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The SOOR Wave Energy Converter
MAE 342 (Principles of Mechanical Design)
Scott Olson- [email protected] _____________________________________________________
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Contents
Introduction .................................................................................................................................................. 3
Concept ......................................................................................................................................................... 3
Analysis of Deep Water Waves ..................................................................................................................... 5
Final Design ................................................................................................................................................. 27
Structural and Fatigue Analysis ................................................................................................................... 31
Stress Concentration and Critical Location ................................................................................................. 33
SOOR’s Output and Power Produced ......................................................................................................... 34
Appendix ..................................................................................................................................................... 35
.................................................................................................................................................................... 48
.................................................................................................................................................................... 54
References .................................................................................................................................................. 54
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Introduction
In recent year, the movement of renewable energies has been on a record increase. Society has
been drawn towards energies like solar and wind technologies, while slowly turning their back on fossil
fuels. The major benefit of renewables is the fact that they will never run out. Unlike fossil fuels, which
have a finite amount; renewable energies are infinite. The sun will not cease to shine and the wind will
not halt for the foreseeable future. Just like the sun and the wind; the oceans will not stop moving as
well. The Earth’s oceans and seas are constantly stirring and never at a standstill. Waves contain
immense amounts of energy and force, which could be used as an alternative renewable energy source.
A number of different devices have been invented to harness wave energy, however very few of them
utilize the full potential of the waves. As an attempt to exploit wave energy, the design group of Scott
Olson, Matthew O’Donnell, Joel Richardson, and Matthew Smith have proposed a new device that has
the potential to make wave energy a bigger competitor in the renewable energies race. The new device
has been title the SOOR (Smith. Olson. O’Donnell. Richardson) Wave Energy Converter.
Concept
Current wave energy devices float on the surface of waters. These mechanisms are long snakes
filled with hydraulic fluid. The waves move the hydraulic fluid inland, and create electricity in the
process. Figure 1.1 shows a picture of the Pelamis Wave Energy Converter (Pelamis Wave), the most
popular wave energy converter on the market.
An issue with the Pelamis devices is that they are usually placed in shallow waters near the
shores. They have the ability to obscure the ocean view on beaches; many tourist locations refuse to
invest in these converters due to this issue. Unfortunately, these devices do not produce a large amount
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of power. A wave is similar to an iceberg due
to the fact that much of its energy is under
the surface of the water where Pelimas are
unable to reach. The entire concept of SOOR
is to create a device that doesn’t lay on the
surface of the water, but is submerged
underneath the surface.
The SOORS device contains 2 major components; a collapsible pylon and ballast. The pylon is a
diamond shaped component and is where all the energy is converted. As the wave hits the pylon,
collapses, and runs a series of pistons that are located on the inner walls of the pylon. These pistons are
what create the energy. A series of springs are located in the hollowed body of the pylon so that the
structure does not deform too much and so it reforms back to its original shape after the wave has
passed. Rubber stops are also to the internal structure of the pylon in order to protect the pistons from
damage. Slides along the inner wall of the pylon and hinges on the tips of the diamond shape promote
the collapsing of the structure. The ballasts are large structures placed on each side of the pylon. The
ballasts are large air tanks that keep SOOR from sinking. The ballasts also direct the wave to the pylon
and act as channels for the wave force. Figure 1.2 shows a diagram of the SOOR Wave Energy Converter.
Figure 1.2 is the Top View of Soors.
Figure 1.1: Pelimas Wave Energy Converter
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Figure 1.2 SOOR Wave Energy Generator (Top View)
The reason the ballasts and pylon are configured this way is to create a flow channel for the
wave to follow. Since the area of this flow channel decreases as the wave travels to the center of the
pylon, a nozzle is created that increases pressure. This nozzle pressure is the main component that
SOOR uses to create power.
Analysis of Deep Water Waves
Harnessing wave energy for the purposes of generating electrical energy has long been an
objective of the modern alternative energy movement. Various mechanical mechanisms have
been purposed in prior art and research which attempt to efficiently extract energy from ocean
waves; but such mechanisms have continually been unsuccessful in achieving significant wave
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power extraction efficiencies at relatively cheap costs. Other alternative energy sources, such as
wind farms and solar plants, have shifted the alternative energy focus away from wave energy.
Ironically though, huge/consistent quantities of fluidic/mechanical power are readily available in
the form of wave energy; and when compared to wind or solar energy sources, the quantity and
reliability of the power available in wave energy far exceeds that of other alternative energies.
Although various reasons exist for explaining the slow emergence wave energy as had with
regard to the rapidly growing alternative energy industry, a few reasons stand out. Wave motion
through a fluid is fundamentally complex in nature and it is not straightforward or obvious how
to design a mechanism to extract a significant amount of energy from the majority of the
oscillating fluid due to phase and water orbital differences. In addition, wave motion and velocity
vary along with depth, making for a difficult problem to readily analyze.
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-Background
Based on the Airry wave equations for modeling ocean waves, the displacement and
velocity of circulating water orbitals depend on the position, depth, and time at which a specific
point is analyzed. Moreover, since the wave transfers energy through circulating water orbitals, it
is important to note that the net mass and volumetric flow through any given point has a time
average of zero. This is because of the circulating water orbitals created by the wave to transfer
energy. A net flow of energy through each point is not zero, but the net mass/volumetric flow is
zero. The main Airry wave equations for water orbital displacement and velocity are as follows:
1. Horizontal/Vertical Water Orbital Diameter (Circular Motion):
𝑑 = 𝜋𝐻
𝑇𝑒𝑘𝑧
-where H = significant wave height, T = wave period, k = wave number, and z = depth
below average waterline.
2. Horizontal/Vertical Water Orbital Velocity (Circular Motion):
𝑣 = 𝝅𝐻
𝑇𝒆𝑘𝑧𝑐𝑜𝑠(𝑘 𝑥 − 𝜎 𝑡)
-where H = significant wave height, T = wave period, k = wave number, z = depth below
average waterline, x = horizontal position of specific analysis point, σ = 2π/T, and t = the
time at which analysis is taking place.
8
*Included below is an image of the water orbitals along different depths for visual
reference:
9
Conceptual Design Process
To begin a proper mechanical analysis of the wave energy system, several factors were
considered, where wave power extraction efficiency was considered the highest priority. Several
wave extraction devices and the physical mechanism they used to extract power were considered
during the conceptual design process. Examples of the most prevalent physical mechanisms used
to extract wave energy in prior-art include:
1. Pneumatic pressure differentials caused by oscillatory tidal wave motion (i.e. the
PowerBuoy).
2. Tidal reservoir/dam height differentials, high and low tide height differentials (i.e. the
Wave Dragon).
3. Changing height differentials in wave motion relative to the average waterline (i.e.
buoy displacement over time).
4. Flexible physical barriers utilizing sub-surface water orbital displacements present in
ocean waves (i.e. the Anaconda Wave Energy Converter).
5. Oscillating and length-dependent hydraulic fluid tubes/columns utilizing wave
height/pressure differentials (i.e. the Pelamis Wave Energy Converter).
6. Incoming wave front costal channeling systems utilizing and concentrating wave
height differentials.
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-Design Criteria
The first task in determining a more efficient and practical wave energy extraction device design
was to define the shortcomings of documented prior-art devices. Although a numerous quantity
of wave energy extraction devices are documented in prior art, the shortcomings of these devices
were characterized as:
1. Poor overall power efficiency in terms of wave energy content per meter of wave
front versus device energy output per meter of wave front. Efficiency (ϵsys) is defined
as wave energy device output per device width relative to the wave front directional
velocity vector (Eout) divided by total wave energy content (Ewave).
a. Mathematically defined as: ϵsys =Eout
Ewave=
Eout𝜌𝑔2
64𝜋𝐻𝑚𝑜𝑇𝑡
, where ρ is the density of
sea water, g is the gravitational acceleration constant, T is the dominant wave
period, Hmo is the significant wave height observed at the average waterline,
and t is the time interval over which device energy is measured.
2. Device specific location limitations. The majority of devices purposed only are able
to achieve appreciable power output when they are located directly off-shore in
shallow water. This location limitation requires that proposed wave energy devices be
placed close to the shore, and thus, use up prime coastal real-estate, and arguably
reduces the height and increases the turbulence of coastline wave fronts.
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a. Shallow water is defined in industry and oceanographic terms as locations
where the depth of the sea floor (D0) from the average waterline is less than
one-half the wavelength of the dominant wavelength (DDλ or λ): D0 < 1
20𝜆.
3. Underutilization of total wave energy content. As can be shown in research and
analytical equations, the energy content in an ocean wave is not fully concentrated at
the average waterline. The energy content of an ocean wave, depending on the
location (shallow or deep water), is a function of water depth from the average
waterline and its energy is spread across the average waterline height to a depth of
approximately 1
20𝜆 for deep water. If a wave energy device relies solely on a
waterline height differential to produce energy, it is unlikely to extract significant
amounts of power from the wave (Eout is small). The most efficient devices utilize
not only the average wave height differential, but also the orbital motion of the water
underneath the surface.
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-Initial Concept Design Layouts: Collapsible Pylon Structure
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14
15
Simulating Loads & Pressures Applied to Conceptual Design
1. Define fixed coordinate system for X-Y axis and horizontal intervals:
2. Define fixed coordinate system for X-Z axis:
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3. Create genrealized linear equations to model differential length as a function of x
across each horizontal interval:
Interval 1:
𝑳𝒙,𝑰𝟏 =𝒙 (𝒚𝟐 − 𝒚𝟏)
𝒙𝟐 − 𝒙𝟏−
𝒙 𝒃𝟐𝒂𝟐 − 𝒂𝟏
+ 𝒚𝟏
𝑥2 − 𝑥1 ≠ 0 𝑎𝑛𝑑 𝑎2 − 𝑎1 ≠ 0
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Note: 𝐿0 = 𝑦1Interval 2:
𝑳𝒙,𝑰𝟐 = 𝒚𝟐 − (𝒙 (𝒃𝟑 − 𝒃𝟐)
𝒂𝟑 − 𝒂𝟐+ 𝒃𝟐)
𝑎3 − 𝑎2 ≠ 0
Interval 3:
𝑳𝒙,𝑰𝟑 = 𝒚𝟑 − (𝒙 (𝒃𝟒 − 𝒃𝟑)
𝒂𝟒 − 𝒂𝟑+ 𝒃𝟑)
𝑎4 − 𝑎3 ≠ 0
Note: 𝑦3 = 𝑦2
Interval 4:
𝑳𝒙,𝑰𝟒 =𝒚𝟓 − 𝒚𝟒𝒙𝟓 − 𝒙𝟒
𝒙 + 𝒚𝟏 − (−𝒃𝟒
𝒂𝟓 − 𝒂𝟒 𝒙 + 𝒃𝟒)
𝑥5 − 𝑥4 ≠ 0 𝑎𝑛𝑑 𝑎5 − 𝑎4 ≠ 0
Note: 𝑦4 = 𝑦3 = 𝑦2
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4. Calculate relative horizontal velocity of water orbitals by applying ideal nozzle flow
asssumptions to wave passing through ballast and pylon for each horizontal interval
(add initial oribital velocities from Airry Equations):
Interval 1:
{
𝑽𝒙,𝑰𝟏 = 𝑳𝟎𝑽𝟎𝑳𝒙,𝑰𝟏
+ 𝝊𝒕,𝒙,𝒛 = 𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏
) 𝒙 + 𝒚𝟏 − ((𝒃𝟐
𝒂𝟐 − 𝒂𝟏) 𝒙)
+ (𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕))
}
𝒙 = 𝒂𝟏 𝒕𝒐 𝒂𝟐
Interval 2:
{
𝑽𝐱,𝐈𝟐 = 𝑳𝟎𝑽𝟎𝑳𝐱,𝐈𝟐
+ 𝝊𝒕,𝒙,𝒛 = 𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
𝒚𝟐 − ((𝒃𝟑 − 𝒃𝟐𝒂𝟑 − 𝒂𝟐
) 𝒙 + 𝒃𝟐)
+ (𝝅𝑯𝑻 𝒆𝒌𝒛 𝐜𝐨𝐬(𝒌 𝒙 − 𝝈 𝒕))
}
𝒙 = 𝒂𝟐 𝒕𝒐 𝒂𝟑
20
Interval 3:
{
𝑽𝒙,𝑰𝟑 = 𝑳𝟎𝑽𝟎𝑳𝒙,𝑰𝟑
+ 𝝊𝒕,𝒙,𝒛 = 𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
𝒚𝟐 − ((𝒃𝟒 − 𝒃𝟑𝒂𝟒 − 𝒂𝟑
) 𝒙 + 𝒃𝟑)
+ (𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕))
}
𝒙 = 𝒂𝟑 𝒕𝒐 𝐚𝟒
Interval 4:
{
𝑽𝐱,𝐈𝟒 = 𝑳𝟎𝑽𝟎𝑳𝐱,𝐈𝟒
+ 𝝊𝒕,𝒙,𝒛 = 𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟓 − 𝒚𝟒𝒙𝟓 − 𝒙𝟒
) 𝒙 + 𝒚𝟐 − ((−𝒃𝟒
𝒂𝟓 − 𝒂𝟒) 𝒙 + 𝒃𝟒)
+ 𝝅𝑯𝑻 𝒆𝒌𝒛 𝐜𝐨𝐬(𝒌 𝒙 − 𝝈 𝒕)
}
𝒙 = 𝒂𝟒 𝒕𝒐 𝒂𝟓
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5. Calculate relative pressure of wave passing through pylon and ballast structure by
applying ideal nozzle flow asssumptions and Bernoulli’s streamline flow equation
for each horizontal interval (add initial oribital velocities from Airry Equations):
*Pressure is Relative to Atmospheric Pressure at Sea Level
22
Interval 1:
𝑃𝑥,𝐼1 = (1
2) · 𝜌 · (((
𝐿0 𝑉0
𝐿𝑥,𝐼1 ) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝜌𝑔𝑧 =
=𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏
) 𝒙 + 𝒚𝟏 − ((𝒃𝟐
𝒂𝟐 − 𝒂𝟏) 𝒙)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
Interval 2:
𝑃𝑥,𝐼2 = (1
2) · 𝜌 · (((
𝐿0 𝑉0
𝐿𝑥,𝐼2 ) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝜌𝑔𝑧 =
=1
2 𝜌
(
(
𝑦2 (
𝝅𝐻𝑇 𝒆𝑘𝑧)
𝑦2 − ((𝑏3 − 𝑏2𝑎3 − 𝑎2
) 𝑥 + 𝑏2)
− (𝝅𝐻𝑇 𝒆𝑘𝑧) +
𝝅𝐻𝑇 𝒆𝑘𝑧 𝑐𝑜𝑠(𝑘 𝑥 − 𝜎 𝑡)
)
2
− (𝝅𝐻𝑇 𝒆𝑘𝑧)
2
)
− 𝜌𝑔𝑧
Interval 3:
𝑃𝑥,𝐼3 = (1
2) · 𝜌 · (((
𝐿0 𝑉0
𝐿𝑥,𝐼3 ) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝜌𝑔𝑧 =
=𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
𝒚𝟐 − ((𝒃𝟒 − 𝒃𝟑𝒂𝟒 − 𝒂𝟑
) 𝒙 + 𝒃𝟑)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
*Note: 𝜌𝑔𝑧 = relative pressure contribution of water weight above point.
23
Interval 4:
𝑃𝑥,𝐼4 = (1
2) · 𝜌 · (((
𝐿0 𝑉0
𝐿𝑥,𝐼4 ) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝜌𝑔𝑧 =
=𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟓 − 𝒚𝟒𝒙𝟓 − 𝒙𝟒
) 𝒙 + 𝒚𝟐 − ((−𝒃𝟒
𝒂𝟓 − 𝒂𝟒) 𝒙 + 𝒃𝟒)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
6. Calculate the average pressure across each horiztonal interval using a double
integral with relative pressure and dividing by each horizontal intervals total
exposed area.
Interval 1:
𝑃𝑎𝑣𝑔(𝑥,𝐼1) =1
𝐴1∫ ∫ 𝑃𝑥,𝐼1 ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧
𝑑
𝑐
=
=1
𝐴1∫ ∫ [(
1
2) · 𝜌 · (((
𝐿0 𝑉0𝐿𝑥,𝐼1
) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝝆𝒈𝒛] ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧𝑑
𝑐
=
=1
𝐴1∫ ∫
[ 𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟐 − 𝒚𝟏𝒙𝟐 − 𝒙𝟏
)𝒙 + 𝒚𝟏 − ((𝒃𝟐
𝒂𝟐 − 𝒂𝟏) 𝒙)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
]
∗ 𝒅𝒙 ∗𝒈
𝒇
𝒅𝒛𝒅
𝒄
24
Interval 2:
𝑃𝑎𝑣𝑔(𝑥,𝐼2) =1
𝐴2∫ ∫ 𝑃𝑥,𝐼2 ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧
𝑑
𝑐
=
=1
𝐴2∫ ∫ [(
1
2) · 𝜌 · (((
𝐿0 𝑉0𝐿𝑥,𝐼2
) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝝆𝒈𝒛] ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧𝑑
𝑐
=
=1
𝐴2∫ ∫
[ 𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
𝒚𝟐 − ((𝒃𝟑 − 𝒃𝟐𝒂𝟑 − 𝒂𝟐
) 𝒙 + 𝒃𝟐)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
]
∗ 𝒅𝒙 ∗𝒈
𝒇
𝒅𝒛𝒅
𝒄
Interval 3:
𝑃𝑎𝑣𝑔(𝑥,𝐼3) =1
𝐴3∫ ∫ 𝑃𝑥,𝐼3 ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧
𝑑
𝑐
=
=1
𝐴3∫ ∫ [(
1
2) · 𝜌 · (((
𝐿0 𝑉0𝐿𝑥,𝐼3
) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝝆𝒈𝒛] ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧𝑑
𝑐
=
=1
𝐴3∫ ∫
[ 𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
𝒚𝟐 − ((𝒃𝟒 − 𝒃𝟑𝒂𝟒 − 𝒂𝟑
) 𝒙 + 𝒃𝟑)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
]
∗ 𝒅𝒙 ∗𝒈
𝒇
𝒅𝒛𝒅
𝒄
25
Interval 4:
𝑃𝑎𝑣𝑔(𝑥,𝐼4) =1
𝐴4∫ ∫ 𝑃𝑥,𝐼4 ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧
𝑑
𝑐
=
=1
𝐴4∫ ∫ [(
1
2) · 𝜌 · (((
𝐿0 𝑉0𝐿𝑥,𝐼4
) − 𝑉0 + 𝜐𝑡,𝑥,𝑧 )
2
− (𝑉02)) − 𝝆𝒈𝒛] ∗ 𝑑𝑥 ∗
𝑔
𝑓
𝑑𝑧𝑑
𝑐
=
=1
𝐴4∫ ∫
[ 𝟏
𝟐 𝝆
(
(
𝒚𝟐 (
𝝅𝑯𝑻 𝒆𝒌𝒛)
(𝒚𝟓 − 𝒚𝟒𝒙𝟓 − 𝒙𝟒
)𝒙 + 𝒚𝟐 − ((−𝒃𝟒
𝒂𝟓 − 𝒂𝟒) 𝒙 + 𝒃𝟒)
− (𝝅𝑯𝑻 𝒆𝒌𝒛) +
𝝅𝑯𝑻 𝒆𝒌𝒛 𝒄𝒐𝒔(𝒌 𝒙 − 𝝈 𝒕)
)
𝟐
− (𝝅𝑯𝑻 𝒆𝒌𝒛)
𝟐
)
− 𝝆𝒈𝒛
]
∗ 𝒅𝒙 ∗𝒈
𝒇
𝒅𝒛𝒅
𝒄
7. Once the average pressure is calculated for each interval, average the first and second
intervals as well as the third and fourth. The two new resultant averaged values will be
the average pressures loads applied to the FEA model in the program, Ansys. Note: the
complex double integration formulas listed in the previous steps were evaluated using
Mathematica’s numeric integrand solver.
8. A complex Microsoft Excel program was also used to catalogue and update various
coordinate and deflection values during the FEA and Mathematica pylon-structural
analyses.
26
-Excel Equation Constants and Reference Tables:
Physical & Buoy Data
ConstantsSymbol Units Value Assumptions/Notes Initial Data Source
Buoy Dominant Wave Period
(x-axis)T s 11.06388889
*Combined Average Calculated in "Buoy Data"
WorksheetBuoy Data
Buoy Sea Floor (z-axis) D0 m -3684*Combined Average Calculated in "Buoy Data"
WorksheetBuoy Data
Buoy Average Significant Wave
Height (z-axis)H m 2.75
*Combined Average Calculated in "Buoy Data"
WorksheetBuoy Data
Gravitational Acceleration g m/s^2 9.807 Constant Wolfram Alpha
PI Constant π - 3.141592654 Constant Excel Internal Function
Initial Incoming Fluid Pressure P0 Pa 0Wave Entering Flow Channel (no area
constraint, ΔL = 0)Derived
Averaged Waterline Height z0 m 0 See Pylon Analysis Coordinates Diagram
Ocean Salt Water Density ρ kg/m^3 20.9620°C and atmospheric pressure and 30 g/kg
salinity
http://www.kayelaby.npl.co.uk/ge
neral_physics/2_7/2_7_9.html
Calculated
Constants/VariablesSymbol Units Value Assumptions/ Notes Assumption Validation Equation Source
Wavelength (x-axis) λ m 191.0609436 Assume Deep Water (H > 1/2*λ) OK g*T^2/(2π) airry pdf
Wave Number k 1/m 0.005233932 1/λ airry pdf
Horizontal Intervals Channeling BallastHorizontal Interval 1 x1-x2
Horizontal Interval 2 x2-x3
Horizontal Interval 3 x3-x4
Horizontal Interval 4 x4-x5
Coordinate Symbol Coordinate Value (m)
x1 -5
y1 5.5
x2 -3.149
y2 3.5
x3 0
y3 3.5
x4 3.149
y4 3.5
x5 5
y5 5.5
CHANNELING BALLAST
(X, Y) COORDINATES
POSITION FROM ORIGIN
(CONSTANT OVER TIME
& DEPTH)
CHANNELING BALLAST PARAMETERSHorizontal Intervals Pylon
Horizontal Interval 1 a1-a2
Horizontal Interval 2 a2-a3
Horizontal Interval 3 a3-a4
Horizontal Interval 4 a4-a5
Coordinate Symbol (a = x-
coordinate, b = y-
coordinate)
Coordinate
Value (m)
a1 -5
b1 5.5
a2 -3.149
b2 3.5
a3 0
b3 3.5
a4 3.149
b4 3.5
a5 5
b5 5.5
LENGTH OF PYLON SIDE WALL
(HYPONTENUSE)
ζ
5.830951895
INITIAL (X, Y) PYLON POSITION
COORDINATES FROM ORIGIN, t =
0 s (CHANGE OVER TIME)
PYLON PARAMETERS
27
Final Design
Figure 3.3 shows a solid model, created in Solidworks and imported into ANSYS, of the
pylon structure. Figure 3.3 does not include the springs of the structure.
The pylon is
supported by an
internal support
structure that runs
down the length of the
pylon. This support
structure also is where
the pistons are housed.
The outer shell of the
structure is shaped as a
diamond in order to
create a flow channel
with decreasing area
between the pylon and the ballast. Springs are connected to the support structure and the inner
face of the shell. The spring constant of these springs are equal to 150 Nm2 (Please see
structural analysis section for spring calculations). Hinges connect the sides of the shells to each
other to promote structural compression. Springs and the dampening feature of the pistons will
insure that the structure forms back to its original state. Dimensioned drawings of the pylon and
Figure 3.3: Pylon structure
28
of major components of the pylon are included in the appendix. Figure3. 4 include a top view of
the pylon structure. Please note that no springs have been added in figure 3.4.
Figure 34: Pylon with no springs (Top view)
29
The original design only included one tier of pistons and springs, the final design of 5
tiers was made to promote more power output. The original design of the pylon also consisted of
4 springs connecting the support to the shell, two springs on each side of the support. This idea
was also discarded and 6 springs were added to reduce the amount of stress on each spring, this
reduction in stress eliminated the springs as a critical location.
Figure 3.5 consists of a diagram of the ballast tank. The ballast tanks design did not
change at all during the design process. Due to its simplistic functions of directing waves and
water towards the pylon and to keep the entire structure from sinking, there was no need to create
a detailed design. Similar to the pylon, a dimensioned drawing of the ballast is included in the
appendix of the report.
30
As shown in figure 2, ballasts will be
welded to each side of the pylon in order to
create a flow channel for the water. This flow
channel will create a nozzle for and increase the
pressure acting on the pylon
Deciding materials for SOOR was not a very
difficult design step. A lightweight material was
needed for the shell of the pylon and the ballast.
A heavier material may have made structural
compress more difficult, and a more heavy
material would make the structure sink. The
material must have a strong yield strength to
protect against yielding. With this taken into
account, the structures were designed to be made out of 2024 Aluminum. Stronger materials
were needed for the support structure of the pylon; the final decision was to build these
components out of 1060 hot rolled steel. The reason for choosing this type of steel for the
support structure was because of the increased weight excreted on the support due to the pistons.
All materials and material properties were referenced from Shigley’s Mechanical Engineering
Design 9th Edition. Figure A-3.10 shows a materials list used for SOOR.
Figure 3.5: Ballast Diagram
31
Structural and Fatigue Analysis
In order to find all the spring constants, 3 equations needed to be solved. The momentum
equation, the energy equation, and the force balance equation. All three equations are listed in
the equation appendix as equation 4.1, 4.2, and 4.3. With these 3 equations, 3 unknowns are left;
one of which is the spring constant. We found the spring constant to be 148Nm, which we
rounded up to 150 Nm. This was the spring constant used for all springs in the structure.
The major forces and pressures SOOR experiences are exerted from the wave. Please see
the dead water wave analysis section for the derivation of the wave force and pressure. In order
to increase accuracy, the force from the wave was divided into 4 equal parts. Please see figure A-
4.1 for the free body diagram of the SOOR structure. Please note that wherever force is applied
in the free body diagram, there is an accompanied pressure. The pressure is the main component
that SOOR uses to create power. Using the equations and coordinate system discussed in the
deep wave analysis section, we discovered the maximum pressure for the four intervals to be the
following: P1= 14693.1 Pa, P2=14696.1 Pa, P3=15338.7 Pa, P4= 14592.9 Pa. The forces that
accompany these maximum pressures are as follows: F1=416.225 Pa, F2=416.225, F3=434.426,
F4=396.795. Notice that the maximum pressures and stresses occur in section where the flow
channel area is the smallest. The initial pressures that are exerted on the structure are the
following: P1=11852.9 Pa, P2=11872.3 Pa, P3= 12791.3 Pa, P4=11891.7 Pa. These initial
pressures are considered to be dead loads and were used during the dead load calculation. Since
waves are fluctuant, there is a fluctuant force that occurs between the initial pressure and the
maximum pressure. Using these variable loads, a fatigue analysis using the Modified Goodman
32
equation was completed for the shell of the pylon. For all criteria, including Von Mises and
Modified Goodman the desired factor of safety was 2.
The initial plan to solve all these different criteria was to import the Solidworks solid
model into ANSYS to perform the analysis. This unfortunately did not work due to the massive
size of the pylon structure. Over eight hundred parts make the pylon structure, all with different
mates and connections to them. This makes ANSYS models extremely slow to solve. The
secondary plan was to simplify the pylon structure by only analyzing one layer of the support
structure. We then created a ANSYS model that was 2m in height, but still included one tier of
pistons and supports; this again did not work in ANSYS. Even with the simplified model, the
time needed to solve the ANSYS model was still insufficient. In order to create a simpler model,
only the shell was analyzed. To make the model more accurate, the shell was fitted with a
springs and dampers to mimic the support springs and pistons. These were shown as two large
elements connecting the sides together in the ANSYS model. See figure A-4.2 for a picture of
the model. The maximum pressures were added to the shell and the maximum deformation
contour and the Von Mises Stress contour were created (Figures 4.3 and A-4.4). The deformation
contour is used in the power calculation. While the Von Mises stress is used to find the factor of
safety (Equation A-4.4). The maximum von mises stress comes out to be 2.7x10^8 Pa. Using
equation A-4.5 and the yield strength for 2024 aluminum. We found our factor of safety for von
mises criteria to be 2.8 which is in inside our criteria for design factor. The equation for
Modified Goodman is listed as Equation A-4.6. The value for Se’ is equal to 93, which is half the
ultimate strength of 2024 aluminum. Unfortunately we must use Se’ as our value for endurance
limit. We decided to solve the factor of safety for the third pressure since it is the largest and will
33
have the smallest factor of safety. If this pressure meets our criteria than the rest of the pressures
will. The values for σa and σm are 14065 Pa and 1276.7 Pa. Using the Modified Goodman
equation, we can solve for our factor of safety; the factor of safety for fatigue comes out to be
4.86, which is a very large factor of safety but is still adequate to our design factor.
Stress Concentration and Critical Location
Due to the hollowed structure and the constant stress that is applied, the inner slide is
considered the critical location of the structure. To find the point with the highest stress
concentration in the adjustable slide, we analyzed each part of it individually at its maximum
extension to find the moment at the end of each section. We then superimposed them onto each
other in order to calculate the point where the maximum stresses occur. In order to analyze each
section, we needed to know all forces acting on the each segment. For each section, there were
six forces. Each force acting on it was caused by a compressed spring. In order to find the
corresponding force, we used EquationA- 5.1. All the springs on the top side of the slide had the
same stiffness constant of 150, and the springs on the bottom had a value of 100. The springs on
the bottom of the slide also had the same displacement. Each spring on the top side had a
different displacement, calculated with Equation A-5.2, where θ1 is the angle between the side
wall and the slide while uncompressed, θ2is the same angle after compression, and L is the
length from the slide to the spring along the wall. From there we inserted each part into an
analysis software to calculate the deflection, von Mises stresses, and safety factor of each part.
The results for the larger cross section beam can be seen in figures A-5.1, A-5.2, and A-5.3
34
respectively, and the results for the smaller cross section are shown in figures A- 5.4, A-5.5, and
A-5.6. From these results, we were able to calculate the moment at the end of each part, using
Equation A-5.3, where l is the length of the part, E=2(10¹¹), and I is calculated by Equation A-
5.4, where ho is the outside cross section height and hi is the inside cross section height. From
the calculated moments, we were able to find the equivalent forces at the end of each beam using
Equation 5.5. After superimposing these new forces onto the longer, smaller cross section part,
we ran the simulation again to find the deflection, von Mises stresses, and the factor of safety.
These results are shown in figures 5.7, 5.8, and 5.9. Analyzing these new results after super
imposing the parts onto each other, we can see where the highest stress concentration occurs.
This occurs at the top and bottom of the bar, just after the thickness is reduced.
35
SOOR’s Output and Power Produced
The power output from the pistons is based on the deformation of the entire pylon
structure. The maximum deformation is equal to .111m which was calculated from the
deformation contour. Each tier, due the six pistons on each tier, has the potential to generate
significant amounts of power which can be converted into electrical energy.
36
Appendix
Equations:
Equation A-4. 1: Energy Equation
(1
2 𝑚 𝑡2 (𝐹𝑎𝑣𝑔1 + 𝐹𝑎𝑣𝑔2 + 𝐹𝑎𝑣𝑔3 + 𝐹𝑎𝑣𝑔4)
= 𝑘 ((𝜁1sin(𝜃1)− 𝜁1 sin(𝜃2))
2+ (𝜁
2sin(𝜃1) − 𝜁2 sin(𝜃2))
2
+ (𝜁3sin(𝜃1)− 𝜁3 sin(𝜃2))
2) + 6 𝐹𝑎𝑝𝑝 (𝜁𝑠𝑙𝑖𝑑𝑒 sin(𝜃1) − 𝜁𝑠𝑙𝑖𝑑𝑒 sin(𝜃2)))
Equation A-4.2: Force Balance Equation
(𝐹𝑎𝑣𝑔1 + 𝐹𝑎𝑣𝑔2 + 𝐹𝑎𝑣𝑔3 + 𝐹𝑎𝑣𝑔4 = 2 𝑘 ((𝜁1sin(𝜃1) − 𝜁1 sin(𝜃2)) + (𝜁2 sin(𝜃1) − 𝜁2 sin(𝜃2)) +
(𝜁3sin(𝜃1) − 𝜁3 sin(𝜃2))) + 6 𝜌 𝑔 ℎ 𝐴)
Equation A-4.3 Momentum Equation
(𝑡 (𝐹𝑎𝑣𝑔1 + 𝐹𝑎𝑣𝑔2 + 𝐹𝑎𝑣𝑔3 + 𝐹𝑎𝑣𝑔4) = 𝑚𝑤𝑎𝑙𝑙 (𝜁𝑤 sin(𝜃1) − 𝜁𝑤 sin(𝜃2)) + 2 (𝑚𝑠𝑝𝑟𝑖𝑛𝑔1 (𝜁1 sin(𝜃1) −
𝜁1sin(𝜃2)) + 𝑚𝑠𝑝𝑟𝑖𝑛𝑔2 (𝜁2 sin(𝜃1) − 𝜁2 sin(𝜃2)) + 𝑚𝑠𝑝𝑟𝑖𝑛𝑔3 (𝜁3 sin(𝜃1) − 𝜁3 sin(𝜃2))) +
6 (𝑚𝑝𝑖𝑠𝑡𝑜𝑛 + 𝑚𝑠𝑙𝑖𝑑𝑒 + 𝜌 ℎ 𝐴) (𝜁𝑠𝑙𝑖𝑑𝑒 sin(𝜃1) − 𝜁𝑠𝑙𝑖𝑑𝑒 sin(𝜃2)))
37
Equation A-4.4: Distortion Energy Theory
𝜎′ =𝑆𝑦
𝑛
Equation A-4.5: Modified Goodman
1
𝑛=𝜎𝑎𝑆𝑒+𝜎𝑚𝑆𝑢𝑡
Equation A-5.1: Spring Force
F = kΔx
Equation A-5.2: Change in X
2: Δx = L(sin(𝜃1) − sin(𝜃2))
Equation A-5.3: Bending Stress
3: 𝛿 =𝑀𝑙2
2𝐸𝐼
38
Equation A-5.4: Second- Area Moment of hollow rectangular area
4: 𝐼 =1
12(ℎo
4 − ℎi4)
Equation A-5.5: Force equation
5: F = 𝑀
𝑙
39
Diagrams:
Figure A-3.1: General Pylon Assembly dimensioned drawing
40
Figure A-3.2: Assembly of hydraulic (Piston Casing) dimensioned drawing
41
Appendix Figure 3.3: Piston Rod Dimensioned Drawing (meters)
42
Figure A-3.4: Ballast dimensioned drawing (meters)
43
Figure A-3.5: Fixed support Structure dimensioned Drawing (meters)
44
Figure A-3.6: Support Shaft (outer slide) Assembly Dimensioned Drawing (meters)
45
Figure A-3.7 Shell (1 Side) Dimensioned Drawing (meters)
Figure A-3.8: Inside Hinge Assembly Dimensioned Drawing (meters)
46
Figure A-3.9: Slide Dimensioned Drawing (meters)
47
Figure A-3.10: Materials List
Figure A-4.1: SOOR free body Diagram
48
Figure A-4.2: Pylon Shell ANSYS Model
Figure A-4.3: Maximum Deformation Contour of Pylon
49
Figure A-4.4: Maximum Von Mises Stress Contour
50
Figure A-5.1, A-5.2, A-5.3: Critical Location Large Slide
51
52
Figures A-5.4, A-5.5, A5.6: Stress Concentration of smaller slide
53
54
Figures A-5.7, A-5.8, A5.9: Factors of safety of critical locations.
55
References
WWW.PELAMISWAVE.COM
Shigley’s Mechanical Engineering Design 9th Edition
