2 + L/2 sin q3)[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2
,
(3–49)
∂EP
∂q3= Cq3
+ kA
(
[
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2 − LA0
)
· [(q1 − L/2 cos q3)L/2 sin q3 − (q2 − L/2 sin q3)L/2 cos q3][
(q1 − L/2 cos q3)2 + (q2 − L/2 sin q3)
2]
1
2
+ kB
(
[
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2 − LB0
)
· [− (q1 + L/2 cos q3 − L0)L/2 sin q3 + (q2 + L/2 sin q3)L/2 cos q3][
(q1 + L/2 cos q3 − L0)2 + (q2 + L/2 sin q3)
2]
1
2
.
(3–50)
Equations (3–48), (3–49) and (3–50) can be written in matrix form as
∂EP
∂q= G(q) =
[
∂EP
∂q1
∂EP
∂q2
∂EP
∂q3
]T
. (3–51)
Equations (3–40), (3–41) and (3–51) represent the first and third terms of (3–34).
52
3.4.4 Friction Vector
As described in Section (2.6.2.2), the most important characteristic in the fluid-structure
interaction phenomenon is the presence of damping due to viscous effects and radiation of
waves. Viscous damping is considered in both surging and heaving motions, and radiation
damping is considered for heaving and pitching motions. The friction vector is then given
by
F (q) =
bvxq1
(brz + bvz)q2
brθq3
, (3–52)
where the damping coefficients in the heave/surge directions are given by (2–58) and
(2–61), and the radiation damping coefficient due to pitching motion is given by
brθ = b′
r
L3
12, (3–53)
where b′
r is the radiation damping coefficient per unit length given in (2–58).
3.4.5 Generalized Forces
The generalized force vector is formed by the wave-induced forces that act over the
generalized coordinates. Since the generalized coordinates were chosen to describe directly
the motion of the float, the generalized force vector is given by
Q(t) =
0
Fzo cos(ωt)
Mθo sin(ωt)
. (3–54)
The complete equation of motion is then given by
m+ awx 0 0
0 m+ awz 0
0 0 Iy + Aw
q +
bvxq1
(brz + bvz)q2
brθq3
+
∂EP
∂q1∂EP
∂q2∂EP
∂q3
=
0
Fzo cos(ωt)
Mθo sin(ωt)
.
(3–55)
53
CHAPTER 4OCEAN WAVE ENERGY HARVESTING
4.1 Electrical Generators for Wave Energy Harvesting
Several works have been done with electrical generators for wave energy harvesting.
Baker and Muller [110] and Muller [111] investigated the concept of direct drive wave
energy converters. Salter et al. [104] described a range of different control strategies
for wave energy power conversion mechanisms. Danielsson [112] designed a three-phase
permanent magnet linear generator for direct coupling to a floating buoy. Baker et al.
[113] outlined the performance and modeling of a prototype linear tubular permanent
magnet machine with an air cored stator. Polinder et al. [114] described the main
characteristics of the Archimedes Wave Swing. Thorburn et al. [115] presented different
topologies for the electrical system transmitting power to the grid. Mueller and Baker
[116] investigated the issues associated with converting the energy produced by marine
renewable energy converters using direct drive electrical power take-off. Leijon et al.
[117] presented a novel approach for electric power conversion discussing also the
economical and some environmental considerations. Rhinefrank et al. [118] described
the research, design, construction and prototype testing process of a novel ocean energy
direct drive permanent magnet linear generator buoy. Danielsson [119] studied the
electromagnetic properties, built a laboratory prototype and analyzed the performance of
linear synchronous permanent magnet generators. Muller et al. [120] described different
power take off mechanisms and described how some disadvantages shown by conventional
rotary generators can be overcome with direct drive systems. Szabo et al. [121] proposed a
novel modular permanent magnet tubular linear generator, analyzed by means of numeric
field computations. Trapanese [122] performed the optimization of a permanent magnet
linear generator directly coupled to sea waves. Elwood et al. [123] developed a hybrid
numerical/physical modeling approach for the design of a 10kW energy conversion system.
Liu et al. [124] presented an analytical model for predicting the electromagnetic and
54
electromechanical characteristics of a slotless tubular linear generator. Ruellan et al. [125]
presented a design methodology for the all-electric solution adapted to SEAREV. Cheng et
al. [126] presented a study of a multi-pole magnetic generator for energy harvesting at low
frequencies.
Since this research involves the first analysis of a tensegrity system for energy
harvesting under the influence of sea waves, the work is concentrated into the behavior
of the connecting linkage which transmits the wave-generated forces to the electrical
generators. Therefore, the electrical model is not included and is considered as damping
due to power take off, as suggested in [123] and described in (2–52), where bpz is the
power take-off damping coefficient. The efficiency of permanent magnet linear generators
used in ocean wave energy conversion is about 75-85%. [70, 112, 112].
4.2 Sea State Selection
Although the best wave climates are found in zones from 30 to 60 degrees latitude
as stated in Section 1.1.2, attractive wave climates are found also within ±30 degrees
latitude, where the lower power level is compensated by smaller power variability.
Additionally, there are several isolated places that can not be connected to continental
power grids in a lot of countries around the world, making ocean energy a feasible
renewable energy source. Therefore, wave energy can be especially useful for small
communities living near shore or in islands, avoiding the transportation and utilization of
fossil fuels.
Osorio et al. [127] developed a road map for harnessing marine renewable energy
for Colombia, pointing out that a profound knowledge of the available resource and
oceanographic conditions is required. In that sense, Ortega et al. [128] designed a
methodology for estimating wave power potential in places lacking instrumentation by
using reanalysis winds and wave generation models.
Ortega [129] performed a study on the exploitation of wave energy for Isla Fuerte, a
small Colombian island, located in the Caribbean Sea that does not have access to electric
55
power from the national grid and generates its power using fossil fuels. The wave available
resource and its time behavior was identified using third generation wave generation
models, with bathymetries and reanalysis winds as inputs. Ortega selected a place with
coordinates 9.408◦ N and 76.180◦ W, whose depth is around thirty meters, i.e. h = 30 m.
Such place was chosen taking in account environmental issues in order to protect the coral
reef, which is found even near places with higher wave power availability, Figure 4-1.
A Isla Fuerte. Taken from Google EarthTM.
LAT
LON
25
30
35
40
45
50
45
40
35
30
25
10
5
10 15
20
10
20
15
5
-76.22 -76.21 -76.2 -76.19 -76.18 -76.17 -76.16 -76.15
9.34
9.35
9.36
9.37
9.38
9.39
9.4
9.41
9.42
9.43
B Bathymetry of Isla Fuerte [129].
Figure 4-1. Location and bathymetry of Isla Fuerte.
4.2.1 Wave Power and Wave Kinematic Properties
At the selected location the wave height and period vary from 0.2 to 1.2 meters
and from 3 to 10 seconds respectively; and the joint probability for H and T shows that
H ∈ [0.4 0.6] m and T ∈ [4 6] s [129]. Using (2–48) with h = 30 m, the available power
per meter of wave front can be computed over the range of T and H. From Figure 4-2, the
maximum available power, P = 2.125 kW/m, is found at H = 0.6 m, T = 6 s.
The wave properties for h = 30 m, H = 0.6 m, T = 6 s can be found by solving
(2–25), and substituting the result into (2–20), (2–22) and (2–21), yields
λ = 56.1081 m, k = 0.1120 rad/m, c = 9.3514 m/s, ω = 1.0472 rad/s.
The ratio h/λ = 0.5347 ≥ 1/2, indicates that we have deep water conditions.
56
4
5
6
0.4
0.5
0.6
1
1.5
2
T (s)H (m)
P(k
W/m
)
Figure 4-2. Wave power variation for the sea state at 9.408◦ N, 76.180◦ W.
4.3 Direct Drive Heaving System
Several direct drive wave energy converters behave as a purely heaving body,
connected to electrical generators, see [110, 111, 117, 118, 121, 122] for reference. The
dimensions of the heaving float, as the one shown in Figure 2-6A, are L = 2 m, Z = 0.5 m,
B = 1 m, and d = 0.25 m.
The coefficients of (2–52) can be calculated using equations described in Section 2.6.2
and are listed in Table 4-1.
Table 4-1. Direct drive heaving float coefficients.
Coefficient Value Units Equationm = 515 kg (2–54)awz = 1.09×103 kg (2–56)brz = 2.04×103 N-s/m (2–58)bvz = 695.09 N-s/m (2–61)bpz = 0 N-s/m AssumedAwp = 2 m2 (2–55)N = 0 −− Assumedks = 0 N/m AssumedFzo = 1.19×104 N (2–63)αz = 0 rad Symmetric body in long waveswnz = 3.55 rad/s (2–68)
bcz = 1.14×104√
N (2–69)εz = 0.2090 rad (2–70)Zo = 0.6327 m (2–67) & (2–61)
57
Figure 4-3 shows the force that the wave induces over the body, and Figure 4-4 shows
the position, velocity and instant power dissipated by the float.
0 2 4 6 8 10 12−1.5
−1
−0.5
0
0.5
1
1.5x 10
4
t (s)
Fz(t
)(N
)
Figure 4-3. Heaving body simulation: wave-induced force.
0 2 4 6 8 10 12−1.5
−1
−0.5
0
0.5
1
1.5
t (s)
z(t)
(m),
z(t
)(m
/s)
z(t)z(t)
A Position and velocity.
0 2 4 6 8 10 12−5
0
5
10
15
t (s)
P(t
)(k
W)
B Instant dissipated power.
Figure 4-4. Heaving body simulation response.
The power of the heaving body is given by
Pz(t) = Fz(t)dz(t)
dt. (4–1)
The average power over one period of time is given by
58
Pave =1
T
∫
T
Pz(t)dt. (4–2)
Applying (4–2) over one wave period (in steady-state) of the function shown in
Figure 4-4B, gives an average power Pave = 0.82 kW. Since the float’s breadth is 1 m, then
we can compare this result with the power contained in one meter of wave front.
The available power for take-off, Ppz, is given by the difference between the wave
power and the power dissipated due to radiation (Prz) and viscous effects (Pvz)
Ppz = Pwz − Prz − Pvz =1
2ω2bpzZ
2o , (4–3)
where Pwz is the available wave power.
From Figure 4-2 the average power per meter of wave front is P = 2.125 kW.
Therefore, 38.6% of the wave energy is dissipated as radiation and viscous damping; hence
1.3 kW are available to be harvested with electrical generators (61.4% of the wave power).
From (4–3), the maximum damping coefficient due to power take-off is given by
bpz =2Ppz
ω2Z2o
. (4–4)
Equation (4–4) gives a rough estimate of the maximum additional damping that can
be added to the system using an electrical generator to harvest energy. This equation
becomes useful in design stages where the physical parameters of the system have to be
determined in order to develop a wave energy harvesting device.
4.4 Tensegrity System
Let the dimensions of the top platform be the same of the float in Section 4.3, i.e.
L = 2 m, Z = 0.5 m, B = 1 m, and d = 0.25 m. The additional constant mechanism
parameters are L0 = 6 m and Lpxz = 1/2L m. The base of the mechanism is located at
a depth hm = 6 m. The coefficients in all the terms of (3–55) can be calculated using
equations described in Chapters 2 and 3. Table 4-2 contains the values of the coefficients.
59
Table 4-2. Tensegrity harvesting system coefficients.
Coefficient Value Units Equationm = 515 kg (2–54)awx = 50.56 kg (3–37)awz = 1.09×103 kg (2–56)Aw = 240.45 kg (2–57)Iy = 182.4 kg-m2 (3–37)C = 7.15×103 N-m/rad (3–44)
LA0= 4.32 m Reverse kinematics
LB0= 4.32 m Reverse kinematics
kA = 200 N/m AssumedkB = 200 N/m Assumedbrz = 2.04×103 N-s/m (2–58)brθ = 679.24 N-m-s/rad (3–53)bvx = 6.95 N-s/m (2–61)bvz = 6.95 N-s/m (2–61)bpz = 0 N-s/m AssumedAwp = 2 m2 (2–55)Fzo = 1.19×104 N (2–63)Mθo = 445 N-m (2–65)αz = 0 rad Symmetric body in long waves
Figure 4-5 shows the forces that the wave induces over the body. Figure 4-6 shows the
position and velocity response of the top platform the three directions of motion: surge,
heave and pitch.
0 2 4 6 8 10 12−2
−1
0
1
2x 10
4
t (s)
Fz(t
)(N
)
0 2 4 6 8 10 12−500
0
500
t (s)
Mθ(t
)(N
-m)
Figure 4-5. Tensegrity simulation: wave-induced forces.
60
0 5 10
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
t (s)
x(t
)(m
),x(t
)(m
/s)
x(t)x(t)
A Surging motion.
0 5 10−1.5
−1
−0.5
0
0.5
1
1.5
t (s)
z(t)
(m),
z(t
)(m
/s)
z(t)z(t)
B Heaving motion.
0 2 4 6 8 10 12−6
−4
−2
0
2
4
6
8
10
t (s)
θ(t)
(deg
),θ(t
)(d
eg/s)
θ(t)
θ(t)
C Pitching motion.
Figure 4-6. Tensegrity simulation: surging, heaving and pitching motions.
Figure 4-7 shows the instantaneous power of the body. This power is a measure of
the energy that is being dissipated due to radiation and viscous effects. Applying (4–2)
over one wave period (in steady-state) gives an average power Pave = 0.6231 kW. Since
the float’s breadth is 1 m, then we can compare this result with the power contained
in one meter of wave front. The available wave power for take-off was computed as
P = 2.125 kW. Therefore, 29.31% of the available energy is being dissipated due to
radiation and viscous effects; hence, 1.5 kW are available to be harvested by electrical
generators (70.69% of the wave energy).
The solution of the differential equation of motion of the tensegrity mechanism was
performed for a set of generalized coordinates defined in the Cartesian space. Hence, the
reverse kinematic analysis developed in Chapter 3 allows one to compute the values of any
variable in the joint space.
Figure 4-8A shows the behavior of the hypothetical electrical generators; the required
displacement of such generators is less than 2 m, a value that can be achieved for modern
linear permanent magnet generators. The position and velocity data can be used as inputs
in the analysis of the electrical devices in future work.
61
0 2 4 6 8 10 12−10
−5
0
5
10
15
20
t (s)
P(t
)(k
W)
Figure 4-7. Tensegrity simulation: instant dissipated power.
Figure 4-8B shows the lengths of the springs, which are always greater than the free
length, a requirement to maintain the tensegrity configuration of the mechanism.
0 2 4 6 8 10 126
7
8
9
10
11
t (s)
Posi
tion
(m)
1S 2
4S 5
0 2 4 6 8 10 12−2
0
2
4
t (s)
Vel
oci
ty(m
/s)
1v2
4v5
A Motion of generators.
0 2 4 6 8 10 122
4
6
8
t (s)
Len
gth
(m)
LA
LA0
0 2 4 6 8 10 122
4
6
8
t (s)
Len
gth
(m)
LB
LB0
B Length of elastic ties.
Figure 4-8. Tensegrity simulation: motion in generators and springs.
4.5 Discussion of Results
From sections 4.3 and 4.4, we see that the tensegrity configuration dissipates less
energy than the same float in pure heaving motion. The direct drive heaving system allows
to harvest 1.3 kW (61.4% of the wave power) while the tensegrity mechanism allows to
harvest 1.5 kW (70.69% of the wave energy). This value is about 10% more than the
62
value given by the configuration that is used in most direct drive systems such the ones in
[110, 111, 117, 118, 121, 122].
Additionally, no parameters, other than the dimensions of the float, can be modified
in the purely heaving device. On the other hand, several parameters can be changed in
the tensegrity configuration to modify the energy dissipation due to radiation and viscous
effects. For instance, Figure 4-9 shows the variation of the percentage of the available
power that is dissipated due to radiation and viscous damping, as a function of the base
length (Figure 4-9A) and the stiffness of the elastic ties (Figure 4-9B).
2 4 6 8 1028
28.5
29
29.5
30
30.5
31
31.5
32
L0 (m)
Pave
(%)
A Variation of Pave with L0.
500 1000 1500 200029
30
31
32
33
34
35
36
kA, kB (N/m)
Pave
(%)
B Variation of Pave with kA, kB .
Figure 4-9. Variation of power dissipation with mechanism parameters.
63
CHAPTER 5CONCLUSIONS AND FUTURE WORK
5.1 Conclusions
The idea of getting energy from ocean waves is not new, and only a few references
in literature have stated the possibility of harvesting energy from external disturbances
in a tensegrity structure. This research presented the first approximation to the energy
harvesting potential of a tensegrity system interacting with ocean waves.
This work addressed the analysis of a tensegrity mechanism for ocean wave energy
harvesting. A planar tensegrity morphology was selected for the system, and the dynamic
analysis was performed considering a two-dimensional linear model of the ocean surface
waves, considering the added mass, radiation damping and viscous damping phenomena.
The forward and reverse position analyses were performed using a geometric
approach. The forward and reverse velocity analyses were performed using theory of
screws. Numerical examples are provided in both cases. The Lagrangian approach
was used to deduce the equations of motion of the mechanism subject to the action of
wave-induced forces and moments.
The tensegrity potential for wave energy harvesting was demonstrated by comparison
with a purely heaving configuration that is commonly used in harvesting devices. It
was shown that the tensegrity configuration allows to harvest about 70% of the energy
contained in a linear wave, against 60% that was allowed by the conventional system using
the same floating body as input into the system. It was shown how the change in some
reconfigurable parameters of the tensegrity system affect the power dissipation due to
radiation and viscous damping.
The interaction between ocean waves, a multi-degree-of-freedom linkage and electrical
generators poses challenging problems in terms of mathematical modeling and simulation.
Nonetheless, the ideas presented in this document will be useful for the analysis and
testing of more advanced and complex energy harvesting devices.
64
This research constitutes an interesting approach to show how the extensive
knowledge acquired in the analysis and design of mechanism can be used in new
applications that contribute to the world’s sustainable development.
5.2 Future Work
One important stage in the development of wave energy converters is the experimental
phase. Since this research included only simulation results, it would be interesting to
develop a physical scale model to be tested in a tank. Appropriate power scaling methods
are well referenced in literature and must be taken in account.
In this research we used a linear model for the sea waves, and the analysis was
performed for a planar mechanism. However, more realistic nonlinear and random models
for the behavior of ocean waves can be used. Such models would allow one to use not only
a planar linkage, but spatial tensegrity mechanisms and several different morphologies.
Optimization tools can be used to improve the performance of the analyzed
mechanism, and to to find a more suitable, or maybe optimal, tensegrity morphology
for ocean wave energy harvesting.
Because there are several options for electrical generators, further studies must
be carried out in order to determine which configuration offer better conditions to be
used with different tensegrity configurations. Additionally, the control methods and the
connection to a power grid or energy storage options shall be addressed in order to develop
a commercial device.
Future projects can include the development of mobile ocean energy harvesting
stations. Such stations can serve as complementary energy sources to relatively small
ocean devices such as underwater remotely operated vehicles.
65
REFERENCES
[1] Johansson, T., McCormick, K., Neij, L., and Turkenburg, W., 2004. “The potentialsof renewable energy”. In Proceedings of the 2004 International Conference forRenewable Energies.
[2] Favre–Perrod, P., Geidl, M., Klockl, B., and Koeppel, G., 2005. “A vision of futureenergy networks”. In Proceedings of the 2005 IEEE Power Engineering SocietyInaugural Conference and Exposition in Africa, IEEE.
[3] Redondo–Gil, C., Esquibel, L., Alonso Sanchez, A., and Zapico., P., 2009. “Europeanstrategic energy technology plan”. In Proceedings of the 2009 InternationalConference on Renewable Energy and Power Quality ICREPQ09, EA4EPQ.
[4] Pontes, M., and Falcao, A., 2001. “Ocean energies: Resources and utilisation”.In Proceedings of the 18th Congress of the World Energy Council, Buenos Aires,Argentina.
[5] Scruggs, J., and Jacob, P., 2009. “ENGINEERING: Harvesting Ocean WaveEnergy”. Science, 323(5918), pp. 1176–1178.
[6] Murray, R., 2006. Review and analysis of ocean energy systems, development andsupporting policies. Tech. rep., Prepared by AEA Energy & Environment on thebehalf of Sustainable Energy Ireland for the IEAs Implementing Agreement onOcean Energy Systems.
[7] Clement, A., McCullenc, P., Falcao, A., Fiorentino, A., Gardner, F., Hammarlund,K., Lemonis, G., Lewish, T., Nielseni, K., Petroncini, S., Pontes, M., Schild, P.,Sjostromm, B., Sorensen, H., and Thorpe, T., 2002. “Wave energy in Europe:Current status and perspectives”. Renewable Sustainable Energy Rev., 6(5), Oct,pp. 405–431.
[8] Technology Future Energy Solutions AEA, 2003. Wave and marine current energy.Tech. rep., Department of Trade and Industry United Kingdom, New and RenewableEnergy Programme.
[9] Previsik, M., Hagerman, G., and Bedard, R., 2004. Offshore wave energy conversiondevices. report E2I EPRI WP–004–US. Tech. rep., Electric Power Research Institute,Palo Alto, CA.
[10] Pontes, M., Aguiar, R., and Oliveira Pires, H., 2005. “A nearshore wave energy atlasfor Portugal”. J. Offshore Mech. Arct. Eng., 127(3), pp. 249–255.
[11] Cornett, A., and Tabotton, M., 2006. Inventory of Canadian marine renewableenergy resources. Tech. rep., Canadian Hydraulics Centre, National ResearchCouncil Canada and Triton Consultants Ltd., Canada.
[12] Bedard, R., Previsic, M., Hagerman, G., Polagye, B., Musial, W., Klure, J.,von Jouanne, A., Mathur, U., Partin, J., Collar, C., Hopper, C., and Amsden,
66
S., 2007. North American ocean energy status. Tech. rep., Electric Power ResearchInstitute, Palo Alto, CA.
[13] Musial, W., 2008. Status of wave and tidal power technologies for the United States.Tech. rep., National Renewable Energy Laboratory NREL, U.S. Department ofEnergy, Golden, Colorado, Aug.
[14] Khan, J., and Bhuyan, G., 2009. Ocean energy: Global technology developmentstatus. Tech. rep., Prepared by Powertech Labs for the IEA-OES, [Online],Available: www.iea-oceans.org.
[15] Hao, W., Qiang, Y., Donju, W., Yiru, W., Jianhui, Y., and Hao, Z., 2009. “Thestatus and prospect of ocean energy generation in China”. In Proceedings ofthe International Conference on Sustainable Power Generation and Supply, 2009.SUPERGEN ’09., IEEE, pp. 1–6.
[16] Geoscience Australia and ABARE, 2010. Australian energy resource assessment.Tech. Rep. 70142, Prepared by Geoscience Australia and Australian Bureau ofAgricultural and Resource Economics (ABARE) for the Australian GovernmentDepartment of Resources, Energy and Tourism (RET), Canberra, Australia.
[17] Florez, D. A., Correa, J. C., Posada, N. L., Valencia, R. A., and Zuluaga, C. A.,2010. “Marine energy devices for Colombian seas”. In Proceedings of the ASME2010 International Mechanical Engineering Congress & Exposition IMECE2010,ASME.
[18] Blunden, L., and Bahaj, A., 2007. “Tidal energy resource assessment for tidal streamgenerators”. P. I. Mech. Eng. A - J. Pow, 221(2), pp. 137–146.
[19] Henfridsson, U., Neimane, V., Strand, K., Kapper, R., Bernhoff, H., Danielsson,O., Leijon, M., Sundberg, J., Thorburn, K., Ericsson, E., and Bergman, K., 2007.“Wave energy potential in the Baltic Sea and the Danish part of the North Sea, withreflections on the Skagerrak”. Renewable Energy, 32(12), Oct, pp. 2069–2084.
[20] Defne, Z., Haas, K., and Fritz, H., 2009. “Wave power potential along the Atlanticcoast of the southeastern USA”. Renewable Energy, 34(10), Oct, pp. 2197–2205.
[21] Iglesias, G., and Carballo, R., 2009. “Wave energy potential along the Death Coast(Spain)”. Energy, 34(11), Nov, pp. 1963–1975.
[22] Lavrakas, J., Smith, J., and Corporation, A. R., 2009. Wave energy infrastructureassessment in Oregon. Tech. rep., Oregon Wave Energy Trust, Portland.
[23] Mackay, E., Bahaj, A., and Challenor, P., 2009. “Uncertainty in wave energyresource assessment. part 1: Historic data”. Renewable Energy, 35(8),pp. 1792–1808.
67
[24] Pontes, M. T., Bruck, M., and Lehner, S., 2009. “Assessing the wave energy resourceusing remote sensed data”. In Proceedings of the 8th European Wave and TidalEnergy Conference, Uppsala, Sweden.
[25] Seng Lim, Y., and Lee Koh, S., 2010. “Analytical assessments on the potential ofharnessing tidal currents for electricity generation in Malaysia”. Renewable Energy,35(5), pp. 1024–1032.
[26] Iglesias, G., and Carballo, R., 2010. “Wave energy resource in the Estaca de Baresarea (Spain)”. Renewable Energy, 35(7), pp. 1574–1584.
[27] Falcao, A., 2010. “Wave energy utilization: A review of the technologies”. RenewableSustainable Energy Rev., 14(3), pp. 899 – 918.
[28] U.S. Department of Energy. Marine andhydrokinetic technology database, [Online]. Available:http://www1.eere.energy.gov/windandhydro/hydrokinetic/default.aspx.
[29] Fuller, R., 1962. Tensile-integrity structures, US Patent 3,063,521, Nov. 13.
[30] Emmerich, D., 1964. Construction de rseaux autotendants, French patent no.1,377,290, Sep. 28.
[31] Snelson, K., 1965. Continuous tension, discontinuous compression structures, USPatent 3,169,611, Feb. 16.
[32] Pugh, A., 1976. An introduction to tensegrity. University of California Press.
[33] Motro, R., 1992. “Tensegrity systems: the state of the art”. Int. J. Space Struct.,7(2), pp. 75–83.
[34] Skelton, R., and de Oliveira, M. C., 2009. Tensegrity Systems. Springer.
[35] Hanaor, A., 1992. “Aspects of design of double layer tensegrity domes”. Int. J. SpaceStruct., 7(2), pp. 101–103.
[36] Motro, R., 2003. Tensegrity: structural systems for the future. Kogan Page Science,Guildford, UK.
[37] Fu, F., 2005. “Structural behavior and design methods of tensegrity domes”. J.Constr. Steel Res., 61(1), pp. 23 – 35.
[38] Gomez, V., 2010. Tensegrity Structures and their Application to Architecture.PUbliCan – Ediciones de la Universidad de Cantabria, Santander.
[39] Rhode-Barbarigos, L., Bel Hadj Ali, N., Motro, R., and Smith, I. F., 2010.“Designing tensegrity modules for pedestrian bridges”. Eng. Struct., 32(4), pp. 1158– 1167.
68
[40] Bel Hadj Ali, N., Rhode-Barbarigos, L., Pascual Albi, A. A., and Smith, I. F., 2010.“Design optimization and dynamic analysis of a tensegrity-based footbridge”. Eng.Struct., 32(11), pp. 3650 –3659.
[41] Stern, I., 1999. “Development of design equations for self-deployable N-struttensegrity systems”. Master’s thesis, University of Florida, Gainesville, FL.
[42] Duffy, J., Rooney, J., Knight, B., and Crane III, C. D., 2000. “A review of a familyof self-deploying tensegrity structures with elastic ties”. The Shock and VibrationDigest, 32(2), pp. 100–106.
[43] Knight, B., 2000. “Deployable antenna kinematics using tensegrity structure design”.PhD thesis, Mechanical Engineering, University of Florida, Gainesville, FL.
[44] Pellegrino, S., Kukathasan, S., Tibert, G., and Watt, A., 2000. Small satellitedeployment mechanisms. Tech. rep., Prepared by the Defence Evaluation ResearchAgency and the University of Cambridge on behalf of the British National SpaceCentre.
[45] Tibert, G., 2002. “Deployable tensegrity structures for space applications”. PhDthesis, Department of Mechanics, Royal Institute of Technology, Sweden.
[46] Sultan, C., and Skelton, R., 2003. “Deployment of tensegrity structures”. Int. J.Solids Struct., 40(18), pp. 4637 – 4657.
[47] Ingber, D. E., 1998. “The architecture of life”. Sci. Am., 278, pp. 48–57.
[48] Caadas, P., Laurent, V. M., Oddou, C., Isabey, D., and Wendling, S., 2002.“A cellular tensegrity model to analyse the structural viscoelasticity of thecytoskeleton”. J. Theor. Biol., 218(2), pp. 155 – 173.
[49] Ingber, D., 2003. “Tensegrity I. Cell structure and hierarchical systems biology”. J.Cell Sci., 116, Jan, pp. 1157–1163.
[50] Huang, S., Sultan, C., and Ingber, D., 2007. Complex Systems Science inBiomedicine. Springer US, ch. Tensegrity, Dynamic Networks, and ComplexSystems Biology: Emergence in Structural and Information Networks Within LivingCells, pp. 283–310.
[51] Cretu, S., 2009. Modeling, Simulation and Control of Nonlinear EngineeringDynamical Systems. Springer, Netherlands, ch. Tensegrity as a Structural Frameworkin Life Sciences and Bioengineering, pp. 301–311.
[52] Sultan, C., Corless, M., and Skelton, R., 2000. “Tensegrity flight simulator”. J. ofGuid. Control Dynam., 23(6), Nov, pp. 1055–1064.
[53] Tran, T., Crane, C., and Duffy, J., 2002. “The reverse displacement analysis of atensegrity based parallel mechanism”. In Proceedings of the 5th Biannual WorldAutomation Congress, pp. 637–643.
69
[54] Kanchanasaratool, N., and Williamson, D., 2002. “Motion control of a tensegrityplatform”. Communications in Information and Systems, 2(3), pp. 299–324.
[55] Sultan, C., and Skelton, R., 2004. “A force and torque tensegrity sensor”. Sens.Actuators, A, 112(2-3), pp. 220–231.
[56] Marshall, M., and Crane, C., 2004. “Design and analysis of a hybrid parallelplatform that incorporates tensegrity”. In Proceedings of the ASME 2004International Design Engineering Technical Conferences and Computers andInformation in Engineering Conference (IDETC/CIE2004), no. 46954, ASME,pp. 535–540.
[57] Arsenault, M., and Gosselin, C., 2005. “Kinematic, static, and dynamic analysis of aplanar one-degree-of-freedom tensegrity mechanism”. ASME J. Mech. Des., 127(6),pp. 1152–1160.
[58] Arsenault, M., and Gosselin, C., 2006. “Kinematic, static, and dynamic analysisof a spatial three-degree-of-freedom tensegrity mechanism”. ASME J. Mech. Des.,128(5), pp. 1061–1069.
[59] Arsenault, M., and Gosselin, C., 2006. “Kinematic, static and dynamic analysis of aplanar 2-DOF tensegrity mechanism”. Mech. Mach. Theory, 41(9), pp. 1072–1089.
[60] Vasquez, R., and Correa, J., 2007. “Kinematics, dynamics and control of aplanar 3-DOF tensegrity robot manipulator”. In Proceedings of the ASME 2007International Design Engineering Technical Conferences and Computers andInformation in Engineering Conference (IDETC/CIE2007), ASME, pp. 855–866.
[61] Scruggs, J., and Skelton, R., 2006. “Regenerative tensegrity structures for energyharvesting applications”. In Proceedings of the 45th IEEE Conference on Decision &Control, IEEE, pp. 2282–2287.
[62] Jensen, O., Wroldsen, A., Lader, P., Fredheim, A., and Heide, M., 2007. “Finiteelement analysis of tensegrity structures in offshore aquaculture installations”.Aquacult. Eng., 36(3), May, pp. 272–284.
[63] Wroldsen, A., 2007. “Modelling and control of tensegrity structures”. PhD thesis,Department of Marine Technology, Norwegian University of Science and Technology.
[64] National Renewable Energy Laboratory, 2009. Ocean energy technology overview.Tech. Rep. DOE/GO-102009-2823, Prepared for the U.S. Department of Energy,Office of Energy Efficiency and Renewable Energy Federal Energy ManagementProgram, Golden, CO.
[65] Electric Power Research Institute, 2008. Prioritized RDD&D needs: Marine andother hydrokinetic energy. Tech. rep., Electric Power Research Institute, FinalReport 2008, Palo Alto, CA.
70
[66] U.S. Department of Energy, EISA Report to Congress, 2008. Potentialenvironmental effects of marine and hydrokinetic energy technologies. Tech.rep., Prepared in response to the Energy Independence and Security Act of 2007,Section 633(b).
[67] Dalton, G., and Gallachir, B., 2010. “Building a wave energy policy focusing oninnovation, manufacturing and deployment”. Renewable Sustainable Energy Rev.,14(8), pp. 2339 – 2358.
[68] U.S. Department of the Interior, Minerals Management Service, 2009. “Renewableenergy and alternate uses of existing facilities on the outer continental shelf; finalrule”. Fed. Regist., 74(81), Apr, pp. 19637–19871.
[69] Det Norske Veritas, 2008. Certification of tidal and wave energy converters, OffshoreService Specification DNV-OSS-312.
[70] McCormick, M., 2007. Ocean Wave Energy Conversion. Dover Publications.Originally published by Wiley-Interscience, New York in 1981.
[71] Cruz, J., 2008. Ocean Wave Energy: Current Status and Future Prespectives.Springer.
[72] Fugro OCEANOR. Wave power, [Online]. Available:http://www.oceanor.no/index.htm.
[73] Gato, L., and Falcao, A., 1988. “Aerodynamics of the wells turbine”. Int. J. Mech.Sci., 30(6), pp. 383 – 395.
[74] Falcao, A., 2004. “First-generation wave power plants: Current status and R&Drequirements”. J. Offshore Mech. Arct. Eng., 126(4), pp. 384–388.
[75] Arup Energy, 2005. Oscillating water column wave energy converter evaluationreport. Tech. rep., The Carbon Trust.
[76] Nielsen, K., and Pedersen, T., 2009. “A dynamic model for control purposes of awave energy power plant buoyancy system”. In Proceeding of the IEEE InternationalConference on Control and Automation, ICCA 2009., IEEE, pp. 825–830.
[77] Dunnett, D., and Wallace, J. S., 2009. “Electricity generation from wave power inCanada”. Renewable Energy, 34(1), pp. 179 – 195.
[78] Skelton, R., Helton, J., Adhiraki, R., Pinaud, J., and Chan, W., 2001. Handbookof mechanical systems design. CRC Press, ch. An Introduction to the Mechanics ofTensegrity Structures, pp. 316–449.
[79] Michell, A., 1904. “The limits of economy in frame structures”. Philos. Mag., 8(47),Nov, pp. 589–597.
71
[80] Skelton, R., Adhikari, R., Pinaud, J., Chan, W., and Helton, J., 2001. “Anintroduction to the mechanics of tensegrity structures”. In Proceedings of the40th IEEE Conference on Decision and Control, 2001., Vol. 5, pp. 4254 –4259.
[81] Aldrich, J., and Skelton, R., 2003. “Control/structure optimization approach forminimum-time reconfiguration of tensegrity systems”. In Proceedings of the SPIE:Smart Structures and Materials 2003: Modeling, Signal Processing, and Control,R. Smith, ed., SPIE, pp. 448–459.
[82] Pellegrino, S., 1990. “Analysis of prestressed mechanisms”. Int. J. Solids Struct.,26(12), pp. 1329 – 1350.
[83] Calladine, C., and Pellegrino, S., 1991. “First-order infinitesimal mechanism”. Int. J.Solids Struct., 27(4), pp. 505–515.
[84] Djouadi, S., Motro, R., Pons, J., and Crosnier, B., 1998. “Active control oftensegrity systems”. J. Aerosp. Eng., 11(2), April, pp. 37–44.
[85] Sultan, C., 1999. “Modeling, design, and control of tensegrity structures withapplications”. PhD thesis, Aerospace Engineering, Purdue University.
[86] Oppenheim, I. J., and Williams, W. O., 2001. “Vibration of an elastic tensegritystructure”. Eur. J. Mech. A. Solids, 20(6), pp. 1023 – 1031.
[87] Skelton, R. E., Pinaud, J. P., and Mingori, D. L., 2001. “Dynamics of the shell classof tensegrity structures”. J. Franklin Inst., 338(2-3), pp. 255 – 320.
[88] Sultan, C., Corless, M., and Skelton, R., 2001. “The prestressability problem oftensegrity structures: some analytical solutions”. Int. J. Solids Struct., 38(30-31),pp. 5223–5252.
[89] Sultan, C., Corless, M., and Skelton, R. E., 2002. “Linear dynamics of tensegritystructures”. Eng. Struct., 24(6), pp. 671 – 685.
[90] Bossens, F., Callafon, R., and Skelton, R., 2004. Modal analysis of a tensegritystructure: an experimental study. Tech. rep., Department of Mechanical andAerospace Engineering, University of California, San Diego.
[91] Chan, W. L., Arbelaez, D., Bossens, F., and Skelton, R. E., 2004. “Active vibrationcontrol of a three-stage tensegrity structure”. In Proceeding of the SPIE 11thAnnual International Symposium on Smart Structures and Materials, K.-W. Wang,ed., no. 1, SPIE, pp. 340–346.
[92] Crane, C., Duffy, J., and Correa, J., 2005. “Static analysis of tensegrity structures”.ASME J. Mech. Des., 127(2), pp. 257–268.
[93] Bayat, J., and Crane, C., 2007. “Closed-form equilibrium analysis of planartensegrity structures”. In Proceedings of the ASME 2007 International Design
72
Engineering Technical Conferences and Computers and Information in EngineeringConference (IDETC/CIE2007), ASME, pp. 13–23.
[94] Arsenault, M., and Gosselin, C., 2007. “Static balancing of tensegrity mechanisms”.ASME J. Mech. Des., 129(3), pp. 295–300.
[95] Crane, C., Bayat, J., Vikas, V., and Roberts, R., 2008. Advances in Robot Kinemat-ics: Analysis and Design. Springer Netherlands, May, ch. Kinematic Analysis of aPlanar Tensegrity Mechanism with Pre-Stressed Springs, pp. 419–427.
[96] Arsenault, M., and Gosselin, C., 2009. “Kinematic and static analysis of a 3-PUPSspatial tensegrity mechanism”. Mech. Mach. Theory, 44(1), pp. 162–179.
[97] Wroldsen, A. S., de Oliveira, M. C., and Skelton, R. E., 2009. “Modelling andcontrol of non-minimal non-linear realisations of tensegrity systems”. Int. J. Control,82(3), pp. 389–407.
[98] McCormick, M., 2009. Ocean Engineering Mechanics with Applications. CambridgeUniversity Press.
[99] Craik, A. D., 2004. “The origins of water wave theory”. Annu. Rev. Fluid Mech.,36(1), pp. 1–28.
[100] Airy, G. B., 1841. Encyclopaedia Metropolitana (1817-1845), Mixed Sciences, Vol. 3.Part publication, ch. Tides and Waves, p. 396.
[101] Falnes, J., 2002. Ocean Waves and Oscillating Systems: Linear InteractionsIncluding Wave-Energy Extraction. Cambridge University Press.
[102] Vugts, J., 1968. “The hydrodynamics coefficients for swaying, heaving, and rollingcylinders in a free surface”. Int. Shipbg. Prog., 5(167), pp. 251–276.
[103] Szuladzinski, G., 2010. Formulas for Mechanical and Structural Shock and Impact.CRC Press, ch. Aerodynamic Drag Coefficients, pp. 747–748.
[104] Salter, S., Taylor, J., and Caldwell, N., 2002. “Power conversion mechanisms forwave energy”. P. I. Mech. Eng. M - J. Eng., 216(1), pp. 1–27.
[105] Crane III, C. D., Rico, J. M., and Duffy, J., 2009. Screw theory and its applicationto spatial robot manipulators. Tech. rep., Center for Intelligent Machines andRobotics, University of Florida.
[106] Rico, J. M., Gallardo, J., and Duffy, J., 1999. “Screw theory and higher orderkinematic analysis of open serial and closed chains”. Mech. Mach. Theory, 34(4),pp. 559 – 586.
[107] Ball, S. R. S., 1900. A treatise on the theory of screws. Cambridge University Press.
[108] Doughty, S., 1988. Mechanics of machines. John Wiley & Sons, New York.
73
[109] Rao, A., 2006. Dynamics of Particles and Rigid Bodies: A Systematic Approach.Cambridge University Press.
[110] Baker, N. J., and Mueller, M. A., 2001. “Direct drive wave energy converters”. Rev.des Energ. Ren., Power Engineering, pp. 1–7.
[111] Mueller, M., 2002. “Electrical generators for direct drive wave energy converters”.IEEE Proc-C, 149(4), pp. 446 – 456.
[112] Danielsson, O., 2003. “Design of a linear generator for wave energy plant”. Master’sthesis, Uppsala University, School of Engineering.
[113] Baker, N., Mueller, M., and Spooner, E., 2004. “Permanent magnet air-coredtubular linear generator for marine energy converters”. In Proceedings of the SecondInternational Conference on Power Electronics, Machines and Drives PEMD2004,Vol. 2, IEEE, pp. 862 – 867.
[114] Polinder, H., Damen, M., and Gardner, F., 2004. “Linear pm generator system forwave energy conversion in the aws”. IEEE T. Energy. Conver., 19(3), pp. 583 – 589.
[115] Thorburn, K., Bernhoff, H., and Leijon, M., 2004. “Wave energy transmission systemconcepts for linear generator arrays”. Ocean Eng., 31(11-12), Aug, pp. 1339–1349.
[116] Mueller, M., and Baker, N., 2005. “Direct drive electrical power take-off for offshoremarine energy converters”. P. I. Mech. Eng. A - J. Pow, 219(3), pp. 223–234.
[117] Leijon, M., Danielsson, O., Eriksson, M., Thorburn, K., Bernhoff, H., Isberg, J.,Sundberg, J., Ivanova, I., Sjstedt, E., gren, O., Karlsson, K., and Wolfbrandt, A.,2006. “An electrical approach to wave energy conversion”. Renewable Energy, 31(9),Jul, pp. 1309–1319.
[118] Rhinefrank, K., Agamloh, E., von Jouanne, A., Wallace, A., Prudell, J., Kimble,K., Aills, J., Schmidt, E., Chan, P., Sweeny, B., and Schacher, A., 2006. “Novelocean energy permanent magnet linear generator buoy”. Renewable Energy, 31(9),pp. 1279 – 1298.
[119] Danielsson, O., 2006. “Wave energy conversion. linear synchronous permanentmagnet generator”. PhD thesis, Uppsala University, Faculty of Science andTechnology.
[120] Mueller, M., Polinder, H., and Baker, N., 2007. “Current and novel electricalgenerator technology for wave energy converters”. In IEEE International ElectricMachines Drives Conference IEMDC ’07., Vol. 2, pp. 1401 –1406.
[121] Szabo, L., Oprea, C., Viorel, I., and Biro, K. A., 2007. “Novel permanent magnettubular linear generator for wave energy converters”. In Proceeding of the IEEEInternational Electric Machines Drives Conference, IEMDC ’07, IEEE.
74
[122] Trapanese, M., 2008. “Optimization of a sea wave energy harvesting electromagneticdevice”. IEEE T. Magn., 44(11), pp. 4365 – 4368.
[123] Elwood, D., Schacher, A., and Rhinefrank, K., 2009. “Numerical modeling and oceantesting of a direct-drive wave energy device utilizing a permanent magnet lineargenerator for power take-off”. In Proceedings of the ASME 2009 28th InternationalConference on Ocean, Offshore and Arctic Engineering (OMAE2009), ASME,pp. 817–824.
[124] Liu, C.-T., Lin, C.-L., Hwang, C.-C., and Tu, C.-H., 2010. “Compact model of aslotless tubular linear generator for renewable energy performance assessments”.IEEE T. Magn., 46(6), pp. 1467 –1470.
[125] Ruellan, M., BenAhmed, H., Multon, B., Josset, C., Babarit, A., and Clement,A., 2010. “Design methodology for a SEAREV wave energy converter”. IEEE T.Energy. Conver., 25(3), pp. 760 –767.
[126] Cheng, S., and Arnold, D. P., 2010. “A study of a multi-pole magnetic generatorfor low-frequency vibrational energy harvesting”. J. Micromech. Microeng., 20(2),02/2010, p. 025015.
[127] Osorio, A., Agudelo, P., Correa, J., Otero, L., Ortega, S., Hernandez, J., andRestrepo, J., 2011. “Building a roadmap for the implementation of marine renewableenergy in Colombia”. In Proceedings of OCEANS 2011 IEEE - Spain, pp. 1 –5.
[128] Ortega, S., Osorio, A., Agudelo-Restrepo, P., and Velez, J., 2011. “Methodologyfor estimating wave power potential in places with scarce instrumentation in theCaribbean Sea”. In Proceedings of OCEANS 2011 IEEE - Spain, pp. 1 –5.
[129] Ortega, S., 2010. “Estudio de aprovechamiento de la energıa del oleaje en Isla Fuerte(Caribe colombiano)”. Master’s thesis, School of Geosciences and Environment,National University of Colombia.
75
BIOGRAPHICAL SKETCH
Rafael E. Vasquez received his B.S. in mechanical engineering in 2002 and his
M.Sc. in engineering with emphasis in automation in 2007, both from the Universidad
Pontificia Bolivariana UPB, Medellin, Colombia. He joined the faculty of the UPB
in 2003 where he is associate professor in the area of dynamics, systems and control.
He is currently completing a doctoral degree in mechanical engineering, under a
Fulbright-Colciencias-DNP scholarship, at the Center for Intelligent Machines and
Robotics (CIMAR) at the University of Florida, Gainesville, Florida. His research
interests are theory of mechanisms; design, analysis and control of dynamic systems
and new technologies for energy harvesting. He is member of the American Society of
Mechanical Engineers (ASME) since 2005. After completing his PhD, Rafael will go back
to Colombia to conduct research and teach in the Department of Mechanical Engineering
at the Universidad Pontificia Bolivariana UPB, Medellin, Colombia.
76